Examining Mirror Image

[Frogreaver]

It's often said that mirror image and high AC don't work well together. Intuitively it sounds correct as the higher your AC the more likely it is that an image will vanish on a hit that would have ordinarily missed you. However, while technically true, that assessment is a bit misleading. You see, often the best way to compare how much more survivable a defensive ability makes you is by looking at what that ability does for effective hp. When looking at Mirror Image's impact on effective hp, the math seems to show that it is independent of AC, except for ac from dex mod. That is with all other parameters equal, it has the same impact on effective hp no matter what your AC.

I'll post some math in a bit (it's fairly complicated). I make no promises of a complete proof, only that I can show this for a few cases and am able to show the principles behind my reasoning and intuitively it seems they should apply to the whole problem. I wanted to get this out there to get thoughts, opinions and see if anyone else had done any work on Mirror Image.

Thanks,
Frogreaver

[Segev]

Mirror image adds less defense the higher your non-dex AC is, true, but it always adds something to your chances of not being hit. The question is mostly whether the action to cast it and the first level spell slot are worth it.

It also takes increasing investment to get the spell the more likely you are to have high AC from armor or the like. Sure, an Eldritch Knight is an obvious one who could get it and sacrifice nothing in terms of AC, but that’s the corner case. And I believe war magic would even let him attack and cast it in the first round in a fight he doesn’t need bigger buffs right away.

Further, the higher the enemies’ attack bonuses, the more effectiveness it has in increasing your defense.

I would be interested in seeing the math for various AC vs to hit scenarios, because it is a question of opportunity cost over whether this defense is worth the price, but it never is rendered useless.

[Frogreaver]

Mirror image adds less defense the higher your non-dex AC is, true

The whole point of this thread is that if 2 character have the same dex and take the same number of attacks then their effective hp will increase by the same percentage when casting mirror image regardless of the players AC values. That sounds quite a bit different than your claim above.

[Segev]

The whole point of this thread is that if 2 character have the same dex and take the same number of attacks then their effective hp will increase by the same percentage when casting mirror image regardless of the players AC values. That sounds quite a bit different than your claim above.
I look forward to the math on that, then.

If I had to guess, it might be true due to AC independently changing “effective hp,” but my gut says that the increased odds of losing an image when the real one wouldn’t have been hit reduces the over-the-duration effective hp increase.

But like I said, I look forward to the math. This is a very complicated probability interaction, and I could be missing important details.

[Frogreaver]

I look forward to the math on that, then.

If I had to guess, it might be true due to AC independently changing “effective hp,” but my gut says that the increased odds of losing an image when the real one wouldn’t have been hit reduces the over-the-duration effective hp increase.

But like I said, I look forward to the math. This is a very complicated probability interaction, and I could be missing important details.

Thanks. I will be posting the math. I like to see peoples thoughts and logic before I do. So I'm going to let it simmer a bit longer.

That said, consider the scenario of barbarian rage and AC values. Is Barbarian rage more effective the lower or higher your AC? It would seem one could make the argument that higher AC means you get hit less which means your barbarian rage damage resistance applies less often and is therefore less effective. I think we would agree that's the wrong conclusion to draw though. I think we would both agree that damage resistance gives you twice as much effective hp regardless of your chance to be hit. Does it make sense that this will be similar when it comes to mirror image?

[Mellack]

It would seem to me that mirror image loses value as your AC increases. Take it to an extreme as a mental exercise. If the caster is only hit on a 20, then even if attacked 6 times, there is over 70% chance that all those attacks missed the caster, meaning mirror image had no effect. If instead the caster is hit on anything but a 1, the caster is struck by all of those attacks over 70% of the time, with the mirror image taking several of them.

[Frogreaver]

It would seem to me that mirror image loses value as your AC increases. Take it to an extreme as a mental exercise. If the caster is only hit on a 20, then even if attacked 6 times, there is over 70% chance that all those attacks missed the caster, meaning mirror image had no effect.

But isn't that the same with Barbarian rage damage resistance. We don't hear that it gets less effective the more AC you have. Intuitively we know it doubles your effective hp no matter what your chance to be hit.


If instead the caster is hit on anything but a 1, the caster is struck by all of those attacks over 70% of the time, with the mirror image taking several of them.

And isn't this also the same with barbarian rage damage resistance. If you are hit virtually all the time then it still just doubles your effective hp.

[Mellack]

Yes, barbarian resistance also loses effectiveness as their AC increases. If they never get hit, it has no benefit. Also, it has no benefit if they are not doing BPS damage. Rage does nothing for your effective HP if you are fighting a fire elemental.

Is your question not that the effectiveness changes, but that some people perceive the amount of change differently?

[Satori01]

But isn't that the same with Barbarian rage damage resistance. We don't hear that it gets less effective the more AC you have.

That is because for it's Duration a Barbarian Rage has unlimited uses.

That is not the case for Mirror Image.

Thought Experiment:

If the damage reduction portion of a Barbarian rage was limited to applying to just 20 attacks -Hit or Miss..(DM tells the player they are being attacked, the player then indicates if the Damage Reduction applies for example),

......then absolutely, from a certain perspective, having a high AC, would work against the absolute effectiveness of Barbarian Rage damage reduction.

Of course, Barbarian Rage works nothing like this.

Might as well hold onto your numbers, and save the theatrics....I'm afraid your premise is flawed.

(Actually I am interested in your numbers)

[Segev]

Thanks. I will be posting the math. I like to see peoples thoughts and logic before I do. So I'm going to let it simmer a bit longer.

That said, consider the scenario of barbarian rage and AC values. Is Barbarian rage more effective the lower or higher your AC? It would seem one could make the argument that higher AC means you get hit less which means your barbarian rage damage resistance applies less often and is therefore less effective. I think we would agree that's the wrong conclusion to draw though. I think we would both agree that damage resistance gives you twice as much effective hp regardless of your chance to be hit. Does it make sense that this will be similar when it comes to mirror image?

The difference is that barbarian rage isn’t going away as you use it.

I doubt the comment on mirror image would be brought up if the images agree your AC entirely or if they didn’t disappear when hit.

It’s not when you would have been missed anyway due to AC that makes it less useful. It’s when you would have been missed anyway but one of them is hit and destroyed that it feels wasted.

Imagine if barbarian rage didn’t give resistance, but instead let you spend a reaction when attacked (but before the attack roll is made) to gain 20 temporary hit points that go away at the start of your next turn. This becomes a wasted reaction (especially if you have only, say, 5 uses per day) if the attacks all miss you anyway.

The mathematical increase in effective hp Of this hypothetical barbarian ability would be equal to the number of throes temp hp that were eaten by attacks. If the barabrian’s AC is high enough to be missed all the time, there are better things to do with that reaction.

If the wizard’s AC is sufficiently higher than that if his images, the chance that they will protect none of his hit points before they all vanish to attacks that would have missed him anyway becomes so great that there are better things to do with the action and spell slot.

The percent of the wizard’s hp that are protected decrease with faster rate as the difference between AC grows because it makes the spell end proportionately faster and wastes hits that might otherwise have been absorbed.

Remember, it only protects against a number of hits equal to the number of images, maximum, unless there’s something strange going on where the images have higher AC than the wizard.

If a hit on an image would have missed the wizard anyway, that is a wasted hit.

[LudicSavant]

It's often said that mirror image and high AC don't work well together. Intuitively it sounds correct as the higher your AC the more likely it is that an image will vanish on a hit that would have ordinarily missed you. However, while technically true, that assessment is a bit misleading. You see, often the best way to compare how much more survivable a defensive ability makes you is by looking at what that ability does for effective hp. When looking at Mirror Image's impact on effective hp, the math seems to show that it is independent of AC, except for ac from dex mod. That is with all other parameters equal, it has the same impact on effective hp no matter what your AC.

I'll post some math in a bit (it's fairly complicated). I make no promises of a complete proof, only that I can show this for a few cases and am able to show the principles behind my reasoning and intuitively it seems they should apply to the whole problem. I wanted to get this out there to get thoughts, opinions and see if anyone else had done any work on Mirror Image.

Thanks,
Frogreaver

We've done work on Mirror Image before; it basically loses value the higher your non-Dex AC is.

[heavyfuel]

If I may present some evidence that high AC does interfere with MI: A friend was playing a Hexadin with 22 AC (+1 full plate and +1 shield) and one of his spells was MI, which he really liked to use.

A lot of the time, I (the DM) would attack him and roll something like 14 to hit. So, when he was rolling the d20 to see if the attack hit him or one of his images, the players would hope it hit him, because that would mean no damage and no images destroyed.

Most enemies could only hit his high AC on a 15+, but could easily destroy his 10 AC images on a 3+. As such, the majority of times he used MI, his images were destroyed before he took a single instance of damage. They almost never saved him from actual damage.

Diminishing returns is definitely a thing when it comes from (non Dex) AC and Mirror Image.

[Hellpyre]

Having run the math on my own prior, it does in fact lose value as your AC investments outside of Dex go up, relative to the amount of damage you take without it.

I think I know where in your numbers you're getting your claim - the idea is that if it hits the duplicate, it didn't matter what your AC was anyways, you still prevented an attack. The thing is, it only increases your effective HP if it hits a duplicate and would also have hit you. If it hits the duplicate but wouldn't have hit you anyways, you just wasted magic and actions.

[Frogreaver]

Yes, barbarian resistance also loses effectiveness as their AC increases. If they never get hit, it has no benefit.

That feels alot like saying "if enemies always roll 1's then AC offers no benefit". While technically true in both instances, it's irrelevant. Probabilistically speaking, we can't base the effectiveness of something on a single case unless that case is guaranteed to happen.


Also, it has no benefit if they are not doing BPS damage. Rage does nothing for your effective HP if you are fighting a fire elemental.

Maybe you can explain the relevance of this rather obvious "inisght" to the rest of the discussion?


Is your question not that the effectiveness changes, but that some people perceive the amount of change differently?

Nope.

[Satori01]

I would find it interesting to see numbers ran not on now Mirror Image works by RAW, but rather, how I think it is run at many tables.

I have certainly seen Mirror Image ran where the DM tells the player that they are hit, forgetting that the player has an active Mirror Image spell, and let's the player see if an Image is hit instead, and the method just sticks.

This method certainly cuts down losing an Image to a "false positive" hit, but isn't RAW.

Mirror Image is always a good counter to Faerie Fire or your opponents that have Advantage on attacks.

[Frogreaver]

Having run the math on my own prior, it does in fact lose value as your AC investments outside of Dex go up, relative to the amount of damage you take without it.

I think I know where in your numbers you're getting your claim - the idea is that if it hits the duplicate, it didn't matter what your AC was anyways, you still prevented an attack. The thing is, it only increases your effective HP if it hits a duplicate and would also have hit you. If it hits the duplicate but wouldn't have hit you anyways, you just wasted magic and actions.

In absolute terms I have no doubt that's the case. But effective hp is a relative measurement. If your taking 2 DPR then reducing that by 1 DPR doubles your effective hp. If you take 20 DPR then reducing that by 10 DPR doubles your effective hp.

This is the same methodology that claims that increasing your AC against an attack such that you go from 10% chance to be hit to 5% chance to be hit doubles your effective hp. Even though in absolute terms you are only taking 1 less DPR (assuming monster damage on a hit is 20).


If I may present some evidence that high AC does interfere with MI: A friend was playing a Hexadin with 22 AC (+1 full plate and +1 shield) and one of his spells was MI, which he really liked to use.

A lot of the time, I (the DM) would attack him and roll something like 14 to hit. So, when he was rolling the d20 to see if the attack hit him or one of his images, the players would hope it hit him, because that would mean no damage and no images destroyed.

Most enemies could only hit his high AC on a 15+, but could easily destroy his 10 AC images on a 3+. As such, the majority of times he used MI, his images were destroyed before he took a single instance of damage. They almost never saved him from actual damage.

Diminishing returns is definitely a thing when it comes from (non Dex) AC and Mirror Image.

In the absolute sense I agree, but effective hp is a relative measurement. If the number of hits you would take goes down as AC increases then mirror image needs to affect proportionally fewer number of hits in order for it's impact on effective hp to remain constant.

[Vegan Squirrel]

I'm interested in seeing what your math looks like, as intuitively I agree with the other posters that a higher AC would seem to make mirror image less useful. I'm also interested in your focus on effective HP.

It occurs to me that HP decrease in value as AC increases, as a higher AC makes it less likely you'll lose as many HP. A high-AC character has more effective HP, because they can make each HP go farther. HP are subject to diminishing returns; the first 50 HP are worth more than the next 50, as you might not need that second 50.

This post is getting confusing, so I'm going to add an example. Alice and Bob both have 45 HP and a Dexterity of 10, but Alice has an AC of 20 and Bob has an AC of 10. They exclusively fight Green Monsters that attack at +4 for 20 damage. They hit Alice on a roll of 16 or higher, or 25% of the time, dealing DPR of 5. They hit Bob on a roll of 6 or higher, or 75% of the time, dealing DPR of 15. Bob can survive 3 rounds of attacks, enough that he may or may not go unconscious during a given fight; since Green Monsters deal 20 damage, we can say that his effective HP against them is 60 eHP. Alice can survive 9 rounds of attacks, which is plenty; since Green Monsters deal 20 damage, we can say that her effective HP against them is 180 eHP.

Perhaps your point is that mirror image gives you fewer HP when you have a high AC, but because your higher AC makes each HP go farther, the smaller number of HP is effectively greater. But then you're talking about adding a similar number of effective HP to a character that's starting with a larger pool of effective HP to begin with, due to their high AC. And my response is that the same number of effective HP is still less valuable to the high AC character than it would be to the low AC character, because of the diminishing returns of HP.

Alice and Bob both cast mirror image. Their mirror duplicates have an AC of 10, meaning they are hit 75% of the time, like Bob. When Bob's duplicate is hit, that attack would have hit Bob 100% of the time, meaning three duplicates grant him 60 effective HP, bringing his total to 120 eHP. He can now survive the fight easily. When Alice's duplicate is hit, that attack would have hit Alice 33% of the time, meaning three duplicates grant her 20 effective HP, bringing her total to 200 eHP. That... doesn't do much.

Granted, that was an extreme example, with a wide disparity in AC, but even so, the spell was much more useful to the lower AC character. Alice did not receive as large of a boost in eHP, and the extreme difference highlighted how much less useful each additional HP would be to her anyway. With a smaller disparity, the difference will be smaller, but I don't see how it would completely disappear.

Note: on previewing this post, I see I've missed several posts since I started typing. I'll go ahead and submit this now anyway, as the points stand.

[Frogreaver]

We've done work on Mirror Image before; it basically loses value the higher your non-Dex AC is.

Whose to say that you looked at Mirror Image in context of effective hp (a measurement you commonly use to tout the impact of impact of high AC).

In fact that would be another great thread, "A true examination of increasing AC on effective hp" - because what's often left out of those comparisons is that monster attack comes in a range and increasing AC to take 1 enemy from .1 to .05 chance to hit means you get no effective benefit on that already had lower attack.

[Mellack]

That feels alot like saying "if enemies always roll 1's then AC offers no benefit". While technically true in both instances, it's irrelevant. Probabilistically speaking, we can't base the effectiveness of something on a single case unless that case is guaranteed to happen.



Maybe you can explain the relevance of this rather obvious "inisght" to the rest of the discussion?



Nope.

I wasn't basing the effectiveness on their roll, but on the AC. If your AC is so high that it is impossible to be hit, then resistance has no benefit.

You said that you assume resistance doubles the effective HP. That should not be assumed. When you are fighting things that do not do BPS damage, barbarian resistance gives nothing. It was a comment that went against your assumption.

[Nhorianscum]

Mirror image stops the same % of final damage after defense regardless of AC.

This can mean it blocks "less" damage but it's a static modifier not a diminishing one.

[x3n0n]

IThat is with all other parameters equal, it has the same impact on effective hp no matter what your AC.

Is there a definition somewhere for effective HP?

Something like "number of attacks required to kill you on average"?

[Mellack]

Is there a definition somewhere for effective HP?

Something like "number of attacks required to kill you on average"?

I have to agree with X3n0n. It might be a good idea to first define the terms. What exactly is being meant by "effective hp" in this discussion? How many times you can be hit before dropping? How many hp that actually hit you can you absorb? How many times you can be attacked before hitting zero? Those are not the same.

[Chugger]

I agree with those who have brought up the action-economy consideration of whether or not to cast MI.

5e's peculiar AE and mastering it is a major factor in good optimization/winning conflicts. Fights tend to be very fast and over in a few rounds. A caster landing a control spell is usually far, far more effective than burning a turn to cast a buff. Getting an enemy controlled or damaged is usually far more important than other things, because an enemy that can't fight (controlled or dead) is not harming the party - and this is the best way to deal with the lack of effective healing aspect of 5e - not taking damage in the first place. An enemy not being able to control or cast nasty spells and so on is also obviously a high priority. A debuff is also usually more effective than casting a buff in combat - provided the debuff has impact.

Now before a fight, that's another matter. Buffing is fine.

For a very long fight, maybe buff - except controlling or killing enemies is better. If what the enemy does every time it hits is super nasty - like it poisons - then MI might make a lot of sense. There are exceptions, but overall I'd keep the basic AE of 5e in mind when trying to pick good tactics.

The other thing to keep in mind re MI, in case someone hasn't mentioned it, is that as we level up, the creatures we face tend to get several to even many attacks. A smart DM will burn his lower-damage attacks zotzing all the mirror images first, then hit with his big attack(s) when the images are gone or mostly gone. Mobility, blur, an effective tank (with compel duel, cavalier mark, ancestral barb feature, or something like that) can be better than the caster doing MI. A sorc can always quicken cast Blade Ward if he's in trouble and still use his action to be effective; sometimes this makes tons of sense.

The most insane use for MI comes from a Treantmonk video on Arcane Trickster. You take sentinel feat, you cast MI, and if an enemy attacks one of your images, Sentinel triggers, giving you a reaction attack, at least a RAW reading of it works (some DMs will not accept this, but if you took the feat, I don't see why not). This gives the AT a second sneak attack on the round (not the turn, but you can have two s.a.'s on a round if one is a reaction attack) - if he qualifies for a s.a. (an ally is near the target, he's cast shadow blade, the target has been faerie fired, and so on).

[Frogreaver]

I'm interested in seeing what your math looks like, as intuitively I agree with the other posters that a higher AC would seem to make mirror image less useful. I'm also interested in your focus on effective HP.

I mean if the question is, should I cast something that doubles my effective hp by preventing say 1 attack on average - I'm with you that using the slot and action for that is likely not "worthwhile"


It occurs to me that HP decrease in value as AC increases, as a higher AC makes it less likely you'll lose as many HP. A high-AC character has more effective HP, because they can make each HP go farther. HP are subject to diminishing returns; the first 50 HP are worth more than the next 50, as you might not need that second 50.

I tend to agree. I would say effective hp / defenses in general are important up to a certain threshold but after that such things are basically useless. Effective hp is a good expected value measure, but we still need to ensure worst case risk scenarios are minimized. For example a wizard with a 5% chance to be hit and 10 hp has 200 effective hp vs attacks (excluding crit damage). That's alot of effective hp but 1 hit from most anything will down him.


Perhaps your point is that mirror image gives you fewer HP when you have a high AC, but because your higher AC makes each HP go farther, the smaller number of HP is effectively greater. But then you're talking about adding a similar number of effective HP to a character that's starting with a larger pool of effective HP to begin with, due to their high AC. And my response is that the same number of effective HP is still less valuable to the high AC character than it would be to the low AC character, because of the diminishing returns of HP.

This is very close. The only difference is I'm saying it adds an amount of effective hp proportional to a variable that depends only on your dex and the number of attacks you take. Which means that the higher your effective hp the greater impact mirror image has on your effective hp. In terms of effective hp, mirror image doesn't have diminishing returns. (Although I think you are right that effective hp itself diminishes in importance the more you have of it - but that isn't a reflective of mirror image having diminishing returns on effective hp)


Granted, that was an extreme example, with a wide disparity in AC, but even so, the spell was much more useful to the lower AC character. Alice did not receive as large of a boost in eHP, and the extreme difference highlighted how much less useful each additional HP would be to her anyway. With a smaller disparity, the difference will be smaller, but I don't see how it would completely disappear.

Note: on previewing this post, I see I've missed several posts since I started typing. I'll go ahead and submit this now anyway, as the points stand.

I plan to come back to your example scenario

[heavyfuel]

In the absolute sense I agree, but effective hp is a relative measurement. If the number of hits you would take goes down as AC increases then mirror image needs to affect proportionally fewer number of hits in order for it's impact on effective hp to remain constant.

But that's my point. A Wizard with no (mage) armor has four times their HP in Effective HP (EHP) upon casting. Since only 1 in 4 attacks will hit them, their effective HP is quadrupled. After an image is destroyed, their EHP is now three times larger than their HP. After another image is destroyed, it's now only twice as large.

However, once you start putting on armor, the likelyhood that an attack that wouldn't hit you anyway but can still destroy an image increases.

Let's say a single attack missed your AC but still destroyed your image. You wouldn't be hit by this attack anyway, so the spell offered no EHP. But now, you only have 2 images left, so even if the images do avoid attacks, you're still only gettin thrice your HP in EHP (as opposed to quadruple)

You keep saying you have the math to back up your claims. I say it's time you presented it.

[Lunali]

I think the biggest reason mirror image is "weakened" by high AC is that the higher your AC, the less you need x% more effective hp.

[LudicSavant]

Non-Dex AC has a direct impact on the probability that an image gets expended without mitigating a hit.

[Hellpyre]

I'm going to go ahead and echo the other posters: please show your math. When I ran the numbers (I want to say two years or so ago), I both worked through a pure math example and ran an excel-simulated batch of 1000 attacks against AC 10-20, with Dex values for armor at +0, +2, and +4, with and without MI. The results reflected what the general consensus on MI is, in terms of it scaling the amount of hits avoided by casting it.

[Frogreaver]

Non-Dex AC has a direct impact on the probability that an image gets expended without mitigating a hit.

Yes. But the higher your AC the fewer hits you take and thus the fewer hits you need to mitigate in order to have the same proportional impact on effective hp. You are leaving out half the effective hp calculation only talking about fewer hits being mitigated.

[Mellack]

Yes. But the higher your AC the fewer hits you take and thus the fewer hits you need to mitigate in order to have the same proportional impact on effective hp. You are leaving out half the effective hp calculation only talking about fewer hits being mitigated.

1. Define how you are determining "effective hp".
2. Show your math.

Otherwise the conversation is just going in circles.

[Frogreaver]

I'm going to go ahead and echo the other posters: please show your math. When I ran the numbers (I want to say two years or so ago), I both worked through a pure math example and ran an excel-simulated batch of 1000 attacks against AC 10-20, with Dex values for armor at +0, +2, and +4, with and without MI. The results reflected what the general consensus on MI is, in terms of it scaling the amount of hits avoided by casting it.

That's not something I'm disagreeing with. But when I'm talking effective hp granted by MI and you are talking hits avoided via MI... I think it should be pretty obvious what the problem is.

[Vegan Squirrel]

Going back to my previous post with Alice and Bob...


I'm interested in seeing what your math looks like, as intuitively I agree with the other posters that a higher AC would seem to make mirror image less useful. I'm also interested in your focus on effective HP.

It occurs to me that HP decrease in value as AC increases, as a higher AC makes it less likely you'll lose as many HP. A high-AC character has more effective HP, because they can make each HP go farther. HP are subject to diminishing returns; the first 50 HP are worth more than the next 50, as you might not need that second 50.

This post is getting confusing, so I'm going to add an example. Alice and Bob both have 45 HP and a Dexterity of 10, but Alice has an AC of 20 and Bob has an AC of 10. They exclusively fight Green Monsters that attack at +4 for 20 damage. They hit Alice on a roll of 16 or higher, or 25% of the time, dealing DPR of 5. They hit Bob on a roll of 6 or higher, or 75% of the time, dealing DPR of 15. Bob can survive 3 rounds of attacks, enough that he may or may not go unconscious during a given fight; since Green Monsters deal 20 damage, we can say that his effective HP against them is 60 eHP. Alice can survive 9 rounds of attacks, which is plenty; since Green Monsters deal 20 damage, we can say that her effective HP against them is 180 eHP.

Perhaps your point is that mirror image gives you fewer HP when you have a high AC, but because your higher AC makes each HP go farther, the smaller number of HP is effectively greater. But then you're talking about adding a similar number of effective HP to a character that's starting with a larger pool of effective HP to begin with, due to their high AC. And my response is that the same number of effective HP is still less valuable to the high AC character than it would be to the low AC character, because of the diminishing returns of HP.

Alice and Bob both cast mirror image. Their mirror duplicates have an AC of 10, meaning they are hit 75% of the time, like Bob. When Bob's duplicate is hit, that attack would have hit Bob 100% of the time, meaning three duplicates grant him 60 effective HP, bringing his total to 120 eHP. He can now survive the fight easily. When Alice's duplicate is hit, that attack would have hit Alice 33% of the time, meaning three duplicates grant her 20 effective HP, bringing her total to 200 eHP. That... doesn't do much.

Granted, that was an extreme example, with a wide disparity in AC, but even so, the spell was much more useful to the lower AC character. Alice did not receive as large of a boost in eHP, and the extreme difference highlighted how much less useful each additional HP would be to her anyway. With a smaller disparity, the difference will be smaller, but I don't see how it would completely disappear.

Note: on previewing this post, I see I've missed several posts since I started typing. I'll go ahead and submit this now anyway, as the points stand.

Mirror Image, in the example I used, gave Bob 60 eHP, or 3 Green Monster attacks, which is equal to the number of Green Monster attacks that would have hit him over 4 rounds. Alice gained only 20 eHP, or 1 Green Monster attack, but that is also equal to the number of Green Monster attacks that would have hit her over 4 rounds. So, in effect, they each gained 4 attacks' worth of effective HP from the spell.

This does bring me back to the same previous argument, however. Those 4 attacks' worth of effective HP are still worth more to the low AC character than they are worth to the high AC character. Bob could only withstand 3 attacks before; 4 more is a big deal (adding 60 eHP to 45 HP is great!). Alice could already withstand 9 attacks, so adding 4 more is much less likely to make a difference (adding 20 eHP to 180 HP is underwhelming). The effect of diminishing returns with each marginal HP (or effective HP) is very real.

[bid]

The whole point of this thread is that if 2 character have the same dex and take the same number of attacks then their effective hp will increase by the same percentage when casting mirror image regardless of the players AC values. That sounds quite a bit different than your claim above.
The easy non-math is this:

1 - you have no AC other than Dex and 2 mirror images: average damage for first hit goes from 1-in-1 to 1-in-3.

2 - you have good AC (twice as good as no AC) and 2 mirror images: average damage goes from 1-in-2 to 1-in-4.

One is 3 times better while the other 2 times.

Even if it adds the same amount, there's a big difference between adding 10 to 10 and adding 10 to 90. One doubles the result, the other is negligible.

[x3n0n]

That's not something I'm disagreeing with. But when I'm talking effective hp granted by MI and you are talking hits avoided via MI... I think it should be pretty obvious what the problem is.

Can you help me, then? I still don't know what "effective HP granted" is.

I would assume that "expected number of hits avoided by using MI" times "expected damage per hit" is "expected damage avoided by using MI", which is my best guess at a definition for "effective HP from MI".

If not, what am I missing?

[cutlery]

I would find it interesting to see numbers ran not on now Mirror Image works by RAW, but rather, how I think it is run at many tables.



I think you're right - I have certainly played when attacks were rolled against my character with mirror up and I wasn't told about it or how many there were.

I think mirror image that works like it is often played would not be that bad a spell; you can conceivably burn though them in one round of attacks from one enemy in Tier 2; in which case it is an action to counter an action.


Full caster wizards that are not bladesingers, it's probably worth using even until tier 3.

For bladesingers, it's an excellent way to train your DM not to attack your character, which can have other interesting effects.

[Hellpyre]

That's not something I'm disagreeing with. But when I'm talking effective hp granted by MI and you are talking hits avoided via MI... I think it should be pretty obvious what the problem is.

I'm not so sure. The eHP gain from mirror image it based solely on the percentage of hits you avoid with it versus without it. If you think avoiding, say 20% of hits with MI is going to give you different eHP multipliers for different AC, you are just incorrect. 20% more hits avoided gives you 20% more eHP, no matter what your percentage of avoiding other hits was. And that's the point I'm getting at here.

The percentage of hits avoided as a result of using MI vs not using MI decreases as your non-DEX AC increases. This is the same as saying that the eHP multiplier of MI decreases. I'm not looking at some absolute value here when I say hits avoided - I'm looking at what difference MI itself brings to the table. And the math disagrees with what you are saying.

[Frogreaver]

Some Math:

C = chance to be hit
C' = chance image is hit
M' = chance you are targeted instead of image (3 images)
M'' = chance you are targeted instead of image (2 images)
M''' = chance you are targeted instead of image (1 images)
D = average damage of attack

Against 1 attack let's calculate the effective Damage you take with mirror image.
1. M'*C*D

Now let's calculate the effective Damage you take without mirror image.
2. C*D

Now let's calculate the effective Damage factor
3. (M'*C*D) / (C*D) = M'

Since we know M' = .25 then mirror reduces effective damage by .25 against all AC's in the case of 1 attack. (*Note there is no dependency on AC or chance to hit here - probably not surprising).

------------------------------------------------------------------------------------------------------------

Against 2 attacks let's calculate the effective Damage you take with mirror image (this is more complicated)
There are 7 parts:
1: 2*D*[M'*C]^2
2: (M'*C*D)*(M'*[1-C])
3: (M'*C*D)*([1-M']*C')
4: (M'*C*D)*([1-M']*[1-C'])
5: (M'*[1-C])*(M'*C*D)
6: ([1-M']*C')*(M''*C*D)
7: ([1-M']*[1-C'])*(M'*C*D)

Adding these together and rearranging a bit we get
= (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C') M'(1-C)] + (M'*C*D)*[M'C + M'(1-C) + (1-M')(M''/M')C' + (1-M')(1-C')]
= (M'*C*D)*(1) + (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C') M'(1-C)] + (M'*C*D)*[(1-M')(M''/M')C' - (1-M')C']
= 2M'*C*D + (M'*C*D)(1-M')(C')(M''/M' - 1)

Now let's calculate the effective Damage you take without mirror image for the 2 attack scenario.
2. 2*C*D

Now let's calculate the effective Damage factor
3. [2M'*C*D + (M'*C*D)(1-M')(C')(M''/M' - 1)] / [2CD]
=(M')(1+(0.5)(1-M')(C')(M''/M'-1)

As can be seen from this the effective Damage factor doesn't depend on C. Therefore, we have found that the effective Damage factor for mirror image is independent of C.

Edited: had incorrect variable accidently type in 2 places. Nothing changes with calc, was a keying error.

[Hellpyre]

Now let's calculate the effective Damage factor
3. (M'*C*D) / (C*D) = M'
Well there's your problem right there. You've dropped your C' already on the assumption that it isn't relevant. This is where errors are creeping in on you.

To put in some numbers, lets assume say, AC 15+2DEX for final AC 17. We'll go against a +2 attack to make math nice and simple here. Say 10 damage on a hit.

At +2, they have a hit on 15+, so C=.3 here

C*D=3 dpr
M' is .25, as it works on a 6+, but your enemy hits the image on a 10+ for a C' of .55

C' - C= .25, which looks nice for a second, except that's the percentage of times when you MI does nothing. In this case, M'*(1-[C'-C]) is getting your actual reduction from MI

Lets bump up AC by two points, non-Dex

C here is a hit on 17+ for a total of .2
C' remains at .55
C'-C=.35, and we see that the MI chance of doing nothing increases, decreasing the actual eHP gained accordingly.

[Mellack]

Some Math:

C = chance to be hit
C' = chance image is hit
M' = chance you are targeted instead of image (3 images)
M'' = chance you are targeted instead of image (2 images)
M''' = chance you are targeted instead of image (1 images)
D = average damage of attack

Against 1 attack let's calculate the effective Damage you take with mirror image.
1. M'*C*D

Now let's calculate the effective Damage you take without mirror image.
2. C*D

Now let's calculate the effective Damage factor
3. (M'*C*D) / (C*D) = M'

Since we know M' = .25 then mirror reduces effective damage by .25 against all AC's in the case of 1 attack. (*Note there is no dependency on AC or chance to hit here - probably not surprising).

------------------------------------------------------------------------------------------------------------

Against 2 attacks let's calculate the effective Damage you take with mirror image (this is more complicated)
There are 7 parts:
1:
2:
3:
4:
5:
6:
7:

Will fill them in shortly.

Your calculations completely ignore the odds that an image is hit when the character would not be and are therefore incorrect.

[Edea]

Somewhat off-topic, but what sorts of actions are available that normally put your AC in the crapper in exchange for a benefit?

I know of Reckless Attack (and mirror image is actually not a concentration spell, a barbarian could hold onto it while raging), but barbarian + spellcaster is just discouraged by the devs at every turn.

[Frogreaver]

Your calculations completely ignore the odds that an image is hit when the character would not be and are therefore incorrect.

Effective Damage is based on the chance you get hit times the average damage you take when you are hit. That was calculated correctly. It seems the factor you want to apply happily gets cancelled out.


Well there's your problem right there. You've dropped your C' already on the assumption that it isn't relevant. This is where errors are creeping in on you.


Incorrect. C' isn't relevant in the case you take a single attack. M' is the factor for you being targeted instead of the image and that is accounted for.

[Valmark]

Somewhat off-topic, but what sorts of actions are available that normally put your AC in the crapper in exchange for a benefit?

I know of Reckless Attack (and mirror image is actually not a concentration spell, a barbarian could hold onto it while raging), but barbarian + spellcaster is just discouraged by the devs at every turn.

Mmm... Pinning someone through the Grappler feat? If that can be considered a benefit.

Dropping Prone probably. Can't think of more at the moment.

As for the matter at hand, as others have said: your calculations completely ignore C' (Frogreaver's I mean).

