The math of Advantage and Disadvantage

[Satyrn]

I wonder if I could pose a challenge to y'all.

I've been curious about the usefulness of my Hyperion inspired gun. In Borderlands, Hyperion guns become more accurate the more you fired them, so I gave them this feature:

Attack rolls with an Hyperion gun are made with disadvantage. When you hit a creature with this gun, you have advantage on further attacks with this gun against that creature until the end of your next turn

So what happens is that once you hit a target you get advantage against that target round after round so long as you keep hitting at least once each round. I have no idea how to judge this mathematically, unlike my Torgue gun, which is brutally simple:

Bolts fired from a Torgue gun EXPLODE on contact, dealing +1d6 EXPLOSIVE damage. Which is LOUD. On a critical hit, creatures adjacent to the target suffer EXPLOSIVE damage equal to half of what the target took.

So, like, what's the math that shows how Hyperion compares to that?


(By the way, the guns are essentially repeating crossbows, and apart from the special feature, a Torgue gun has identical stats to the Hyperion version.)

 

[Blue]

It is less about the frequency of needing a natural 18 or higher, it is about when these situations come up, the advantage vanishes. Oppositely, advantage is only strong when the result of the roll has less consequence.

Advantage is a great mechanic to represent being reliable at average things. However, it is of diminishing value to achieve anything difficult.

Advantage never increases the maximum number of what you can roll - a d20+1 is infinitely better than a d20 with advantage if you need a DC of 21. It doesn't make you able to do thing you couldn't before. What it does is make you more reliable at things.

Earlier editions of D&D you could stack so many bonuss that low-skilled characters couldn't even reach the DC with a 20. Advantage and disadvantage avoid all of that by never changing the min and max of what you can roll.

So I agree with what you are describing, but see that as a intended outcome of the system.

 

[Blue]

I wonder if I could pose a challenge to y'all.

I've been curious about the usefulness of my Hyperion inspired gun. In Borderlands, Hyperion guns become more accurate the more you fired them, so I gave them this feature:


So what happens is that once you hit a target you get advantage against that target round after round so long as you keep hitting at least once each round. I have no idea how to judge this mathematically, unlike my Torgue gun, which is brutally simple:



So, like, what's the math that shows how Hyperion compares to that?


(By the way, the guns are essentially repeating crossbows, and apart from the special feature, a Torgue gun has identical stats to the Hyperion version.)

Not knowing the damage they do on base, it's hard to say how much of a bonus the Torgue gun is doing.

Now, if you are keeping to bounded accuracy in what you need to hit, I'm going to say that advantage is turning 20% of your misses into hit, with disadvantage doing the opposite.

So before you hit, you are, over a statistically large amount of rounds, doing 20% less average damage. After you hit, 20% more until you switch targets (death, out of range/sight, out of ammo, etc.).

Torgue does +d6 ... but I don't know how much that is compared to the base weapon. I'm going to take a guess that they are d8 weapons, and say +4 from ability score mod or other things. So average damage is 8.5 and explosive grants +3.5. That's about +41% damage on a hit.

So, even if Hyperior always granted advantage AND Torgue never did bonus damage to bystanders, Torgue would be better. With just +4 bonus damage. If you start cranking that up it could end up differently.

 

[Satyrn]

Not knowing the damage they do on base, it's hard to say how much of a bonus the Torgue gun is doing.

Now, if you are keeping to bounded accuracy in what you need to hit, I'm going to say that advantage is turning 20% of your misses into hit, with disadvantage doing the opposite.

So before you hit, you are, over a statistically large amount of rounds, doing 20% less average damage. After you hit, 20% more until you switch targets (death, out of range/sight, out of ammo, etc.).

Torgue does +d6 ... but I don't know how much that is compared to the base weapon. I'm going to take a guess that they are d8 weapons, and say +4 from ability score mod or other things. So average damage is 8.5 and explosive grants +3.5. That's about +41% damage on a hit.

