The Math of Advantage/Disadvantage

[Jack of Spades]

So, I just did a bunch of math on some index cards, and I've come up with some pertinent probability math regarding advantaged/disadvantaged rolls.

I'll update this post with spiffy graphs when I make them sometime later.

So, as we're mostly aware, if you perfectly randomize a perfectly balanced d20 in a vacuum (all other things being equal yadda yadda), the chance of each result coming up is exactly %5. So here's a table showing the various probabilities compared to that baseline.



Result
1d20
Advantage
Disadvantage


1
5%
0.25%
9.75%


2
5%
0.75%
9.25%


3
5%
1.25%
8.75%


4
5%
1.75%
8.25%


5
5%
2.25%
7.75%


6
5%
2.75%
7.25%


7
5%
3.25%
6.75%


8
5%
3.75%
6.25%


9
5%
4.25%
5.75%


10
5%
4.75%
5.25%


11
5%
5.25%
4.75%


12
5%
5.75%
4.25%


13
5%
6.25%
3.75%


14
5%
6.75%
3.25%


15
5%
7.25%
2.75%


16
5%
7.75%
2.25%


17
5%
8.25%
1.75%


18
5%
8.75%
1.25%


19
5%
9.25%
0.75%


20
5%
9.75%
0.25%



So if you look at the table, a roll on which the player has Advantage is more likely to roll 11 or higher, whereas one where the player has disadvantage is more likely to roll 10 or lower. That's pretty expected. What's a bit more interesting is that critical rolls (natural 1's and 20's) come up nearly twice as often on their respective modified rolls. Which means that one is not only more likely to succeed or fail, one is more often to succeed or fail big.

The other bit of math that I did was to compare the Advantage/Disadvantage mechanic with the more traditional modifier of +/-2 from older editions. Assuming that the die you keep is considered the 'unmodified' die, rolling twice and keeping the better/worse result averages to a modifier of +/-3.125. So, Advantage/Disadvantage could be considered to be a bit more potent than the +/-2 modifier on average. Of course, that's assuming that the die you keep is considered to be the 'modified' result.

Which brings me to an important point, I think: When you play 5e, make sure you roll Advantage/Disadvantage dice simultaneously. This is important because rolling 1 and 20 at the same time is generally much more fun and dramatic than rolling 1 then 20 or vice versa. I'm serious: in the name of fun, please roll both of those d20s at the same time. That way, there's less chance of the idea of "wasting" Advantage/Disadvantage rolls becoming a thing.

[Yuki Akuma]

I'd be interested to see your work, if you've got it laying around (or are willing to reproduce it).

I find it interesting that Advantage/Disadvantage are apparently worth approximately +/-3, while the rule for Advantage/Disadvantage on passive checks is +/-5. This means that, if you have Advantage on Perception, you are actually better off relying on your passive Perception. Meanwhile, if you had Disadvantage, you should roll for it.

[Seppo87]

The average improvement is 4.75

[TheOOB]

This math is interesting and informative, and it leads to an interesting follow up question, namely, how much does advantage/disadvantage influenced your chance of success of a given roll?

Here are a few examples.

If you need to roll an 11 or higher, you normally have a 50% chance of success.
With advantage you'd have a 75% chance of success, which is a 25 point change
With disadvantage you'd have a 25% chance of success, which is a 25 point change

If you need to roll a 16 or higher, you normally have a 25% chance of success
With advantage you'd have a 44% chance of success, which is 19 point change
With disadvantage you'd have a 6% chance of success, which is a 19 point change

If you need to roll a 6 or higher, you normally have a 75% chance of success
With advantage you have a 94% chance of success, which is a 19 point change
With disadvantage you have a 56% chance of success, which is a 19 point change.

[Yorrin]

I'm pretty sure this has already been discussed to death in about three other threads. Long story short the difference it makes has an essentially parabolic curve, with a greater effect at the center of the distribution.

[Windscion]

If you needed N or more to succeed on a d20, the bonus or malus is N*(20-N)/20.