[Mellack]

Effective Damage is based on the chance you get hit times the average damage you take when you are hit. That was calculated correctly. It seems the factor you want to apply happily gets cancelled out.

I have asked multiple times, as have others, exactly how do you calculate this effective hit points? Where did you come up with that?

How do you determine the number of images still in existence for an attack? You have nothing that is accounting for the changing amount of images you will have from attacks that were not part of the times you would be hit. Ignoring that means you calculations are not a proper measure of the events.

[Lunali]

Somewhat off-topic, but what sorts of actions are available that normally put your AC in the crapper in exchange for a benefit?

I know of Reckless Attack (and mirror image is actually not a concentration spell, a barbarian could hold onto it while raging), but barbarian + spellcaster is just discouraged by the devs at every turn.

Actually, I can't think of any offhand, all the ones I can think of, reckless attack included, don't affect AC, just chance to get hit.

[Frogreaver]

As for the matter at hand, as others have said: your calculations completely ignore C' (Frogreaver's I mean).

And it's posts like these why I don't start with math. 3 of you all in a row haven't even realized that C' doesn't impact the 1 attack case. It's the simplest case and if you can't even criticize it correctly I have little hope that the more complex case will be read, understood and properly criticized if something is actually incorrect.

[Valmark]

And it's posts like these why I don't start with math. 3 of you all in a row haven't even realized that C' doesn't impact the 1 attack case. It's the simplest case and if you can't even criticize it correctly I have little hope that the more complex case will be read, understood and properly criticized if something is actually incorrect.

Why it doesn't? Let's say MI AC is 13 and caster's AC is 15.

There are three cases:
- Attack rolls 12- (this is included in both C and C');
- Attack rolls 13+ (C')
- Attack rolls 15+ (C)

There is clearly a disparity between C and C' which matters on a single attack.

[Hellpyre]

And it's posts like these why I don't start with math. 3 of you all in a row haven't even realized that C' doesn't impact the 1 attack case. It's the simplest case and if you can't even criticize it correctly I have little hope that the more complex case will be read, understood and properly criticized if something is actually incorrect.

We have. You need to run the numbers instead of pure math and you'll see where you missed the sanity check. If the hit goes to the image, doesn't exceed C, but does exceed C', it hasn't mitigated anything. You are dropping it earlier than you need to and that inflates the end result. The value of C'-C is important, and dropping it out of your equation is where you end up without C impacting your final function.

[Frogreaver]

Why it doesn't? Let's say MI AC is 13 and caster's AC is 15.

There are three cases:
- Attack rolls 12- (this is included in both C and C');
- Attack rolls 13+ (C')
- Attack rolls 15+ (C)

There is clearly a disparity between C and C' which matters on a single attack.

For a single attack it doesn't matter if any images die or not -> Also known as: C' doesn't matter in the case of taking a single attack

Or trying this a different way.
In the case of you taking 1 attack only: How much damage do you take if your image is targeted instead of you? Answer: 0

[Vegan Squirrel]

I think the crucial point most of your critics are trying to get at is that the single attack case isn't what matters to them, in the final analysis.

[Frogreaver]

I think the crucial point most of your critics are trying to get at is that the single attack case isn't what matters to them, in the final analysis.

Maybe. Doesn't really excuse messing up the trivial single attack case that badly though, does it?


You need to run the numbers instead of pure math and you'll see where you missed the sanity check.

Wow...

[Hellpyre]

For a single attack it doesn't matter if any images die or not -> Also known as: C' doesn't matter in the case of taking a single attack

Or trying this a different way.
In the case of you taking 1 attack only: How much damage do you take if your image is targeted instead of you? Answer: 0

Yes - however, not every hit that kills the image will have hit you. If you take, say 10 attacks, which I've rolled with random.org and the ACs I mentioned earlier

13 - Hits MI misses PC <- point where your eHP gains drop because the effect prevents no damage but is used up
5 - Misses both
13 - Hits MI misses PC <- point where your eHP gains drop because the effect prevents no damage but is used up
8 - Misses both
17 - Hits both
19 - Hits both
4 - Misses both
15 - Hits both
4 - Misses both
1 - Misses both

You can see that there are points where the .75 dodge chance prevents no damage but can still be invoked. You running that as damage prevention skews all of your later numbers.

[Frogreaver]

Yes - however, not every hit that kills the image will have hit you. If you take, say 10 attacks, which I've rolled with random.org and the ACs I mentioned earlier

13 - Hits MI misses PC <- point where your eHP gains drop because the effect prevents no damage but is used up
5 - Misses both
13 - Hits MI misses PC <- point where your eHP gains drop because the effect prevents no damage but is used up
8 - Misses both
17 - Hits both
19 - Hits both
4 - Misses both
15 - Hits both
4 - Misses both
1 - Misses both

You can see that there are points where the .75 dodge chance prevents no damage but can still be invoked. You running that as damage prevention skews all of your later numbers.

I've accounted for that by looking at the effective damage you would have taken without the mirror image and comparing the effective damage taken with mirror image to it via ratio. Maybe the problem is you aren't familiar with using ratios to make such comparisons?

[JNAProductions]

I wouldn’t go calling others stupid when you’re making mistakes in your math like that.

Bad assumptions, even with technically correct math, do not reflect real gameplay.

[Vegan Squirrel]

Maybe. Doesn't really excuse messing up the trivial single attack case that badly though, does it?

I meant that their quibble wasn't with the math of the single attack case, but on the implications you seemed to be preparing to draw from it. I'll now withdraw from this discussion, as I don't find it worthwhile to read through your hostile tone to get to the merits of the underlying math. Good luck with the rest of this discussion.

[Valmark]

For a single attack it doesn't matter if any images die or not -> Also known as: C' doesn't matter in the case of taking a single attack

Or trying this a different way.
In the case of you taking 1 attack only: How much damage do you take if your image is targeted instead of you? Answer: 0

I still disagree but I think I'll wait for the math to be finished, to see if it makes sense later.

[Frogreaver]

I still disagree but I think I'll wait for the math to be finished, to see if it makes sense later.

It's been finished for a bit now.

[JNAProductions]

Why would you only check for one attack?

I certainly wouldn’t bother casting Mirror Image if I anticipated the combat being over before I took more than one attack.

[Frogreaver]

Why would you only check for one attack?

I certainly wouldn’t bother casting Mirror Image if I anticipated the combat being over before I took more than one attack.

Why wouldn't one start with the case of 1 attack and work your way up to more complex cases from there?

[Valmark]

Can you add some explanations to those calculations? For example how did you go from "M'C + M'(1-C) + (1-M')C' + M'(1-C)" to "1"?

You can get to "M'C+(1-M')+2M'(1-C)", but I don't see how you get 1.

[JNAProductions]

Because you can trivially prove that you’re wrong, with actual hard examples.

Take a PC of Dex 20, with Mage Armor. AC 18, Image AC 15.
The enemy makes four attacks-they roll (with bonuses) 15, 16, or 17 for all hits.
You lose all images, on average, with only a slim chance of retaining one. But they saved you no damage.

Take another PC-Dex 8, with Defensive Fighting, +3 full plate and +3 shield. AC 27, Image AC 9.
The enemy has +7 to-hit, so hits you on a 20, but your images on a 2.
The odds of your spell doing ANYTHING are pretty slim, since most attacks kill an Image without touching you.

[Frogreaver]

Because you can trivially prove that you’re wrong, with actual hard examples.

Take a PC of Dex 20, with Mage Armor. AC 18, Image AC 15.
The enemy makes four attacks-they roll (with bonuses) 15, 16, or 17 for all hits.
You lose all images, on average, with only a slim chance of retaining one. But they saved you no damage.

Take another PC-Dex 8, with Defensive Fighting, +3 full plate and +3 shield. AC 27, Image AC 9.
The enemy has +7 to-hit, so hits you on a 20, but your images on a 2.
The odds of your spell doing ANYTHING are pretty slim, since most attacks kill an Image without touching you.

There's one glaring problem. You've stipulated 1 specific combination of attack rolls out of like 80,000+. Your scenario reveals next to nothing probabilistically speaking.

[Valmark]

There's one glaring problem. You've stipulated 1 specific combination of attack rolls out of like 80,000+. Your scenario reveals next to nothing probabilistically speaking.

There are 80000+ scenarios like that one. And even if there weren't, that is a possible scenario that your math doesn't account for, thus proving it flawed. Besides the fact that some of that equation looks wrong as I pointed out in the previous post (I might be mistaken of course, which is why I asked).

[Frogreaver]

Can you add some explanations to those calculations? For example how did you go from "M'C + M'(1-C) + (1-M')C' + M'(1-C)" to "1"?

Good questions.

For this specific question figure out all the probabilities for everything to happen in the 1 attack case. You'll note each of those values represent the probability corresponding to 1 of those cases. Since the sum of all cases in probability always = 1 then that must =1.


You can get to "M'C+(1-M')+2M'(1-C)", but I don't see how you get 1.

Another great question.

The premise is that I can add some variable as long as I also subtract the same thing. So I simply add the variable I need to make 1 and subtract them out on the end. However, I was reviewing there and I think there is an error in that step as the 4th variable also isn't what I need that in order for it to become 1.

Good eye. I'll recalc that.

EDIT: Ended up being variable in first series at end was written incorrect. That variable was changed via copying pasting the first series down there but since the first was wrong it messed up that part as well. I've updated this. No actual changes in results.


There are 80000+ scenarios like that one. And even if there weren't, that is a possible scenario that your math doesn't account for, thus proving it flawed. Besides the fact that some of that equation looks wrong as I pointed out in the previous post (I might be mistaken of course, which is why I asked).

Expected value is pretty basic probability stuff. Comparing 2 algebraic ratios is common practice to determine what variables actually matter between them.

I'm at a loss that either of those basic concepts are being questioned. I'm at a loss that there's even a question about 1 out of 80,000 cases as if mentioning that case has any bearing on any results, especially when comparing via ratio has already cared for all the cases. I don't know what else to say on that. I'm at a complete loss that these incorrect counterpoints keep getting brought up again and again. I don't know how not to sound frustrated about that. Maybe you'll notice how I respond to actually valid points as yours about the algebra involved.

[Hellpyre]

https://docs.google.com/spreadsheets/d/1fl_Ni1wI7m2CXn0ZEbwhIsJ5rLaP08_owV4kPTZT1GI/edit?usp=sharing

Here, I've put together a rough spreadsheet (sorry in advance about the output being at the bottom). You can play with both target ACs and get results 500 simulated rolls at a time. You'll also note that as PC target AC increases but MI target stays the same, the expected and actual damage will change at different rates, trending towards 25% damage taken at PC's AC = MI's AC, and increasing as the values diverge.

[HolyDraconus]

I always thought mirror image was an ok low level defensive spell, that worked until you got up to higher levels where the damage dealers aren't stupid and can just close their eyes.

[Valmark]

Can you add some explanations to those calculations? For example how did you go from "M'C + M'(1-C) + (1-M')C' + M'(1-C)" to "1"?

You can get to "M'C+(1-M')+2M'(1-C)", but I don't see how you get 1.
Funny enough, I got it wrong here. It's not "M'C+(1-M')+2M'(1-C)" but "M'C+(1-M')C'+2M'(1-C)". Forgot a C'. Still doesn't make 1 (please when you modified it say so that I don't miss it like earlier).

This also shows me one thing- why are you considering "(1-M')C' "? That is the case where the mirror image isn't targeted AND its AC is hit. Why do we care of MI's AC when it's not getting targeted?

And thinking about it, why do we consider M'(1-C) twice? (I'll double check to have copied correctly after this).




Expected value is pretty basic probability stuff. Comparing 2 algebraic ratios is common practice to determine what variables actually matter between them.

I'm at a loss that either of those basic concepts are being questioned. I'm at a loss that there's even a question about 1 out of 80,000 cases as if mentioning that case has any bearing on any results, especially when comparing via ratio has already cared for all the cases. I don't know what else to say on that. I'm at a complete loss that these incorrect counterpoints keep getting brought up again and again. I don't know how not to sound frustrated about that.

...so, according to the OP you say that MI's effectiveness stays the same regardless of AC. Your math is meant to prove that.

I have 15 AC and 13 as MIAC (which has been cast before the fight), and take, say, 7 attacks.

Let's say attacks are 14 14 14 15 15 15 15. Let's say MI triggers on the first three.

MI blocked no damage that I took, proving itself to be a waste of slots.

If my AC was, say, 14 then MI would have blocked three attacks out of the seven that hit me. It would have blocked nearly half of the damage assuming equal damage.

MI's effectiveness changed a LOT (from 0% to 100%) by varying AC by 1.

In terms of effective hp, I lost effective hp from something that normally would have done nothing.

Where is this accounted in your math?

[Frogreaver]

Funny enough, I got it wrong here. It's not "M'C+(1-M')+2M'(1-C)" but "M'C+(1-M')C'+2M'(1-C)". Forgot a C'. Still doesn't make 1 (please when you modified it say so that I don't miss it like earlier).

This also shows me one thing- why are you considering "(1-M')C' "? That is the case where the mirror image isn't targeted AND its AC is hit. Why do we care of MI's AC when it's not getting targeted?

And thinking about it, why do we consider M'(1-C) twice? (I'll double check to have copied correctly after this).

Made an update to a couple of keying errors I made in calcs. Maybe that will help you walk through it.


...so, according to the OP you say that MI's effectiveness stays the same regardless of AC. Your math is meant to prove that.

I have 15 AC and 13 as MIAC (which has been cast before the fight), and take, say, 7 attacks.

Let's say attacks are 14 14 14 15 15 15 15. Let's say MI triggers on the first three.

MI blocked no damage that I took, proving itself to be a waste of slots.

If my AC was, say, 14 then MI would have blocked three attacks out of the seven that hit me. It would have blocked nearly half of the damage assuming equal damage.

MI's effectiveness changed a LOT (from 0% to 100%) by varying AC by 1.

A few comments:

A given specific case as you describe here is nearly never going to match up with the expected value which is essentially the weighted average of all cases.

Mirror Image AC will affect the outputs just not player AC.


In terms of effective hp, I lost effective hp from something that normally would have done nothing.

Where is this accounted in your math?

The comparison ratio cares for that. (Step 3 in the formula)

[Valmark]

Adding these together and rearranging a bit we get
= (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C') M'(1-C)] + (M'*C*D)*[M'C + M'(1-C) + (1-M')(M''/M')C' + (1-M')(1-C')]
= (M'*C*D)*(1) + (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C') M'(1-C)] + (M'*C*D)*[(1-M')(M''/M')C' - (1-M')C']
= 2M'*C*D + (M'*C*D)(1-M')(C')(M''/M' - 1)

Now let's calculate the effective Damage you take without mirror image for the 2 attack scenario.
2. 2*C*D

Now let's calculate the effective Damage factor
3. [2M'*C*D + (M'*C*D)(1-M')(C')(M''/M' - 1)] / [2CD]
=(M')(1+(0.5)(1-M')(C')(M''/M'-1)

As can be seen from this the effective Damage factor doesn't depend on C. Therefore, we have found that the effective Damage factor for mirror image is independent of C.

Edited: had incorrect variable accidently type in 2 places. Nothing changes with calc, was a keying error.

Whoa, wait, no. You can't add a variable in a parenthesis then subtract it from another parentesis. You need to subtract it from the same.

[Frogreaver]

Whoa, wait, no. You can't add a variable in a parenthesis then subtract it from another parentesis. You need to subtract it from the same.

Okay baby steps!

M'CD(A + B)
= M'CD(A) + M'CD(B) +M'CD(B') - M'CD(B')
= M'CD(A+B') + M'CD(B-B')

That's all I did. I just didn't baby step through it.

[JNAProductions]

To be proved valid in this instance, it has to be true ALWAYS.

Doesn’t matter how likely the opponent is to roll 15-17 only. It’s possible.

And what about the Image AC 9 and PC AC 27 situation?
That means that any given attack at +7 has a 57/80 chance of hitting an Image, but only a 1/80 chance of hitting you.

[Frogreaver]

To be proved valid in this instance, it has to be true ALWAYS.

Doesn’t matter how likely the opponent is to roll 15-17 only. It’s possible.

And what about the Image AC 9 and PC AC 27 situation?
That means that any given attack at +7 has a 57/80 chance of hitting an Image, but only a 1/80 chance of hitting you.

If someone else wants to explain why what you are doing is incorrect that's fair game. I'm tired of fending off egregious fallacies. I think from now on I'll just note when it happens and move on.

[Valmark]

Made an update to a couple of keying errors I made in calcs. Maybe that will help you walk through it.

A few comments:

A given specific case as you describe here is nearly never going to match up with the expected value which is essentially the weighted average of all cases.

Mirror Image AC will affect the outputs just not player AC.



To be proved valid in this instance, it has to be true ALWAYS.


This.

On the other point, alright but assuming I understood correctly and you put +(1-M')(1-C') where is -(1-M')(1-C')?

You really need to go through the baby steps if you want to prove your point. At least the starting point of the equation needs to show all the variables- for example, where did the "2*D*[M'*C]^2" go? Can we have the full starting equation?

[JNAProductions]

If someone else wants to explain why what you are doing is incorrect that's fair game. I'm tired of fending off egregious fallacies. I think from now on I'll just note when it happens and move on.

Your stance is, correct me if I’m wrong, that the relative AC between your PC and your images is irrelevant in terms of damage blocked.

Which means that, over, say, five attacks against our foolish AC 27 Fighter, there is normally an about...
77% chance of being hit by one of those five attacks.
Any given attack has about a 71% chance of killing an Image when there are three left.
Meaning that there’s going to be about a 20% chance that the first three attacks hit images every time, and actually hit those images, rendering the spell useless.

And again-your stance is, as far as I can tell, a UNIVERSAL statement. Meaning a SINGLE example of it being wrong disproves it.

[Frogreaver]

This.

On the other point, alright but assuming I understood correctly and you put +(1-M')(1-C') where is -(1-M')(1-C')?

There is no -(1-M')(1-C'). It's from the 4th and 7th cases in the cases I list out above the main algebra.


You really need to go through the baby steps if you want to prove your point. At least the starting point of the equation needs to show all the variables- for example, where did the "2*D*[M'*C]^2" go? Can we have the full starting equation?

M'CD got pulled out to the front leaving 2M'C. That got broken out to M'C + M'C. You'll see the frist M'C in the first long set of parenthesis. The 2nd you will see in the 2nd long set of parenthesis.

The 2 is on front of that one because that case only happens once but it's the cases when both attacks hit the PC. Thus it gets weighted for 2 attacks instead of 1 attack like the other cases.

Apologies if not enough is present for you to step through on your own. I had thought I had included enough but if you've ever posted math I'm sure you know how that goes. I'm happy to answer any questions or help with any steps that you find confusing.

[Hellpyre]

If someone else wants to explain why what you are doing is incorrect that's fair game. I'm tired of fending off egregious fallacies. I think from now on I'll just note when it happens and move on.

For what it's worth, I agree with you here. It's ludicrous to try and pull single data points to disprove statistical trends.

That said, I've found something odd in messing with the data on my own some more. Given a uniform distribution, the data agrees with you that PC AC is irrelevant to the first attack. That, in fact, MI AC is basically irrelevant as well. But as soon as I go back to a randomised distribution, the trend towards divergence reappears. It may be a data artifact. I concede to you that AC is irrelevant as far as single attacks, and apologize that my data was flawed. Running more in-depth on my copy of Excel, I still see statistical divergence on later attacks relative to the AC gap.

[Frogreaver]

Your stance is, correct me if I’m wrong, that the relative AC between your PC and your images is irrelevant in terms of damage blocked.

Then this should settle the confusion, that is not my stance.

Further, even if someone were to make such a statement, it would obviously be about the AVERAGE damage blocked per trial after running infinite trials and thus the specific actual damage blocked in any given trial would be irrelevant to disproving the claim.


For what it's worth, I agree with you here. It's ludicrous to try and pull single data points to disprove statistical trends.

That said, I've found something odd in messing with the data on my own some more. Given a uniform distribution, the data agrees with you that PC AC is irrelevant to the first attack. That, in fact, MI AC is basically irrelevant as well. But as soon as I go back to a randomised distribution, the trend towards divergence reappears. It may be a data artifact. I concede to you that AC is irrelevant as far as single attacks, and apologize that my data was flawed. Running more in-depth on my copy of Excel, I still see statistical divergence on later attacks relative to the AC gap.

Thanks, and no problem. I technically went up through case 2 attack and extrapalated that the others should also be that way. I think it's a fair extrapolation but if you are getting something quite different in a montecarlo sim then can you drop it down to a 2 attack case and see what happens? I'm curious if it will show independent of the AC gap in that case (please make sure that you are leaving the mirror image ac static and only changing the relative player ac). If it doesn't lets delve into your methodology a bit.

[Valmark]

There is no -(1-M')(1-C'). It's from the 4th and 7th cases in the cases I list out above the main algebra.

M'CD got pulled out to the front leaving 2M'C. That got broken out to M'C + M'C. You'll see the frist M'C in the first long set of parenthesis. The 2nd you will see in the 2nd long set of parenthesis.

The 2 is on front of that one because that case only happens once but it's the cases when both attacks hit the PC. Thus it gets weighted for 2 attacks instead of 1 attack like the other cases.

Apologies if not enough is present for you to step through on your own. I had thought I had included enough but if you've ever posted math I'm sure you know how that goes. I'm happy to answer any questions or help with any steps that you find confusing.

If there is no -(1-M')(1-C') why did you talk about adding a variable as long as you subtract that same variable when I pointed it out then corrected putting that? Or rather, which variable did you add/subtract?

Yeah, the 1st case I got it. Looking at it, I'm still missing where you get (1-M')(M''/M')C'. Mainly the division.

[Frogreaver]

If there is no -(1-M')(1-C') why did you talk about adding a variable as long as you subtract that same variable when I pointed it out then corrected putting that? Or rather, which variable did you add/subtract?

The full added and subtracted variable was (M'CD)(1-M')C' or if your looking in the parenthesis it will read (1-M')C' in the 2nd set of parenthesis (3rd value) and at the very end.


Yeah, the 1st case I got it. Looking at it, I'm still missing where you get (1-M')(M''/M')C'. Mainly the division.

I multiply that by M'/M' and carry the numerator M' to the front of the parenthesis. This allows the variable to keep the M'CD prefix since basically all my terms have that. (both preceded by M'CD)

[Hellpyre]

Okay, having gone down the rabbit hole and created a fixed statistical set instead of a random set, I think I see where Frogreaver is coming from here. In terms of absolute damage blocked by MI, you can model the attacks blocked by each MI in terms of how long it lasts (I.E. its personal AC) and the percentage of attacks diverted to it (I.E. the number of images remaining). These two correlate without reference to the AC of the PC caster in question. It can block attacks without bias based on the player's AC. The issue I had communicating here appears to be based entirely in different methods of eHP calculation leading to different conclusions about the final multiplier.

Please correct me if I'm wrong here, Frogreaver, but your contention here is that the number of attacks, and therefore damage, absorbed by MI disregards the AC of the PC, correct? And so the absolute eHP it offers is the same regardless of non-Dex boosts to AC.

I believe most of the rest of us are looking at what it does multiplicatively to the eHP of the PC, and seeing that the multiplier goes down as non-Dex AC grows. So we would say that it gives less effective boost relative to the non-MI eHP as player AC grows.

[Valmark]

The full added and subtracted variable was (M'CD)(1-M')C' or if your looking in the parenthesis it will read (1-M')C' in the 2nd set of parenthesis (3rd value) and at the very end.

I multiply that by M'/M' and carry the numerator M' to the front of the parenthesis. This allows the variable to keep the M'CD prefix since basically all my terms have that. (both preceded by M'CD)

Alright, the division makes sense. Now... You added (M'CD)(1-M')C' and subtracted it. You brought M'CD out and in front, yes.

This still doesn't explain why it's two +(1-M')C' instead of a +etc. And a -etc., Correct?

[Frogreaver]

Okay, having gone down the rabbit hole and created a fixed statistical set instead of a random set, I think I see where Frogreaver is coming from here.

You are getting closer but I think still some flaws in our communication.


In terms of absolute damage blocked by MI, you can model the attacks blocked by each MI in terms of how long it lasts (I.E. its personal AC) and the percentage of attacks diverted to it (I.E. the number of images remaining).

I don't think I agree with your "absolute part".

I think it's a given that the amount of time MI lasts is independent of your non dex AC. I also think that the amount of attacks diverted to it is as well. But as has been noted numerous times, not every attack directed to MI is an attack that would have hit you. So I agree that not every attack diverted to it is reduced damage. In fact I would go a step further, the higher the non-dex ac and AC disparity is the less absolute damage it is going to reduce.

However, relative damage (ie final multipler) is a different story, because while you prevent less damage you are already taking less damage. My contention (and I believe the algebra) show that those less effects are independent of non-dex ac.


These two correlate without reference to the AC of the PC caster in question. It can block attacks without bias based on the player's AC. The issue I had communicating here appears to be based entirely in different methods of eHP calculation leading to different conclusions about the final multiplier.

I don't believe there are 2 ways of accurately calculating eHP?


Please correct me if I'm wrong here, Frogreaver, but your contention here is that the number of attacks, and therefore damage, absorbed by MI disregards the AC of the PC, correct? And so the absolute eHP it offers is the same regardless of non-Dex boosts to AC.

If I'm reading you correctly, I would be saying the opposite. The absolute damage reduction lowers as non-dex ac increases but the relative (ie multiplier) stays constant regardless of non-dex ac.


I believe most of the rest of us are looking at what it does multiplicatively to the eHP of the PC, and seeing that the multiplier goes down as non-Dex AC grows. So we would say that it gives less effective boost relative to the non-MI eHP as player AC grows.

Then the problem isn't the terminology but different conclusions.

In my equation I take the eDmg with mirror image and divide by the eDmg without. If we took the inverse that would be the factor this ability can raise our eHP by provided the encounter we are in takes us right down to 0 hp (possibly a large assumption - but i'm not sure there's another way to calculate it).


Alright, the division makes sense. Now... You added (M'CD)(1-M')C' and subtracted it. You brought M'CD out and in front, yes.

This still doesn't explain why it's two +(1-M')C' instead of a +etc. And a -etc., Correct?


Some Math:

Against 2 attacks let's calculate the effective Damage you take with mirror image (this is more complicated)
There are 7 parts:
1: 2*D*[M'*C]^2
2: (M'*C*D)*(M'*[1-C])
3: (M'*C*D)*([1-M']*C')
4: (M'*C*D)*([1-M']*[1-C'])
5: (M'*[1-C])*(M'*C*D)
6: ([1-M']*C')*(M''*C*D)
7: ([1-M']*[1-C'])*(M'*C*D)

Adding these together and rearranging a bit we get
= (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C') M'(1-C)] + (M'*C*D)*[M'C + M'(1-C) + (1-M')(M''/M')C' + (1-M')(1-C')]
= (M'*C*D)*(1) + (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C') M'(1-C)] + (M'*C*D)*[(1-M')(M''/M')C' - (1-M')C']
= 2M'*C*D + (M'*C*D)(1-M')(C')(M''/M' - 1)


Orange is the 1 I had originally due to case 2. Red is for the added and substracted. There shouldn't be any others.

[Hellpyre]

I think it's a given that the amount of time MI lasts is independent of your non dex AC. I also think that the amount of attacks diverted to it is as well. But as has been noted numerous times, not every attack directed to MI is an attack that would have hit you. So I agree that not every attack diverted to it is reduced damage. In fact I would go a step further, the higher the non-dex ac and AC disparity is the less absolute damage it is going to reduce.

However, relative damage (ie final multipler) is a different story, because while you prevent less damage you are already taking less damage. My contention (and I believe the algebra) show that those less effects are independent of non-dex ac.

Alright. I see where that conclusion would come from. The issue we seem to be having is that as far as I can tell from working the numbers on my end, the reduction from MI is independent on a per-attack basis, but the relative modifiers from the AC gap are generally larger than the general damage modifier that the high AC provides, so they end up interdependent across the whole of the spell.



If I'm reading you correctly, I would be saying the opposite. The absolute damage reduction lowers as non-dex ac increases but the relative (ie multiplier) stays constant regardless of non-dex ac.

Then I have misunderstood you here.



In my equation I take the eDmg with mirror image and divide by the eDmg without. If we took the inverse that would be the factor this ability can raise our eHP by provided the encounter we are in takes us right down to 0 hp (possibly a large assumption - but I'm not sure there's another way to calculate it).


I do take the inverse factor there, but that's a matter of personal taste and won't affect the math any.

[Valmark]

Oh good I was starting to think that I was seeing things. Yeah I'll check anything else when I next wake up.

One last question before I turn in: did you try doing the math for more attacks then Mirror Image can stop?

After this long I think I agree that If you only take one attack in a minute, Mirror Image's effectivness doesn't care about AC.

Same thing for two. Basically, If you don't take more attacks then the three Mirror Image can stop in a minute, AC doesn't matter for it.

Math should change once you consider 4+ attacks. Because in that case MI can definitely be wasted on attacks that would do nothing while other attacks that could have been stopped hit the character.

Note: I'm using logic, I didn't do any calculation. But if you look at my earlier example with seven attacks, it proves you wrong unless we consider the distinction between 3- attacks and 4+ attacks since your calculations were based on 1 and 2 attacks.

It seemed before that you said that you assumed that the math for two attacks would hold true for many more. This is probably why then we disagree and provided several examples for that. And also why Hellpyre's found that discrepancy.

Also, small note: a single case can be used to disprove statistical trends if the trends are based on wrong basis. But I think we are using different basis, rather then Frog's math being wrong.

[Hellpyre]

Also, small note: a single case can be used to disprove statistical trends if the trends are based on wrong basis. But I think we are using different basis, rather then Frog's math being wrong.

Disprove axioms, sure. But trends really don't ever apply to single cases in-and-of-themselves.

[Frogreaver]

Oh good I was starting to think that I was seeing things. Yeah I'll check anything else when I next wake up.

One last question before I turn in: did you try doing the math for more attacks then Mirror Image can stop?

Unless I do 3 attacks and find some awesome pattern to simplify the problem then 4 is almost assuredly to large a problem for me to want to attempt. Talking probably over 50 cases to consider.


After this long I think I agree that If you only take one attack in a minute, Mirror Image's effectivness doesn't care about AC.

Well that's at least a start :smallsmile:


Same thing for two. Basically, If you don't take more attacks then the three Mirror Image can stop in a minute, AC doesn't matter for it.

That's progress! :smallsmile: I'm not sure why you think having the chance to run out of images would change that. I thought we already all agreed that images would expire independently of non-dex AC?


Math should change once you consider 4+ attacks. Because in that case MI can definitely be wasted on attacks that would do nothing while other attacks that could have been stopped hit the character.

Note: I'm using logic, I didn't do any calculation. But if you look at my earlier example with seven attacks, it proves you wrong unless we consider the distinction between 3- attacks and 4+ attacks since your calculations were based on 1 and 2 attacks.

Logic is good. I'm not seeing why having the potential that all images have been used is going to change anything?

I also challenge the assertion that any example proved me wrong. Everything I've seen up to now has been seriously flawed. Most common reasonable mistake was to prove that "absolute" damage reduction dropped as non dex AC increased. The mistake there was to not evaluate how much damage taken was dropping due to the higher AC.


It seemed before that you said that you assumed that the math for two attacks would hold true for many more. This is probably why then we disagree and provided several examples for that. And also why Hellpyre's found that discrepancy.

Maybe. I think if we dug more into Hellpyre's we could resolve that discrepancy. I'm still curious about what happens on his in the case of 2 attacks.

By the way I would say it's more than an assumption - it's more of a logical deduction, but of course without proof there's always the possibility that it's wrong.

[Satori01]

For a single attack it doesn't matter if any images die or not -> Also known as: C' doesn't matter in the case of taking a single attack

Or trying this a different way.
In the case of you taking 1 attack only: How much damage do you take if your image is targeted instead of you? Answer: 0

The glaring flaw in this formulation is that MI doesn't just function per hit, but rather the spells future effectiveness is determined by how many images are left.
The spells effectiveness is a function of the outcome of the three hits.

So when Mirror Image negates a "false positive" hit...a hit that would not have hit the AC of the PC, but still strikes the Image....the real harm is the spell's ability to intercept a Real future hit is diminished.

A Missile defense system that air bursts on a false target, preventing no actual Missile strikes, and now is more likely to NOT trigger when the Critical Nuclear Stike is going to land in your capital is a bad defense system.

You are ignoring this.

[Segev]

Mirror image negates 4 hits, maximum.

If your AC and the mirror image's AC are the same (or, somehow, yours is lower, which I think is impossible), it always negates exactly 4 hits, unless nothing would have hit you at all during the duration.

If your AC is higher than the mirror image's, then it is possible for mirror image to negate fewer than 4 hits, as an image disappears to an attack that would not have hit you. This probability increases the greater the difference between your AC and that of your images.

If you wish to calculate "effective hp" that mirror image is adding, you have to calculate how many hp of damage each image negates. You do this by rolling the damage dice of the attacks that hit the images instead of hitting you. This is the effective hp the image gave you.

If the hit would have missed you but still hit the image, the effective hp the image gave you is 0, because the hit would have been a miss and done 0 damage to you.

High AC is still preferable to low AC. You having higher AC than your images will still net you more protection overall than you having the same AC as your images. Mirror image never results in worse outcomes for you just by being up; the worst that can happen is all 4 images are destroyed on attacks that would have missed you, so the spell adds 0 effective hp. But as the difference in AC increases, the amount of hp mirror image effectively adds decreases. If you're counting "effective hp" of increased AC and effective hp of mirror image together, increased AC reduces the amount of effective hp that mirror image adds, but mirror image is still adding 0 or more effective hp.

[bid]

When looking at Mirror Image's impact on effective hp, the math seems to show that it is independent of AC, except for ac from dex mod.


