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A tweak for the Battlemaster fighter
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<blockquote data-quote="Esker" data-source="post: 7789191" data-attributes="member: 6966824"><p>It's not only the first hit against an enemy that matters, though. Any time the enemy is within a potential average hit of dying, a character with a low chance of doing a very high amount of damage is not as effective as a character with a high chance of doing just enough damage, if the averages are close. </p><p></p><p></p><p></p><p>I'm not talking about comparing two PCs with very different average DPR, I'm talking about comparing two PCs with similar DPRs, but different variances. The bigger the difference in variance, the bigger the difference in DPR has to be for the high-mean-high-variance character to be more effective.</p><p></p><p>Take an extreme example, much more pronounced than any difference you'd see in reality, but presented as a proof-of-concept. Character 1 only hits on a natural 20 but does 20d6 damage when they do hit. Character 2 always hits, but only does 1d6 damage on a hit. So both have 3.5 DPR but achieved in very different ways. Suppose each character is going one-on-one with an enemy with 70 HP. No real battle has an expected length of 20 rounds, but again, we're taking an extreme which is far, far away from the one-to-two-shot enemy scenario. </p><p></p><p>Character 1 has a chance to one-shot it, and is virtually guaranteed not to need more than two hits, but also has good chance they'll whiff round after round. Meanwhile, Character 2 has almost no chance to kill the enemy in fewer than 15 hits, but makes steady progress every round. Who would you bet on finishing the job faster?</p><p></p><p>[SPOILER]</p><p>Character 2 finishes first about 53% of the time; Character 1 about 45% of the time, with the other 2% being ties.</p><p>[/SPOILER]</p><p></p><p>In a long battle like this, the matched DPR characters are pretty well matched for who finishes first, though the low variance character has a non-trivial edge. So who would you rather be? </p><p></p><p>If the odds are stacked against you such that you're likely to die "on average", then you'd take the Hail Mary and hope for that early crit. But most encounters are balanced such that the PCs are unlikely to die unless things go badly for them. So typically, guarding against tail risk is more important than trying to blow through encounters faster than expected. In that typical case, you stand to lose more by taking longer than average than you stand to gain by finishing sooner than average.</p><p></p><p>In this toy example, Character 1 has about a 1/4 chance of needing more than 40 rounds to finish the enemy, whereas Character 2 is nearly guaranteed to finish in 25 rounds or fewer.</p><p></p><p>So in terms of guarding against tail risk, you'd much rather be the low variance character.</p><p></p><p></p><p></p><p>You're right about that, assuming the paladin has a high chance of hitting, and therefore being able to expend their smite slots when they want to. Smite is in some ways like an even more flexible action surge: a single smite is not worth as much damage as a whole new attack action, but since you use it when you know you hit, and since you can use it twice in a row, it's a more flexible and reliable form of damage (when you're facing a single big enemy). This actually supports my point.</p><p></p><p></p><p></p><p>Let's say the non-SS PC has a 60% chance to do 1d8+4, whereas the SS-PC has a 35% chance to do 1d8+14. So the regular archer has a DPR of 5.1, compared with the SS archer's DPR of 6.5. So the SS archer is about 25% better from an average DPR perspective. And indeed, against a foe with exactly 15 HP, such that on a hit the SS archer is guaranteed to kill them, they're very likely to do the job faster, and also less likely to take a dangerously long time.</p><p></p><p>But suppose the enemy has 18 HP, so the SS archer still has a better than 50% chance of one-shotting them if they hit, but no longer a guarantee.</p><p></p><p>The SS archer has a lower expected number of turns needed to finish the enemy (3.9 vs 4.3), and a much better chance of doing it within two rounds (41% vs 16%), but they also have nearly double the chance of taking a "dangerously long time" (say 8 rounds or more; about twice the average), at 12% vs 6.5%. Even if your threshold for "dangerously long" is more liberal, at 1.5 times the average, or 6 rounds, the non-SS character has a slightly better chance of staying clear of that danger zone.</p><p></p><p>So if you're optimizing for the odds of impressive breezy battles, yes, the extra 1.4 DPR is worth the added variance. But if you're hedging against catastrophe, you'd rather not use the -5/+10.</p></blockquote><p></p>