What the 'average' is depends upon what you expect to be the difficulty of an 'average' roll. However, for N=8,9,10,11,12 it is at least +4.8 (24%), and I expect that most rolls will be in that range. So saying it amounts to +5 is a fair rule of thumb. YMMV, obviously, depending upon campaign challenge/optimization levels.

[Lokiare]

Actually, this page here (http://anydice.com/program/168e)1 highlights the same thing you did.

The thing to take away from Advantage and Disadvantage is that its worth more when you are aiming for lower numbers than when you are aiming for higher numbers. For instance if you need to roll an 11 on the d20 before modifiers to succeed, you have a 50% chance normally, but you have a 75% chance if you have advantage. That is an increase of 25% or +5. If on the other hand you need a 19 before modifiers to succeed then on a normal d20 roll you have a 10% chance, whereas with advantage you have a 19% chance. A difference of 9% rounded up that's equivalent to a +2 bonus.

This is one of the things the developers didn't understand in their play test. When they told the rogue they could trade out advantage for extra damage from their sneak attack (or whatever it was) in some cases for an optimized rogue it was just better to keep advantage and not trade it for sneak attack damage.

In the case of the barbarian taking disadvantage to gain +5 to the damage of their attack, they didn't understand that for an optimized barbarian, they would lose more than 5 damage per round on average using this feature (I think it was something like 9 damage per round).

So this is my problem with the (dis)advantage system. It creates hidden issues that only probability statisticians can find.

1Note: You have to hit the "at least" button and then calculate to see the percent chance of hitting at least the target number.

[Sartharina]

This is one of the things the developers didn't understand in their play test. When they told the rogue they could trade out advantage for extra damage from their sneak attack (or whatever it was) in some cases for an optimized rogue it was just better to keep advantage and not trade it for sneak attack damage.Actually, I think they did understand. You could trade higher precision for higher damage - and given that a sneak attack could deal the same damage as 5-6 hits, that's not a problem. Especially because at the time, rogues could get advantage a lot easier as well. We didn't have a problem with the tradeoff at my tables.

[Lokiare]

Actually, I think they did understand. You could trade higher precision for higher damage - and given that a sneak attack could deal the same damage as 5-6 hits, that's not a problem. Especially because at the time, rogues could get advantage a lot easier as well. We didn't have a problem with the tradeoff at my tables.

The play test rogue lost damage if they targeted an AC that they needed a specific range of numbers on the d20 to hit. It seems you are falling into the same trap as the developers. They played a few sessions and said 'I don't see a problem, it seems to work.' when in fact on average it lowers the damage of the rogue. It went away at higher levels but at lower levels an optimized rogue would deal less damage if they gave up advantage in order to deal sneak attack damage.

In other words you wouldn't spot it until several months into the game the rogue seems to be doing less damage than the rest of the group on a consistent basis, or if you ran some probability math.

[Sartharina]

The play test rogue lost damage if they targeted an AC that they needed a specific range of numbers on the d20 to hit. It seems you are falling into the same trap as the developers. They played a few sessions and said 'I don't see a problem, it seems to work.' when in fact on average it lowers the damage of the rogue. It went away at higher levels but at lower levels an optimized rogue would deal less damage if they gave up advantage in order to deal sneak attack damage.

In other words you wouldn't spot it until several months into the game the rogue seems to be doing less damage than the rest of the group on a consistent basis, or if you ran some probability math.Again - it depends on the target AC.

[Jack of Spades]

I'd be interested to see your work, if you've got it laying around (or are willing to reproduce it).

I find it interesting that Advantage/Disadvantage are apparently worth approximately +/-3, while the rule for Advantage/Disadvantage on passive checks is +/-5. This means that, if you have Advantage on Perception, you are actually better off relying on your passive Perception. Meanwhile, if you had Disadvantage, you should roll for it.

For the percentages, I just compared the number of cases that would produce a specific roll with the total of 400 possible rolls.

For the average improvement, I added up the 'bonus' from each possible case (rolling a 19 and a 20 would yield a 'bonus' of 20, for example) and then averaged them. I'm not entirely sure why my number comes out so much lower than Seppo87's here:

The average improvement is 4.75
But I'm always willing to bet against my own math.