If the hit would have missed you but still hit the image, the effective hp the image gave you is 0, because the hit would have been a miss and done 0 damage to you.
Those 2 statements are in contradiction. The second one is most likely true, ergo the first one is false.

Even if you apply statistics...
A mirror image that covers all the hit range has the effective hp of 100% of the damage.
A mirror image that covers twice the hit range has the effective hp of 50% of the damage, since half the time we are in Segev's case.

[Xetheral]

By the way I would say it's more than an assumption - it's more of a logical deduction, but of course without proof there's always the possibility that it's wrong.

Your specific first-attack and first-two-attacks cases don't appear to have the same number of degrees of freedom as the general n-attack case. I think that's why your conclusions aren't generally applicable and you're getting different results than the numerical models.

Specifically, there is a variable that can be described as "odds Mirror Image prevents no damage on a particular attack because there are no images left". When you only look at the first (up to) three attacks, this variable is zero. But the moment n>3 , that variable becomes non-trivial and changes the pattern of the results. (That variable is also non-trivial in the case where n=1, but the single attack being looked at isn't necessarily the first attack made after casting the spell.)

Instances where the number of (effective) degrees of freedom depends on the values of the inputs can be subtle and hard to spot, and their implications even less clear. So it's possible I'm off base here, but from a first look that appears to be the source of the disagreement between your approach and the numerical model.

[Chugger]

Head-spinning, but we really shouldn't be ignoring the Action Economy issue.

The question is not only "is Mirror Image worth casting at higher levels when my AC is very high" (or something like that), it's also very much "is Mirror Image worth casting in combat instead of doing something else?"

Prep rounds were normal in older versions of DnD (I started in the 70s) - but for some reason fights tend to "just start" in 5e with no prep rounds. And fights tend to be over in not too many rounds.

If it's a three round fight and you mirror image on one of those rounds, you've reduced your offensive output capacity by 33%, at least in a simplistic way of thinking about this. You could have controlled; you could have AoE'd; you could have debuffed. Usually all of those things are better than casting Mirror Image.

I don't think you can mathematically model this Action Economy issue, at least not easily. It's an instinct thing most likely.

[Valmark]

Your specific first-attack and first-two-attacks cases don't appear to have the same number of degrees of freedom as the general n-attack case. I think that's why your conclusions aren't generally applicable and you're getting different results than the numerical models.

Specifically, there is a variable that can be described as "odds Mirror Image prevents no damage on a particular attack because there are no images left". When you only look at the first (up to) three attacks, this variable is zero. But the moment n>3 , that variable becomes non-trivial and changes the pattern of the results. (That variable is also non-trivial in the case where n=1, but the single attack being looked at isn't necessarily the first attack made after casting the spell.)

Instances where the number of (effective) degrees of freedom depends on the values of the inputs can be subtle and hard to spot, and their implications even less clear. So it's possible I'm off base here, but from a first look that appears to be the source of the disagreement between your approach and the numerical model.

So, this seems to be everything I wanted to say but in a much cleaned and sensible way.

@Hellpyre that thing you used to check for two attacks, can you do it for three, four and five? Something should change past three.

[Frogreaver]

Those 2 statements are in contradiction. The second one is most likely true, ergo the first one is false.

Even if you apply statistics...
A mirror image that covers all the hit range has the effective hp of 100% of the damage.
A mirror image that covers twice the hit range has the effective hp of 50% of the damage, since half the time we are in Segev's case.

I've only explained like 10 times how they aren't contradictions...


The glaring flaw in this formulation is that MI doesn't just function per hit, but rather the spells future effectiveness is determined by how many images are left.
The spells effectiveness is a function of the outcome of the three hits.

The glaring flaw as you put it is a reply to a post about a specific case that was identified at least 3 times in said post as if that case is all i've ever posted or talked about.

In the 2 attack case which I worked out the future effectiveness was included in M''.


So when Mirror Image negates a "false positive" hit...a hit that would not have hit the AC of the PC, but still strikes the Image....the real harm is the spell's ability to intercept a Real future hit is diminished.

The ability to block hits can be diminished. I've no where denied that. In fact, that falls right under my explanation of how it works. You block fewer hits but because you are being hit less the effective hp multipler remains constant. You are only focused on half the effective hp equation. You are ignoring the rest.


A Missile defense system that air bursts on a false target, preventing no actual Missile strikes, and now is more likely to NOT trigger when the Critical Nuclear Stike is going to land in your capital is a bad defense system.

Let's talk about what would happen if the attack in question triggered mirror image but the player had the same AC as the image. The image would either have dissappeared or not. If disappared (this is your image wasted on a false positive case) then the true positive would have disappeared also - leaving no future advantage over the false positive case. If the false positive wouldn't disappear then the true positive wouldn't either, thus leaving both back at square one.


You are ignoring this.

Dealt with the same exact objections worded in different ways about 10 times now. Nothing is being ignored.


Your specific first-attack and first-two-attacks cases don't appear to have the same number of degrees of freedom as the general n-attack case. I think that's why your conclusions aren't generally applicable and you're getting different results than the numerical models.

So you agree with my 2 attacks case? If so no one has shown that the numerical model actually aligns in the 2 attacks case...


Specifically, there is a variable that can be described as "odds Mirror Image prevents no damage on a particular attack because there are no images left". When you only look at the first (up to) three attacks, this variable is zero. But the moment n>3 , that variable becomes non-trivial and changes the pattern of the results.

Sure. But what makes you believe that the C in that variable won't be cancelled out similar to how the other Cs and (1-C)s were cancelled out? Believing that is quite a leap isn't it?


(That variable is also non-trivial in the case where n=1, but the single attack being looked at isn't necessarily the first attack made after casting the spell.)

Since I'm the one that set up the single attack case I can tell you with certainty that the single attack is the first attack made against the MI character after MI was cast.


Instances where the number of (effective) degrees of freedom depends on the values of the inputs can be subtle and hard to spot, and their implications even less clear. So it's possible I'm off base here, but from a first look that appears to be the source of the disagreement between your approach and the numerical model.

It's not clear the numerical approach works out correctly for the case of 2 attacks. Until we are sure that is in alignment how can you even begin to believe the higher cases shown by the numerical approach are unflawed?


Head-spinning, but we really shouldn't be ignoring the Action Economy issue.

The question is not only "is Mirror Image worth casting at higher levels when my AC is very high" (or something like that), it's also very much "is Mirror Image worth casting in combat instead of doing something else?"

Prep rounds were normal in older versions of DnD (I started in the 70s) - but for some reason fights tend to "just start" in 5e with no prep rounds. And fights tend to be over in not too many rounds.

If it's a three round fight and you mirror image on one of those rounds, you've reduced your offensive output capacity by 33%, at least in a simplistic way of thinking about this. You could have controlled; you could have AoE'd; you could have debuffed. Usually all of those things are better than casting Mirror Image.

I don't think you can mathematically model this Action Economy issue, at least not easily. It's an instinct thing most likely.

I'm all for action economy discussion, but maybe we agree on the basic principles first before moving to Action Economy?

[x3n0n]

Consider MI with only one image remaining. (Eventually, we can work upwards to full MI.)

Then, for a specific attack, consider the d20s: M is the mirror image die, H is the raw to-hit roll (that is, without modifiers).

If M is low, the attack targets you, and MI is unaffected (preserving its "future eHP").
If H is 1 or H + enemy to-hit bonus is less than image AC, the attack misses (preserving future eHP).
Otherwise (M is high, H is high enough to hit the image), this attack will generate the last burst of "eHP" for this casting of MI.

If H is 20, it generates one crit's worth of eHP.
If H + enemy to-hit bonus is greater than or equal to your actual AC, it generates one normal hit's worth of eHP.
Otherwise (H too low to hit you, high enough to hit the image), the image is destroyed while generating 0 eHP.

Frogreaver, does that analysis reflect your definition of eHP?

If not, what am I missing?

If so, the amount of expected eHP for the one-image-remaining case is determined by the number of d20 rolls of H that fit into each category: crit, hit you, hit image but not you. Then, note that the relevant factors do not depend on the number of remaining images, so you can just multiply by 3 to get the expected eHP of full MI.

(This analysis ignores the action economy and the possibility of not receiving enough attacks to hit the image at all, neither of which improves the eHP value of MI.)

[KorvinStarmast]

Rage does nothing for your effective HP if you are fighting a fire elemental. With the notable exception of Bear Totem level three feature. (Though this seems to make Mind Flayers an especially nasty counter to a barbarian...)

Is your question not that the effectiveness changes, but that some people perceive the amount of change differently? My problem is that it takes an action to cast. If it were casts with a bonus action I'd prefer it.
However, the two times I ever used it from level 1 through 7 (I got rid of it at 8) the party had done their due diligence and scouted out the enemy, and I cast it before we entered into battle.

That was handy, to be sure.

[heavyfuel]

Despite OP's numerous (doubtful) calculations and claims, did I miss something or did OP still not mention what exactly do they mean by "effective HP"?

[Frogreaver]

Despite OP's numerous (doubtful) calculations and claims,

Despite all the attempts to knock it down, my work is still standing. Do you have anything constructive to add? Maybe even some specific error you find with it?


did I miss something or did OP still not mention what exactly do they mean by "effective HP"?

Should be obvious. It seems to be for others.

[x3n0n]

Frogreaver, does that analysis reflect your definition of eHP?


Should be obvious. It seems to be for others.

I guessed at your definition (because it was still unclear to me) and did analysis from scratch based on my guess in the post above.

The implied definition of eHP in that analysis is something like "the number of damage that you didn't take, but would have if you had not cast MI."

If my guess at your definition doesn't match your actual definition, please correct it, as requested.

If you believe my analysis does not reflect my implied definition given here, feel free to explain how.

[Reynaert]

Let's try some math of my own:

eHP = our HP, divided by the chance of being hit. E.g. if we have 10 hp and an AC giving us a 25% chance of being hit, our eHP is 40.

Suppose that Mirror Image only gives one image, for simplicity.

Let's say the average attack damage coming your way is D.

First case: Image AC gives 50% hit chance, our own AC gives 50% hit chance.
That means exactly one attack is prevented, i.e. we prevented D damage.
This means we mitigated D actual hp by casting. If our hp is 10, our eHP is 20. With Image, we mitigated D/50% = 2*D effective HP.
Second case: Image AC gives 50% hit chance, our own AC gives 25% hit chance.
That means there's only a 50% chance the one attack prevented D damage, so we say it prevented D*0.5 damage.
This means we mitigated D*0.5 actual hp by casting. If our hp is 10, our eHP is 40. With Image, we mitigate (D*0.5)/25% = 2*D effective HP.

The same with variables:

First case: Image AC gives X hit chance, our own AC gives X hit chance.
That means exactly one attack is prevented, i.e. we prevented D damage.
This means we mitigated D actual hp by casting. If our hp is 10, our eHP is 10/X. With Image, we mitigated D/X effective HP.
Second case: Image AC gives X hit chance, our own AC gives Y hit chance.
That means there's only a Y/X chance the one attack prevented D damage, so we say it prevented D*Y/X damage.
This means we mitigated D*Y/X actual hp by casting. If our hp is 10, our eHP is 10/Y. With Image, we mitigate (D*Y/X)/Y = D/X effective HP.

Why Y/X chance:
For the damage to be prevented the attack has to be in the range that hit the X chance.
So the random number is between 0 and X instead of 0 and 1
This means that we have to scale Y to get the chance that it would have hit the Y chance as well.
Or for statisticians: The chance that the attack hit Y, given that it hit X, is Y/X


So yes, the added effective HP is the same.
But it's smaller in relation to the amount of effective HP you have from the larger AC.
So with larger AC it is less useful.

[Frogreaver]

Let's try some math of my own:

eHP = our HP, divided by the chance of being hit. E.g. if we have 10 hp and an AC giving us a 25% chance of being hit, our eHP is 40.

Yea, that sounds like the normal definition of eHP. The same definition should be used in the case of mirror image as well right?


Suppose that Mirror Image only gives one image, for simplicity.

Let's say the average attack damage coming your way is D.

First case: Image AC gives 50% hit chance, our own AC gives 50% hit chance.
That means exactly one attack is prevented, i.e. we prevented D damage.
This means we mitigated D actual hp by casting. If our hp is 10, our eHP is 20. With Image, we mitigated D/50% = 2*D effective HP.
Second case: Image AC gives 50% hit chance, our own AC gives 25% hit chance.
That means there's only a 50% chance the one attack prevented D damage, so we say it prevented D*0.5 damage.
This means we mitigated D*0.5 actual hp by casting. If our hp is 10, our eHP is 40. With Image, we mitigate (D*0.5)/25% = 2*D effective HP.

Why aren't you using the same formula for effective hp in the case of mirror image as you use for AC?


The same with variables:

First case: Image AC gives X hit chance, our own AC gives X hit chance.
That means exactly one attack is prevented, i.e. we prevented D damage.
This means we mitigated D actual hp by casting. If our hp is 10, our eHP is 10/X. With Image, we mitigated D/X effective HP.
Second case: Image AC gives X hit chance, our own AC gives Y hit chance.
That means there's only a Y/X chance the one attack prevented D damage, so we say it prevented D*Y/X damage.
This means we mitigated D*Y/X actual hp by casting. If our hp is 10, our eHP is 10/Y. With Image, we mitigate (D*Y/X)/Y = D/X effective HP.

It's nice to see some independent validation that non dex ac doesn't affect mirror images effective hp in the slightest.


Why Y/X chance:
For the damage to be prevented the attack has to be in the range that hit the X chance.
So the random number is between 0 and X instead of 0 and 1
This means that we have to scale Y to get the chance that it would have hit the Y chance as well.
Or for statisticians: The chance that the attack hit Y, given that it hit X, is Y/X

Well said.


So yes, the added effective HP is the same.
But it's smaller in relation to the amount of effective HP you have from the larger AC.
So with larger AC it is less useful.

It seems to me that you aren't actually using the same formula to assess the eHP provided via mirror image as you are for AC. I believe that's the issue. Thinking back I think that's been the issue with most posters methodology. The correct methodology is to compute your chance to be hit with mirror image and your chance to be hit without mirror image. You then divide that and it gives you the effective hp contribution of mirror image.

[Segev]

Despite all the attempts to knock it down, my work is still standing. Do you have anything constructive to add? Maybe even some specific error you find with it?



Should be obvious. It seems to be for others.

“My assumptions and definitions should be obvious, so please address my math by my assumptions and definitions,” is not good argumentation and is terrible proofing when people are actively asking you to clarify definitions.

Can you please look at x3n0n’s post and see if his definition of eHP reflects yours, and then address whether his analysis seems to come to correct or incorrect conclusions based on your math?

It is clear that there is difficulty for some of us (myself included) to determine what the exact take-away claim of your proof is, because we’re not sure if what we are defining as “effective hit points” matches your definition or not.

It also looks to me like, if it does, your math doesn’t do a good job supporting your position, because I cannot connect your equation to the claim.

“The proof of my somewhat controversial claim is left as an exercise for the reader” is not convincing. (It’s irritating in a text book with uncontroversial claims, and a guaranteed flop in a conference paper.)

[Satori01]

I've only explained like 10 times how they aren't contradictions...



The glaring flaw as you put it is a reply to a post about a specific case that was identified at least 3 times in said post as if that case is all i've ever posted or talked about.

In the 2 attack case which I worked out the future effectiveness was included in M''.



The ability to block hits can be diminished. I've no where denied that. In fact, that falls right under my explanation of how it works. You block fewer hits but because you are being hit less the effective hp multipler remains constant. You are only focused on half the effective hp equation. You are ignoring the rest.



Let's talk about what would happen if the attack in question triggered mirror image but the player had the same AC as the image. The image would either have dissappeared or not. If disappared (this is your image wasted on a false positive case) then the true positive would have disappeared also - leaving no future advantage over the false positive case. If the false positive wouldn't disappear then the true positive wouldn't either, thus leaving both back at square one.



Dealt with the same exact objections worded in different ways about 10 times now. Nothing is being ignored.


That is the second problem..the EHP value you are developing is a not the goal you should be striving for to illuminate the true value of Mirror Image.

The "true value" or Effective Value of a HP is influenced by so many situational variables that the Effective Value of HP even for the same character is not constant.

Rather then focusing on Effective Hit points the metric needs to be turning Hits to Misses.

Then MI at all it's various Image stages (having 1 image, 2 images etc), need to be compared to the the Miss chance value that
1): Taking the Dodge action adds through disadvantage
2): The static +2 AC bonus that Shield of Faith grants

Shield of Faith, as it successfully turns Hit to Misses, does not have the same diminishment of effectiveness that Mirror Image has.

You also seem to be ignoring the largest defense value that Mirror Image has: the ability to turn a Critical Hit, to turn a roll of a Natural 20 into a miss.

Frogreaver I would like to propose a thought experiment to you:

What would be the impact to the Mirror Image's ability to turn Hits to Misses if the AC of the Image was changed instead to be: "Caster's AC +2".

Your contention is the effectiveness of Mirror Image is not impacted by AC disparities between the Image and the Caster.

Does your contention hold true that AC disparities between the Image and the Caster are irrelevant, when an Image is less likely to be struck, and thus as a whole Mirror Image is more likely to redirect an attack from striking the caster to the Image track?

In terms of Mirror Image and Action Economy...as a player of a Cleric of Trickery, the Cast a Spell action required for Mirror Image, ain't nothing but a thing, for a cleric.

[Segev]

Let D be the expected damage of an attack.
Let N be the number of attacks the wizard ("you") has been targeted with since casting mirror image.
Let M(N) be the number of images left.
M(0)=3 by definition.

The maximum damage that a single casting of mirror image can negate is 3D. This requires that each image intercept a hit that would have hit you and dealt D damage.

For simplicity's sake, I will assume we can roll an (M+1)-sided die to determine if an image intercepts a hit or the hit targets the real wizard, rather than the d20 method in the spell. The d20 method can come close to the same thing, but the slight differences would complicate the math.

Let A be the probability that the d20 lands such that an attack would miss the wizard, but would hit an image. A = ([Wizard AC] - [image AC])*5%

Let P be the probability that the d20 lands such that an attack would hit the wizard.

The probability that the d20 lands such that an attack would miss an image even if it targeted the image is thus 1-(P+A).

Against a single attack, there are four possibilities while M(N) > 0:
The attack targets the real wizard, and no damage is mitigated: 1/(M+1) probability
Note: In this case, M remains the same.
The attack targets an image, and D damage is mitigated: P*[M/(M+1)] probability
Note: In this case, M is reduced by 1.
The attack targets an image, and hits the image, but no damage is mitigated: A*[M/(M+1)] probability
Note: In this case, M is reduced by 1.
The attack targets an image, and misses the image, so no damage is mitigated: [1-(P+A)]*[M/(M+1)] probability
Note: In this case, M remains the same.

M is reduced by 1 with probability (P+A)*[M/(M+1)] with each attack.

This can be used to calculate expected value of M(N); it will be a function of a probability raised to the Nth power, and trend towards zero without ever QUITE getting there.

There is a chance, every time you cast mirror image, that it does nothing. The probability that a given image does nothing (assuming duration of the spell is inconsequential) is related to the probability that that specific image is hit, and the probability that the hit would have missed the real wizard.

The probability that a specific image is the target is 1/(M+1), and the probability that, if it is the target, it is hit when the real wizard would not have been is A. Therefore, the probability that, on a specific attack, a specific image is determined to mitigate 0 damage is A/(M+1).

While there are three images, the probability that a given image does nothing before the number of images is reduced to 2 is (A/4)N3.

The probability that a given image does prevent D damage while there are 3 images is, therefore, 1-{(A/4)N3}.

The only real change is the expected value of NM and the denominator under A as M goes down.

As we can see, it remains dependent on A. The higher A is, the greater the
chance that mirror image does nothing. Remember: A is a function of the difference between the AC of the real wizard and the AC of the images. So the higher the real wizard's non-dex-based AC, the greater the probability that mirror image does nothing.


None of this is to say that increasing your AC is a bad thing. Just that mirror image does technically offer diminishing returns the greater the difference between your AC and [your Dex Mod +10] is.

[Frogreaver]

Let D be the expected damage of an attack.
Let N be the number of attacks the wizard ("you") has been targeted with since casting mirror image.
Let M(N) be the number of images left.
M(0)=3 by definition.

The maximum damage that a single casting of mirror image can negate is 3D. This requires that each image intercept a hit that would have hit you and dealt D damage.

For simplicity's sake, I will assume we can roll an (M+1)-sided die to determine if an image intercepts a hit or the hit targets the real wizard, rather than the d20 method in the spell. The d20 method can come close to the same thing, but the slight differences would complicate the math.

Let A be the probability that the d20 lands such that an attack would miss the wizard, but would hit an image. A = ([Wizard AC] - [image AC])*5%

Let P be the probability that the d20 lands such that an attack would hit the wizard.

The probability that the d20 lands such that an attack would miss an image even if it targeted the image is thus 1-(P+A).

Against a single attack, there are four possibilities while M(N) > 0:
The attack targets the real wizard, and no damage is mitigated: 1/(M+1) probability
Note: In this case, M remains the same.
The attack targets an image, and D damage is mitigated: P*[M/(M+1)] probability
Note: In this case, M is reduced by 1.
The attack targets an image, and hits the image, but no damage is mitigated: A*[M/(M+1)] probability
Note: In this case, M is reduced by 1.
The attack targets an image, and misses the image, so no damage is mitigated: [1-(P+A)]*[M/(M+1)] probability
Note: In this case, M remains the same.

M is reduced by 1 with probability (P+A)*[M/(M+1)] with each attack.

This can be used to calculate expected value of M(N); it will be a function of a probability raised to the Nth power, and trend towards zero without ever QUITE getting there.

There is a chance, every time you cast mirror image, that it does nothing. The probability that a given image does nothing (assuming duration of the spell is inconsequential) is related to the probability that that specific image is hit, and the probability that the hit would have missed the real wizard.

The probability that a specific image is the target is 1/(M+1), and the probability that, if it is the target, it is hit when the real wizard would not have been is A. Therefore, the probability that, on a specific attack, a specific image is determined to mitigate 0 damage is A/(M+1).

While there are three images, the probability that a given image does nothing before the number of images is reduced to 2 is (A/4)N3.

The probability that a given image does prevent D damage while there are 3 images is, therefore, 1-{(A/4)N3}.

The only real change is the expected value of NM and the denominator under A as M goes down.

As we can see, it remains dependent on A. The higher A is, the greater the
chance that mirror image does nothing. Remember: A is a function of the difference between the AC of the real wizard and the AC of the images. So the higher the real wizard's non-dex-based AC, the greater the probability that mirror image does nothing.


None of this is to say that increasing your AC is a bad thing. Just that mirror image does technically offer diminishing returns the greater the difference between your AC and [your Dex Mod +10] is.

I don’t know how many times I can say - I agree.

The absolute damage mitigated by mirror image decreases the higher the difference between you dex ac and your non dex ac. That’s the diminishing returns you are taking about. I agree with that.

However, The factor mirror image modifies effective hp by does not depend on your non dex ac.

These statements aren’t contradictory. They can both be true. I’m not sure why you and others keep trying to disprove me by repeatedly bringing up a point I agree with and acting like I don’t agree with it?

[x3n0n]

However, The factor mirror image modifies effective hp by does not depend on your non dex ac.

These statements aren’t contradictory. They can both be true. I’m not sure why you and others keep trying to disprove me by repeatedly bringing up a point I agree with and acting like I don’t agree with it?

FWIW, I am not trying to "disprove you". I still just don't know what "The factor mirror image modifies effective hp by does not depend on your non dex ac" means, mostly because I don't know what "effective hp" is.

Exploring the uses of MI is interesting, and I'm glad you brought it up. Before this thread, I had uncritically assumed it was among the best defensive spells without thinking about what it actually does.

From what I can tell, it does 2 things really well (under favorable conditions):
* It prevents critical hits (as mentioned by Satori01) and
* It prevents up to 3 hits overall, which can preserve your concentration on some other effect.

Some favorable conditions include
* A high Dex mod,
* Enough incoming attacks to actually benefit, and
* A small enough "bonus AC" to have the image-destroying hits not be "wasted".

Those conditions apply more to some situations than others, in which case it's a great tool for maintaining your concentration on some other effect. (Perhaps a light armor user with no other AC bonus who is trying to maintain concentration on Shadow Blade, for example.)

For others, it's subpar (in the extreme, a heavy armor user with no concentration effect to protect).

[heavyfuel]

Despite all the attempts to knock it down, my work is still standing. Do you have anything constructive to add? Maybe even some specific error you find with it?



Should be obvious. It seems to be for others.

Clearly you didn't do a good enough job at proving your claim that attacks that would miss due to armor but still destroy the image somehow also help your effective HP since people are still contesting that. You claim you've knocked it down plenty of times, but you're failing to specify the parameters you're using, which leads me to my next point.

Considering others have also asked for your definition of effective HP, it's definitely not obvious.

I'm no longer sure if you're making these claims in good faith or if you're just Challenge 5 Large Giant

[JNAProductions]

I'm going to try to plug some actual numbers in.

I'll use the Fighter with AC 27 and a Dex of 8, just to see if it's true in the extreme. We'll say they have 224 HP, assuming level 20 and Con 20.
The enemy will have +7 to-hit and do 1d10+5 damage (average of 10.5, 16 on a crit).

Without Mirror Image, it takes an average of 280 attacks to kill them.
Mirror Image will last an average of less than 10 attacks, in more than 99% of cases. The odds of one of those ten attacks being a hit is about 40%.
Taken together, that means we can add 40% of 20 (or 8) to the number of attacks needed to kill on average, for 288 attacks.
This is rough math, and the actual benefit is likely smaller.

If the Fighter is naked (AC 9 and Image AC 9) then the math changes, like so:
Without Mirror Image, it takes an average of 22 attacks to kill them.
Mirror Image will absorb exactly three attacks, since there's only a minute chance that you'll have an Image remaining after more than 10 attacks, let alone the more than 20 needed to kill the Fighter.
Taken together, that means the Fighter lasts 25 hits with Mirror Image, as compared to 22.
This math is a lot less rough, since it's easier.

So, the full armor Fighter gains about 3% on the number of hits needed to kill them. (Which is an overestimation, but works for this example.)
The naked Fighter gains about 14% on the number of hits needed to kill them.

So if EHP is based on the number of attacks needed to kill a PC, you gain a much larger benefit from Mirror Image when your AC is the same as compared to a much higher AC.

[Bobthewizard]

Some Math:

C = chance to be hit
C' = chance image is hit
M' = chance you are targeted instead of image (3 images)
M'' = chance you are targeted instead of image (2 images)
M''' = chance you are targeted instead of image (1 images)
D = average damage of attack

Against 1 attack let's calculate the effective Damage you take with mirror image.
1. M'*C*D

Now let's calculate the effective Damage you take without mirror image.
2. C*D

Now let's calculate the effective Damage factor
3. (M'*C*D) / (C*D) = M'

Since we know M' = .25 then mirror reduces effective damage by .25 against all AC's in the case of 1 attack. (*Note there is no dependency on AC or chance to hit here - probably not surprising).

Here's the issue I see with this math. Yes. No matter your AC, mirror image decreases your effective damage by 25%. The problem is that your expected damage is higher if you have a low AC. At higher AC, your expected damage per hit is lower so when you reduce it by 25% you get less benefit in terms of total HP. If mirror image lasted forever, the 25% reduction would be stable across AC's. But since it only lasts 3 hits, a 25% reduction is more important for someone with a low AC.

[Hellpyre]

Clearly you didn't do a good enough job at proving your claim that attacks that would miss due to armor but still destroy the image somehow also help your effective HP since people are still contesting that. You claim you've knocked it down plenty of times, but you're failing to specify the parameters you're using, which leads me to my next point.
...
I'm no longer sure if you're making these claims in good faith or if you're just Challenge 5 Large Giant

I mean, you're clearly misrepresenting his stance here, as he's made it clear that he feels they don't, but scale at the same rate that eHP from increasing AC on the player are scaling. He's been arguing in good faith (if occasionally somewhat heatedly, on both sides)

Anyways, after a lot of jiggering to model in a way that wouldn't crash Excel, I worked out a way to model the ratios that didn't expand my worksheets exponentially. So onto raw data:

A few notes first: I write it as AC, but what we really measure here is the to-hit number for the attacker.
All numbers are rounded off to a whole percentage.
Later clones take fewer hits, so they tend to stick around a bit longer. My ratios here are rough, but I woke up with the functions to run it - may reevaluate after I'm more awake

In the format of eHP effectiveness per clone (measured by hits the image takes without preventing a hit to the PC versus hits the PC would take) at 3, 2, and 1 images remaining, followed by the added eHP percentage caused by the AC gap:

AC 10 vs 10 - 100 100 100 0
AC 10 vs 11 - 93 94 95 10
AC 10 vs 12 - 86 87 90 22
AC 10 vs 13 - 78 80 84 37
AC 10 vs 14 - 70 73 78 57
AC 10 vs 15 - 62 65 70 83
AC 10 vs 16 - 53 56 62 120
AC 10 vs 17 - 43 47 53 175
AC 10 vs 18 - 33 37 43 267
AC 10 vs 19 - 23 25 30 550
AC 10 vs 20 - 12 13 17 1100


AC 12 vs 12 - 100 100 100 0
AC 12 vs 13 - 91 92 94 13
AC 12 vs 14 - 82 84 88 29
AC 12 vs 15 - 72 75 80 50
AC 12 vs 16 - 63 66 71 80
AC 12 vs 17 - 52 55 62 125
AC 12 vs 18 - 40 43 50 200
AC 12 vs 19 - 28 31 36 350
AC 12 vs 20 - 14 16 20 800

It becomes clear that the eHP boost from AC scales up at a different rate than the losses from wasted hits. At the extreme ends the difference grows pretty large, but within a few points the ratio stays pretty tight. Overall, no matter how much you invest in AC you can expect the amount of your eHP multiplier gained over a cast of MI scales down by at most around 30%. This doesn't go into what happens if you take more attacks than your images survive between casts, which is an obvious case that further reduces the value of MI in combat later on, but it does show that while both eHP mods progress, they don't do so at rates that end up cancelling out even over the course of a cast. Criticals, incidentally, push the ratios back closer for extreme cases (to-hit only on a 19 or 20 and such), but don't have much of an effect at more reasonable assumptions on hitting, so I've left them out of this. I can fully re-evaluate with them if you guys feel it's needed.

[heavyfuel]

I mean, you're clearly misrepresenting his stance here, as he's made it clear that he feels they don't, but scale at the same rate that eHP from increasing AC on the player are scaling. He's been arguing in good faith (if occasionally somewhat heatedly, on both sides)


I admit it's a bit confusing following this thread with a bunch of math, so I apologize if I've misrepresenting OP's stance.

Still, post #106 clearly shows that armor reduces the effectiveness of MI.

[Segev]

I don’t know how many times I can say - I agree.

The absolute damage mitigated by mirror image decreases the higher the difference between you dex ac and your non dex ac. That’s the diminishing returns you are taking about. I agree with that.

However, The factor mirror image modifies effective hp by does not depend on your non dex ac.

These statements aren’t contradictory. They can both be true. I’m not sure why you and others keep trying to disprove me by repeatedly bringing up a point I agree with and acting like I don’t agree with it?

I've read your posts and posts agreeing with you (I think they were, anyway), and I'm having trouble seeing what you mean by "The factor mirror image modifies effective hp by."

Let me see if I can reword things and ask if it's what you mean: Are you saying that mirror image increases the expected number of hits to kill you by the same amount regardless of how big the gap between your AC and the AC of the images is?

i.e., if mirror image increases it from 10 hits to 15 hits to kill you, it will increase it from 10 to 15 hits regardless of how far apart your AC is from that of your images?

If that's not what you're saying, I still am not sure what your "factor mirror image modifies effective hp by" is, nor what it translates to in real play terms.

[Frogreaver]

I'm going to try to plug some actual numbers in.

I'll use the Fighter with AC 27 and a Dex of 8, just to see if it's true in the extreme. We'll say they have 224 HP, assuming level 20 and Con 20.
The enemy will have +7 to-hit and do 1d10+5 damage (average of 10.5, 16 on a crit).

Without Mirror Image, it takes an average of 280 attacks to kill them.
Mirror Image will last an average of less than 10 attacks, in more than 99% of cases. The odds of one of those ten attacks being a hit is about 40%.
Taken together, that means we can add 40% of 20 (or 8) to the number of attacks needed to kill on average, for 288 attacks.
This is rough math, and the actual benefit is likely smaller.

If the Fighter is naked (AC 9 and Image AC 9) then the math changes, like so:
Without Mirror Image, it takes an average of 22 attacks to kill them.
Mirror Image will absorb exactly three attacks, since there's only a minute chance that you'll have an Image remaining after more than 10 attacks, let alone the more than 20 needed to kill the Fighter.
Taken together, that means the Fighter lasts 25 hits with Mirror Image, as compared to 22.
This math is a lot less rough, since it's easier.

So, the full armor Fighter gains about 3% on the number of hits needed to kill them. (Which is an overestimation, but works for this example.)
The naked Fighter gains about 14% on the number of hits needed to kill them.

So if EHP is based on the number of attacks needed to kill a PC, you gain a much larger benefit from Mirror Image when your AC is the same as compared to a much higher AC.

Your example varies the number of attacks taken by each character. I’ve already stated that The eHP multiplier depends on your number of attacks. Thus, Showing the eHP multiplier changes in an example where attacks aren’t the same doesn’t disprove me.


*snip*

I'm curious if you've actually ran the case where you take 2 attacks only? Does it align with the algebra I posted?


I've read your posts and posts agreeing with you (I think they were, anyway), and I'm having trouble seeing what you mean by "The factor mirror image modifies effective hp by."

Let E be your effective hp without mirror image. Then provided you take N attacks, D Damage, have C chance to be hit and have C' dex based AC there is some value F such that for N,C,C',D that f(N,C, C', D) = F, such that FE is your effective hp with mirror image. F is the multiplier. My contention is that f(N,C,C',D) = f(N,C') = F.


