Examining Mirror Image

[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/comment..._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.