So, even if Hyperior always granted advantage AND Torgue never did bonus damage to bystanders, Torgue would be better. With just +4 bonus damage. If you start cranking that up it could end up differently.

There are 4 different guns, doing 1d6 (pistol) 1d8 (rifle) and 1d10 (sniper rifle) and 2d8 within 20 ft. (Shotgun)*.

If other people agree with your analysis, I'm gonna have to give Hyperion something more. And possibly cut Torgue's critical splash damage . . . Oh but there is one piece of that Torgue feature that matters that gets readily overlooked. When I say it's LOUD, I mean that it increases the chance of a random encounter. (And a critical hit actually prompts a wandering monster check)


*and for those wondering about smgs, I really can't find enough design space to make them different enough than a pistol or rifle to bother with.

 

[5ekyu]

What's "maximum impact"? If I need a 20 then advantage almost doubles my chance to hit.

If I need a 1 to hit then advantage reduces my chance to miss by 95%. Almost guaranteed success seems maximally impactful.

If the enemy needs a 20 to hit then disadvantage reduces damage by a factor of 20. Seems impactful to me. But here you are saying reducing incoming damage by 95% isn't impactful.

Two offers to a pair of PCs of yours - one has 1gp the other 10.

Show me 1 gp of yours and I will give you 1 gp a 100% reward.

Show me 10gp and I will give you 5gp a 50% reward.

Only one can take the deal.

Which would you take for "maximum impact?"

Impact is not measured by change multiplier but by change total.

If we assume an expected damage on a hit of 20...

Taking a needs 20 roll to advantage takes expected damage of an attack from 1 to just shy of 2. Gain +1 hp.

Taking a needs 11 roll to advantage takes expected dmg kon attack from 10 to 15. Gain +5.

The "% increase" is often (used to be) misleading when presented without the net gain as well.

 

[Krachek]

The largest effect is the smallest reduction in times hit? At 11, disadvantage will reduce the number of hits by a factor of 2. At 20, disadvantage reduces the number of hits by a factor of 20. "Almost invulnerable" seems like a big effect to me.

Giving the disadvantage protection to the big AC guy is flashy but ineffective.
It will avoid one hit in my example, avoiding one hit is a small effect and a poor tactical choice.

Otherwise protecting to soft AC guy will help him survive the fight. Being hit 10 times vs 5 times may make the difference between life or death. It is the good call.

Your analysis is obsess with the math, but ignore the real impact on the game.

 

[Ovinomancer]

Giving the disadvantage protection to the big AC guy is flashy but ineffective.
It will avoid one hit in my example, avoiding one hit is a small effect and a poor tactical choice.

Otherwise protecting to soft AC guy will help him survive the fight. Being hit 10 times vs 5 times may make the difference between life or death. It is the good call.

Your analysis is obsess with the math, but ignore the real impact on the game.

Nope. We disagree on impact. Making the tank impossible to hit is often far more important that having things miss the mage twice as often. You're blinded by big numbers.

 

[Krachek]

Dead mage, or mage loosing concentration are indeed not important.
You’re right.

 

[Ovinomancer]

Dead mage, or mage loosing concentration are indeed not important.
You’re right.

Your right, we've devolved into arguing specific uses where either of us can create a situation where it's more useful to apply disadvantage or advantage. That's a losing game. As you note above, mathematically the tails are the biggest effect. I'm content to leave it there.

 

[Blue]

There are 4 different guns, doing 1d6 (pistol) 1d8 (rifle) and 1d10 (sniper rifle) and 2d8 within 20 ft. (Shotgun)*.

If other people agree with your analysis, I'm gonna have to give Hyperion something more. And possibly cut Torgue's critical splash damage . . . Oh but there is one piece of that Torgue feature that matters that gets readily overlooked. When I say it's LOUD, I mean that it increases the chance of a random encounter. (And a critical hit actually prompts a wandering monster check)


*and for those wondering about smgs, I really can't find enough design space to make them different enough than a pistol or rifle to bother with.

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.