Let me see if I can reword things and ask if it's what you mean: Are you saying that mirror image increases the expected number of hits to kill you by the same amount regardless of how big the gap between your AC and the AC of the images is?

i.e., if mirror image increases it from 10 hits to 15 hits to kill you, it will increase it from 10 to 15 hits regardless of how far apart your AC is from that of your images?

That's not all i'm saying but that certainly is something that is true if I am correct.


If that's not what you're saying, I still am not sure what your "factor mirror image modifies effective hp by" is, nor what it translates to in real play terms.

hopefully, that's answered it.

[Hellpyre]

I'm curious if you've actually ran the case where you take 2 attacks only? Does it align with the algebra I posted?


I have, and it doesn't quite line up. After the first attack I see divergence based on the player AC outgrowing the divergence based on wasted hits, so the eHP isn't scaling without regard to it even before we hit the 4+ attacks case.

EDIT: Does anyone know a way to pull from an Excel sheet into something I can plug for regression analysis (in Wolframalpha or some such) Excel hangs when I try to use its internal regression function.

[x3n0n]

Let E be your effective hp without mirror image. Then provided you take N attacks, D Damage, have C chance to be hit and have C' dex based AC there is some value F such that for N,C,C',D that f(N,C, C', D) = F, such that FE is your effective hp with mirror image. F is the multiplier. My contention is that f(N,C,C',D) = f(N,C') = F.


I still don't even understand what the units are for E!

Can you give an example of two non-identical characters that have the same eHP and explain how you calculated that they have the same eHP? I don't even understand what the inputs are into such a calculation.

[Mellack]

Let me see if I can reword things and ask if it's what you mean: Are you saying that mirror image increases the expected number of hits to kill you by the same amount regardless of how big the gap between your AC and the AC of the images is?

i.e., if mirror image increases it from 10 hits to 15 hits to kill you, it will increase it from 10 to 15 hits regardless of how far apart your AC is from that of your images?


That's not all i'm saying but that certainly is something that is true if I am correct.




Then it would appear you are incorrect, as it is not true.

Our AC 29 caster is attacked by someone with a +9 to hit and does 10 damage. The caster has 100 hp. It takes 10 hits to kill them. Anything but a 1 hits an image, only a 20 hits the caster. Over 50% of the time, all the images will be destroyed before blocking a single actual hit to the caster. The vast majority of the time, the caster will take 10-11 hits to be killed.
Compare that to an AC 10 caster against the same opponent. Every hit taken by an image is one that would have hit the caster. That means on average it will take almost 13 hits to kill the caster.

[Hellpyre]

I still don't understand what the units even are for E!

Can you give an example of two non-identical characters that have the same eHP and explain how you calculated that they have the same eHP? I don't even understand what the inputs are into such a calculation.

E (and Ef) are both going to be integers, in terms of the raw damage your character can expect to take and survive. Well, rational numbers that are rounded up to integers anyways.

For example, a fighter with 100 HP and a barbarian in rage with 50 hp both have the same eHP against a sword, but would have differing eHP against a fireball. In this case, we're mainly concerned with what increasing the number your opponent needs to roll to hit you, and increasing your AC increases your eHP for that calculation by the difference in their required to-hit roll before and after the AC adjustment. So if you have 100 HP and 11 AC, against a monster with a +0 to hit, you have a baseline 200 eHP (100 hp / .5 chance to land the hit). If you up that AC to 16, you reach 400 eHP (100 / .25 chance to hit). We will usually want to reference eHP in regards to a monster's Damage Per Round, because eHP accounts for its chance to not deal damage to you (or to deal reduced damage).

[Segev]

Let E be your effective hp without mirror image. Then provided you take N attacks, D Damage, have C chance to be hit and have C' dex based AC there is some value F such that for N,C,C',D that f(N,C, C', D) = F, such that FE is your effective hp with mirror image. F is the multiplier. My contention is that f(N,C,C',D) = f(N,C') = F.



That's not all i'm saying but that certainly is something that is true if I am correct.



hopefully, that's answered it.

Well, let's test this with some experimental examples.

Let's go with 10 damage per hit from an attacker (we'll call him a fighter) with +9 to hit, with a 100 hp wizard who has a 20 AC and a Dex-based AC of 10. (The wizard has spent feats and resources to get plate mail and a shield, maybe.)

The fighter hits the wizard 50% of the time without mirror image, yielding 10*.5 = 5 expected damage per attack, and 20 hits to drop the wizard.

The wizard casts mirror image.

While there are three images, the expected damage per attack that is targeting the wizard is: .25*.5*10 = 1.25 damage per attack.
While there are two images, the expected damage per attack that is targeting the wizard is: .33*.5*10 = 1.66 damage per attack.
While there is one image, the expected damage per attack that is targeting the wizard is: .5*.5*10 = 2.5 damage per attack.

I'm going to use some slightly poor probability math for the next part; it works for "back of the envelope" but technically is going to be a bit off.

Our next goal is to figure out the expected number of attacks before the first image is destroyed. Then the expected number after that before the second one is, and then the expected number after that before the last one is.

Actually, I'm going to stop doing math, here, because I think I see what Frogreaver is arguing:

The expected number of attacks before an image is destroyed is dependent strictly on the dex-based AC. Since the above calculations are expected values with each count of images present, all you need to do is determine when each image is expected to vanish, which will not depend at all on whether its vanishing has protected the caster.

The reason this seems so misleading is because mirror image will have an increasingly high variance on its effectiveness as the gap between wizard AC and image AC grows. Essentially, its diminishing contribution means only its increasingly high variance keeps it relevant at all. AC overshadows mirror image for defensive purposes as it goes up without increasing the image AC. The expected value may stay the same, but it will vary more greatly in its actual effectiveness, and the number of times mirror image winds up doing nothing versus having an effect will grow.

[x3n0n]

E (and Ef) are both going to be integers, in terms of the raw damage your character can expect to take and survive. Well, rational numbers that are rounded up to integers anyways.

For example, a fighter with 100 HP and a barbarian in rage with 50 hp both have the same eHP against a sword, but would have differing eHP against a fireball. In this case, we're mainly concerned with what increasing the number your opponent needs to roll to hit you, and increasing your AC increases your eHP for that calculation by the difference in their required to-hit roll before and after the AC adjustment. So if you have 100 HP and 11 AC, against a monster with a +0 to hit, you have a baseline 200 eHP (100 hp / .5 chance to land the hit). If you up that AC to 16, you reach 400 eHP (100 / .25 chance to hit). We will usually want to reference eHP in regards to a monster's Damage Per Round, because eHP accounts for its chance to not deal damage to you (or to deal reduced damage).

Thank you! So the inputs are actual HP and probability of some attempt causing damage, and the units are "attempted damage" to KO the PC.

Given that, for a fixed enemy/PC pair, "expected number of attacks to KO the PC" is eHP/damage-per-hit. Yes?

If so, the OP claim should be equivalent to:

Attacks to KO unarmored PC with MI / Attacks to KO unarmored PC without MI
equals
Attacks to KO armored PC with MI / Attacks to KO armored PC without MI

All true?

[Frogreaver]

Then it would appear you are incorrect, as it is not true.

Our AC 29 caster is attacked by someone with a +9 to hit and does 10 damage. The caster has 100 hp. It takes 10 hits to kill them. Anything but a 1 hits an image, only a 20 hits the caster. Over 50% of the time, all the images will be destroyed before blocking a single actual hit to the caster. The vast majority of the time, the caster will take 10-11 hits to be killed.

so 20ish attacks here


Compare that to an AC 10 caster against the same opponent. Every hit taken by an image is one that would have hit the caster. That means on average it will take almost 13 hits to kill the caster.

and 13ish attacks here...

eHP multiplier will not be the same when looking at different numbers of attacks. That's something I've stated from the beginning. Based on this, why do you think an example that has 1 character take 20 attacks and and another take 13 is even relevant?

[Mellack]

so 20ish attacks here



and 13ish attacks here...

eHP multiplier will not be the same when looking at different numbers of attacks. That's something I've stated from the beginning. Based on this, why do you think an example that has 1 character take 20 attacks and and another take 13 is even relevant?

Why are you changing from hit to attacks? The original quote from you was about hits. In the first case it is an average of less than 11 hits. In the second case it is almost 13 hits. You said they would be the same, according to the quotes.

[Frogreaver]

Thank you! So the inputs are actual HP and probability of some attempt causing damage,

Doesn't account for barbarian rage, but seems correct.


and the units are "attempted damage" to KO the PC.

That seems correct. However, note you cannot change the scenario presented, so if that scenario doesn't yield PC death you can't just add a few more attacks on the end until PC is dead. Instead you must repeat that same exact scenario again and again till PC death. Not doing this is of the most common mistakes I'm seeing.


Given that, for a fixed enemy/PC pair, "expected number of attacks to KO the PC" is eHP/damage-per-hit. Yes?

Only if that's the scenario you are wanting to run. You could run a PC gets attacked one time scenario and compute the effective hp for that as well.


If so, the OP claim should be equivalent to:

Attacks to KO unarmored PC with MI / Attacks to KO unarmored PC without MI
equals
Attacks to KO armored PC with MI / Attacks to KO armored PC without MI

All true?

That doesn't appear to be a claim to me?


Why are you changing from hit to attacks? The original quote from you was about hits. In the first case it is an average of less than 11 hits. In the second case it is almost 13 hits. You said they would be the same, according to the quotes.

Without you pointing to the specific quote I have no idea if you are misremembering or if I mispoke or if the topic of the post you are referencing is something else entirely. I did go back and find what I think is most recent quote by me on the matter.


Your example varies the number of attacks taken by each character. I’ve already stated that The eHP multiplier depends on your number of attacks. Thus, Showing the eHP multiplier changes in an example where attacks aren’t the same doesn’t disprove me.

[Mellack]

Without you pointing to the specific quote I have no idea if you are misremembering or if I mispoke or if the topic of the post you are referencing is something else entirely. I did go back and find what I think is most recent quote by me on the matter.

I had already given your specific quote. See my post #114.

[Reynaert]

It seems to me that you aren't actually using the same formula to assess the eHP provided via mirror image as you are for AC. I believe that's the issue. Thinking back I think that's been the issue with most posters methodology.

Yes I am, maybe just in a slightly different way. I first calculate actual prevented damage, and then scale that number according using the same formula.

What I do is first calcuate how much actual damage is prevented by one image, then assume that's the equivalent of giving yourself that much extra actual hp, and then scale that to effective hp.

For example, suppose that in a specific given situation, mirror image prevents 20 actual hp of damage. That's exactly the same as casting a spell that gives you 20 actual temp-hp.

So if you scale that to effective hp according to your AC/chance to be hit, then *after* that scaling, it turns out the *absolute* effective HP granted by mirror image does not depend on your armour AC.

However, if you have higher AC you also have more effective HP, so the relative impact of the extra effective HP given by mirror image is less.


The correct methodology is to compute your chance to be hit with mirror image and your chance to be hit without mirror image. You then divide that and it gives you the effective hp contribution of mirror image.

It certainly is not. By that methodology, if you have twice the amount of hitpoints, mirror image would be twice as effective!
But it isn't, because it only lasts for a fixed number of attacks against you. Therefore it prevents a fixed amount of damage (depending on a lot of factors, but not your total hp).

Basically, once you realise these points, the math becomes a lot more simple:

1 - Any attack that missed Mirror Image's AC can be ignored
2 - Mirror Image prevents at most three(*) of the attacks that we did not ignore in step 1

*) it will only be less in the somewhat unlikely case that you're not attacked enough times in the duration

[Frogreaver]

I had already given your specific quote. See my post #114.

Then you are correct. I should not have universally agreed with that question about my position. I do not universally agree with that and my other repeated statements have obviously contradicted it.

[x3n0n]

Thank you! So the inputs are actual HP and probability of some attempt causing damage, and the units are "attempted damage" to KO the PC.

Given that, for a fixed enemy/PC pair, "expected number of attacks to KO the PC" is eHP/damage-per-hit. Yes?

If so, the OP claim should be equivalent to:

Attacks to KO unarmored PC with MI / Attacks to KO unarmored PC without MI
equals
Attacks to KO armored PC with MI / Attacks to KO armored PC without MI


Doesn't account for barbarian rage, but seems correct.



That seems correct. However, note you cannot change the scenario presented, so if that scenario doesn't yield PC death you can't just add a few more attacks on the end until PC is dead. Instead you must repeat that same exact scenario again and again till PC death. Not doing this is of the most common mistakes I'm seeing.



Only if that's the scenario you are wanting to run. You could run a PC gets attacked one time scenario and compute the effective hp for that as well.



That doesn't appear to be a claim to me?


In order:

You're right about barbarian rage. Modified inputs: actual HP, probability of the enemy hitting/delivering the damage, and the ratio of damage "delivered" to damage actually taken.

Given that each attempted attack is statistically identical to each other attempted attack from the same enemy, I don't understand how adding more attacks on the end is different in any way from repeating the same initial sequence of attacks.

I don't understand the third statement at all. If eHP is the measure of attempted damage via some method to KO the PC, what is the meaning of the eHP of a single attack? (Is it the expected reduction in PC eHP due to that attack? If so, in the absence of MI and resistance/vulnerability, it would just be actual accuracy-adjusted damage per attack, yes?)

Sorry if I was unclear. I was saying that your original claim is equivalent to the claim "A/B = C/D" (that the ratio of added survivability in one scenario is equal to the ratio of added survivability in the other).


Given all that, the claim has become uninteresting to me. Thanks again for bringing this up and forcing me to think about it, but I do not think this measure of MI's value will be useful to me in my games. I choose to think about it in terms of hits avoided, critical hits avoided, and actual expected damage avoided, which I think have been summarized nicely in several of these posts.

[Frogreaver]

Yes I am, maybe just in a slightly different way. I first calculate actual prevented damage, and then scale that number according using the same formula.

Then you have ceased to be comparing the same scenario as I am. If it took player A on average 10 attacks to be KO'ed with mirror image then your case would be attacks equal 10. If it took player B on average 20 attacks to be KO'ed with mirror image then your 2 cases have varying numbers of attacks and I've already said from the beginning that number of attacks will cause the eHP multiplier to differ.

This is the most common issue I find, you refuse to hold number of attacks constant between player A and B. Trying to calculate effective hp of 2 characters with the same hp and different AC's will always lead to this everytime.


What I do is first calcuate how much actual damage is prevented by one image, then assume that's the equivalent of giving yourself that much extra actual hp, and then scale that to effective hp.

For example, suppose that in a specific given situation, mirror image prevents 20 actual hp of damage. That's exactly the same as casting a spell that gives you 20 actual temp-hp.

So if you scale that to effective hp according to your AC/chance to be hit, then *after* that scaling, it turns out the *absolute* effective HP granted by mirror image does not depend on your armour AC.

However, if you have higher AC you also have more effective HP, so the relative impact of the extra effective HP given by mirror image is less.

I understand what you are doing. Do you understand now why doing it that way doesn't fit the necessary requirement in my work of equal attacks for both characters?


Basically, once you realise these points, the math becomes a lot more simple:

1 - Any attack that missed Mirror Image's AC can be ignored
2 - Mirror Image prevents at most three(*) of the attacks that we did not ignore in step 1

*) it will only be less in the somewhat unlikely case that you're not attacked enough times in the duration

I realize your points and I'm saying that you aren't limited to looking at effective hp in only the scenarios where you take enough attacks to kill you .

[Reynaert]

Then you have ceased to be comparing the same scenario as I am. If it took player A on average 10 attacks to be KO'ed with mirror image then your case would be attacks equal 10. If it took player B on average 20 attacks to be KO'ed with mirror image then your 2 cases have varying numbers of attacks and I've already said from the beginning that number of attacks will cause the eHP multiplier to differ.

How are you reading that my case would have varying numbers of attacks when I SPECIFICALLY STATED I am calculating with a fixed number of attacks?
Like I said: I'm looking at a fixed number of attacks because Mirror Image ends after a fixed number of attacks


I understand what you are doing. Do you understand now why doing it that way doesn't fit the necessary requirement in my work of equal attacks for both characters?

You obviously don't understand what I'm doing because I AM using a fixed number of attacks.


I realize your points and I'm saying that you aren't limited to looking at effective hp in only the scenarios where you take enough attacks to kill you .

What? I never said anything about taking enough attacks to kill you. So you obviously do not realize my points.

Also, saying stuff like 'do you understand now' comes across as very condescending, which is not conductive to a civil discussion.

The simple fact of the matter is that the influence of one casting of Mirror Image is limited to a fixed number of attacks, i.e. the precise (average) number of attacks needed to hit the Mirror Image AC three times. However, with higher armour AC, each single attack has less impact on your health, so the positive effect of Mirror Image on your health is less with better armour. QED.

[LudicSavant]

Whose to say that you looked at Mirror Image in context of effective hp

Me. Rather obviously.

[Frogreaver]

Me. Rather obviously.

Why do you expect me to just take your word for things?

[LudicSavant]

Why do you expect me to just take your word for things?

I don't.

You asked who you could ask about whether or not I looked at it in the context of effective HP. Rather obviously, I am the one you could have asked. It would be pretty trivial to show. Instead you didn't ask and just did this whole 'whose to say' thing. :smallconfused:

[Frogreaver]

How are you reading that my case would have varying numbers of attacks when I SPECIFICALLY STATED I am calculating with a fixed number of attacks?
Like I said: I'm looking at a fixed number of attacks because Mirror Image ends after a fixed number of attacks

Let's step through this.

You look at the effective damage mirror image prevents in X attacks. You call that eHP and add it to something else you call eHP that was derived using some other calculation.

I'm saying we should compute the eHP of your AC in the same way as you are computing it for mirror image. Since you are using damage prevented for mirror image then lets do the same for AC. So how much damage does your AC prevent in N attacks (no mirror image). Assuming your chance to be hit is C and the number of attacks you face is N and the damage you take is D then you prevent N(1-C)D worth of damage via AC. The effective hp you used was eHP = hp/C. This implies your AC is preventing hp/C - hp damage. However, N(1-C)D only = hp/C - hp when N = (hp/C - hp) / [(1-C)D]. So unless you take N = (hp/C - hp) / [(1-C)D] number of attacks then your AC isn't actually preventing hp/C - hp damage. Thus, because in both cases you let AC prevented damage be N(1-C)D = hp/C - hp and D remained constant but C varied then you weren't actually using the same number of attacks N for both characters (at least not for the eHP without mirror image part of the calc).

Hopefully this highlights the problem of simultaneously using 2 differing definitions of eHP.


I don't.



You asked who you could ask about whether or not I looked at it in the context of effective HP. Rather obviously, I am the one you could have asked, and had you actually asked, I would have answered you.

When someone asks "whose to say you did X", coming back and saying "I can say I did X" isn't really addressing the concern...

[LudicSavant]

Here is an AnyDice program I wrote for calculating the expected damage against you over X attacks using Mirror Image.

https://anydice.com/program/1e0c2

Here is another AnyDice program I wrote that does the inverse -- instead of calculating the damage against you, it calculates how much damage the images blocked that you otherwise would have taken.

https://anydice.com/program/1e0c1

Thanks to Stealth_Elephant and MaxWilson and AureusFulgens for helping to check for bugs.

If you're confused about how the function works, Stealth_Elephant has a great tutorial about how to use state systems in AnyDice: https://www.reddit.com/r/3d6/comments/gf111s/anydice_tutorial_part_3_state_the_great_weapon/

The results are, unsurprisingly, basically what folks have been trying to tell the OP since the first page.

If you have any questions I'd be happy to answer them. And if you can find any mistakes please do let me know.

[Segev]

You do realize that a very obvious conclusion to be drawn from responding to somebody saying “I did this” with “whose to say you did that?” Is that you’re calling them a liar, right?

Maybe that’s not what you mean to convey, or maybe it is. But if not, you may wish to rethink how you express whatever you’re trying to say. Because as it is, it comes off very personally insulting, and as an as hominem attack. “I don’t believe you when you say you did this, because you’re not trustworthy.”

This is implied by the “who’s to say...?” line, which strongly implies that somebody more trustworthy than the person you’re addressing would have to be found to vouch for them. That no evidence they might present is sufficient due to in how little esteem you hold their word.

Again, you may not mean this. But it certainly can come off that way. And I frankly am not sure what else you might mean, though I hope it is a miscommunication.

[LudicSavant]

You do realize that a very obvious conclusion to be drawn from responding to somebody saying “I did this” with “whose to say you did that?” Is that you’re calling them a liar, right?

Maybe that’s not what you mean to convey, or maybe it is. But if not, you may wish to rethink how you express whatever you’re trying to say. Because as it is, it comes off very personally insulting, and as an as hominem attack. “I don’t believe you when you say you did this, because you’re not trustworthy.”

This is implied by the “who’s to say...?” line, which strongly implies that somebody more trustworthy than the person you’re addressing would have to be found to vouch for them. That no evidence they might present is sufficient due to in how little esteem you hold their word.

Again, you may not mean this. But it certainly can come off that way. And I frankly am not sure what else you might mean, though I hope it is a miscommunication.

This is pretty much how I felt about the matter. Thanks Segev. I don't think I could have articulated it so well as you did.

[Frogreaver]

You do realize that a very obvious conclusion to be drawn from responding to somebody saying “I did this” with “whose to say you did that?” Is that you’re calling them a liar, right?

Or, you could just as easily interpret as an ask for evidence of the claim. I'm not sure why you would be using the worst interpretation possible when there's an equally valid 1 that isn't "bad"?

It also strikes me that you can dispute my claims and debate with me for over a 100 post thread but the moment Ludic says "I did this" without posting a single shred of evidence I must take her at her word? That feels an awful lot like a double standard to me.

Out of curiosity, do you also consider your posts in this thread challenging my claims that I did Y to be "calling me a liar"?


Maybe that’s not what you mean to convey, or maybe it is. But if not, you may wish to rethink how you express whatever you’re trying to say. Because as it is, it comes off very personally insulting, and as an as hominem attack. “I don’t believe you when you say you did this, because you’re not trustworthy.”

I don't believe anyone when they say they did X just because they say it. Just like you all didn't believe me when I said I did Y in this thread just because I said it. That's how things are normally supposed to work I thought?

Also, telling someone you won't just take their word for something is asking them for proof. If that's an ad hominen attack then I've been ad hominen attacked about every other post this thread.


This is implied by the “who’s to say...?” line, which strongly implies that somebody more trustworthy than the person you’re addressing would have to be found to vouch for them. That no evidence they might present is sufficient due to in how little esteem you hold their word.

Wasn't asking anyone to vouch for anyone. Why would I have accepted any one elses word on the matter? What I would have accepted would be a link to the work done so that I could review it, a summary of the methodology used, really anything that doesn't require me to just accept that someone did X because someone said so. Any of these things could have been offered with the original claim or after the challenge I made, but instead the response I get seemed to be one of the most pedantic replies I've experienced in a while.


Again, you may not mean this. But it certainly can come off that way. And I frankly am not sure what else you might mean, though I hope it is a miscommunication.

I'll say this: I think you are taking it wrong more than what I said actually being wrong, but I apologize if anyone felt they were being called a liar. That was not my intent.

[LudicSavant]

Or, you could just as easily interpret as an ask for evidence of the claim.

When you say "who's to say" that suggests that you do not have a person available who you could ask for evidence. As well as... pretty much everything else Segev pointed out to you.

When I pointed out that I was right here, you could ask me for my evidence... you then argued with that.


the moment Ludic says "I did this" without posting a single shred of evidence I must take her at her word?

This is not something that I, or Segev, or anyone else, has asked of you.

You asked "who's to say" (e.g. who is to say a thing that provides evidence of this claim?) The very simple answer to which is "me. I would be the one you would ask for evidence."

Anyways, even though you never actually asked me to show my work, I did so anyways. It's right here (https://forums.giantitp.com/showsinglepost.php?p=24730398&postcount=131).

Note:
I edited this part while Frogreaver was in the middle of replying. For continuity's sake, it originally said:

"Nobody ever asked you to 'just take someone at their word.'

The issue is that, instead of asking someone for evidence, you chose to go 'whose to say you did this?' When I said "I would be the one to say. You could ask me for evidence of my claims and if you had done so, I would have given it to you." And then you started arguing with even that statement."

[Frogreaver]

Nobody ever asked you to 'just take someone at their word.'

The issue is that, instead of asking someone for evidence, you chose to go 'whose to say you did this?' When I said "I would be the one to say. You could ask me for evidence of my claims and if you had done so, I would have given it to you." And then you started arguing with even that statement.

I'm happy you brought this up.

Why do you demand I ask you for evidence? Why don't you just provide it when making your claim - especially when it's a claim that's disputing a claim made in the OP?

Why didn't you just provide your evidence when I asked "whose to say you did X"?

Why didn't you just provide your evidence when I asked "why do you expect me to take your word for these things?"

Any of those instances would have been a perfect opportunity to provide the evidence. And in case it's not clear, all of those statements/questions were meant as prods to get you to volunteer your evidence.

It doesn't seem fair that you expect me to have to ask for your evidence in situations where you are disputing something I've said.


This is not something that I, or Segev, or anyone else, has asked of you.

You asked "who's to say" (e.g. who is say a thing that provides evidence of this claim?) The very simple answer to which is "me. I would be the one you would ask for evidence."

Anyways, even though you never actually asked me to show my work, I did so anyways. It's right here. (https://forums.giantitp.com/showsinglepost.php?p=24730398&postcount=131)

Trying to take every single phrase hyper literally as you do here is a good way of talking past people.

No where is the phrase "Whose to say you did X" meant to be directly responded to. It's kind of like a figure out speech meant to drive home the point that actual evidence is required.

[LudicSavant]

Why do you demand I ask you for evidence?

I 'demanded' you ask me, did I? :smallconfused:

I pointed out that you did not ask me. And then I provided the evidence anyways, even though you did not ask for it.

[Frogreaver]

I 'demanded' you ask me, did I? :smallconfused:

I pointed out that you did not ask me.

I think your hyper literalism is the issue. When you are telling me the problem is that I didn't ask for evidence and that i could have and you would have given it - then that's a defacto demand that I ask for evidence before you provide it.


And then I provided the evidence anyways, even though you did not ask for it.

Way to late.

[Reynaert]

Let's step through this.

You look at the effective damage mirror image prevents in X attacks. You call that eHP and add it to something else you call eHP that was derived using some other calculation.

I'm saying we should compute the eHP of your AC in the same way as you are computing it for mirror image. Since you are using damage prevented for mirror image then lets do the same for AC. So how much damage does your AC prevent in N attacks (no mirror image). Assuming your chance to be hit is C and the number of attacks you face is N and the damage you take is D then you prevent N(1-C)D worth of damage via AC. The effective hp you used was eHP = hp/C. This implies your AC is preventing hp/C - hp damage. However, N(1-C)D only = hp/C - hp when N = (hp/C - hp) / [(1-C)D]. So unless you take N = (hp/C - hp) / [(1-C)D] number of attacks then your AC isn't actually preventing hp/C - hp damage. Thus, because in both cases you let AC prevented damage be N(1-C)D = hp/C - hp and D remained constant but C varied then you weren't actually using the same number of attacks N for both characters (at least not for the eHP without mirror image part of the calc).

Except AC does not end after a fixed number of attacks. It works for all attacks. Mirror image only works on a fixed, limited number of attacks.

Hopefully this highlights the problem of fixating on using the same definitions of eHP, even though the cases are different.

Compare these cases:
1 - We cast Mirror Image in the first round of a fight. During the fight we are attacked 10 times. During the fight, all three Images took a hit.
2 - We cast Mirror Image in the first round of a fight. During the fight we are attacked 50 times. During the fight, all three Images took a hit.

In case 2, our AC prevented 5 times as much damage as case 1. Agreed?

So, how much more damage did Mirror Image prevent in case 2, compared to case 1?

[Valmark]

Except AC does not end after a fixed number of attacks. It works for all attacks. Mirror image only works on a fixed, limited number of attacks.

Hopefully this highlights the problem of fixating on using the same definitions of eHP, even though the cases are different.

Compare these cases:
1 - We cast Mirror Image in the first round of a fight. During the fight we are attacked 10 times. During the fight, all three Images took a hit.
2 - We cast Mirror Image in the first round of a fight. During the fight we are attacked 50 times. During the fight, all three Images took a hit.

In case 2, our AC prevented 5 times as much damage as case 1. Agreed?

So, how much more damage did Mirror Image prevent in case 2, compared to case 1?

Note that Frogreaver isn't saying that MI is as effective as AC, but that MI's effectiveness isn't influenced by AC. This question and these cases don't seem to address that. I think.

[LudicSavant]

Note that Frogreaver isn't saying that MI is as effective as AC, but that MI's effectiveness isn't influenced by AC. This question and these cases don't seem to address that. I think.

The Anydice program I posted should address that pretty directly.

https://anydice.com/program/1e0c1

This is specifically measuring how much damage would be prevented by Mirror Image. So for example, the "effective hit points granted by Mirror Image" in example case 1 (21 AC vs 5 Bandit attacks) is 3.29, and in case 2 (11 AC vs 5 Bandit attacks) is 11.91.

You should be able to test basically any cases you want with this.

It'd be great if someone would double check by hand, though. :smallsmile:

[Valmark]

The Anydice program I posted should address that pretty directly.

https://anydice.com/program/1e0c1

This is specifically measuring how much damage would be prevented by Mirror Image. So for example, the "effective hit points granted by Mirror Image" in example case 1 (21 AC vs 5 Bandit attacks) is 3.29, and in case 2 (11 AC vs 5 Bandit attacks) is 11.91.

You should be able to test basically any cases you want with this.

It'd be great if someone would double check by hand, though. :smallsmile:

I can do it better later, but for now it seems to be correct. For each istance of the image (as in, 3 remaining, 2 remaining, etc.) It controls if it hits image and then if it would have hit AC, including natural 1s and 20s.

And has varying amounts of damage (calculated in the functions) with percentages rapresenting the chance of blocking a certain amount of damage.

If that's what it should do then apparently going from AC 11 to 21 the chance of MI being fundamentally useless rises by 57% (as in, from 3% to 60%) keeping bonus to attack roll and damage fixed (with critical damage considered). Or did I miss damage rolls?

[LudicSavant]

I can do it better later, but for now it seems to be correct. For each istance of the image (as in, 3 remaining, 2 remaining, etc.) It controls if it hits image and then if it would have hit AC, including natural 1s and 20s.

Yeah. It takes a normal "DPR program" (accounting for accuracy, damage, crits, all that good stuff) and adds a state system which checks whether the enemy hit you, your image, or neither.

If it hits an image, it decreases the number of images you have left for the rest of the loop. If an attack that would otherwise have hit you hits the image, it measures the damage of the prevented attack. And then it continues through the loop for however many attacks there are.

It'll give you the average, the distribution, everything.

You can set the attacks / damage / etc to be whatever you want.

[MinotaurWarrior]

The Anydice program I posted should address that pretty directly.

https://anydice.com/program/1e0c1

This is specifically measuring how much damage would be prevented by Mirror Image. So for example, the "effective hit points granted by Mirror Image" in example case 1 (21 AC vs 5 Bandit attacks) is 3.29, and in case 2 (11 AC vs 5 Bandit attacks) is 11.91.

You should be able to test basically any cases you want with this.

It'd be great if someone would double check by hand, though. :smallsmile:

I'm reasonably confident that this is correct, but it's only looking at the practically relevant portion of the function, not the entire scope, because you are constraining the number of attacks.

The factor that makes mirror image less effective for higher AC is that the mirror image goes away after preventing fewer hits. The factor that makes it more effective with higher AC is that each HP saved this way is worth more EHP due to your higher AC. With a finite number of attacks, you capture more of the first effect than the second.

For a fairly stupid example, imagine a fight where the enemy cannot crit, and hits the image on a 19 or 20 and the PC on either just 20 or 19/20, and keeps attacking infinitely. In the case where the enemy needs a 20 to hit the PC, each image absorbs 1/2 of a hit. In the situation where the PC is hit on a 19 or a 20 each image absorbs a full hit. By the time the images are exhausted, they have prevented more damage to the pc with ac 19. But for the PC with AC 20, each HP goes twice as far.

Does it math out that the second effect perfectly counterbalances the first overall (not just in that dumb scenario)? I don't feel like doing the calculus to check, honestly. Practically, your simulation addresses the area under the part of the curve that actually shows up in play.

EHP, like DPR, is a very white room number that doesn't really directly matter in the game as it is played, but unlike DPR it doesn't have the advantage of being so easily calculated with basic arithmetic.

To be clear, the definition of EHP I'm using is ~ HP/(%Avoiding being damaged).

[fat.hampster]

1. We can ignore attacks that are lower than the mirror image AC, assuming we hold it constant, we just need to increase the number of attacks taken to compensate. This has already been noted earlier in the thread. I will remove these misses from any calculations so MI AC 11/Char AC11 is equivalent to a 100% hit chance above the MI AC threshold.

2. I'll assume a 100% chance that all images will be hit. The probability tends to this as we increase the number of attacks, so the analysis tends towards correctness by increasing the number of attacks (and the HP).

2. Lets consider some characters.
A: 10hp 100% hit chance
B: 10hp 50% hit chance

Given 6 attacks each dealing 1 damage:
a1: A w/o MI: takes 6 damage
a2: A w/ MI: takes 3 damage
b1: B w/o MI: takes 3 damage
b2: B w/ MI: takes 1.5 damage

Now clearly A has avoided more damage from MI, however a1/a2 (3/6) == b1/b2 (1.5/3) == 1/2. That is, both A and B have avoided half the damage they would've taken. Given more attacks, the ratio will decrease, but it will remain constant between the two. This is the sense that MI is equally effective between the 2 cases.

Of course this doesn't exist in a Vacuum, and avoiding the same proportion of damage might not be equally useful.