[QUOTE="Esker, post: 7789191, member: 6966824"] It's not only the first hit against an enemy that matters, though. Any time the enemy is within a potential average hit of dying, a character with a low chance of doing a very high amount of damage is not as effective as a character with a high chance of doing just enough damage, if the averages are close. I'm not talking about comparing two PCs with very different average DPR, I'm talking about comparing two PCs with similar DPRs, but different variances. The bigger the difference in variance, the bigger the difference in DPR has to be for the high-mean-high-variance character to be more effective. Take an extreme example, much more pronounced than any difference you'd see in reality, but presented as a proof-of-concept. Character 1 only hits on a natural 20 but does 20d6 damage when they do hit. Character 2 always hits, but only does 1d6 damage on a hit. So both have 3.5 DPR but achieved in very different ways. Suppose each character is going one-on-one with an enemy with 70 HP. No real battle has an expected length of 20 rounds, but again, we're taking an extreme which is far, far away from the one-to-two-shot enemy scenario. Character 1 has a chance to one-shot it, and is virtually guaranteed not to need more than two hits, but also has good chance they'll whiff round after round. Meanwhile, Character 2 has almost no chance to kill the enemy in fewer than 15 hits, but makes steady progress every round. Who would you bet on finishing the job faster? [SPOILER] Character 2 finishes first about 53% of the time; Character 1 about 45% of the time, with the other 2% being ties. [/SPOILER] In a long battle like this, the matched DPR characters are pretty well matched for who finishes first, though the low variance character has a non-trivial edge. So who would you rather be? If the odds are stacked against you such that you're likely to die "on average", then you'd take the Hail Mary and hope for that early crit. But most encounters are balanced such that the PCs are unlikely to die unless things go badly for them. So typically, guarding against tail risk is more important than trying to blow through encounters faster than expected. In that typical case, you stand to lose more by taking longer than average than you stand to gain by finishing sooner than average. In this toy example, Character 1 has about a 1/4 chance of needing more than 40 rounds to finish the enemy, whereas Character 2 is nearly guaranteed to finish in 25 rounds or fewer. So in terms of guarding against tail risk, you'd much rather be the low variance character. You're right about that, assuming the paladin has a high chance of hitting, and therefore being able to expend their smite slots when they want to. Smite is in some ways like an even more flexible action surge: a single smite is not worth as much damage as a whole new attack action, but since you use it when you know you hit, and since you can use it twice in a row, it's a more flexible and reliable form of damage (when you're facing a single big enemy). This actually supports my point. Let's say the non-SS PC has a 60% chance to do 1d8+4, whereas the SS-PC has a 35% chance to do 1d8+14. So the regular archer has a DPR of 5.1, compared with the SS archer's DPR of 6.5. So the SS archer is about 25% better from an average DPR perspective. And indeed, against a foe with exactly 15 HP, such that on a hit the SS archer is guaranteed to kill them, they're very likely to do the job faster, and also less likely to take a dangerously long time. But suppose the enemy has 18 HP, so the SS archer still has a better than 50% chance of one-shotting them if they hit, but no longer a guarantee. The SS archer has a lower expected number of turns needed to finish the enemy (3.9 vs 4.3), and a much better chance of doing it within two rounds (41% vs 16%), but they also have nearly double the chance of taking a "dangerously long time" (say 8 rounds or more; about twice the average), at 12% vs 6.5%. Even if your threshold for "dangerously long" is more liberal, at 1.5 times the average, or 6 rounds, the non-SS character has a slightly better chance of staying clear of that danger zone. So if you're optimizing for the odds of impressive breezy battles, yes, the extra 1.4 DPR is worth the added variance. But if you're hedging against catastrophe, you'd rather not use the -5/+10. [/QUOTE]
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