The math of Advantage and Disadvantage

Ovinomancer said:

Okay, I'm not beating on you, but this is... wrong. Understandably so, but wrong. Soapbox time!

Converting the difference in odds that occurs when using ad/disadvantage into a flat bonus leads to erroneous thinking. It's (for nerdy math reasons) just wrong to do this.

Click to expand...

I disagree that there is anything mathematically wrong about looking at an absolute percentage-point change in addition to looking at the corresponding proportional change. They measure different things, and thus are both valuable.

When considering a binary event, such as whether a d20+bonus roll hits a DC, the absolute percentage-point change provided by an Effect (e.g. advantage) is an accurate measure of how often the Effect will be outcome-determinitive. This is useful information and is not in any way "wrong".

Furthermore, the absolute percentage-point change in success rate of a binary event often provides more information than the proportional change when either are presented alone. This is because the practical significance (in the colloquial sense) of the absolute change is dependent only on the practical significance of success. By contrast, the practical significance of the proportional change is dependent on both the practical significance of success and on the base rate of success.

Blue said:

Where Hyperion has an advantage (heh, pun not intended but appreciated) is the characters that can take advantage of crits. GWM for a bonus action attack, Half-Orc Savage Attacks, and paladin's Divine Smite are melee so they are out. Worst I could see a rogue having fun with it - until the target dies once they get a hit they can auto sneak attack, and sneak attack damage goes up nicely with a crit. Hmm, offset the initial disad with adv by attacking from hiding, and take elven accuracy to make more crits. Still, not a huge deal.

Click to expand...

Thank you for the feedback.

I'm cool with the idea of the Hyperion guns being more appealing only to rogues eager to sneak attack with a gun, but I think you've convinced me to drop the disadvantage part so that there's no penalty to use. Especially if you're right that advantage averages out to a roughly 20% damage increase over time.

The crit fishers will want a Jakobs instead, since I gave them an expanded crit range (that stacks with the champion's) to capture the essence of that gun in the game: "If it took more than one shot, then you weren't using a Jakobs."

Xetheral said:

I disagree that there is anything mathematically wrong about looking at an absolute percentage-point change in addition to looking at the corresponding proportional change. They measure different things, and thus are both valuable.

When considering a binary event, such as whether a d20+bonus roll hits a DC, the absolute percentage-point change provided by an Effect (e.g. advantage) is an accurate measure of how often the Effect will be outcome-determinitive. This is useful information and is not in any way "wrong".

Click to expand...

The "wrong" part was converting a normal distribution to a mean-shifting a flat distribution. The is wrong.

Evaluating how often a the "Effect" (a nebulous term that I can easily define in a way that makes this wrong, but am choosing to assume that you're mostly referring to ad/disadvantage) is useful has little to do with the size of the absolute percentage change. To consider needing an 11, disadvantage would mean that the normal 200/400 chance drops to 100/400, or the Effect is determines the outcome 100/400 times more than the normal. If you need a 20, though, the normal 20/400 chance becomes 399/400, or the Effect determines the outcome 379/400 times more than the normal.

Furthermore, the absolute percentage-point change in success rate of a binary event often provides more information than the proportional change when either are presented alone. This is because the practical significance (in the colloquial sense) of the absolute change is dependent only on the practical significance of success. By contrast, the practical significance of the proportional change is dependent on both the practical significance of success and on the base rate of success.

Click to expand...

It does not provide more information. You're assuming facts, such as what is needed to be rolled for the absolute change while ignoring them for the proportional change. On even footing, neither present more information than the other. Frex, If I have an absolute change of -10%, and need a 20, then I will never succeed. You must also know the target number for the absolute change to have any meaning.

Luckily, the point I was making is not that the absolute change lacked information, or that the proportional change conveys more, but rather that the full sum of the impact requires BOTH to be understood. You may recall that I said, in the part you snipped of my post above, that you have to go deeper to get the full understanding and that stopping at the absolute change will confuse you? Because, as illustrated above, you do. Proportional changes are not sufficient by themselves, either, as they require the additional information to ground them and provide the needed context.

As I conceded to [MENTION=6811643]Krachek[/MENTION] above, even this doesn't determine optimal usage in game, as the context of the situation is even more important than the math. However, the math does clearly show that the tails have the greatest impact absent that context. In game, it depends.

Yaarel said:

The problem is, the straightforward math fails to take into account the following consideration.

Advantage is beneficial when you dont need it − when the AC is already low.

Advantage is unhelpful when you really need it − when the AC is too high.

Click to expand...

What really the math doesn't take into account is that a flat bonus allows skills and saves to beat DCs that are otherwise not possible with advantage, and similarly a flat penalty prevents to beat DCs that are still possible with disadvantage. This consideration doesn't apply to attacks since they have the autohit/automiss rule.

So as an example, all the advantage in the world including Lucky et al won't help a Rogue pick a lock with a DC beyond her maximum possible roll result, while a flat bonus might.

This doesn't devalue a bit the excellent analysis of the OP...

About AC, I add the following thoughts:

- I think in general 5e monsters AC is quite low, so you are more often in the situation when advantage is equivalent to a bigger probability boost
- it is not true that you "don't need it", you ALWAYS need it because in the course of the adventuring day any extra hit you score makes you go further; +25% when your rate is 50% means 50% more hits on average, so possibly 1 extra encounter every 2 -> that's a very rough gauge since obviously the dynamics of encounters are far more complex than just rolling attacks, but it's just to give a general idea of what it could mean if you could have advantage all the time
- lower AC monsters are often huge bags of HP, easy to hit but slow to take down, in which case 50% more hits equates to taking 33% less time to kill it
- OR lower AC monsters might mean lower-level minions which often come in hordes, in which case dropping them faster is important to decrease the number of attacks (or actions) that the party could be targeted with