To me, the most obvious definition of eHP would be the number of rounds in this type of scenario which a character could survive. This is *not* the definition which Frogreaver has been using, hence a large part of the confusion (alongside his inability to define eHP, and his condescending tone). To quickly run through:
a1: 10 attacks
a2: 13 attacks
b1: 20 attacks
b2: 23 attacks
(and quickly to throw it in, define C as 100% hit chance, 20hp):
c1: 20 attacks
c2: 23 attacks

So, MI increases survivability by a constant number of attacks (accounting point 1, i.e. constant number of attacks that surpass MI AC), namely 3, but with more eHP that is proportionally less (3/20 vs 3/10). As your eHP increases either by increased HP or increased (non-MI-applicable) AC, the relative number of rounds MI keeps you alive decreases, but the absolute number remains the same. (Obviously if you're not going to die anyway before the next long rest, then MI is pointless).

Frogreaver uses a different definition of eHP, which I will call fHP to distinguish. He requires the number of attacks to be kept constant, which we can do by increasing the damage for the higher AC case, or we can sort of shoehorn it into eHP format by decreasing the HP to keep the eHP constant across the base cases. In this case we'd have:
A: 10 hp 100% hit chance
B: 5 hp 50% hit chance

Which keeps the hits the same to death/means MI is equally useful.

now eHP and fHP are both interesting metrics. In particular, fHP draws attention to the fact that, if higher AC is due to levelling, then one might expect damage/attack to increase at a comensurate rate, in which case MI retains its effectiveness. Otoh, my impression is that higher damage at higher levels often scales through means other than per attack damage, but through extra damage etc. Additionally, AC through armour is likely to be available from 1st level, and much discussion will be comparing the effectiveness of 2 characters at the same level, in which case eHP makes more sense.

Footnotes:
* the assumption of 3 hits is based on receiving enough hits to get through all the images, if this is unlikely to be the case, a character who is likely to be attacked more may get more value than a character who will be hit less during the duration.
* One way that a character might not get enough attacks is by being dead.

[Icewind]

So I'm going to try and figure this out without getting too advanced with the math, because to be completely honest I find it hard to keep up when a lot of variables are involved.

Let's say that we have Wizard A and Wizard B.

Wizard A has an AC of 30.

Wizard B has an AC of 10.

The monster has an attack bonus of +9, and deals 10 damage a hit.

Without any spells, the monster hits Wizard B 95% of the time, and Wizard A 5% of the time, because it only misses Wizard B on a 1 and only hits Wizard A on a 20.

Let's assume both Wizards have 100 HP.

This makes Wizard A's effective HP roughly 2000 (slightly less, because crits do more damage than a normal hit, but close enough) against this monster, and Wizard B's effective HP roughly 105 against this monster, before accounting for Mirror Image.

With that established, let's throw Mirror Image into the mix.


Assume the Mirror Images of both Wizard A and Wizard B have an AC of 10, and so are hit on everything except a natural 1. Let's also assume all the Mirror Images are triggered before either Wizard would drop unconscious.

Wizard A's Mirror Image, with the above assumptions, has a 5% chance per image to block an attack that would've hit him, and a 95% chance to be wasted on an attack that would've missed anyway. Because each attack does 10 damage, each image blocks an average of 0.5 damage. This increases his effective HP to 2001.5, or by less than 1%.

Wizard B's Mirror Image, with the above assumptions, has a 100% chance per image to block an attack that would've hit him (and a 0% chance to be wasted on an attack that would've missed anyway) - because each attack does 10 damage, and each image blocks all of that, this increases his effective HP to 135, or by slightly less than 30%.


If anything I've said is mistaken, or if there's some sort of faulty premise behind what I wrote, please let me know! Otherwise, I think this checks out, and shows that Mirror Image has a significantly more useful effect the lower AC the caster has.

EDIT: Calculations incorrect, see posts below, but I think the conclusion is still right in that it’s still better to use when you have lower AC.

[fat.hampster]

Wizard A's Mirror Image, with the above assumptions, has a 5% chance per image to block an attack that would've hit him, and a 95% chance to be wasted on an attack that would've missed anyway. Because each attack does 10 damage, each image blocks an average of 0.5 damage. This increases his effective HP to 2001.5, or by less than 1%.




You're not multiplying the damage blocked by the same multiplier as you multiplied the HP. i.e. it saves 0.5 damage, but that translates into 0.5 * 20 ehp, leaving 2030 eHP.

(I think you're also slightly miscounting the hits in this case because it's 1/19 and 18/19 (rather than 1/20 and 19/20 as you're using). however this doesn't make a huge difference (I think turning off natural 1s makes the analysis easier without much else). and I'm feeling there must be something else I'm missing that it cancels with (probably the 20 in the multiplier in the ehp, should actually be a 19 or something))

[MinotaurWarrior]

Wizard A's Mirror Image, with the above assumptions, has a 5% chance per image to block an attack that would've hit him, and a 95% chance to be wasted on an attack that would've missed anyway. Because each attack does 10 damage, each image blocks an average of 0.5 damage. This increases his effective HP to 2001.5, or by less than 1%.

The image prevents 0.5 actual damage. You then need to translate that actual HP back into EHP with the same formula you used to get the first 2000 EHP.

Making my prior example more concrete:

Guy 1, only hit on 20s, immune to crits, being attacked with a 1 damage attack (blow gun), 10HP. Image hit on 19 or 20.

No mirror image ehp: 200 (10/0.05)
1 mirror image ehp: 210 (200 + 0.5/0.05)

Guy 2, hit on 19-20, immune to crits, being attacked with a 1 damage attack (blow gun), 10HP. Image hit on 19 or 20.

No mirror image ehp: 100 (10/0.1)
1 mirror image ehp: 110 (100 + 1/0.1)

[Segev]

Frogreaver, it’s not the literalness of the phrase “who’s to say” that’s the problem. It’s that there is no colloquial way to take it that doesn’t come off insulting when you use it as you did.

The phrase is commonly used only to suggest that what follows has no available authority who could be trusted to provide verifiable knowledge on the subject. Both literally and colloquially, it is not the right phrase to use if you are simply asking for evidence, because both literally and colloquially, it suggests that you don’t know of anybody who has or could be trusted to provide believable evidence.

What I take it you means was something along the lines of, “I don’t know that that’s true; please provide your evidence of that assertion.” This is an entirely different phrase than “who’s to say that’s true?” It accepts the possibility that the person making the assertion might have something to support it. “Who’s to say...?” instead suggests - again, both colloquially and literally - that you don’t trust the speaker to be able to provide evidence to the point that if they try, you wouldn’t trust it. After all, they are not an obvious answer to the question of “who’s to say what you say is true?” if you feel the need to ask the question.

I say this because you’re coming off as potentially much more rude than you mean to, and I suspect (and apologize if I’m mistaken) that either English is not your first language or that you at least don’t know how that phrase is taken in colloquial American parlance. It’s not even that it’s automatically insulting, but used as you did, it always will seem so.

It’s more commonly used in statements like, “Who’s to say there’s no intelligent life on other planets?” It points out that something is unknowable with our current ability to gather evidence, and that we know of no subject matter expert who can believably claim to have sure knowledge.”

That’s not just the literal meaning, but the common usage. There’s no figurative usage that carries a simple request for more information. When used in a context of whether something somebody says about themselves or their activities is true, the only colloquial interpretation is that you’re expressing that you lack sufficient evidence to believe anything that person might say on the subject.

“I’m a PhD in Computer Engineering.” “Who’s to say that you are? Anybody can claim anything on the internet, and even if you showed us your diploma, we can’t trust that you are really the person you’re claiming to be.”

While quite true, this is suggesting that there’s reason to doubt the veracity of the speaker’s personal claims. This can be non-insulting when it’s serious enough. For instance, if someone claims to be a Nigerian Prince, and wants you to make financial decisions based on that, not being willing to take their word is more a matter of safety than anything else. But in a case where the speaker says they’ve done something and proving it is as easy as showing the work, suggesting that there’s no way they can prove it (which is what “who’s to say...?” does) is very insulting. It implies and denotes that you don’t think they are a person who could say whether their claim is true or not with any believable authority.

I say all of this in hopes it helps you with your communication in the future. Using language and idioms to convey what you mean in both implication and decoration is important. And it’s always going to impede communication when you think there exists a figurative colloquial sense in which something is taken, but that colloquial sense doesn’t actually exist.

Ludic Savant isn’t being hyper-literal; you seem to have thought there was a figurative way to take it that doesn’t exist in colloquial English. Thus, it came off as saying, “You’re a lying liar that lies,” and then, when called out on it, your defense is, “stop being so literal. I was just saying that I haven’t seen your proof. I wasn’t really calling you a liar, geeze.” I hope you can see why that would offend people.

I understand you didn’t mean it that way, and I hope this overly-long dissertation on what the phrase means both literally and colloquially helps you better convey your meaning in the future.

[Icewind]

Alright, so from what I’m reading I definitely made a mistake in the math, but the end result is that the flat amount of eHP Mirror Image gives is constant regardless of AC, but the percentage it increases your eHp by is much higher if you have lower AC, as a result of the amount of eHP it gives being flat.

Kind of the same concept (but not quite as extreme) of how a +1 to hit on d20 roll isn’t equal to a 5% damage boost, right? E.G. if you need a 20 to hit without it but only need 19 with it, it effectively doubles your damage (discounting crits), whereas if you only miss on a 1 already it gives you 0 extra damage.

[Frogreaver]

Frogreaver, it’s not the literalness of the phrase “who’s to say” that’s the problem. It’s that there is no colloquial way to take it that doesn’t come off insulting when you use it as you did.

The phrase is commonly used only to suggest that what follows has no available authority who could be trusted to provide verifiable knowledge on the subject. Both literally and colloquially, it is not the right phrase to use if you are simply asking for evidence, because both literally and colloquially, it suggests that you don’t know of anybody who has or could be trusted to provide believable evidence.

What I take it you means was something along the lines of, “I don’t know that that’s true; please provide your evidence of that assertion.” This is an entirely different phrase than “who’s to say that’s true?” It accepts the possibility that the person making the assertion might have something to support it. “Who’s to say...?” instead suggests - again, both colloquially and literally - that you don’t trust the speaker to be able to provide evidence to the point that if they try, you wouldn’t trust it. After all, they are not an obvious answer to the question of “who’s to say what you say is true?” if you feel the need to ask the question.

I say this because you’re coming off as potentially much more rude than you mean to, and I suspect (and apologize if I’m mistaken) that either English is not your first language or that you at least don’t know how that phrase is taken in colloquial American parlance. It’s not even that it’s automatically insulting, but used as you did, it always will seem so.

It’s more commonly used in statements like, “Who’s to say there’s no intelligent life on other planets?” It points out that something is unknowable with our current ability to gather evidence, and that we know of no subject matter expert who can believably claim to have sure knowledge.”

That’s not just the literal meaning, but the common usage. There’s no figurative usage that carries a simple request for more information. When used in a context of whether something somebody says about themselves or their activities is true, the only colloquial interpretation is that you’re expressing that you lack sufficient evidence to believe anything that person might say on the subject.

“I’m a PhD in Computer Engineering.” “Who’s to say that you are? Anybody can claim anything on the internet, and even if you showed us your diploma, we can’t trust that you are really the person you’re claiming to be.”

While quite true, this is suggesting that there’s reason to doubt the veracity of the speaker’s personal claims. This can be non-insulting when it’s serious enough. For instance, if someone claims to be a Nigerian Prince, and wants you to make financial decisions based on that, not being willing to take their word is more a matter of safety than anything else. But in a case where the speaker says they’ve done something and proving it is as easy as showing the work, suggesting that there’s no way they can prove it (which is what “who’s to say...?” does) is very insulting. It implies and denotes that you don’t think they are a person who could say whether their claim is true or not with any believable authority.

I say all of this in hopes it helps you with your communication in the future. Using language and idioms to convey what you mean in both implication and decoration is important. And it’s always going to impede communication when you think there exists a figurative colloquial sense in which something is taken, but that colloquial sense doesn’t actually exist.

Ludic Savant isn’t being hyper-literal; you seem to have thought there was a figurative way to take it that doesn’t exist in colloquial English. Thus, it came off as saying, “You’re a lying liar that lies,” and then, when called out on it, your defense is, “stop being so literal. I was just saying that I haven’t seen your proof. I wasn’t really calling you a liar, geeze.” I hope you can see why that would offend people.

I understand you didn’t mean it that way, and I hope this overly-long dissertation on what the phrase means both literally and colloquially helps you better convey your meaning in the future.

We disagree on this.

[JNAProductions]

We disagree on this.

Communication is a two-way street. Even if you intend to be perfectly kind and polite, if you do not communicate that properly (which can be difficult via text-everyone has trouble, at least sometimes, getting their intent misunderstood when typing) then it's certainly not solely the fault of the person receiving the message.

[MinotaurWarrior]

Alright, so from what I’m reading I definitely made a mistake in the math, but the end result is that the flat amount of eHP Mirror Image gives is constant regardless of AC, but the percentage it increases your eHp by is much higher if you have lower AC, as a result of the amount of eHP it gives being flat.

It's more complicated than that due to the factors we're all simplifying away. But yes, at the extreme MI gives a constant absolute EHP increase, and so is equally as good on a high and low AC character, for an absurd white room combat.

However, I don't think the % increase is ever a relevant metric. The EHP tells you something like, "I can expect to survive 10 more rounds of this blow gun attack, or being attacked for 1 round by ten more blowgunners" The % EHP increase doesn't really tell you anything directly meaningful.

[Segev]

We disagree on this.On which part? You're going to have to provide examples of "Who's to say...?" being used as you intend it to be taken if you want to convince me that it can be used as you seem to think.

Or, to demonstrate by example, "Who's to say the phrase can be used the way you claim?"

(Please note that I am not actually insulting anybody, here, but am trying to point out how using the phrase in this way comes off as insulting.)


Communication is a two-way street. Even if you intend to be perfectly kind and polite, if you do not communicate that properly (which can be difficult via text-everyone has trouble, at least sometimes, getting their intent misunderstood when typing) then it's certainly not solely the fault of the person receiving the message.Precisely. My purpose here is to give some insight as to how the phrase comes off to - I believe - most English-speakers, as Frogreaver used it. It is my hope that this will help him communicate what he means better. A great deal of communication is implication. And if he believes there's implication present that isn't when he uses a phrase whose denotation is VERY insulting in the context in which he used it, he's going to unintentionally insult a great many people.


Alright, so from what I’m reading I definitely made a mistake in the math, but the end result is that the flat amount of eHP Mirror Image gives is constant regardless of AC, but the percentage it increases your eHp by is much higher if you have lower AC, as a result of the amount of eHP it gives being flat.

Kind of the same concept (but not quite as extreme) of how a +1 to hit on d20 roll isn’t equal to a 5% damage boost, right? E.G. if you need a 20 to hit without it but only need 19 with it, it effectively doubles your damage (discounting crits), whereas if you only miss on a 1 already it gives you 0 extra damage.

There are a few factors at play.

Expected damage per attack varies depending on caster AC and number of images.
Expected number of attacks images are around for depends solely on image AC.
Images actual expected contribution to eHP would be number of attacks they're expected to be around for times the expected damage per attack.

Also, this is entirely deceptive in analyzing image contribution to effect, because expected damage per attack while images are up is really, really poor as a metric with something like mirror image. Each image negates up to one attack. If caster AC is higher than image AC, the chance that an image disappears without negating an attack is 5% times the difference between the ACs (provided we're not at a point where only critical hits or misses matter).

In the formulation where you simply look at expected damage per attack, you're using averages in a super-high-variance situation, which tends to be unreliable without an enormous number of samples. Far more than you're going to get in a typical session. Mirror image undeniably has a lower impact the higher the AC of the caster is. (For evidence of this, consider a caster with very high AC that is strictly due to his dexterity. The damage mitigation from having M images is the same as if the images have very low AC, but the images won't disappear as soon. However, they still only last for one combat, at most, so if the caster is never hit based on his AC, the images contributed nothing even though none of them vanished to attacks that would have missed the caster anyway.) This is reflected in expected damage per attack with 0 images being low, so M images dividing expected damage per attack by (M+1) divides a much smaller number, making the mitigation of having those images much smaller.

When the "effective hp" added by mirror image is significantly less than the expected damage of a successful attack (i.e. "What is the expected damage of an attack given that it hits?" as opposed to the expected damage per attack without knowing whether it hit or not), then mirror image will do nothing a significantly higher percentage of the time than one might expect. And since eHP goes down with image AC due to reducing the expected number of attacks that M images will last, a high difference between caster AC and image AC results in a much smaller eHP contribution relative to the expected damage of a successful attack, yielding mirror image doing nothing on a significantly larger number of castings.

[JNAProductions]

Some Math:

C = chance to be hit
C' = chance image is hit
M' = chance you are targeted instead of image (3 images)
M'' = chance you are targeted instead of image (2 images)
M''' = chance you are targeted instead of image (1 images)
D = average damage of attack

Against 1 attack let's calculate the effective Damage you take with mirror image.
1. M'*C*D

Now let's calculate the effective Damage you take without mirror image.
2. C*D

Now let's calculate the effective Damage factor
3. (M'*C*D) / (C*D) = M'

Since we know M' = .25 then mirror reduces effective damage by .25 against all AC's in the case of 1 attack. (*Note there is no dependency on AC or chance to hit here - probably not surprising).

------------------------------------------------------------------------------------------------------------

Against 2 attacks let's calculate the effective Damage you take with mirror image (this is more complicated)
There are 7 parts:
1: 2*D*[M'*C]^2
2: (M'*C*D)*(M'*[1-C])
3: (M'*C*D)*([1-M']*C')
4: (M'*C*D)*([1-M']*[1-C'])
5: (M'*[1-C])*(M'*C*D)
6: ([1-M']*C')*(M''*C*D)
7: ([1-M']*[1-C'])*(M'*C*D)

Adding these together and rearranging a bit we get
= (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C') M'(1-C)] + (M'*C*D)*[M'C + M'(1-C) + (1-M')(M''/M')C' + (1-M')(1-C')]
= (M'*C*D)*(1) + (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C') M'(1-C)] + (M'*C*D)*[(1-M')(M''/M')C' - (1-M')C']
= 2M'*C*D + (M'*C*D)(1-M')(C')(M''/M' - 1)

Now let's calculate the effective Damage you take without mirror image for the 2 attack scenario.
2. 2*C*D

Now let's calculate the effective Damage factor
3. [2M'*C*D + (M'*C*D)(1-M')(C')(M''/M' - 1)] / [2CD]
=(M')(1+(0.5)(1-M')(C')(M''/M'-1)

As can be seen from this the effective Damage factor doesn't depend on C. Therefore, we have found that the effective Damage factor for mirror image is independent of C.

Edited: had incorrect variable accidently type in 2 places. Nothing changes with calc, was a keying error.

I'm going to actually put numbers in here, and see if they make sense. I will use a Wizard with Mage Armor and 16 Dex, against a foe with +2 to-hit and dealing 2d4+2 damage.

One Attack

Damage with Mirror Image
.25*.35*7=.6125

Damage without
.35*7=2.45

Effective Damage Factor
(.25*.35*7)/(.35*7)=.25
.6125/2.45=.25
.25=.25

So far, everything checks out.


Two Attacks

Damage with Mirror Image, using your added together bit.

(.25*.35*7)*(.25*.35+.25*(1-.35)+(1-.25)*.5+(1-.25)*(1-.5))+(.25*.35+7)*(.25*.35+.25*(1-.35)+(1-.25)*((1/3)/.25)*.5+(1-.25)*(1-.5))=(.6125)*(.0875+.25*(.65)+(.75)*(.5))+(.6125)* (.0875+.25*(.65)*(4/3)*.5+(.75)*(.5)=0.73244791666

(.25*.35*7)*1+(.25*.35*7)*(.25*.35+.25*(1-.35)+(1-.25)*.5+(1-.25)*(1-.5))+(.25*.35*7)*((1-.25)*((1/3)/.25)*.5-(1-.25)*.5)=1.3015625

2*.25*.35*7+(.25*.35*7)*(1-.25)*(.5)*((1/3)/.25-1)=1.3015625

So, having plugged in actual numbers, you clearly did something wrong from your first step to your second. They're supposed to equal one another-and they don't.

(M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C')] + (M'*C*D)*[M'C + M'(1-C) + (1-M')(M''/M')C' + (1-M')(1-C')]
(M'*C*D)*(1) + (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C')] + (M'*C*D)*[(1-M')(M''/M')C' - (1-M')C']

Here are the formulas again, without any bolding or strikethroughs.

(M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C')] + (M'*C*D)*[M'C + M'(1-C) + (1-M')(M''/M')C' + (1-M')(1-C')]
(M'*C*D)*(1) + (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C')] + (M'*C*D)*[(1-M')(M''/M')C' - (1-M')C']

The bolded bit is fine. Removing it.

[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C')] + (M'*C*D)*[M'C + M'(1-C) + (1-M')(M''/M')C' + (1-M')(1-C')]
(1) + (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C')] + (M'*C*D)*[(1-M')(M''/M')C' - (1-M')C']

The bolded bit is...

(.25*.35+.25(1-.35)+(1-.25).5+(1-.25)(1-.5))
(.0875+.25(.65)+(.75).5+(.75)(.5))
(.0875+.1625+.375+.375)
1.4625

1.4625=/=1

This part is wrong.

(M'*C*D)*[M'C + M'(1-C) + (1-M')(M''/M')C' + (1-M')(1-C')]
(M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C')] + (M'*C*D)*[(1-M')(M''/M')C' - (1-M')C']

Bolded part is fine.

[M'C + M'(1-C) + (1-M')(M''/M')C' + (1-M')(1-C')]
[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C')] + (M'*C*D)*[(1-M')(M''/M')C' - (1-M')C']

(.25*.35+.25(1-.35)+(1-.25)((1/3)/.25).5+(1-.25)(1-.5)
(.0875+.25(.65)+(.75)(4/3).5+(.75)(.5))
(.0875+.1625+.5+.375)
1.125

(.25*.35+.25(1-.35)+(1-.25).5+(1-.25)(1-.5))+(.25*.35*7)((1-.25)((1/3)/.25).5-(1-.25).5)
(.0875+.25(.65)+(.75).5+(.75)(.5))+(.6125)((.75)(4/3).5-(.75).5)
(.0875+.1625+.375+.375)+.6125(.5-.375)
1+.6125(.1875)
1+.11484375
1.11484375

These ALSO do not equal.

So, this part:

[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C')]=1

Is wrong.

Let's expand it out a bit.

M'C+M'-C+C'-M'C'+1+1(1-C')-M'(1-C')
M'C+M'-C+C'-M'C'+1+1-C'-M'+M'C'
M'C+M'-C+C'-M'C'+2-C'-M'+M'C'
M'C+M'-C+C'-M'C'+2-C'-M'+M'C'
M'C-C+C'-M'C'+2-C'+M'C'
M'C-C+C'-M'C'+2-C'+M'C'
M'C-C+C'+2-C'
M'C-C+C'+2-C'
M'C-C+2

So it does NOT equal one-it equals M'C-C+2. Since all variables, save for D, in this equation are 0 to 1, this cannot be correct.

[fat.hampster]

When the "effective hp" added by mirror image is significantly less than the expected damage of a successful attack (i.e. "What is the expected damage of an attack given that it hits?" as opposed to the expected damage per attack without knowing whether it hit or not), then mirror image will do nothing a significantly higher percentage of the time than one might expect. And since eHP goes down with image AC due to reducing the expected number of attacks that M images will last, a high difference between caster AC and image AC results in a much smaller eHP contribution relative to the expected damage of a successful attack, yielding mirror image doing nothing on a significantly larger number of castings.

How much of that is the higher variance though, and how much is the lower value of 30eHP when you have 2000 vs 105.

I mean, if we rework Icewind's examples so that the relative eHP is the same, we can change the wizard A such that the monster will 1hit kill the them (either multiply dmg by 10, or divide hp by 10), and we still need to double the health (or the AC) on wizard B.

so 200HP with a guaranteed 3 hits mitigated
vs
10HP with an extra 10%+ chance of not being dead after 4 attacks.

I'm not sure that the improvement in the former is clearly better than the latter, even though it's lower variance.

[Reynaert]

Alright, so from what I’m reading I definitely made a mistake in the math, but the end result is that the flat amount of eHP Mirror Image gives is constant regardless of AC, but the percentage it increases your eHp by is much higher if you have lower AC, as a result of the amount of eHP it gives being flat.

Kind of the same concept (but not quite as extreme) of how a +1 to hit on d20 roll isn’t equal to a 5% damage boost, right? E.G. if you need a 20 to hit without it but only need 19 with it, it effectively doubles your damage (discounting crits), whereas if you only miss on a 1 already it gives you 0 extra damage.

That's the same result I got earlier in this thread.


Having thought about it some more, I can also back that result up with some reasonably simple logic:

Assume that we take enough attacks for all the Mirror Images to be destroyed, and also assume that we have enough hitpoints to survive these attacks.

1: The amount of damage prevented by Mirror Image over the entire fight would be exactly the same if the chance to hit an Image would be 100%.

(To prove this, note that you can take all the attack+damage rolls that happened during the fight and rearrange them in any order without changing how much damage we have taken in total. So therefore we can move all the attacks that were aimed at the Mirror Image, hit or miss, to the beginning.)

2: Effective damage to HP (or effective HP) is equivalent to the number of attacks we take, times damage. This translates to actual HP on account of our chance to be hit (i.e. our AC).

3: The first N attacks on us were (given 1) used to deplete the Mirror Images

4: N does not depend on our own AC but only on the AC of the Mirror Images

5: If we had not cast Mirror Image, we would have taken N more attacks, and therefor N times damage more 'effective-HP' damage.

6: The absolute value of the 'effective-HP' that Mirror Image gives us does not depend on our AC

corollary: If we have more AC, our total 'effective-HP' gets larger, so therefore the ratio of Mirror Image effect to total effective-HP becomes smaller with higher AC.

AIUI, Frogreaver keeps claiming that actually, it is the ratio that remains constant.

[fat.hampster]

AIUI, Frogreaver keeps claiming that actually, it is the ratio that remains constant.

This is because Frogreaver is using a different definition of eHP which requires the number of attacks to be kept constant.

This is a weird way to look at it, because HP is a measure of how much damage and therefore how many attacks you can be hit by.

If you keep the number of attacks constant, then the maths does work out to the same ratio, but it's a questionably useful metric, and he hasn't at any point defined or justified it (leading to much confusion as people assume something more sensible).

[Reynaert]

However, I don't think the % increase is ever a relevant metric. The EHP tells you something like, "I can expect to survive 10 more rounds of this blow gun attack, or being attacked for 1 round by ten more blowgunners" The % EHP increase doesn't really tell you anything directly meaningful.

If it takes N attacks to make all of the images going away (only counting the attacks that the randomizer points at the Images), then we can expect to survive N more attacks than we would without casting Mirror Image.

And this number N, the number of attacks it takes to make the images go away, depends only on the image's chance of getting hit by an attack (as calculated by its AC versus attack bonus), and not on our own AC.

[Mellack]

If it takes N attacks to make all of the images going away (only counting the attacks that the randomizer points at the Images), then we can expect to survive N more attacks than we would without casting Mirror Image.

And this number N, the number of attacks it takes to make the images go away, depends only on the image's chance of getting hit by an attack (as calculated by its AC versus attack bonus), and not on our own AC.

I am not sure that is true because those N attacks on a high AC character could all come from attacks that had missed, meaning they had no effect at all on the number of attacks that are survived. I don't follow how we can expect to survive any more attacks when the only attacks that interacted with the images were ones that had no effect on the actual character.

[Valmark]

If it takes N attacks to make all of the images going away (only counting the attacks that the randomizer points at the Images), then we can expect to survive N more attacks than we would without casting Mirror Image.

And this number N, the number of attacks it takes to make the images go away, depends only on the image's chance of getting hit by an attack (as calculated by its AC versus attack bonus), and not on our own AC.

No, we can expect to survive N more attacks thanks to Mirror Image only if those attacks would have hit us otherwise.

[Satori01]

So brass tacks...how are the maths applied to gameplay?

Mirror Image is the spell you want, when you can't afford to get hit.

The effect will last longer for some then for some others.

Essentially that is what the math boils down to?

[fat.hampster]

No, we can expect to survive N more attacks thanks to Mirror Image only if those attacks would have hit us otherwise.

Expect is used, aiui, in the probabilistic sense.

The extra attacks we don't survive because of something that would miss if it hadn't removed a mirror image, are cancelled out by the fact that if a mirror image succesfully tanks damage, then we might well survive more than N extra attacks -- because only a proportion of the extra attacks we're still alive to face will actually hit.

[Frogreaver]

I'm going to actually put numbers in here, and see if they make sense. I will use a Wizard with Mage Armor and 16 Dex, against a foe with +2 to-hit and dealing 2d4+2 damage.

One Attack

Damage with Mirror Image
.25*.35*7=.6125

Damage without
.35*7=2.45

Effective Damage Factor
(.25*.35*7)/(.35*7)=.25
.6125/2.45=.25
.25=.25

So far, everything checks out.

This is an excellent method to check work.



Two Attacks

Damage with Mirror Image, using your added together bit.

(.25*.35*7)*(.25*.35+.25*(1-.35)+(1-.25)*.5+(1-.25)*(1-.5))+(.25*.35+7)*(.25*.35+.25*(1-.35)+(1-.25)*((1/3)/.25)*.5+(1-.25)*(1-.5))=(.6125)*(.0875+.25*(.65)+(.75)*(.5))+(.6125)* (.0875+.25*(.65)*(4/3)*.5+(.75)*(.5)=0.73244791666

(.25*.35*7)*1+(.25*.35*7)*(.25*.35+.25*(1-.35)+(1-.25)*.5+(1-.25)*(1-.5))+(.25*.35*7)*((1-.25)*((1/3)/.25)*.5-(1-.25)*.5)=1.3015625

2*.25*.35*7+(.25*.35*7)*(1-.25)*(.5)*((1/3)/.25-1)=1.3015625

So, having plugged in actual numbers, you clearly did something wrong from your first step to your second. They're supposed to equal one another-and they don't.

When I plug in your parameters I get the exact same value on each line. I think you've made a typo somewhere. Easy to do in such a long equation. I used excel and defined each addition, subtraction and multiplication term in cells and combined them that way (much less error prone).



[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C')] + (M'*C*D)*[M'C + M'(1-C) + (1-M')(M''/M')C' + (1-M')(1-C')]
(1) + (M'*C*D)*[M'C + M'(1-C) + (1-M')C' + (1-M')(1-C')] + (M'*C*D)*[(1-M')(M''/M')C' - (1-M')C']

The bolded bit is...

(.25*.35+.25(1-.35)+(1-.25).5+(1-.25)(1-.5))
(.0875+.25(.65)+(.75).5+(.75)(.5))
(.0875+.1625+.375+.375)
1.4625

1.4625=/=1

This part is wrong.

But .0875+.1625+.375+.375 = 1 (not 1.4625).

I think you need to rework through your calcs.

[MaxWilson]

So brass tacks...how are the maths applied to gameplay?

Mirror Image is the spell you want, when you can't afford to get hit.

The effect will last longer for some then for some others.

Essentially that is what the math boils down to.

Naw, usually Blur or Protection From Evil is the spell you want when you can't afford to get hit. Or Dimension Door, or Expeditious Retreat.

Mirror Image is the spell you cast when you are unarmored (crummy AC) and have more spell slots than HP, and nothing important to be casting this round, and aren't fighting enemies with blindsight, and yet for some reason can't afford to simply leave combat.

Or to put it the other way: in my experience, Mirror Image is the spell you learn (because hey, Quickened Mirror Image with no concentration cost sounds great for a Paladorc! maybe it will help with big tough monsters) and then never wind up actually casting because something else is always better.

[bid]

I've only explained like 10 times how they aren't contradictions...
I've slept on it and I think I understand where you're going.

There's that specific enemy that does 20 damage every hit. He damages your Dex AC half the time and your full AC 1/4 of the time.

1) You have 100 hp and only Dex AC, therefore 200 effective hp. 10 attacks and you're down. Each mirror image adds 2 extra attacks or 40 effective hp.

2) You have 100 hp and fill AC, therefore 400 effective hp. 20 attacks and you're down. Each mirror image adds 2 extra attacks or 40 effective hp.


Of course, 10+2 attacks has a bigger impact than 20+2 attacks. But both mirror images are 40 effective hp.
So the value of mirror image depends on the enemy, not on your AC.

[Frogreaver]

However, I don't think the % increase is ever a relevant metric. The EHP tells you something like, "I can expect to survive 10 more rounds of this blow gun attack, or being attacked for 1 round by ten more blowgunners" The % EHP increase doesn't really tell you anything directly meaningful.

I think that's a fair criticism. I'm not sure I agree but it's definitely a point worthy of discussion. Maybe approach it form resource expenditure viewpoint. It tends to require a mirror image every combat in the day to approach the eHP evaluation I derive for it. That could be alot of spell slots and alot of "actions" being used. In that perspective, the higher your effective hp is without mirror image the less beneficial it is to use that many slots or actions to add more effective hp.

[Valmark]

Expect is used, aiui, in the probabilistic sense.

The extra attacks we don't survive because of something that would miss if it hadn't removed a mirror image, are cancelled out by the fact that if a mirror image succesfully tanks damage, then we might well survive more than N extra attacks -- because only a proportion of the extra attacks we're still alive to face will actually hit.

...not following. Mirror Image allows us to survive N extra attacks assiming MIAC=AC.

If AC>MIAC then there is a chance to survive less attacks. Never to survive more attacks.

So, given X the number of attacks that hit the MI without having been able to hit us, we can expect to survive N-X attacks with X never bigger then N.

You can't survive more attacks then N, at least not thanks to MI (since Reynaert defined N as the attacks triggering MI required to destroy it).

[Frogreaver]

This is because Frogreaver is using a different definition of eHP which requires the number of attacks to be kept constant.

This is a weird way to look at it, because HP is a measure of how much damage and therefore how many attacks you can be hit by.