[WibbleNZ]

I would say your math is good, but assumes that all target numbers are equally likely. If you use a weighted average based on how likely different target numbers are, you can come up with anything from +/-0 to +/- 5.

[Lokiare]

For the percentages, I just compared the number of cases that would produce a specific roll with the total of 400 possible rolls.

For the average improvement, I added up the 'bonus' from each possible case (rolling a 19 and a 20 would yield a 'bonus' of 20, for example) and then averaged them. I'm not entirely sure why my number comes out so much lower than Seppo87's here:

But I'm always willing to bet against my own math.

My earlier post details the math behind it. The bonus is entirely reliant on what number you need to roll on the d20 and varies from +2 to +6. Furthermore the closer you are to 10 or 11 the more of a bonus you get and the further away the less of a bonus. Which when you add it up turns it into a very interesting mathematical curve, which makes it problematic when swapping it out for something that is static.

It appears www.anydice.com is down at the moment so this should be helpful http://www.wolframalpha.com/input/?i=N*%2820-N%29%2F20


Again - it depends on the target AC.

Yes, my point exactly. There is no easy or obvious way for the player to know when its a good idea to give up advantage for sneak attack damage or just to keep advantage. This is because of the parabolic curve that advantage generates. It changes based on the target d20 roll that is needed. So its obvious that the developers didn't understand the math behind it when they put that in there.

[Seppo87]

So its obvious that the developers didn't understand the math behind it when they put that in there.
Non sequitur.

[Lokiare]

Non sequitur.

There aren't really any other options. Well I mean malfeasance, but I give them the benefit of the doubt here and say they are ignorant rather than malicious.

[Seppo87]

There aren't really any other options. Well I mean malfeasance, but I give them the benefit of the doubt here and say they are ignorant rather than malicious.
But it's not inherently bad.

[Lokiare]

But it's not inherently bad.

It was inherently less damage for optimized rogues targeting a specific range of AC values. In other words the developers didn't know they were giving a trap option in some cases.

This still doesn't address the +5 damage for giving up advantage for the barbarian. No matter how you look at that, it was worse than just attacking with advantage.

There was a few monster traits that worked like that too. Where it was always better for the monster to just ignore the trait and attack normally.

I pointed it out to them and its no longer in the final product, but they still didn't understand it. Which shows they don't understand the fundamental math of the (dis)advantage mechanic.

[Seppo87]

It was inherently less damage for optimized rogues targeting a specific range of AC values. In other words the developers didn't know they were giving a trap option in some cases.
Non sequitur.
4E Power attack can also reduce damage if used badly.
This can be entirely intentional, and is not a bad thing.

[da_chicken]

I don't know if you're aware, but a lot of people like luck and gambling in games. Not everybody wants games to be a pure test of skill. That's why poker is so much more popular than chess. And why Magic is so popular, too. The fact that choosing between advantage and no advantage but sneak attack damage is too complex to evaluate means that it's a gamble. It's clear to the player that they're sacrificing accuracy for damage, but they're going to have to guess which one is actually the better play because it's too complex to evaluate ad hoc. It's a risk. It's a gamble. Many people like that. That's why Power Attack was so popular in 3.x.

[Lokiare]

Non sequitur.
4E Power attack can also reduce damage if used badly.
This can be entirely intentional, and is not a bad thing.

We've already seen that they didn't understand 4E's math either when they kept claiming that 5E is great for 4E players. They did this several times in the various articles that they made about '5E is for everyone'. This showed that they didn't understand what 4E players found enjoyable about 4E which was for the most part the balanced math that allowed for tactical decisions.

Our choices are:
1. They didn't know about its effects and therefore they are incompetent (ignorance)
2. They did know and intentionally used ivory tower design, even after decrying it as bad. (malfeasance)
3. They did know and left it in because they were lazy and didn't want to come up with an alternative system.(lazy)

Take your pick. I choose the ignorance one because at least that has some hope of being corrected.