One can look at effective hp as an enemy attacking you till you die

Or

One can look at effective hp as an enemy attacks you N times and then you repeat the same scenario again and again till you die. (Which also happens to include the case where the enemy attacks you till you are dead, you just need to determine the appropriate N).


If you keep the number of attacks constant, then the maths does work out to the same ratio, but it's a questionably useful metric, and he hasn't at any point defined or justified it (leading to much confusion as people assume something more sensible).

So when the number of attacks you are going to take in a given combat is pretty much the same with or without mirror image, how realistic of a scenario is it to look at the number of attacks it's going to take to kill you?

Isn't it more realistic to look at a scenario with a set number of attacks?

[JNAProductions]

Would it not make the most sense to look at actual damage prevented?

And mea culpa-I dropped a zero in the .0875, making my math wrong.

[Valmark]

One can look at effective hp as an enemy attacking you till you die

Or

One can look at effective hp as an enemy attacks you N times and then you repeat the same scenario again and again till you die. (Which also happens to include the case where the enemy attacks you till you are dead, you just need to determine the appropriate N).

So when the number of attacks you are going to take in a given combat is pretty much the same with or without mirror image, how realistic of a scenario is it to look at the number of attacks it's going to take to kill you?

Isn't it more realistic to look at a scenario with a set number of attacks?

What's the difference between getting attacked until we die and getting attacked an arbitrary number of times and repeating it until we die? In both cases it's the same total number of attacks, no?

EDIT: Well, the second can overkill us I guess, so there are a few wasted attacks more.

[MinotaurWarrior]

If it takes N attacks to make all of the images going away (only counting the attacks that the randomizer points at the Images), then we can expect to survive N more attacks than we would without casting Mirror Image.

And this number N, the number of attacks it takes to make the images go away, depends only on the image's chance of getting hit by an attack (as calculated by its AC versus attack bonus), and not on our own AC.

No, the benefit does depend on your AC.

The image disappears after N attacks. You then suffered N*EV(attack damage) less damage. That is then scaled by your AC (and other defenses) to a higher effective HP.

Again, look at the simplified case:

You have 10HP, the image gets hit on 19, you get hit on a 20, the attacker is using a blow gun for a constant 1 damage per hit.

The image, on average, prevents 0.5 damage - there's a 50% chance it is hit by an attack that would have missed you. But your AC makes it so that 0.5HP translates into 10 effective HP, because you only get hit by one in twenty attacks.

If on the other hand your ac is lower and you get hit on 19s, the image prevents a full 1hp of damage, but that again only translates into 10 effective HP, because you get hit by one in ten attacks.

Practically this doesn't matter because you don't keep fighting until you hit 0hp, you keep fighting until the combat is over, and the number of HP you have left over is a feel-good measure and the EHP calculation is irrelevant to feels. LudicSavants calculation covers the part of the equation that actually comes up in gameplay.

But theoretically in a white room, this is how the math shakes out.

[fat.hampster]

...not following. Mirror Image allows us to survive N extra attacks assiming MIAC=AC.

If AC>MIAC then there is a chance to survive less attacks. Never to survive more attacks.

So, given X the number of attacks that hit the MI without having been able to hit us, we can expect to survive N-X attacks with X never bigger then N.

You can't survive more attacks then N, at least not thanks to MI (since Reynaert defined N as the attacks triggering MI required to destroy it).

mirror image allows us to survive at most N extra hits. But if I can only be hit on a 20, and mirror image absorbs an actual hit, then on average I'll survive an extra 20 attacks than had I not had mirror image up, and I'd taken that hit.

[fat.hampster]

One can look at effective hp as an enemy attacks you N times and then you repeat the same scenario again and again till you die. (Which also happens to include the case where the enemy attacks you till you are dead, you just need to determine the appropriate N).


Sure, but then you're recasting mirror image multiple times, so it's not a fair comparison, assuming it is a limited resource. (and if it's not, then it doesn't matter how good it is -- it's always better than not having it!)



So when the number of attacks you are going to take in a given combat is pretty much the same with or without mirror image, how realistic of a scenario is it to look at the number of attacks it's going to take to kill you?

Isn't it more realistic to look at a scenario with a set number of attacks?


But presumably with more eHP, and the same difficulty of encounters you're either going to have more encounters, or eHP is going to become more irrelevant.

[MaxWilson]

Here is an AnyDice program I wrote for calculating the expected damage against you over X attacks using Mirror Image.

https://anydice.com/program/1e0c2

Here is another AnyDice program I wrote that does the inverse -- instead of calculating the damage against you, it calculates how much damage the images blocked that you otherwise would have taken.

https://anydice.com/program/1e0c1

Thanks to Stealth_Elephant and MaxWilson and AureusFulgens for helping to check for bugs.

If you're confused about how the function works, Stealth_Elephant has a great tutorial about how to use state systems in AnyDice: https://www.reddit.com/r/3d6/comments/gf111s/anydice_tutorial_part_3_state_the_great_weapon/

The results are, unsurprisingly, basically what folks have been trying to tell the OP since the first page.

If you have any questions I'd be happy to answer them. And if you can find any mistakes please do let me know.

@Everyone,

What LudicSavant's program (https://anydice.com/program/1e0c1) does is precisely to calculate the effective HP increase for a given number of specific attacks against a specific defense. As in, if you had this many extra HP, you'd expect the same outcomes as if you had cast Mirror Image. The extra HP and Mirror Image are equivalent.

Feel free to tweak the parameters if you want to make the hypothetical attacks bigger, e.g. if I have an AC 21 Dex 10 Paladorc vs. a Fire Giant it's, my effective HP increases by 48.11 HP on average over three rounds of combat (six attacks from the Fire Giant). But if I increase AC to 23 (in bold) from Shield of Faith or something, my effective HP gain is only 39.92. More attacks get "wasted" on misses. Mirror Image becomes more redundant against high AC (and of course it's completely redundant against anything with blindsight).

If you want to calculate other results, go to https://anydice.com/program/1e0d1 and change the last line in the program, e.g. to change AC from 21 to 23 change the bolded bit below from:

output
[images 3 imageac 10 imageroll 1d20 attacks ATTACKS roll 1d20 plus 11 vs 21 for 6d6+7 crit 6d6 on 20] named "AC 21 Forge Cleric vs. Fire Giant: effective HP gain from Mirror Image over three rounds of combat"

To this:

output
[images 3 imageac 10 imageroll 1d20 attacks ATTACKS roll 1d20 plus 11 vs 23 for 6d6+7 crit 6d6 on 20] named "AC 21 Forge Cleric vs. Fire Giant: effective HP gain from Mirror Image over three rounds of combat"

Tactical note: against soft targets (AC 10-15ish) the Fire Giant can opt to close his eyes while attacking and just accept disadvantage in order to ignore the Mirror Images, but AC 21ish is the break-even point where that tactic stops helping. Either way you gain just under 50 effective HP.

Mirror Image yields fewer additional effective HP for high AC. QED.

[Reynaert]

No, the benefit does depend on your AC.

The image disappears after N attacks. You then suffered N*EV(attack damage) less damage. That is then scaled by your AC (and other defenses) to a higher effective HP.

Again, look at the simplified case:

You have 10HP, the image gets hit on 19, you get hit on a 20, the attacker is using a blow gun for a constant 1 damage per hit.

The image, on average, prevents 0.5 damage - there's a 50% chance it is hit by an attack that would have missed you. But your AC makes it so that 0.5HP translates into 10 effective HP, because you only get hit by one in twenty attacks.

If on the other hand your ac is lower and you get hit on 19s, the image prevents a full 1hp of damage, but that again only translates into 10 effective HP, because you get hit by one in ten attacks.

In both cases that's 10 effective HP. So it does not depend on your AC.

[MinotaurWarrior]

@Everyone,

What LudicSavant's program (https://anydice.com/program/1e0c1) does is precisely to calculate the effective HP increase for a given number of specific attacks against a specific defense.

Snip

Mirror Image yields fewer additional effective HP for high AC. QED.

It is proven, under those constraints. But that's like saying, "Does pushing a car in neutral downhill at the start of a trip extend your range less if you are driving a highly feul efficient vehicle" and answering "If you are only driving 20 miles to the super market, it saves you less feul."

The real answer is that no matter how effecient your car is, the downhill glide extends your range by a constant amount equal to the distance you get in neutral. But also, who cares, because you weren't going to drive until you ran out of gas anyway, you were just going to go to the grocery store.


In both cases that's 10 effective HP. So it does not depend on your AC.

We may be talking past each other with the switch between conventions.

The benefit of MI doesn't depend on AC

The benefit of each hit prevented by MI does vary based on AC

[Valmark]

mirror image allows us to survive at most N extra hits. But if I can only be hit on a 20, and mirror image absorbs an actual hit, then on average I'll survive an extra 20 attacks than had I not had mirror image up, and I'd taken that hit.

This means N is 20 and X is 0. You said that we can survive more then N extra attacks, but you can't survive more attacks thanks to Mirror Image then the ones Mirror Image allows you to survive.

Maybe this wasn't what you meant? If so I apologize (also I misunderstood yet again if that's so).




The benefit of MI doesn't depend on AC

The benefit of each hit prevented by MI does vary based on AC

Doesn't the benefit of hits prevented by MI make up the benefit of MI?

Anyway, the benefit of MI does depend on AC. With an high enough AC MI can be a total waste- with Ludic's program and those paremeters AC 21 meant a 60% chance for MI to do absolutely nothing.

To see it another way, the eHP granted by MI can be lost on misses, making it lower the higher the disparity in AC.

[Hellpyre]

And for what it's worth, Ludic's program outputs agree pretty closely with what I got taking the 1600 outputs from any AC value pair and computing on them. I'm at work, so I can't check against cases I didn't already test for, but those numbers agree with what I saw pretty well.

[fat.hampster]

This means N is 20 and X is 0. You said that we can survive more then N extra attacks, but you can't survive more attacks thanks to Mirror Image then the ones Mirror Image allows you to survive.

Maybe this wasn't what you meant? If so I apologize (also I misunderstood yet again if that's so).



Consider a wizard who has 10 hit points and 40 ac
monster attacks wizard and gets a natural 20, and deals 20 damage, killing the wizard.

If we re-run with a single mirror image, and have the attack hit the image, the the wizard is alive, -- the mirror image has successfully allowed us to survive an extra hit.

But the monster keeps attacking us. Most of these fall of us harmlessly, but 17 rounds later, it rolls another natural 20, killing our wizard.

Now the difference between the 2 scenarios was a single mirror image, but the difference in the number of attacks that our wizard faced was 17. In this case a single image has allowed us to survive more than 1 attack.

And the key thing is that the higher our AC, the longer we expect to last without any recourse to mirror image, so the higher our AC, the more extra attacks we expect to face before dying when our MI successfully saves us from an attack we would otherwise suffer, and this perfectly cancels with the higher chance we have for the mirror image to be wasted, such that our expected number of attacks prevented by each image is 1.

[Reynaert]

...not following. Mirror Image allows us to survive N extra attacks assiming MIAC=AC.

If AC>MIAC then there is a chance to survive less attacks. Never to survive more attacks.

So, given X the number of attacks that hit the MI without having been able to hit us, we can expect to survive N-X attacks with X never bigger then N.

You can't survive more attacks then N, at least not thanks to MI (since Reynaert defined N as the attacks triggering MI required to destroy it).

No, we can expect to survive N attacks. X doesn't factor into it.

To visualize:
Suppose we're in a fight, and as the first action we cast Mirror Image.

Now, each attack falls into one of five cases:

A - The attacker targets the Mirror Image, the attack missed the Mirror Image AC
B - The attacker targets the Mirror Image, the attack hits the Mirror Image AC but misses our AC
C - The attacker targets the Mirror Image, the attack hits the Mirror Image AC and also hits ours
D - The attacker targets us, the attack misses our AC
E - The attacker targets us, the attack hits our AC

N (the number of attacks required to destroy the MI) is the sum of cases A, B and C.

Because there are three images, the sum of cases B and C will be exactly 3. (I believe case B is your X, but read on).
Now, to calculate N you only need to know the ratio of A+B+C to B+C, which is equal to the chance of an attack hitting the Mirror Image AC. So N is basically 3*(chance to hit Mirror Image AC). (I believe you agree with this?)

Suppose we're KO'd at the last attack. So the sum of the five cases is basically the number of attacks we can survive minus one (on average, of course) with Mirror Image.

Now, the number of attacks we could have survived without mirror image up is the sum of D and E (agreed?)

So, the extra number of attacks we can survive because of Mirror Image is A+B+C = N

[Reynaert]

Anyway, the benefit of MI does depend on AC. With an high enough AC MI can be a total waste- with Ludic's program and those paremeters AC 21 meant a 60% chance for MI to do absolutely nothing.

But in the other 40% of cases, it will have done something. Because of our high AC, that something is worth so much eHP that it averages out that 60% chance of it being 0.

[Valmark]

Consider a wizard who has 10 hit points and 40 ac
monster attacks wizard and gets a natural 20, and deals 20 damage, killing the wizard.

If we re-run with a single mirror image, and have the attack hit the image, the the wizard is alive, -- the mirror image has successfully allowed us to survive an extra hit.

But the monster keeps attacking us. Most of these fall of us harmlessly, but 17 rounds later, it rolls another natural 20, killing our wizard.

Now the difference between the 2 scenarios was a single mirror image, but the difference in the number of attacks that our wizard faced was 17. In this case a single image has allowed us to survive more than 1 attack.

And the key thing is that the higher our AC, the longer we expect to last without any recourse to mirror image, so the higher our AC, the more extra attacks we expect to face before dying when our MI successfully saves us from an attack we would otherwise suffer, and this perfectly cancels with the higher chance we have for the mirror image to be wasted, such that our expected number of attacks prevented by each image is 1.

Alright, now I get it. We were talking about two different things.

And even then, those 16 attacks you are surviving are also thanks to AC. If the wizard's AC had been, say, 10 and like, the third attack hit them MI would have allowed us to survive 2 attacks.

[EdenIndustries]

@Everyone,

What LudicSavant's program (https://anydice.com/program/1e0c1) does is precisely to calculate the effective HP increase for a given number of specific attacks against a specific defense. As in, if you had this many extra HP, you'd expect the same outcomes as if you had cast Mirror Image. The extra HP and Mirror Image are equivalent.

Feel free to tweak the parameters if you want to make the hypothetical attacks bigger, e.g. if I have an AC 21 Dex 10 Paladorc vs. a Fire Giant it's, my effective HP increases by 48.11 HP on average over three rounds of combat (six attacks from the Fire Giant). But if I increase AC to 23 (in bold) from Shield of Faith or something, my effective HP gain is only 39.92. More attacks get "wasted" on misses. Mirror Image becomes more redundant against high AC (and of course it's completely redundant against anything with blindsight).

If you want to calculate other results, go to https://anydice.com/program/1e0d1 and change the last line in the program, e.g. to change AC from 21 to 23 change the bolded bit below from:

output
[images 3 imageac 10 imageroll 1d20 attacks ATTACKS roll 1d20 plus 11 vs 21 for 6d6+7 crit 6d6 on 20] named "AC 21 Forge Cleric vs. Fire Giant: effective HP gain from Mirror Image over three rounds of combat"

To this:

output
[images 3 imageac 10 imageroll 1d20 attacks ATTACKS roll 1d20 plus 11 vs 23 for 6d6+7 crit 6d6 on 20] named "AC 21 Forge Cleric vs. Fire Giant: effective HP gain from Mirror Image over three rounds of combat"

Tactical note: against soft targets (AC 10-15ish) the Fire Giant can opt to close his eyes while attacking and just accept disadvantage in order to ignore the Mirror Images, but AC 21ish is the break-even point where that tactic stops helping. Either way you gain just under 50 effective HP.

Mirror Image yields fewer additional effective HP for high AC. QED.

Frogreaver, can we get your analysis of LudicSavant's program too?

[Reynaert]

Alright, now I get it. We were talking about two different things.

And even then, those 16 attacks you are surviving are also thanks to AC. If the wizard's AC had been, say, 10 and like, the third attack hit them MI would have allowed us to survive 2 attacks.

I think that example was just to illustrate that even with ridiculous AC, mirror image could still have an effect. But I think I can use that same scenario and explain it for both cases:

To simplify, assume 1 image left but a 100% chance if hitting that image (I think we can both agree that mirror image is less useful if you get killed while it is still up).

In scenario 1, MI has 10ac, we have 40ac, the monster hits with a +10.
(basically, to kill us, the attacker will have to roll a nat20)
Now, the expected number of times you need to roll to get a nat20 is 20 (I think, but the actual number doesn't matter). But the first attack will certainly miss, so that doesn't count, so the actual number of attacks needed to kill us is 21. In other words, the image gives us, on average, one more.

In scenario 2, MI has 10ac, we have 10ac, the monster hits with a +10
(basically, the monster will hit us and kill us. We ignore nat1s for simplicity)
Now, the expected number of times you need to roll is 1. But the first attack will certainly miss, so that doesn't count, so the actual number of attacks needed to kill us is 2. In other words, the image gives us one more.

The only difference here is the change in variance, but the average stays the same.

[Valmark]

No, we can expect to survive N attacks. X doesn't factor into it.

To visualize:
Suppose we're in a fight, and as the first action we cast Mirror Image.

Now, each attack falls into one of five cases:

A - The attacker targets the Mirror Image, the attack missed the Mirror Image AC
B - The attacker targets the Mirror Image, the attack hits the Mirror Image AC but misses our AC
C - The attacker targets the Mirror Image, the attack hits the Mirror Image AC and also hits ours
D - The attacker targets us, the attack misses our AC
E - The attacker targets us, the attack hits our AC

N (the number of attacks required to destroy the MI) is the sum of cases A, B and C.

Because there are three images, the sum of cases B and C will be exactly 3. (I believe case B is your X, but read on).
Now, to calculate N you only need to know the ratio of A+B+C to B+C, which is equal to the chance of an attack hitting the Mirror Image AC. So N is basically 3*(chance to hit Mirror Image AC). (I believe you agree with this?)

Suppose we're KO'd at the last attack. So the sum of the five cases is basically the number of attacks we can survive minus one (on average, of course) with Mirror Image.

Now, the number of attacks we could have survived without mirror image up is the sum of D and E (agreed?)

So, the extra number of attacks we can survive because of Mirror Image is A+B+C = N

B (or X) does not contribute to N. This because Mirror Image hasn't allowed us to survive B, B are attacks that would have missed us. We would have taken no damage nonetheless. Homewever, it did take away one Mirror Image- so either you define N as A+C or as A+B+C-B.

I was defining N as A+B+C, yes, then subtracting B (which I called X).

In other words, simply targeting MI isn't enough for it to be useful (which would be A+B+C) it also needs to have been able to hit us (which would be only A+C).

For completeness, it is true that B+C is 3. Homewever, since B isn't contributing to N, you could have A and C be 0 and B be 3. N would be 0.


But in the other 40% of cases, it will have done something. Because of our high AC, that something is worth so much eHP that it averages out that 60% chance of it being 0.

How does a lower chance of doing something average out with an higher chance of doing nothing? This question was utterly stupid, ignore it. Homewever what follows is correct.

That 40% chance is composed of various chances of taking varying amount of damage and are still dependant on AC with the chances of MI protecting us from higher amounts of damage being lower due to our AC. Or rather, the disparity between our AC and MI's AC. So, higher AC, less effect.


I think that example was just to illustrate that even with ridiculous AC, mirror image could still have an effect. But I think I can use that same scenario and explain it for both cases:

To simplify, assume 1 image left but a 100% chance if hitting that image (I think we can both agree that mirror image is less useful if you get killed while it is still up).

In scenario 1, MI has 10ac, we have 40ac, the monster hits with a +10.
(basically, to kill us, the attacker will have to roll a nat20)
Now, the expected number of times you need to roll to get a nat20 is 20 (I think, but the actual number doesn't matter). But the first attack will certainly miss, so that doesn't count, so the actual number of attacks needed to kill us is 21. In other words, the image gives us, on average, one more.

In scenario 2, MI has 10ac, we have 10ac, the monster hits with a +10
(basically, the monster will hit us and kill us. We ignore nat1s for simplicity)
Now, the expected number of times you need to roll is 1. But the first attack will certainly miss, so that doesn't count, so the actual number of attacks needed to kill us is 2. In other words, the image gives us one more.

The only difference here is the change in variance, but the average stays the same.

The effect of AC on MI is present only when our AC is higher then MI's AC, so yeah- MI is 100% effective in the second case, due to the lack of disparity. Or 1 more attack for a total of 2.

But the first example is wrong. MI gives on average no benefit. MI has a 5% percent chance of providing benefit at all, because it'll be broken on any hit that is not a crit (and not a nat 1, but that wouldn't hit us either).
Say the enemy rolls 20 only on the 20th attack. MI is broken on the first, maybe second hit and we still die on the 20th attack.

[Reynaert]

@Everyone,

What LudicSavant's program (https://anydice.com/program/1e0c1) does is precisely to calculate the effective HP increase for a given number of specific attacks against a specific defense. As in, if you had this many extra HP, you'd expect the same outcomes as if you had cast Mirror Image. The extra HP and Mirror Image are equivalent.

Feel free to tweak the parameters if you want to make the hypothetical attacks bigger, e.g. if I have an AC 21 Dex 10 Paladorc vs. a Fire Giant it's, my effective HP increases by 48.11 HP on average over three rounds of combat (six attacks from the Fire Giant). But if I increase AC to 23 (in bold) from Shield of Faith or something, my effective HP gain is only 39.92. More attacks get "wasted" on misses. Mirror Image becomes more redundant against high AC (and of course it's completely redundant against anything with blindsight).

If you want to calculate other results, go to https://anydice.com/program/1e0d1 and change the last line in the program, e.g. to change AC from 21 to 23 change the bolded bit below from:

output
[images 3 imageac 10 imageroll 1d20 attacks ATTACKS roll 1d20 plus 11 vs 21 for 6d6+7 crit 6d6 on 20] named "AC 21 Forge Cleric vs. Fire Giant: effective HP gain from Mirror Image over three rounds of combat"

To this:

output
[images 3 imageac 10 imageroll 1d20 attacks ATTACKS roll 1d20 plus 11 vs 23 for 6d6+7 crit 6d6 on 20] named "AC 21 Forge Cleric vs. Fire Giant: effective HP gain from Mirror Image over three rounds of combat"

Tactical note: against soft targets (AC 10-15ish) the Fire Giant can opt to close his eyes while attacking and just accept disadvantage in order to ignore the Mirror Images, but AC 21ish is the break-even point where that tactic stops helping. Either way you gain just under 50 effective HP.

Mirror Image yields fewer additional effective HP for high AC. QED.

As far as I can tell, the program doesn't scale the actual HP to effective HP.

For example, with a +11 attack, the chance to get hit is 55% with 21AC and 45% with 23AC.
(Actually almost 60% and almost 50% to account for crits).

So the effective HP mitigation with 21 AC would be 48.11/0.6 =~ 80.2 and with 23 AC would be 39.92 / 0.5 =~ 79.2 That is well within the expected variance of such a calculation, (especially given the slight skew that crits introduce with only doubling dice and not modifiers). So actually the program demonstrated that for *effective* HP, your own AC does not matter.

(Unless I missed the point in the program where actual HP is scaled to effective HP. I'm reasonably sure it's not there but I might have missed it, so anyone is welcome to point out where this is done in the code).

[Segev]

I've slept on it and I think I understand where you're going.

There's that specific enemy that does 20 damage every hit. He damages your Dex AC half the time and your full AC 1/4 of the time.

1) You have 100 hp and only Dex AC, therefore 200 effective hp. 10 attacks and you're down. Each mirror image adds 2 extra attacks or 40 effective hp.

2) You have 100 hp and fill AC, therefore 400 effective hp. 20 attacks and you're down. Each mirror image adds 2 extra attacks or 40 effective hp.


Of course, 10+2 attacks has a bigger impact than 20+2 attacks. But both mirror images are 40 effective hp.
So the value of mirror image depends on the enemy, not on your AC.

I think part of the problem is that we're not properly discounting the images' contribution to effective hp based on the chance that they contribute 0. Remember, an image (in this example) either protects you from 20 hp or 0 hp.

If the attack misses the image, the image didn't protect you. You wouldn't have been hit, either; your AC is at least the same as the image's.
If the attack hits the image, but would have missed you, the image didn't protect you. You wouldn't have been hit even if it wasn't there.
The second case means that there is a discount to the effective hp an image is granting that is a function of the difference between your AC and the images' AC.

I think the math, if I've been following it correctly, all fails to account for this discount, which is where all claims that mirror image adds "extra hits" to your survivability dependent solely on the images' AC come from. It's flawed: if you figure out that discount factor on the hp they're really adding, you'll discover, I think, that increasing your AC actually decreases the number of hits (or effective hp) granted by mirror image if you do it without increasing the images' AC.

Put another way, I think we're incorrectly attributing effective hit points granted by your own AC to effective hit points granted by an image that is destroyed when the attack would have missed you anyway.

[Satori01]

Naw, usually Blur or Protection From Evil is the spell you want when you can't afford to get hit. Or Dimension Door, or Expeditious Retreat.

Mirror Image is the spell you cast when you are unarmored (crummy AC) and have more spell slots than HP, and nothing important to be casting this round, and aren't fighting enemies with blindsight, and yet for some reason can't afford to simply leave combat.

Or to put it the other way: in my experience, Mirror Image is the spell you learn (because hey, Quickened Mirror Image with no concentration cost sounds great for a Paladorc! maybe it will help with big tough monsters) and then never wind up actually casting because something else is always better.

I was intending to end the statement, quoted by you, with a question mark, to elicit some statements of conclusion....LOL....edited it now.

I don't disagree with you.
Mirror Image is useful if you stumble upon something so far above your pay grade, that you are not sure disadvantage is going to be enough to stop you from being hit.

I also assert, that the first time an Illusionist uses Illusory Reality to turn an Image from MI real....there is a 90% chance you are granted Inspiration, just for the macabre humor.

[Valmark]

I also assert, that the first time an Illusionist uses Illusory Reality to turn an Image from MI real....there is a 90% chance you are granted Inspiration, just for the macabre humor.

I know this is a joke, but I feel like I should point out that Illusory Reality only works on inanimate objects part of the illusion.

[Satori01]

I know this is a joke, but I feel like I should point out that Illusory Reality only works on inanimate objects part of the illusion.

Then forget it then...if you can't literally create a "Meat Shield", the spell is dead to me.

So dead to me in fact, I might go to the spell's funereal in a a red dress.

[Valmark]

Then forget it then...if you can't literally create a "Meat Shield", the spell is dead to me.

So dead to me in fact, I might go to the spell's funereal in a a red dress.

I wonder what would happen if an illusionist did use their ability though.

Say the wizard has a shield and uses Illusory Reality to make one of the Images' shield real.

According to Mirror Image, the copies changes places to make pinpointing the correct one impossible so they probably pass through the person or it would be obvious the real one is the one staying still- so, what happens?

I'd probably refuse it outright as a DM. That's probably bad DMing, but I have no intention of handling such a situation in a serious way (I can think silly outcomes though).

[MaxWilson]

(A) As far as I can tell, the program doesn't scale the actual HP to effective HP.

(B) For example, with a +11 attack, the chance to get hit is 55% with 21AC and 45% with 23AC.
(Actually almost 60% and almost 50% to account for crits).

(C) So the effective HP mitigation with 21 AC would be 48.11/0.6 =~ 80.2 and with 23 AC would be 39.92 / 0.5 =~ 79.2 That is well within the expected variance of such a calculation, (especially given the slight skew that crits introduce with only doubling dice and not modifiers). So actually the program demonstrated that for *effective* HP, your own AC does not matter.

(Unless I missed the point in the program where actual HP is scaled to effective HP. I'm reasonably sure it's not there but I might have missed it, so anyone is welcome to point out where this is done in the code).

I agree with (B). I don't understand (A) or (C) but I get the impression you may be misunderstanding what that 48.11 HP are. That is all the damage (on average) taken by a Mirror Image EXCLUDING damage that would have missed your AC anyway. The +11 vs. AC calculation is factored in already. You don't need to multiply by 55%/45% again.

To see this in action, re-run the program with IMAGEAC set to the same as the actual AC (so that every mirror image hit prevents an actual hit). You'll see effective HP gains go way up, because there isn't any waste.

Note BTW that AnyDice isn't a Monte Carlo simulation--it's not randomly sampling from a probability distribution, it's computing the actual probability (I think by enumerating all possibilities).


I was intending to end the statement, quoted by you, with a question mark, to elicit some statements of conclusion....LOL....edited it now.

I don't disagree with you.
Mirror Image is useful if you stumble upon something so far above your pay grade, that you are not sure disadvantage is going to be enough to stop you from being hit.

I also assert, that the first time an Illusionist uses Illusory Reality to turn an Image from MI real....there is a 90% chance you are granted Inspiration, just for the macabre humor.

There are a handful of big tough monsters whom Mirror Image would be helpful against (Zariel), but also so many more that just ignore it (Demogorgon, Hutjin, Orcus, Krakens, all dragons) due to blindsight or truesight. It's really unfortunate IMO. :( Just where you think it should shine it falls flat.

[fat.hampster]

I think part of the problem is that we're not properly discounting the images' contribution to effective hp based on the chance that they contribute 0. Remember, an image (in this example) either protects you from 20 hp or 0 hp.

If the attack misses the image, the image didn't protect you. You wouldn't have been hit, either; your AC is at least the same as the image's.
If the attack hits the image, but would have missed you, the image didn't protect you. You wouldn't have been hit even if it wasn't there.
The second case means that there is a discount to the effective hp an image is granting that is a function of the difference between your AC and the images' AC.

I think the math, if I've been following it correctly, all fails to account for this discount, which is where all claims that mirror image adds "extra hits" to your survivability dependent solely on the images' AC come from. It's flawed: if you figure out that discount factor on the hp they're really adding, you'll discover, I think, that increasing your AC actually decreases the number of hits (or effective hp) granted by mirror image if you do it without increasing the images' AC.

Put another way, I think we're incorrectly attributing effective hit points granted by your own AC to effective hit points granted by an image that is destroyed when the attack would have missed you anyway.

Lets construct 2 wizards with equal eHP (for a +0 attack, and ignoring crits)
A: AC 16, hp 20
B AC 11, hp 40

A has eHP of 20/(5/20) = 80
B has eHP of 40/(10/20) = 80
so both have an eHP of 80.

And let the Images in both cases have AC 11

Now attack comes in for 10 damage: There are 3 possibilites:
1:(50%) 1-10 rolled no effect. A: 80eHP , B: 80eHP
2:(25%) 11-15 rolled, B hit, A not: A: 80eHP B: 60eHP
3.(25%) 16-20 rolled, both hit: A: 40eHP, B 60eHP

Discarding 1 (because the outcomes are identical) and averaging over 2 + 3, gives both A and B with an expected 60eHP from the attack, but A has higher variance.

If we started A and B with the same health, then A's hp would be higher in all the cases, but the amount lost in case 3 would still be 40, so the average eHP lost is still the same.

This is why actually getting hit with a higher AC is "worse" than when it's a lower AC, and why avoiding that damage is worth more, which councils the "wasted" mirror image which would exist in case 2.

That is, nobody is neglecting the negative value of "wasting" a mirror image when it gets destroyed and the attack wouldn't have hit you, rather you're discounting the extra value obtained when avoiding an actual hit when you have a higher AC. And obviously that extra value is 'because' of the higher AC, but that's the point.

There's another way to look at it which is in expected damage.

an attack against A has expected damage 2.5 (25% * 10), an attack against b has expected damage 5 (50% * 10). Mirror image will negate N attacks (where the derivation for N has been described elsewhere). Against A this "only" negates 2.5 expected damage for each of those attacks, but it's still negating the full N attacks. That's the simpler way of looking at it.

Alternatively, each of those N attacks has a 75% chance (outcome 1+2) of negating no damage, and a 25% chance of negating 10 damage. Averaging those out leads to 2.5 damage.

[Reynaert]

B (or X) does not contribute to N. This because Mirror Image hasn't allowed us to survive B, B are attacks that would have missed us. We would have taken no damage nonetheless. Homewever, it did take away one Mirror Image- so either you define N as A+C or as A+B+C-B.

I was defining N as A+B+C, yes, then subtracting B (which I called X).

In other words, simply targeting MI isn't enough for it to be useful (which would be A+B+C) it also needs to have been able to hit us (which would be only A+C).

For completeness, it is true that B+C is 3. Homewever, since B isn't contributing to N, you could have A and C be 0 and B be 3. N would be 0.

Irrelevant. It's not your definition, it is mine. I define N to be A+B+C
I proved that A+B+C only depends on the mirror image AC and not on your own.


How does a lower chance of doing something average out with an higher chance of doing nothing?

Because 'something' has a weight. If I play a game where I always win something, and on average I win 40 dollars, that's my average.
But if I play another game and have 60% of winning nothing and 40% of winning 100 dollars, then on average I will *still* win 40 dollars.


But the first example is wrong. MI gives on average no benefit. MI has a 5% percent chance of providing benefit at all, because it'll be broken on any hit that is not a crit (and not a nat 1, but that wouldn't hit us either).
Say the enemy rolls 20 only on the 20th attack. MI is broken on the first, maybe second hit and we still die on the 20th attack.

That's not how expected values work.
Let's play a game: Roll a d20 until you get a 20. how many rolls on average do you think you need?
Now play the same game again, but for the first roll you have to use a d10. Now how many rolls on average do you need?

[Reynaert]

I agree with (B). I don't understand (A) or (C) but I get the impression you may be misunderstanding what that 48.11 HP are. That is all the damage (on average) taken by a Mirror Image EXCLUDING damage that would have missed your AC anyway. The +11 vs. AC calculation is factored in already. You don't need to multiply by 55%/45% again.

I'm not misunderstanding anything. 48.11HP is the amount of damage that the cleric will need to heal to get you back to full HP.

You, however, may be misunderstanding what most people in this thread mean by 'effective HP'.

This is calculated by another metric: How many times does a creature need to attack you (and how much damage per attack) before your hitpoint pool goes to zero?

If you multiply the number of attacks and the potential damage of each attack, that is the effective HP you have (against that creature).

So, your effective HP times your chance of being hit (plus some modifier for crits), equals your hitpoint pool.


Edit: Let me ask you this: Do you consider the spell Cure Wounds to be more effective on a person with higher AC?

[Segev]

Lets construct 2 wizards with equal eHP (for a +0 attack, and ignoring crits)
A: AC 16, hp 20
B AC 11, hp 40

A has eHP of 20/(5/20) = 80
B has eHP of 40/(10/20) = 80
so both have an eHP of 80.

And let the Images in both cases have AC 11

Now attack comes in for 10 damage: There are 3 possibilites:
1:(50%) 1-10 rolled no effect. A: 80eHP , B: 80eHP
2:(25%) 11-15 rolled, B hit, A not: A: 80eHP B: 60eHP
3.(25%) 16-20 rolled, both hit: A: 40eHP, B 60eHP

Discarding 1 (because the outcomes are identical) and averaging over 2 + 3, gives both A and B with an expected 60eHP from the attack, but A has higher variance.

If we started A and B with the same health, then A's hp would be higher in all the cases, but the amount lost in case 3 would still be 40, so the average eHP lost is still the same.

This is why actually getting hit with a higher AC is "worse" than when it's a lower AC, and why avoiding that damage is worth more, which councils the "wasted" mirror image which would exist in case 2.

That is, nobody is neglecting the negative value of "wasting" a mirror image when it gets destroyed and the attack wouldn't have hit you, rather you're discounting the extra value obtained when avoiding an actual hit when you have a higher AC. And obviously that extra value is 'because' of the higher AC, but that's the point.

There's another way to look at it which is in expected damage.

an attack against A has expected damage 2.5 (25% * 10), an attack against b has expected damage 5 (50% * 10). Mirror image will negate N attacks (where the derivation for N has been described elsewhere). Against A this "only" negates 2.5 expected damage for each of those attacks, but it's still negating the full N attacks. That's the simpler way of looking at it.

Alternatively, each of those N attacks has a 75% chance (outcome 1+2) of negating no damage, and a 25% chance of negating 10 damage. Averaging those out leads to 2.5 damage.

Alright, that makes sense. Part of the problem with the assumptions is the strange jumps we have to make between "fair comparisons" of different characters with the same eHP, and "decision making" with the same character.

For instance, this example makes it almost look like the wizard with the lower AC is better off because he doesn't need mirror image as much to survive one lucky hit, and it's easy to assume this is because of the wizard's higher AC. In reality, it's because the wizard traded real hp for higher AC to keep the same effective AC.

This does mean that, at the same eHP, a higher-AC wizard benefits more than a lower-AC wizard...but that's because the lower-AC wizard has more real hp, so each hit negated by the images is less likely to be a final blow.

On a wizard who's not trading hp for AC, but instead is simply examining the impact of mirror image on his survivability while having a constant hp but an increasing AC, mirror image does, in fact, drop in effectiveness for him as he gets more AC. This is intuitive to most of us, I think. Because we're used to thinking in terms of build decisions on a character, not in terms of hypothetical balance where eHP is the same no matter your AC.

Note: higher AC never makes mirror image a bad thing to have active, just less valuable compared to having lower AC (and thus lower eHP).

And a big part of that is that mirror image's benefits are in up to M negated hits entirely, rather than in any more steady-state protection. This is, again, why effective hp can be deceptive in terms of figuring out how effective mirror image will be in any particular encounter: the higher variance as caster AC increases further above image AC makes the times it isn't useful more frequent. So the opportunity cost of casting it becomes relatively greater.

[MaxWilson]

I'm not misunderstanding anything. (C) 48.11HP is the amount of damage that the cleric will need to heal to get you back to full HP.

You, however, may be misunderstanding what most people in this thread mean by 'effective HP'.

This is calculated by another metric: How many times does a creature need to attack you (and how much damage per attack) before your hitpoint pool goes to zero?

(A) If you multiply the number of attacks and the potential damage of each attack, that is the effective HP you have (against that creature).

(B) So, your effective HP times your chance of being hit (plus some modifier for crits), equals your hitpoint pool.

(A) Ah, interesting. You're right, I misunderstood how you were using the words. If we call the thing defined by (A) and (B) together ReyneartEffectiveHP for clarity, then I can see why you'd want to make the point that Mirror Image adds a fixed-by-my-AC-but-different-for-every-opponent amount of ReyneartEffectiveHP to your PC.

ReyneartEffectiveHP != SegevEffectiveHP (name chosen arbitrarily because Segev was the first one in this thead, in post #2, to propone the other view of effective HP, (C))

Is this whole thread just one big semantic argument about people using the words "effective HP" to mean different things?

[fat.hampster]

Alright, that makes sense. Part of the problem with the assumptions is the strange jumps we have to make between "fair comparisons" of different characters with the same eHP, and "decision making" with the same character.

For instance, this example makes it almost look like the wizard with the lower AC is better off because he doesn't need mirror image as much to survive one lucky hit, and it's easy to assume this is because of the wizard's higher AC. In reality, it's because the wizard traded real hp for higher AC to keep the same effective AC.

This does mean that, at the same eHP, a higher-AC wizard benefits more than a lower-AC wizard...but that's because the lower-AC wizard has more real hp, so each hit negated by the images is less likely to be a final blow.

On a wizard who's not trading hp for AC, but instead is simply examining the impact of mirror image on his survivability while having a constant hp but an increasing AC, mirror image does, in fact, drop in effectiveness for him as he gets more AC. This is intuitive to most of us, I think. Because we're used to thinking in terms of build decisions on a character, not in terms of hypothetical balance where eHP is the same no matter your AC.

Note: higher AC never makes mirror image a bad thing to have active, just less valuable compared to having lower AC (and thus lower eHP).

And a big part of that is that mirror image's benefits are in up to M negated hits entirely, rather than in any more steady-state protection. This is, again, why effective hp can be deceptive in terms of figuring out how effective mirror image will be in any particular encounter: the higher variance as caster AC increases further above image AC makes the times it isn't useful more frequent. So the opportunity cost of casting it becomes relatively greater.

Yeah, I don't necessarily disagree with you on it's effectiveness, though I think that's there's potentially an interesting discussion to be had on it, which requires everyone to be on the same page about what the OP was talking about.




Is this whole thread just one big semantic argument about people using the words "effective HP" to mean different things?

It's almost like the OP should have defined it when he started the thread. Or in response to one of the many people who asked him to define it. :/

[Reynaert]

(A) Ah, interesting. You're right, I misunderstood how you were using the words. If we call the thing defined by (A) and (B) together ReyneartEffectiveHP for clarity, then I can see why you'd want to make the point that Mirror Image adds a fixed-by-my-AC-but-different-for-every-opponent amount of ReyneartEffectiveHP to your PC.

ReyneartEffectiveHP != SegevEffectiveHP (name chosen arbitrarily because Segev was the first one in this thead, in post #2, to propone the other view of effective HP, (C))


butbutbut... SegevEffectiveHP is actually just your real HP then, isn't it? If you take 10 SegevEffectiveHP of damage, your cleric will need to cast a cure wounds to the tune of 10 to heal you up, right? Or am I mistaken?

If so, I've been calling that 'actual HP'. If not, what is the definition? I went back to post #2 but am not really seeing it.


But given that it gets really muddy with all those definitions, I switched (a few posts back) to a different metric to explain the point. This metric is 'number of attacks needed to deplete number of hitpoints'.

Theorem Of Mirror Image Effectiveness:
Definition of variables (N, H, X): If it takes a given creature N attacks(*) to make our hitpoints go down by H, casting Mirror Image will make it so that that same creature now needs N+X attacks(*) to make our hitpoints go down by H
Claim: The average number of attacks X only depends on the AC of the Mirror Image, and not on our own AC
Proof: See previous posts

*) attacks, not hits. I.E. how many times the creature gets to do an attack roll.


Example:
- We have 17AC
- Our mirror image has 11AC
- The creature has a +0 to hit
- The creature deals 4 damage on a hit (8 on a crit)
- We set H to 10. We want (to see how long it takes) the creature to take away 10 of our hitpoints.
So to calculate:
- The creature will deal an average of 1 damage per hit (0 damage on a 16 or less, 4 on a 17, 18 or 19 and 8 on a 20, so 4+4+4+8/20 = 1)
- So, N=10
- Now we cast Mirror Image and want to calculate X
- The chance the creature destroys an Image if it attacks it is 50% (it needs to roll 11 or higher)
- Therefore, on average 6 of the creature's attacks will target the Mirror Image before it is destroyed(**)
- But to take away 10 of our hitpoints, the creature will need to target us an average of 10 times
- So with Mirror Image, it takes 10+6=16 attacks to lower our hitpoints by 10 I.E. X=6
- Note that X=6 was calculated only from the AC of the mirror image and not ours.

**) I may be off on the statistics, but the point is it only depends on the mirror image AC.

I hope that clears up what I'm claiming.

Edit:


Is this whole thread just one big semantic argument about people using the words "effective HP" to mean different things?

Some of it, yes. But the OP is claiming that the ReynaertEffectiveHP(***) that Mirror Image grants is actually a percentage of your total ReynaertEffectiveHP and not an absolute amount.

***) I'm sure it has been defined like this a lot earlier in the thread by somebody else, before me

[Satori01]

[B](A)
Is this whole thread just one big semantic argument about people using the words "effective HP" to mean different things?

That, accurately, is the TL;DR version.

[LudicSavant]

Effective hit points generally is used to mean “how many hit points this defensive feature is worth in this situation.” It gives us a practical way to compare the value/effectiveness of AC to HP, inflicting Disadvantage, etc.

As Max said, that is precisely what the program I posted measures. So for example if you’re in a situation where it says it’s worth 6.5 eHP, then your expected/average benefit of Mirror Image is the same as having an enemy chew through a level 1 False Life.

Note that the OP specifically referenced my previous work on eHP and AC as the measurement I should be considering it in terms of (“who’s to say you were considering it in terms of ehp, as you’ve done for AC before?“). Under that definition, Frogreaver’s conclusions are wrong — eHP is not independent of AC. Far from it; AC has a very considerable impact on the spell slot’s value for your character.


Whose to say that you looked at Mirror Image in context of effective hp (a measurement you commonly use to tout the impact of impact of high AC). Here’s where he indicated I should be using my definition of eHP, a definition which is quite clearly established through years of math posts to this forum.
Not sure what else there is to be said about the matter. I suppose it’s hypothetically possible you could try to redefine eHP until there’s no correlation between it and AC, but this would remove said definition from any practical valuation of how good the spell is for you in real play, AFAICT. And would do nothing to support the OP's conclusions about how good Mirror Image is for Bladesingers.


In fact that would be another great thread, "A true examination of increasing AC on effective hp" - because what's often left out of those comparisons is that monster attack comes in a range and increasing AC to take 1 enemy from .1 to .05 chance to hit means you get no effective benefit on that already had lower attack.

This appears to be mistaken as well. It’s simply not true that taking an enemy from 10% to hit to 5% to hit somehow has no effective benefit.

[Valmark]

Irrelevant. It's not your definition, it is mine. I define N to be A+B+C
I proved that A+B+C only depends on the mirror image AC and not on your own.

Because 'something' has a weight. If I play a game where I always win something, and on average I win 40 dollars, that's my average.
But if I play another game and have 60% of winning nothing and 40% of winning 100 dollars, then on average I will *still* win 40 dollars.

That's not how expected values work.
Let's play a game: Roll a d20 until you get a 20. how many rolls on average do you think you need?
Now play the same game again, but for the first roll you have to use a d10. Now how many rolls on average do you need?

But... You have defined N as A+B+C, but then you explained it and it doesn't work like that. Your definition includes all attacks that target MI's AC while to describe it's effectiveness you should consider all attacks that hit MI but not the caster's AC.
In fact, I was wrong too because A as you defined it does nothing- it would have missed either way. It did not increase wizard's survivability.
N as you define it is literally just C.

That question you quoted? Yeah I said something stupid. I stroke through it afterwards because of that. Well, what I meant was something else, but I clearly didn't say it. I'll explain together with the game thing.

So, first off, you're right on the game. The averages are 20 and 21. But the game is not equivalent of MI.

So, d20 game, you roll 20 times and hope to get a 20. Obviously, thats 20 rolls on average.
Case 1d10+d20, since you can never get a d20 on a d10, it is simply a +1 to your average. The d10 is not a chance- it's as if the game starts on the second roll. It's a 100% chance of not winning the game.

Why is this different from the Mirror Image example? It is different from the Mirror Image example because there you need to get a 20 on the first roll.
It's like saying that "You must get a 20, but if the first roll is a 20 you must get another one". You have a 0,0025% chance of getting an average of 40 rolls- and you have a 99,9975% chance of getting an average of 20 rolls (the 'normal' situation).

The average between the two situations is 20,0995.

MI is likely to save you once in 40000 fights according to the example made (so only on a crit you die, one image left, it always triggers on the first hit, no autofail on a 1).

If MI's AC was 40 too, then the enemy would have an average of 40 rolls. Do you see now how much AC impacts on MI? By making the roll needed to hit MI's AC and the wizard's AC equivalent the chances of MI triggering raised from 0,0025% to 95%.

In terms of number of attacks survived with the low MI's AC the spell gave us 0,0995 more attacks while with equivalent AC it gave us 20 more attacks.
That is a lot of difference in number of attacks we survive (and equivalent effective hp we get, homewever that is calculated) by varying the AC.

Also, I had to stop writing for stuff and a lot of posts came in meanwhile so this might be irrilevant. Great!

[MaxWilson]

butbutbut... SegevEffectiveHP is actually just your real HP then, isn't it? If you take 10 SegevEffectiveHP of damage, your cleric will need to cast a cure wounds to the tune of 10 to heal you up, right? Or am I mistaken?

It's not real HP, because casting Mirror Image against the Fire Giant doesn't actually give you 48 extra HP. It just gives you the equivalent of 48 extra HP. You are as tough as if you have 48 extra HP, and after combat both Mirror Image guy and +48 HP guy will end the combat with the same average actual HP remaining.


It's almost like the OP should have defined it when he started the thread. Or in response to one of the many people who asked him to define it. :/

You're not wrong. :)

[Valmark]

Theorem Of Mirror Image Effectiveness:
Definition of variables (N, H, X): If it takes a given creature N attacks(*) to make our hitpoints go down by H, casting Mirror Image will make it so that that same creature now needs N+X attacks(*) to make our hitpoints go down by H
Claim: The average number of attacks X only depends on the AC of the Mirror Image, and not on our own AC
Proof: See previous posts

*) attacks, not hits. I.E. how many times the creature gets to do an attack roll.


Example:
- We have 17AC
- Our mirror image has 11AC
- The creature has a +0 to hit
- The creature deals 4 damage on a hit (8 on a crit)
- We set H to 10. We want (to see how long it takes) the creature to take away 10 of our hitpoints.
So to calculate:
- The creature will deal an average of 1 damage per hit (0 damage on a 16 or less, 4 on a 17, 18 or 19 and 8 on a 20, so 4+4+4+8/20 = 1)
- So, N=10
- Now we cast Mirror Image and want to calculate X
- The chance the creature destroys an Image if it attacks it is 50% (it needs to roll 11 or higher)
- Therefore, on average 6 of the creature's attacks will target the Mirror Image before it is destroyed(**)
- But to take away 10 of our hitpoints, the creature will need to target us an average of 10 times
- So with Mirror Image, it takes 10+6=16 attacks to lower our hitpoints by 10 I.E. X=6
- Note that X=6 was calculated only from the AC of the mirror image and not ours.

**) I may be off on the statistics, but the point is it only depends on the mirror image AC.


This is wrong too. I mean, until you cast MI it isn't wrong. It's that flat +6 attacks that is wrong.

It needs a 11 to hit MI's AC, yes, but it would have needed a 17 to hit the wizard. If it rolls between 11 and 16 it does nothing besides wasting MI's defensive power. That's 60% of the attacks not actually adding anything to the wizard's survivability, so it actually only adds 2,4 to the attacks needed to down the wizard.

I'm trusting that before considering the wizard's own AC the calculations were correct (since you said you weren't sure on the statistics).

[Reynaert]

But... You have defined N as A+B+C, but then you explained it and it doesn't work like that. Your definition includes all attacks that target MI's AC while to describe it's effectiveness you should consider all attacks that hit MI but not the caster's AC.
In fact, I was wrong too because A as you defined it does nothing- it would have missed either way. It did not increase wizard's survivability.
N as you define it is literally just C.

I defined N as "The number of times you need to attack in order to deplete Mirror Image". So no it isn't.

Suppose you're attacking a creature. It has an AC of 11 and 3 hitpoints. You can only attack with your fists for 1 damage (crits also do 1) and your attack bonus is +0. Then the number of attacks needed to kill that creature on average is 6. THAT is what N is.


So, first off, you're right on the game. The averages are 20 and 21. But the game is not equivalent of MI.

So, d20 game, you roll 20 times and hope to get a 20. Obviously, thats 20 rolls on average.
Case 1d10+d20, since you can never get a d20 on a d10, it is simply a +1 to your average. The d10 is not a chance- it's as if the game starts on the second roll. It's a 100% chance of not winning the game.

Why is this different from the Mirror Image example? It is different from the Mirror Image example because there you need to get a 20 on the first roll.
It's like saying that "You must get a 20, but if the first roll is a 20 you must get another one". You have a 0,0025% chance of getting an average of 40 rolls- and you have a 99,9975% chance of getting an average of 20 rolls (the 'normal' situation).

"You must get a 20, but if the first roll is a 20 you must get another one" is the same as "You must get a 20, but if the first roll is a 20, we pretend that it wasn't".

[Valmark]

I defined N as "The number of times you need to attack in order to deplete Mirror Image". So no it isn't.

Suppose you're attacking a creature. It has an AC of 11 and 3 hitpoints. You can only attack with your fists for 1 damage (crits also do 1) and your attack bonus is +0. Then the number of attacks needed to kill that creature on average is 6. THAT is what N is.

"You must get a 20, but if the first roll is a 20 you must get another one" is the same as "You must get a 20, but if the first roll is a 20, we pretend that it wasn't".

Oh, alright, sorry. But then N isn't indicative of MI's defensive power.

And... Yeah, that is the same thing. That's not an objection to what I said. Rolling 1d10 first then d20s isn't the same thing as having to roll a 20 twice if the first is a 20. For all those reasons I wrote earlier.

And thinking about it, those 6 attacks in the last example aren't even 6. It takes 6 attacks to defeat MI, but three of those would have missed anyway- so it's 3 attacks, and then only 40% of those actually hit the wizard. So 1,2 is what is added to the wizard's survivability.
This does not consider crits.

[Reynaert]

It's not real HP, because casting Mirror Image against the Fire Giant doesn't actually give you 48 extra HP. It just gives you the equivalent of 48 extra HP. You are as tough as if you have 48 extra HP, and after combat both Mirror Image guy and +48 HP guy will end the combat with the same average actual HP remaining.

The OP kept talking about a scaling factor from hp to 'effective HP' (derived from AC). So from that I deduced that his definintion could not be such that 1 'effective HP' was worth 1 actual hitpoint. (Also, this is the first time I've had to be aware of the actual definition). I tried to come up with the most logical definition that would fit his claims. If I'm debunking somebody's maths, I want to be using whatever defintions that person is using. ^^ Sorry for the confusion.

[Reynaert]

Oh, alright, sorry. But then N isn't indicative of MI's defensive power.

It is indicative of how many more times the creature will have to attack you to get the same result as without Mirror Image.

Here's a specific scenario:
- You are put in an arena with a monster.
- This monster will keep attacking you, once per second, until you go down.
- You will then be healed (and you get 100 gold pieces for every second you stay up).
- Without Mirror Image, you expect that on average you will go down after 20 seconds.
- The monster has an attack bonus that will hit your Mirror Image 50% of the time.

The question is: How many more seconds do you expect to stay up if you cast Mirror Image ?

I claim it's 6 seconds more.


And... Yeah, that is the same thing. That's not an objection to what I said. Rolling 1d10 first then d20s isn't the same thing as having to roll a 20 twice if the first is a 20. For all those reasons I wrote earlier.

So "You must get a 20, but if the first roll is a 20 you must get another one" is the same as "You must get a 20, but if the first roll is a 20, we pretend that it wasn't"

But that is the same as "You must get a 20, but for the first roll we give you a d20 where the 20 is crossed out"

Which is the same as "You must get a 20, but for the first roll we give you a dice that can't roll a 20"

Which is the same as "You must get a 20, but for the first roll we give you a d10"

Where in this line of 'is the same' do you believe it is not the same, and why?

[JNAProductions]

It is indicative of how many more times the creature will have to attack you to get the same result as without Mirror Image.

Here's a specific scenario:
- You are put in an arena with a monster.
- This monster will keep attacking you, once per second, until you go down.
- You will then be healed (and you get 100 gold pieces for every second you stay up).
- Without Mirror Image, you expect that on average you will go down after 20 seconds.
- The monster has an attack bonus that will hit your Mirror Image 50% of the time.

The question is: How many more seconds do you expect to stay up if you cast Mirror Image ?

I claim it's 6 seconds more.

If you're hit 50% of the time, along with the Mirror Image, and take 1/10th of your HP in damage each hit, then you'd expect to last six more seconds.
If you're hit 5% of the time, but take your entire HP in damage each hit, then you'd expect to last zero to two more seconds.

Edit: I made a thread for dice rolls for this. As a practical test.

Find it here (https://forums.giantitp.com/showthread.php?619700-Mirror-Image-Rolls&p=24731760#post24731760).

In the first case, 50% hit rate on PCs and Images, you last 25 dice rolls.
In the second case, you last 19.

[bid]

I think part of the problem is that we're not properly discounting the images' contribution to effective hp based on the chance that they contribute 0. Remember, an image (in this example) either protects you from 20 hp or 0 hp.
That's also true of AC. Either you get hit or you don't.

You could rewrite "applying damage" as a 2-step process:
1) either you hit Dex AC or you don't
2) either you hit full AC or you don't
Each step halves the chance of getting hit (or double your effective hp) {in the explicit situation I used}.
Your initial 100 hp, your 200 Dex-only-effective-hp, or your 400 full-AC-effective-hp are different POV of the same entity. All of those say that it will take 20 attacks to down you.

Stopping 2 hits with images is worth 80 full-AC-effective-hp.
Stopping 1 hit with 20 thp is also worth 80 full-AC-effective-hp.


But I'm not sure if that's what frogreaver saw or not, so I'll refrain from going further down the rabbit hole.

[Segev]

That's also true of AC. Either you get hit or you don't.

You could rewrite "applying damage" as a 2-step process:
1) either you hit Dex AC or you don't
2) either you hit full AC or you don't
Each step halves the chance of getting hit (or double your effective hp) {in the explicit situation I used}.
Your initial 100 hp, your 200 Dex-only-effective-hp, or your 400 full-AC-effective-hp are different POV of the same entity. All of those say that it will take 20 attacks to down you.

Stopping 2 hits with images is worth 80 full-AC-effective-hp.
Stopping 1 hit with 20 thp is also worth 80 full-AC-effective-hp.


But I'm not sure if that's what frogreaver saw or not, so I'll refrain from going further down the rabbit hole.
The part I was bringing up is when both defenses are overlapping. If AC is protecting you, is the MI protecting you as well from that hit? Or is that double-counting?

There was already a response that I acknowledged was probably better analysis than mine, though (apologies) I don't recall whose it was.

[Edea]

I did not expect this spell to get so much attention.

[MaxWilson]

The OP kept talking about a scaling factor from hp to 'effective HP' (derived from AC). So from that I deduced that his definintion could not be such that 1 'effective HP' was worth 1 actual hitpoint. (Also, this is the first time I've had to be aware of the actual definition). I tried to come up with the most logical definition that would fit his claims. If I'm debunking somebody's maths, I want to be using whatever defintions that person is using. ^^ Sorry for the confusion.

I applaud your instinct to look for the intent behind someone's words instead of jumping to conclusions or attacking a straw man. It isn't easy.

[Frogreaver]

Frogreaver, can we get your analysis of LudicSavant's program too?

Yes, see below.


@Everyone,

What LudicSavant's program (https://anydice.com/program/1e0c1) does is precisely to calculate the effective HP increase for a given number of specific attacks against a specific defense. As in, if you had this many extra HP, you'd expect the same outcomes as if you had cast Mirror Image. The extra HP and Mirror Image are equivalent.

Let N = the number of attacks input into Ludic's program

From what I can tell, the program accurately calcs the damage reduction in terms of HP for a given scenario f(C,C',N,D), *other variables defined elsewhere. Directionally, it shows exactly what I would expect, that the higher your non dex AC the less "absolute" damage reduced that mirror image provides. Please note though that this is "absolute" damage reduced, whereas the relative damage reduced still needs to be calculated.

So where does a character actually stand after taking N attacks? In context of the 21 AC PC vs giant you cite below, the PC takes 98.7 hp worth of damage. Mirror image you calculated reduces hp damage by 48.11. (98.7 - 48.11) / 98.7 = 51.3%

In context of the 23 AC pc vs giant, the PC takes 81.9 hp worth of damage. Mirror image you calculated reduces the hp damage you took by 39.92. (81.9 - 39.92) / 81.9 = 51.3%

In both cases mirror image reduces the hp worth of damage that you took in those 6 attacks via 51.3%, showing that it's relative value is independent of your non-dex AC.

Hopefully, that helps explain the difference.


If you want to calculate other results, go to https://anydice.com/program/1e0d1 and change the last line in the program, e.g. to change AC from 21 to 23 change the bolded bit below from:

output
21 [/B]for 6d6+7 crit 6d6 on 20] named "AC 21 Forge Cleric vs. Fire Giant: effective HP gain from Mirror Image over three rounds of combat"

To this:

output
[images 3 imageac 10 imageroll 1d20 attacks ATTACKS roll 1d20 plus 11 vs 23 for 6d6+7 crit 6d6 on 20] named "AC 21 Forge Cleric vs. Fire Giant: effective HP gain from Mirror Image over three rounds of combat"

left for easy reference


Mirror Image yields fewer additional effective HP for high AC. QED.

If you want to talk additional then sure. I've no where disagreed with that. However, it yields the same relative amount to whatever damage you would have taken in a given scenario though.


It's not real HP, because casting Mirror Image against the Fire Giant doesn't actually give you 48 extra HP. It just gives you the [I]equivalent of 48 extra HP. You are as tough as if you have 48 extra HP, and after combat both Mirror Image guy and +48 HP guy will end the combat with the same average actual HP remaining.

I think the point was that Ludic's program is computing effective damage prevented which equates to adding 48 hp, not adding 48 effective hp.

Out of curiosity, say 1 character in your 21 AC giant example uses mirror image and has 100 hp. Another character has 48 more hp for a total of 148 hp but doens't use mirror image. You've already agreed those characters effective hp's should be the same.

(In your example chance to be hit was 55%)

So eHp = 148/.55 = 269
But eHp as others calc it for mirror image is 48 + 100/.55 = 229
269 does not equal 229.

[Valmark]

It is indicative of how many more times the creature will have to attack you to get the same result as without Mirror Image.

Here's a specific scenario:
- You are put in an arena with a monster.
- This monster will keep attacking you, once per second, until you go down.
- You will then be healed (and you get 100 gold pieces for every second you stay up).
- Without Mirror Image, you expect that on average you will go down after 20 seconds.
- The monster has an attack bonus that will hit your Mirror Image 50% of the time.

The question is: How many more seconds do you expect to stay up if you cast Mirror Image ?

I claim it's 6 seconds more.

So "You must get a 20, but if the first roll is a 20 you must get another one" is the same as "You must get a 20, but if the first roll is a 20, we pretend that it wasn't"

But that is the same as "You must get a 20, but for the first roll we give you a d20 where the 20 is crossed out"

Which is the same as "You must get a 20, but for the first roll we give you a dice that can't roll a 20"

Which is the same as "You must get a 20, but for the first roll we give you a d10"

Where in this line of 'is the same' do you believe it is not the same, and why?

You need to specify how often the monster would have hit the wizard too in that example.

Also no, "You must get a 20, but if the first roll is a 20, we pretend that it wasn't" is different from "You must get a 20 but for the first roll we give you a d20 where the 20 is crossed out". In the first case you can roll a 20, you just ignore it. That means that if you were to get a 20 you'd need another one.
In the second case, you cannot roll a 20. Period.
In the first case you have a 95% chance of not getting a 20 on the first roll. If the second, you have a 100% chance of not getting a 20 on the first roll.
Any following calculations is dependant on this difference. If you can't understand this specific difference I'll have to drop this specific part of the conversation in this thread because I can't talk probabilities if you don't see the difference between a sure event and a possible event.

[Frogreaver]

Regarding Max's giant example, I think this is an important point.

The moment we say the giant makes 6 attacks, that means after those 6 attacks there is no more opponent left to attack us. You can't compute effective hp without an opponent. That's why instead of having said giant keep on attacking after his 6 attacks, I compute effective hp by repeating the same scenario. This is also the way you would do it if you were going to have differing scenarios throughout an adventuring day and wanted to weight those to come up with your average effective hp in that specified adventuring day.

What others are doing is claiming they are looking at the 6 attack case but really mixing cases with the "attacks to KO you case" and predominately looking at the case where you use mirror image and then take enough attacks to kill you by a given enemy.

[bid]

The part I was bringing up is when both defenses are overlapping. If AC is protecting you, is the MI protecting you as well from that hit? Or is that double-counting?
One is a strict subset of the other. It's like distributing {x * (x+1)} into (x^2 + x}.

[fat.hampster]

Also no, "You must get a 20, but if the first roll is a 20, we pretend that it wasn't" is different from "You must get a 20 but for the first roll we give you a d20 where the 20 is crossed out". In the first case you can roll a 20, you just ignore it. That means that if you were to get a 20 you'd need another one.
In the second case, you cannot roll a 20. Period.
In the first case you have a 95% chance of not getting a 20 on the first roll. If the second, you have a 100% chance of not getting a 20 on the first roll.
Any following calculations is dependant on this difference. If you can't understand this specific difference I'll have to drop this specific part of the conversation in this thread because I can't talk probabilities if you don't see the difference between a sure event and a possible event.

But in the first case you pretend that it wasn't a 20, so effectively the probability of rolling a 20 is 0, i.e. in the first round the chance of getting a 20 that is meaningful on the first round is 0, because if a 20 is rolled, it is ignored.

Normally our expectancy number of rolls is given by the fact there is a 1/20 chance of rolling a 20 in one roll, plus a 19/20 chance that roll is irrelevant, and we're back to the start plus one roll so:
E(r20) = (1/20 * 1 + 19/20*(1 + E(r20))
simplifying:
E(r20) = 1 + 19/20*E(r20)
1/20*E(r20) = 1
E(r20) = 20

If we disregard the first roll if it's a 20, then there's a 1/20 chance of rolling a 20 and it being ignored (and we're back to the start with one additional roll), or a 19/20 chance that roll is irrelevant, and we're back to the start plus one roll, so: (call it R20 to distiguish)
E(R20) = 1/20 * (1 + E(r20)) + 19/20 * (1 + E(r20))
simplifying:
E(R20) = 1 + E(r20)
E(R20) = 1 + 20 = 21

[Reynaert]

Also no, "You must get a 20, but if the first roll is a 20, we pretend that it wasn't" is different from "You must get a 20 but for the first roll we give you a d20 where the 20 is crossed out". In the first case you can roll a 20, you just ignore it. That means that if you were to get a 20 you'd need another one.
In the second case, you cannot roll a 20. Period.
In the first case you have a 95% chance of not getting a 20 on the first roll. If the second, you have a 100% chance of not getting a 20 on the first roll.
Any following calculations is dependant on this difference. If you can't understand this specific difference I'll have to drop this specific part of the conversation in this thread because I can't talk probabilities if you don't see the difference between a sure event and a possible event.

In the second case you also have a 95% chance of not getting a 20 on the first roll. It's just conveniently crossed out to remind you you should be ignoring it.
Suppose that instead of crossing out the 20, I just put a red dot on the 20 face to remind you that you should ignore it for the first roll, does that make a difference?
Or if I give you a red d20 the first time and a blue d20 for the others, so the colour of the dice tells you "If I roll a 20 on a red d20, I need to roll another one"

This is the central point of my argument. You're welcome to discuss this. The question if there is a difference or not is central to the whole point I'm making, so going "If you disagree with my view on how probability works then I can't talk probabilities" is a cop out. I am contesting your view on how probabilities work. It is plain wrong.



If I roll a d20 until I get a 20, the expected number of rolls wil be 20
Here's how that expected value of 20 rolls comes about:

Let's call the number of rolls I expect to make X. There are two cases:
1 - I roll a 20 on my first try (5% chance), the number of rolls is 1
2 - I don't roll a 20 on my first try (95% chance).
I'm back where I started so I still expect to roll X more times to get a 20. The d20 doesn't have memory.
That first roll counts as well, so that is X+1 rolls in total.

So with these two cases, the average expectation is 1*0.05 + (X+1)*0.95
However, we started with saying that the expectation is X. So we have an equation X = (X+1)*0.95 + 1*0.05.
Solving for X, we get that X = 20

Do you agree that this is how it works? If not, where you you think the problem lies?
And if you do agree, try to apply this same reasoning on "Roll a d20 until you get a 20, but if the first one is a 20 you have to get another one"

Edit: ninja'd

[Valmark]

It's a cop out if after three pages we don't agree on something that isn't open to interpretation? I could understand if it was something that has multiple stances one can take but math has hard rules.

Anyway, let's see if you understand this way.
You are supposed to get a single 20 out of twenty rolls, alright? We all agree on that.

This means that if you get a 20 at the beginning you are expected to not roll a 20 in the next nineteen rolls- then, in the following twenty rolls you are expected to get another 20.