I don't know if you're aware, but a lot of people like luck and gambling in games. Not everybody wants games to be a pure test of skill. That's why poker is so much more popular than chess. And why Magic is so popular, too. The fact that choosing between advantage and no advantage but sneak attack damage is too complex to evaluate means that it's a gamble. It's clear to the player that they're sacrificing accuracy for damage, but they're going to have to guess which one is actually the better play because it's too complex to evaluate ad hoc. It's a risk. It's a gamble. Many people like that. That's why Power Attack was so popular in 3.x.

Regardless, that says the developers didn't know the effects or they were intentionally leaving it in for ivory tower optimizers to spot.

[Seppo87]

4. They did know the implications but they left it as it was because they believed players would like it and have fun with it.
Ivory tower is a small price to pay: if a player is so obsessed with perfect damage optimization to spend his time calculating this, so be it. Everybody has the right to approach the game as they like it the most.

[Lokiare]

4. They did know the implications but they left it as it was because they believed players would like it and have fun with it.
Ivory tower is a small price to pay: if a player is so obsessed with perfect damage optimization to spend his time calculating this, so be it.

Except when the non-optimizers start noticing: "Why can't I take down hobgoblins as fast as the fighter? Something fishy is going on here."

The idea that it takes an optimizer to notice is false. The optimizer will be able to track down where the problem lies because of their system mastery, but most players will notice given a decent amount of time and interaction with the rules. Which is why WotC's "play it a few times and hope it doesn't break once it hits the shelves" testing method is a failure.

[Seppo87]

Except when the non-optimizers start noticing: "Why can't I take down hobgoblins as fast as the fighter? Something fishy is going on here."
Nope. Most players will be very happy to trade some virtual average damage to have a chance at hitting really hard. Even if you explained the matter to them in detail, most people will just not care.
And, in fact, the difference is so small that an evaluation "on the fly" will be sufficient to make the average results very similar, to the point you can't notice it.
It doesn't take a genius to understand that if an enemy is difficult to hit, you don't want to lower your chances even more, does it?
So even the person with little mastery will mostly make an efficient use of this potentially trap option.
The difference between the optimizer and the common guy shrinks a lot if you consider this.
So what if the non optimizer gets a lucky hit? His average damage will skyrocket for that one fight, and this chance is enough of a psychological motivator to convince some people to take this kind of risk.
And it's perfectly okay. It's good design. Makes the game fun and rewarding.

[da_chicken]

Except when the non-optimizers start noticing: "Why can't I take down hobgoblins as fast as the fighter? Something fishy is going on here."

I think most players when confronted with this rule would say "always sneak attack unless I need a 17/18 or better to hit" based on whatever they felt comfortable risking. I think the reaction of most players to their spreadsheet wielding party member would be (after rolling their eyes) that they should adjust their rule of thumb +1/-1. The d20 is a hugely variant die. It takes hundreds of attack rolls to see any kind of pattern emerge, meaning a reasonable rule of thumb like the one above likely to be 90% correct. Furthermore, confirmation bias means that players will overemphasize the memory of the time that they crit their sneak attack roll while Mr. Excel missed his attack with advantage. So even in error they will still be happy.

Never mind that some players like to feel like they're beating the odds. They would always sneak attack here just because they want to roll those sneak attack dice. Would they kill fewer? Perhaps, but that's not relevant to them. They want to go balls out because that's fun, even when it's not the wisest attack pattern.

I'm not saying this particular rule is a good one. I don't think it is. I'm just saying that complexity so high that it's effectively luck doesn't make it a bad rule. It just makes it largely a luck-based rule.


The idea that it takes an optimizer to notice is false. The optimizer will be able to track down where the problem lies because of their system mastery, but most players will notice given a decent amount of time and interaction with the rules. Which is why WotC's "play it a few times and hope it doesn't break once it hits the shelves" testing method is a failure.

You mean, other than the fact that their "play it a few times" method apparently decided that this rule wasn't worth keeping? Or are you suggesting that you have knowledge that they eliminated the rule because they're idiots, too, and they're just arriving at the right answers for the wrong reasons?

[Lokiare]

I think most players when confronted with this rule would say "always sneak attack unless I need a 17/18 or better to hit" based on whatever they felt comfortable risking. I think the reaction of most players to their spreadsheet wielding party member would be (after rolling their eyes) that they should adjust their rule of thumb +1/-1. The d20 is a hugely variant die. It takes hundreds of attack rolls to see any kind of pattern emerge, meaning a reasonable rule of thumb like the one above likely to be 90% correct. Furthermore, confirmation bias means that players will overemphasize the memory of the time that they crit their sneak attack roll while Mr. Excel missed his attack with advantage. So even in error they will still be happy.

Never mind that some players like to feel like they're beating the odds. They would always sneak attack here just because they want to roll those sneak attack dice. Would they kill fewer? Perhaps, but that's not relevant to them. They want to go balls out because that's fun, even when it's not the wisest attack pattern.

I'm not saying this particular rule is a good one. I don't think it is. I'm just saying that complexity so high that it's effectively luck doesn't make it a bad rule. It just makes it largely a luck-based rule.



You mean, other than the fact that their "play it a few times" method apparently decided that this rule wasn't worth keeping? Or are you suggesting that you have knowledge that they eliminated the rule because they're idiots, too, and they're just arriving at the right answers for the wrong reasons?

They Didn't have a clue about it until I started a thread on it in their forums.

As to the rest, the house always wins. If players want to take risks, that's their business. They should at least know what their chances are.

[Seppo87]

They Didn't have a clue about it until I started a thread on it in their forums.No proof


They should at least know what their chances are
This is your opinion.

[Elkad]

They Didn't have a clue about it until I started a thread on it in their forums.

LOL. It's immediately obvious.

It takes about 5 seconds to determine that...
If you need a 20 (5%), you get basically double (10%) the chance to succeed. About the same as +1.
Or if you have disadvantage, it becomes very unlikely (1 in 400), about the same as -1 and not having "nat 20 always hits"
But in the middle it gets you a much bigger range. 50% becomes 75% or 25%. For easy stuff (8-12 needed), it was about the same as +4

I didn't bother plot any curves, just glanced at it and said "hey, it makes pushing for one extreme end of the scale (like high AC) worth it". The farther off-center you get, the less effect giving up advantage to your enemy matters.

[Lokiare]

LOL. It's immediately obvious.

It takes about 5 seconds to determine that...
If you need a 20 (5%), you get basically double (10%) the chance to succeed. About the same as +1.
Or if you have disadvantage, it becomes very unlikely (1 in 400), about the same as -1 and not having "nat 20 always hits"
But in the middle it gets you a much bigger range. 50% becomes 75% or 25%. For easy stuff (8-12 needed), it was about the same as +4

I didn't bother plot any curves, just glanced at it and said "hey, it makes pushing for one extreme end of the scale (like high AC) worth it". The farther off-center you get, the less effect giving up advantage to your enemy matters.

Its immediately obvious to us math nerds, but its not obvious to the developers, which is why they put those mechanics into the game in the first place.

[Seppo87]

Its immediately obvious to us math nerds, but its not obvious to the developers, which is why they put those mechanics into the game in the first placeYour emotions interfere with your ability to think. You're clearly biased because it hurts that they never listened as you believe you deserved.

[Lokiare]

Your emotions interfere with your ability to think. You're clearly biased because it hurts that they never listened as you believe you deserved.

Lol, that came out of nowhere...

Sorry no. I leave my emotions at the door when I do analyses. The thing is before I mentioned it they had no clue. Only after I started a massive thread on it did they remove all effects that traded out advantage or disadvantage with the exception of the barbarian which trades advantage on attacks for advantage for creatures that attack the barbarian. This is fine because its trading on two different scales. The one doesn't affect the other.

You are probably right though, they completely ignored my epic thread that got lots of player attention and promises to put a note in the survey by many of the participants, and completely came up with the idea that they needed to remove it right after that completely on their own.

[Seppo87]

Okay, can you link this thread you're talking about and prove that they didn't know that this mechanic could lead to reduced average damage?