What you are saying is that if you get a 20 on the first roll you are likely to get a 20 in the next twenty rolls. You are weighing that first 20 as if it has a 100% chance of happening. This is obviously not true- we all agree that 20 has a 5% of happening on the first roll. Getting that 20 up front doesn't mean that it's chance is boosted to 100% after it happened- what it means is that the next nineteen rolls should fall in the 95% chance of not getting a 20. Then the next twenty rolls have a 95% chance of getting a 20.

[MinotaurWarrior]

So I think everyone now understands two things:

1) There is a limit at which Mirror Image increases your HP/(succeptibility to damage) the same amount regardless of your AC

2) There is another sense in which it is less effective at higher AC

The important next step to understand is that because MI is in no sense better with higher AC, if the sense in which it is not better is in any way relevant, it is actually overall worse.

So, going back to my easy example of the two guys facing the blowgunner, let's add in that after a certain finite amount of time, they have to pass through a hazard that doesn't target AC. Let's use the simplest example. 100 rounds in, they face a flame from an oil flask dealing 5 unavoidable constant fire damage.

Guy 1: The image is hit on 19 or 20, he is hit on 20, and has 10HP. The image saves him 0.5 HP. To survive the flame, he needs to take less than 5 total damage. Plug it into a cdf calculator, you get ~53% survival with the MI, ~44% without. So MI increases the survival rate by a factor of 1.17

Guy 2: The image is hit on a 19 or 20, he is hit on a 19 or 20, and had 10HP. The image saves him 1HP. To survive the flame, he needs to take less than 5 total damage. Plug it into a cdf calculator, you get ~6% survival with the MI, ~2% without. So MI increases the survival rate by a factor of 2.43

Replace oil fire with literally any threat that doesn't target AC, and you see the practical application. Is there a monster in that very combat that inflicts damage on a failed save? How about a trap in the next room? An ability check to avoid a hard fall?

MI is exactly as good on a high AC character as a low AC character in the white room scenario of enduring a lot of attacks that target AC. But because that is a perfectly even split, and it is worse if you ever are subject to damage from a source your higher AC doesn't help against, it is strictly speaking less useful for a higher AC character.

[fat.hampster]

Anyway, let's see if you understand this way.
You are supposed to get a single 20 out of twenty rolls, alright? We all agree on that.

You are expected to get one 20 out of twenty rolls. This includes a ~36% chance of getting no 20s, a ~38% chance of getting a single twenty, and a ~19% chance of getting 2 twos, with the remaining ~7% being covered by progressively higher numbers of 20s with progressively lower probabilities.

Separately, you also expect to require, on average 20 rolls before rolling a 20. This includes a 5% chance of getting a 20 on your first roll, and a ~13% chance of not getting a 20 on your first 40 rolls.






This means that if you get a 20 at the beginning you are expected to not roll a 20 in the next nineteen rolls- then, in the following twenty rolls you are expected to get another 20.

What you are saying is that if you get a 20 on the first roll you are likely to get a 20 in the next twenty rolls. You are weighing that first 20 as if it has a 100% chance of happening. This is obviously not true- we all agree that 20 has a 5% of happening on the first roll. Getting that 20 up front doesn't mean that it's chance is boosted to 100% after it happened- what it means is that the next nineteen rolls should fall in the 95% chance of not getting a 20. Then the next twenty rolls have a 95% chance of getting a 20.

I'm not sure I'm following what you're saying here. But it's worth noting that each roll is entirely independent of the others. If we pick any set of 20 rolls (without first knowing what those rolls are), we expect to get one 20.

So we expect one 20 in the first 20 rolls, but we also expect one 20 in the 2nd-21st rolls (inclusive).

So in rolls 2-21 (inclusive), we expect to get one 20, and we do so regardless of whether we rolled a 20, or whether we rolled a 7, for the first roll. i.e. observing the outcome of the first roll doesn't effect our expectancy for the remaining rolls.

Once we make that roll, our expected number of rolls before we get a 20 is either:
* 1 if the roll was a 20 (i.e. the one that we already did)
* 1 + the expected rolls before we get a 20, if we get anything else (i.e. the one we already did, which has now wasted it's chance of being a 20).

If we ignore the 20 if it happens on the first roll, or cross out the 20 then the first case turns into the second case, and we just always add 1 to our expected number of rolls.

[Valmark]

So I think everyone now understands two things:

1) There is a limit at which Mirror Image increases your HP/(succeptibility to damage) the same amount regardless of your AC

2) There is another sense in which it is less effective at higher AC

The important next step to understand is that because MI is in no sense better with higher AC, if the sense in which it is not better is in any way relevant, it is actually overall worse.

So, going back to my easy example of the two guys facing the blowgunner, let's add in that after a certain finite amount of time, they have to pass through a hazard that doesn't target AC. Let's use the simplest example. 100 rounds in, they face a flame from an oil flask dealing 5 unavoidable constant fire damage.

Guy 1: The image is hit on 19 or 20, he is hit on 20, and has 10HP. The image saves him 0.5 HP. To survive the flame, he needs to take less than 5 total damage. Plug it into a cdf calculator, you get ~53% survival with the MI, ~44% without. So MI increases the survival rate by a factor of 1.17

Guy 2: The image is hit on a 19 or 20, he is hit on a 19 or 20, and had 10HP. The image saves him 1HP. To survive the flame, he needs to take less than 5 total damage. Plug it into a cdf calculator, you get ~6% survival with the MI, ~2% without. So MI increases the survival rate by a factor of 2.43

Replace oil fire with literally any threat that doesn't target AC, and you see the practical application. Is there a monster in that very combat that inflicts damage on a failed save? How about a trap in the next room? An ability check to avoid a hard fall?

MI is exactly as good on a high AC character as a low AC character in the white room scenario of enduring a lot of attacks that target AC. But because that is a perfectly even split, and it is worse if you ever are subject to damage from a source your higher AC doesn't help against, it is strictly speaking less useful for a higher AC character.

I'm don't think I'm sure what you are getting at with adding the unavoidable damage, can you please clarify?

Also yeah, I'm done explaining that chance. I'm clearly not the one capable of explaining it to you (meaning those I addressed until now, not MinotaurWarrior).

[Reynaert]

It's a cop out if after three pages we don't agree on something that isn't open to interpretation? I could understand if it was something that has multiple stances one can take but math has hard rules.

True, but we seem to be disagreeing on what those rules are. I'm trying to convince you that my view is correct and you're doing the same for your view.


Anyway, let's see if you understand this way.
You are supposed to get a single 20 out of twenty rolls, alright? We all agree on that.

This means that if you get a 20 at the beginning you are expected to not roll a 20 in the next nineteen rolls- then, in the following twenty rolls you are expected to get another 20.

That's not how it works. Dice don't have memory. What you're syaing is the same as saying "If you didn't get a 20 for the first roll, you are expected to roll a 20 in the next nineteen rolls".
But if you say that, then you're also saying that "If you didn't get a 20 in the first two rolls, you are expected to roll a 20 in the next eighteen rolls".
Et cetera until you get to "If you didn't get a 20 in the first nineteen rolls, you are expected to roll a 20 in the next one roll."

(edit: phrasing)

[MinotaurWarrior]

I'm don't think I'm sure what you are getting at with adding the unavoidable damage, can you please clarify?


1) Mirror image prevents fewer real HP of damage the higher your AC.

2) Each of those real HP is worth more EHP with higher AC, if that higher AC can mitigate the damage you take.

3) Against sources of damage that are not mitigated by AC, those real HP are not worth any more because of your higher AC.

1 and 2 perfectly balance out, so that if you are in some weird white room scenario where your HP are only used to prevent death by attacks that target AC, MI helps a character the same amount no matter their AC.

But the fact that the third point exists means that MI is practically worse for a character with MI.

[fat.hampster]

So I think everyone now understands two things:

1) There is a limit at which Mirror Image increases your HP/(succeptibility to damage) the same amount regardless of your AC

2) There is another sense in which it is less effective at higher AC

The important next step to understand is that because MI is in no sense better with higher AC, if the sense in which it is not better is in any way relevant, it is actually overall worse.

So, going back to my easy example of the two guys facing the blowgunner, let's add in that after a certain finite amount of time, they have to pass through a hazard that doesn't target AC. Let's use the simplest example. 100 rounds in, they face a flame from an oil flask dealing 5 unavoidable constant fire damage.

Guy 1: The image is hit on 19 or 20, he is hit on 20, and has 10HP. The image saves him 0.5 HP. To survive the flame, he needs to take less than 5 total damage. Plug it into a cdf calculator, you get ~53% survival with the MI, ~44% without. So MI increases the survival rate by a factor of 1.17

Guy 2: The image is hit on a 19 or 20, he is hit on a 19 or 20, and had 10HP. The image saves him 1HP. To survive the flame, he needs to take less than 5 total damage. Plug it into a cdf calculator, you get ~6% survival with the MI, ~2% without. So MI increases the survival rate by a factor of 2.43

Replace oil fire with literally any threat that doesn't target AC, and you see the practical application. Is there a monster in that very combat that inflicts damage on a failed save? How about a trap in the next room? An ability check to avoid a hard fall?

MI is exactly as good on a high AC character as a low AC character in the white room scenario of enduring a lot of attacks that target AC. But because that is a perfectly even split, and it is worse if you ever are subject to damage from a source your higher AC doesn't help against, it is strictly speaking less useful for a higher AC character.


Doesn't that logic apply equally to AC?

i.e. an extra point of AC is worth less at higher AC, because damage which can be mitigated by AC is a smaller proportion of total damage received.

(Maybe that's a completely uncontentious point, though?)

In this sense maybe it makes more sense to talk about MI in terms of eAC than eHP.

assuming 20 attacks, and [dex AC] - [monster attack bonus] <= 2, Mirror image is worth ~3 eAC.

(slightly more from nat 1s, slightly less from not all images getting used up).

doubling the attacks roughly halves the eAC, and vice-versa. (Except with few attacks, the likelihood of not utilising all of the images becomes a significant factor).

[Valmark]

1) Mirror image prevents fewer real HP of damage the higher your AC.

2) Each of those real HP is worth more EHP with higher AC, if that higher AC can mitigate the damage you take.

3) Against sources of damage that are not mitigated by AC, those real HP are not worth any more because of your higher AC.

1 and 2 perfectly balance out, so that if you are in some weird white room scenario where your HP are only used to prevent death by attacks that target AC, MI helps a character the same amount no matter their AC.

But the fact that the third point exists means that MI is practically worse for a character with MI.
I'm not sure I get what you mean? MI doesn't trigger on anything that doesn't target AC (so anything that isn't an attack). Is that what you mean?

Doesn't that logic apply equally to AC?

i.e. an extra point of AC is worth less at higher AC, because damage which can be mitigated by AC is a smaller proportion of total damage received.

(Maybe that's a completely uncontentious point, though?)

In this sense maybe it makes more sense to talk about MI in terms of eAC than eHP.

assuming 20 attacks, and [dex AC] - [monster attack bonus] <= 2, Mirror image is worth ~3 eAC.

(slightly more from nat 1s, slightly less from not all images getting used up).

doubling the attacks roughly halves the eAC, and vice-versa. (Except with few attacks, the likelihood of not utilising all of the images becomes a significant factor).

I thought it was the opposite actually, that each point of AC is worth more then the previous one because the damage you take tends to 0 (I saw this being said on another thread, I made no calculations to prove or disprove this).

[MinotaurWarrior]

Doesn't that logic apply equally to AC?

i.e. an extra point of AC is worth less at higher AC, because damage which can be mitigated by AC is a smaller proportion of total damage received.


No, because AC doesn't reduce less actual damage the higher your AC is (excluding when you're already at "need 20s". It's always ~0.05 * EV(damage dice), with the ~ representing complications like (dis)advantage.

The trick with MI is that it reduces less real HP damage the higher your AC is.


I'm not sure I get what you mean? MI doesn't trigger on anything that doesn't target AC (so anything that isn't an attack). Is that what you mean?

No, not at all.

Ignore MI for a second. You've got two guys - one gets hit on 20s, the other gets hit on 19s, and they're being shot at by a guy with a blowgun (which has no damage dice, and thus deals no bonus damage on a crit)

Against this blowgunner, every HP the guy who gets hit on 20s has is twice as valuable as the HP the guy who gets hit on 19s, right?

But against something like an oil fire that deals a flat 5 damage, both of their HP are equally valuable.

MI saves half as much actual damage for the guy who gets hit on 20s, but each of those actual HP is worth twice as much against more blowgun attacks.

But against anything else, they are worth less.



I thought it was the opposite actually, that each point of AC is worth more then the previous one because the damage you take tends to 0 (I saw this being said on another thread, I made no calculations to prove or disprove this).

The sense in which that is true is as follows:

EHP = HP/(succeptibility to damage)

As your overall succeptibility to damage goes down (due to AC, saves, resistances, etc) EHP asymptotically approaches positive infinity. Every bit of defense thus scales at an "exponential" rate the more you have.

[Frogreaver]

I thought it was the opposite actually, that each point of AC is worth more then the previous one because the damage you take tends to 0 (I saw this being said on another thread, I made no calculations to prove or disprove this).

I think this is a good talking point and is related to the MI discussion.

A point of AC provides you a flat damage reduction of .05*Damage per attack, provided you are within the domain where chance to hit is between .1 and .9. That is, the actual amount of damage reduction a +1 AC provides is independent of your AC.

That, said if you look at relative damage taken, you can see that you take .1*Damage at 10% chance to be hit and you take .05*Damage with +1 AC. That +1 point of AC halved the amount of damage you are taking in the scenario. In the relative view, your damage reduction from AC depends on your current AC.

Mirror image is kind of the opposite, it provides the same relative damage reduction, but it depends on AC when looking at the actual amount of damage reduced.

[fat.hampster]

No, because AC doesn't reduce less actual damage the higher your AC is (excluding when you're already at "need 20s". It's always ~0.05 * EV(damage dice), with the ~ representing complications like (dis)advantage.

The trick with MI is that it reduces less real HP damage the higher your AC is.


Ah right, of course, I was being stupid. As you point out later in your post, going from 19->20 relative AC doubles your eHP, whereas going from 2->3 only increases it by 1/18.

So you can look at it in 2 ways:
* As AC goes up, the value of MI remains constant, and the marginal value of AC increases.
* As AC goes up, the marginal value of AC remains the same, and the value of MI decreases.


And as you already pointed out, in most situations the truth is somewhere between those 2 extremes, so marginal AC slowly gets better, and MI slowly gets worse.

[Valmark]

No, not at all.

Ignore MI for a second. You've got two guys - one gets hit on 20s, the other gets hit on 19s, and they're being shot at by a guy with a blowgun (which has no damage dice, and thus deals no bonus damage on a crit)

Against this blowgunner, every HP the guy who gets hit on 20s has is twice as valuable as the HP the guy who gets hit on 19s, right?

But against something like an oil fire that deals a flat 5 damage, both of their HP are equally valuable.

MI saves half as much actual damage for the guy who gets hit on 20s, but each of those actual HP is worth twice as much against more blowgun attacks.

But against anything else, they are worth less.

The sense in which that is true is as follows:

EHP = HP/(succeptibility to damage)

As your overall succeptibility to damage goes down (due to AC, saves, resistances, etc) EHP asymptotically approaches positive infinity. Every bit of defense thus scales at an "exponential" rate the more you have.

Oh, alright, now I get it thank you.

[MaxWilson]

So where does a character actually stand after taking N attacks? In context of the 21 AC PC vs giant you cite below, the PC takes 98.7 hp worth of damage. Mirror image you calculated reduces hp damage by 48.11. (98.7 - 48.11) / 98.7 = 51.3%

In context of the 23 AC pc vs giant, the PC takes 81.9 hp worth of damage. Mirror image you calculated reduces the hp damage you took by 39.92. (81.9 - 39.92) / 81.9 = 51.3%

In both cases mirror image reduces the hp worth of damage that you took in those 6 attacks via 51.3%, showing that it's relative value is independent of your non-dex AC.

Hopefully, that helps explain the difference.



left for easy reference



If you want to talk additional then sure. I've no where disagreed with that. However, it yields the same relative amount to whatever damage you would have taken in a given scenario though.



I think the point was that Ludic's program is computing effective damage prevented which equates to adding 48 hp, not adding 48 effective hp.

Out of curiosity, say 1 character in your 21 AC giant example uses mirror image and has 100 hp. Another character has 48 more hp for a total of 148 hp but doens't use mirror image. You've already agreed those characters effective hp's should be the same.

(In your example chance to be hit was 55%)

So eHp = 148/.55 = 269
But eHp as others calc it for mirror image is 48 + 100/.55 = 229
269 does not equal 229.

I'm struggling to understand your definitions here. We've established at this point that when you say "effective HP" you mean something different than I do. What I call effective HP (MaxWilsonEffectiveHP) you call "damage prevented."

I think but am not sure that your definition of effective HP (FrogReaverEffectiveHP) is the same as ReyneartEffectiveHP from upthread: expected damage divided by hit rate. Can you confirm? Would you mind also explaining why you call this thing "effective HP" in the first place? To me it seems a misnomer since it is not in any way equivalent to actual HP.

The concept of MaxWilsonEffectiveHP is for calculating which spells are best at keeping you alive and, and e.g. whether it's worth casting Blur or Warding Bond before a fight, or if it's better to just heal the damage after.

What decisions is the concept of FrogReaverEffectiveHP designed to support?

Edit:

P. S. Also just to be clear I agree that Mirror Image's value as measured in FrogReaverEffective does not increase or decrease with personal non-Dex AC. You're not wrong. I just need help understanding what implications that has, because I don't understand yet what FrogReaverEffectiveHP is for.

[Frogreaver]

I'm struggling to understand your definitions here. We've established at this point that when you say "effective HP" you mean something different than I do. What I call effective HP (MaxWilsonEffectiveHP) you call "damage prevented."

Don't take this rudely but MaxWilsonEffectiveHP isn't effective hp in any sense - it's a misnomer.


I think but am not sure that your definition of effective HP (FrogReaverEffectoveHP) is the same as ReyneartEffectiveHP from upthread: expected damage divided by hit rate. Can you confirm?

The traditional method for calculating effective hp is based solely on AC and is eHP = (hp)/(hit rate). We have to expand that definition out a bit for mirror image - but it's not incredibly hard to do so, although we have to be a bit careful in how we do so.


Would you mind also explaining why you call this thing "effective HP" in the first place? To me it seems a misnomer since it is not in any way equivalent to actual HP.

The whole concept of effective hp is to have a quantity whereby we can compare the survivability of characters who have different AC's and HP's in a given situation. It's not intended to be equivalent to actual HP in any sense.


The concept of MaxWilsonEffectiveHP is for calculating which spells are best at keeping you alive and, and e.g. whether it's worth casting Blur before a fight, or if it's better to just heal the damage after.

I think your metric likely does a great job of that, but effective hp it is not.


What decisions is the concept of FrogReaverEffectiveHP designed to support?

It allows for comparison of survivability for characters.

[MaxWilson]

Don't take this rudely but MaxWilsonEffectiveHP isn't effective hp in any sense - it's a misnomer.

You can't mean that literally. You've already acknowledged that gaining MaxWilsonEffectiveHP is equivalent to gaining actual HP.



It allows for comparison of survivability for characters.

Oh, I see. Thanks.

[Frogreaver]

You can't mean that literally. You've already acknowledged that gaining MaxWilsonEffectiveHP is equivalent to gaining actual HP.



Oh, I see. Thanks.

Your welcome. However, I meant that very much literally. Effective hp is not actual hp, nor is it value equivalent to actual hp (except in the case for something like magic missile)

Edit: Maybe an example helps. A character with 50 hp that gets hit 10% of the time has 500 effective hp.

We can also look at effective healing. That would be healed hp / chance to be hit. In this case healing 10 hp grants 100 effective hp.

[MaxWilson]

Your welcome. However, I meant that very much literally. Effective hp is not actual hp, (A) nor is it value equivalent to actual hp (except in the case for something like magic missile)

(A) is wrong.

[Frogreaver]

(A) is wrong.

No it isn’t. I added some examples above after you already replied. Should be pretty obvious after reading those.

[MaxWilson]

No it isn’t. I added some examples above after you already replied. Should be pretty obvious after reading those.

Those are about FrogReaverEffectiveHP, but you're claiming that MaxWilsonEffectiveHP is not effective HP in any sense, when in fact they are interchangeable with actual HP in many situations, not just vs. Magic Missile. That's what makes them effectively the same thing as HP, hence the name.

I think you'll find that this is what many but not all people in this thread mean when they say "effective HP." This whole thread is just a big semantic argument.

[x3n0n]

Your welcome. However, I meant that very much literally. Effective hp is not actual hp, nor is it value equivalent to actual hp (except in the case for something like magic missile)

Is there some other community where "the amount of attempted damage from a specific enemy/method that it takes to KO a target" has the name "effective HP"?

It seems that we've established that it's not a commonly-used definition on this forum.

Since it was not common parlance here, then it seems presumptuous to
* Assume that others are familiar with that definition (and refuse to define it) and
* Inform them that their own definitions for the same term, established in their own discussion groups, are "incorrect".

[Frogreaver]

Those are about FrogReaverEffectiveHP, but you're claiming that MaxWilsonEffectiveHP is not effective HP in any sense, when in fact they are interchangeable with actual HP in many situations, not just vs. Magic Missile. That's what makes them effectively the same thing as HP, hence the name.

I think you'll find that this is what many but not all people in this thread mean when they say "effective HP." This whole thread is just a big semantic argument.

I don’t think so. I think they are confused. For example let’s look at the following question:

Does a character with 50hp and a 50% chance to be hit have the same effective hp as a character with 50hp and a 5% chance to be hit. If effective ho are actual hp then they must right? Yet, you won’t find 1 person here who will say that. Not even you.


Is there some other community where "the amount of attempted damage from a specific enemy/method that it takes to KO a target" has the name "effective HP"?

It seems that we've established that it's not a commonly-used definition on this forum.

Since it was not common parlance here, then it seems presumptuous to
* Assume that others are familiar with that definition (and refuse to define it) and
* Inform them that their own definitions for the same term, established in their own discussion groups, are "incorrect".

I think I can make that case because they aren’t using internally consistent definitions. I mean consider the above example.

[MaxWilson]

Does a character with 50hp and a 50% chance to be hit have the same effective hp as a character with 50hp and a 5% chance to be hit. If effective ho are actual hp then they must right? (A) Yet, you won’t find 1 person here who will say that. Not even you.

(A) is wrong.

They do have the same effective HP. They even have the same actual HP. They have different survivability though, against that enemy. One will last approximately 6 times as long as the other. (11/2 because of crits, rounding down because static mods aren't doubled.) To be more precise I'd need to know the enemy's damage.

[x3n0n]

I think I can make that case because they aren’t using internally consistent definitions. I mean consider the above example.

I understand that the concept of "durability against a specific delivery method for damage" is a useful concept. Calling it "effective HP" with no "subscript" (which could indicate the method in question) is unconventional and unexpected at best, and potentially misleading.

I regret having involved myself with this discussion. You seem to have useful insights, but I have found this exchange in particular to be emotionally exhausting, and I am losing hope that you want to make these exchanges productive.

Best wishes.

[Reynaert]

I don’t think so. I think they are confused. For example let’s look at the following question:

Does a character with 50hp and a 50% chance to be hit have the same effective hp as a character with 50hp and a 5% chance to be hit. If effective hp are actual hp then they must right? Yet, you won’t find 1 person here who will say that. Not even you.

'Effective HP', by this definition, is not something a character has.

Instead, it is how much a certain action or feature provides.
It's worth the same as actual HP, but because you don't actually have it, you just get the same benefit.
So that's why it is called 'Effective HP'.

Take, for example, Arcane Ward. Suppose that ward has 7hp. You dont have 7hp extra, but when you take damage, it's as if you did. So, like, effectively you had 7hp more. So that's 7 'Effective HP'.

[Frogreaver]

'Effective HP', by this definition, is not something a character has.

Instead, it is how much a certain action or feature provides.
It's worth the same as actual HP, but because you don't actually have it, you just get the same benefit.
So that's why it is called 'Effective HP'.

Take, for example, Arcane Ward. Suppose that ward has 7hp. You dont have 7hp extra, but when you take damage, it's as if you did. So, like, effectively you had 7hp more. So that's 7 'Effective HP'.

I see. But doesn’t that mean that ac provides effective hp as well? And that higher ac provides more effective hp than lower ac?

For example. You have 100 hp and a 20 damagr attack is made at you. If your ac gives that attack a 50% chance to hit then your ac prevents 10 damage on average and so it’s like you had +10 effective hp? A higher ac might give you a 25% Chance to be hit and thus prevent 15 damage on average, giving you +15 effective hp?

[Satori01]

I understand that the concept of "durability against a specific delivery method for damage" is a useful concept. Calling it "effective HP" with no "subscript" (which could indicate the method in question) is unconventional and unexpected at best, and potentially misleading.

I regret having involved myself with this discussion. You seem to have useful insights, but I have found this exchange in particular to be emotionally exhausting, and I am losing hope that you want to make these exchanges productive.

Best wishes.

I am certainly a dubious character witness, given my social reception in various threads, but I have interacted with frogreaver for years on other D&D boards.

He is not a troll. He has worked on this very question regarding Mirror Image, for at least 1 year.

So, even if frogreaver's manner might strike one as gruff, I at least feel certain his intentions are honest.

This is where I renew my objection, that EHP of any flavor, stripe or name are just a poor tool of inquiry, for this topic at least.

As an artificial measurement, EHP, have not brought clarity to the inner workings of how Mirror Image interacts in the 5e system....I think it has brought the opposite.

EHP boils down to survivability....whatever name, flavor, or stripe it has.

My sense is something of a consensus is forming around a statement like this:

The ability for the Mirror Image spell to turn a confirmed critical hit to a miss, is not affected by AC.

There is a more elegant way to phrase this...which someone shall provide.

[Frogreaver]

I understand that the concept of "durability against a specific delivery method for damage" is a useful concept. Calling it "effective HP" with no "subscript" (which could indicate the method in question) is unconventional and unexpected at best, and potentially misleading.

I regret having involved myself with this discussion. You seem to have useful insights, but I have found this exchange in particular to be emotionally exhausting, and I am losing hope that you want to make these exchanges productive.

Best wishes.

Weird how most of this thread I’ve felt the same way. For what it’s worth, my goal is productive discussion.

Best wishes.


I am certainly a dubious character witness, given my social reception in various threads, but I have interacted with frogreaver for years on other D&D boards.

He is not a troll. He has worked on this very question regarding Mirror Image, for at least 1 year.

So, even if frogreaver's manner might strike one as gruff, I at least feel certain his intentions are honest.

Thanks. I think I result to gruffness out of frustration - character flaw perhaps. It's definitely not how I intend to portray myself, though it's a common observation so there must be quite a bit of truth to it.


This is where I renew my objection, that EHP of any flavor, stripe or name are just a poor tool of inquiry, for this topic at least.

Maybe. I'm not ready to give up on it yet but you may very well be correct.


As an artificial measurement, EHP, have not brought clarity to the inner workings of how Mirror Image interacts in the 5e system....I think it has brought the opposite.

Sadly it has so far. Maybe that will change.


EHP boils down to survivability....whatever name, flavor, or stripe it has.

That's been disputed though. Which I think is the problem for me, I've never fathomed anyone would use the term eHP with any other meaning. Apparently many others never realized anyone would use it in relation to survivability.


My sense is something of a consensus is forming around a statement like this:

The ability for the Mirror Image spell to turn a confirmed critical hit to a miss, is not affected by AC.

There is a more elegant way to phrase this...which someone shall provide.

Might be getting your hopes up to much ;)

[x3n0n]

Weird how most of this thread I’ve felt the same way. For what it’s worth, my goal is productive discussion.

If so, in the future, please consider defining your terms in advance and being willing to clarify those definitions when asked instead of saying "It seems like some other people understand," implying that anyone who asks is an idiot.

[MaxWilson]

I see. But doesn’t that mean that ac provides effective hp as well?

Nope. AC changes the rate at which you lose effective HP (edit to clarify: specifically MaxWilsonEffectiveHP, which = SegevEffectiveHP and I think many other posters on this thread, and common usage on this forum). DPR calculations are commonplace, so it's straightforward to apply DPR to actual and/or effective HP to compute survivability.

An AC 23 character with 50 HP, attacked by a Guard at +3 for d8+1, gets hit 5% of the time and loses 0.5 HP per round on average. He'll last 100 guard-rounds. (E.g. against 10 guards he'll last 10 rounds, on average, if no guards die and the situation never changes.) This is your "5% chance to hit" scenario from post #241.

Give those guards a +9 to hit, so they hit 50% of the time instead of 5%, and now they attack at +12 to hit for d8+1, resulting in 2.98 damage per guard. Now he'll last 50/2.98 = 16.8 guard-rounds. Those 10 guards will kill him in under 2 rounds, on average. This is your "50% chance to hit" scenario from post #241.

Notice how FrogReaverEffectiveHP would have implied the wrong prediction here BTW--it would have predicted a 10x difference in survivability, but it's actually only 6x.

Also, FrogReaverEffectiveHP can't handle situations with mixed opponents, e.g. 10 guards, but 5 of them are shooting with disadvantage due to long range. However, calculating survivability as actual HP (or MaxWilsonEffectiveHP)/DPR handles that easily. You just add the DPR of the guards without disadvantage (0.5 per guard, if they have the normal +3 to hit) to the DPR of the guards with disadvantage (0.025). 50/(5*0.5 + 5*0.025) = 19 rounds until death.


I am certainly a dubious character witness, given my social reception in various threads, but I have interacted with frogreaver for years on other D&D boards.

He is not a troll. He has worked on this very question regarding Mirror Image, for at least 1 year.

I don't think anyone here said FrogReaver was a troll?

But the idea he's proponing was explained poorly in the OP, opportunities to clarify were missed, and turns out to be less useful and less flexible than a different concept by the same name which other people are already using. It's not wrong for someone to feel frustrated with this thread.


This is where I renew my objection, that EHP of any flavor, stripe or name are just a poor tool of inquiry, for this topic at least.

As an artificial measurement, EHP, have not brought clarity to the inner workings of how Mirror Image interacts in the 5e system....I think it has brought the opposite.

EHP boils down to survivability....whatever name, flavor, or stripe it has.

My sense is something of a consensus is forming around a statement like this:

The ability for the Mirror Image spell to turn a confirmed critical hit to a miss, is not affected by AC.

There is a more elegant way to phrase this...which someone shall provide.

I don't have a good description for FrogReaverEffectiveHP either. They're not actually a good measure of survivability due to neglecting crits and being inconvenient for measuring heterogenous opponents. It makes sense why FrogReaver is interested in them for comparing builds, because they let you avoid picking one of those builds as a baseline while still approximating survivability against a specific opponent, but the ability to avoid choosing a baseline is their only analytic virtue.

FWIW I agree with your statement that <<The ability for the Mirror Image spell to turn a confirmed critical hit to a miss, is not affected by AC.>> but I think FrogReaver's proof is a little bit more than that. He's not wrong that the ratio of damage prevented remains constant as non-Dex AC goes up. Maybe something like "The ratio of expected damage without Mirror Image to expected actual damage after casting Mirror Image is affected solely by ImageAC, and not by your actual AC." Which of course makes total sense, because ImageAC is what determines how long the spell lasts and how many attacks it therefore intercepts, while actual AC is what determines what fraction of the non-intercepted attacks actually apply damage. Of course the ratio isn't affected by actual AC--that would be double-counting it, since it's already been counted when you computed the actual damage in the first place.

[Frogreaver]

If so, in the future, please consider defining your terms in advance and being willing to clarify those definitions when asked instead of saying "It seems like some other people understand," implying that anyone who asks is an idiot.

It wasn't a term I ever fathomed would need explicitly defined. It was true that others were understanding me - though on reflection that was a rude point to bring up. So my apologies.


Nope. AC changes the rate at which you lose effective HP (edit to clarify: specifically MaxWilsonEffectiveHP, which = SegevEffectiveHP and I think many other posters on this thread, and common usage on this forum). DPR calculations are commonplace, so it's straightforward to apply DPR to actual and/or effective HP to compute survivability.

I took the chance to hit provided by AC and computed the MaxWilson Effective hp that provided against an attack. Why are you telling me I can't do that when I just did?


An AC 23 character with 50 HP, attacked by a Guard at +3 for d8+1, gets hit 5% of the time and loses 0.5 HP per round on average. He'll last 100 guard-rounds. (E.g. against 10 guards he'll last 10 rounds, on average, if no guards die and the situation never changes.) This is your "5% chance to hit" scenario from post #241.

Give those guards a +9 to hit, so they hit 50% of the time instead of 5%, and now they attack at +12 to hit for d8+1, resulting in 2.98 damage per guard. Now he'll last 50/2.98 = 16.8 guard-rounds. Those 10 guards will kill him in under 2 rounds, on average. This is your "50% chance to hit" scenario from post #241.

The difference is crits. Remove crit damage and run the numbers. You'll get the 182 rounds and 18.2 rounds respectively when you do so exactly a 10X difference.


Notice how FrogReaverEffectiveHP would have implied the wrong prediction here BTW--it would have predicted a 10x difference in survivability, but it's actually only 6x.

Not when crits are removed. 10X is the exact prediction.


Also, FrogReaverEffectiveHP can't handle situations with mixed opponents, e.g. 10 guards, but 5 of them are shooting with disadvantage due to long range. However, calculating survivability as actual HP (or MaxWilsonEffectiveHP)/DPR handles that easily. You just add the DPR of the guards without disadvantage (0.5 per guard, if they have the normal +3 to hit) to the DPR of the guards with disadvantage (0.025). 50/(5*0.5 + 5*0.025) = 19 rounds until death.

Handles it just as easily. Take the inverse of each the effective hp caused by each opponent. Add them all up. Divide by the number of terms. Take the inverse again. You now have effective hp for a mixed opponent situation.