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Charop DPR and nova: accuracy vs. damage
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<blockquote data-quote="crunchy4e" data-source="post: 8813739" data-attributes="member: 7038414"><p>Here are some thoughts, but I'm interested if anyone disagrees or can find an issue with the line of thinking:</p><p></p><p>As a preliminary note, an n-sided die with an even number of sides averages (n+1)/2. So d6 averages 3.5, d12 averages 6.5, and so on.</p><p></p><p><strong>Step 1: ignoring crits</strong></p><p></p><p>Expected attack damage is E(damage | hit)P(hit). E.g., a 1d4 + 4 attack averages 6.5 on a hit, so the expected DPR is 6.5*(hit chance). Abbreviate this as dh -- d for expected damage, h for hit probability (0 to 1). Ignore crits for now. (I'll include them later.)</p><p></p><p>Increasing accuracy by 1 increases hit probability by 1/20, or 0.05. This changes dh into d(h+0.05) = dh + 0.05d, increasing DPR by d/20. Increasing expected damage by 1 (e.g. increasing a die size) changes dh into (d+1)h = dh + h, increasing DPR by h.</p><p></p><p>Therefore, if h > d/20, then given the choice, it would be better to increase damage by 1. If d/20 > h, it would be better to increase accuracy.</p><p></p><p>Example: a level 1 rogue with 18 dexterity does 1d4 + 4 on several at-will powers. With advantage, a base rogue does an additional 2d6 damage (or 7 on average). Thus, the expected at-will damage with CA is 1d4 + 11, or 13.5. Backstabber increases this expected damage to 15.5, while Nimble Blade increases the hit rate by 5%.</p><p></p><p>Thus: (d+2)h = dh + 2h, and d(h+0.05) = dh + d/20 = dh + 13.5/20 = dh + 0.675.</p><p></p><p>So as a possible answer to my own question, if crits are ignored, Backstabber beats Nimble blade as long as 0.675 > 2h. I certainly expect most rogues (or even any non-optimal PC) to have a hit rate better than 0.675/2 = 33.75%.</p><p></p><p><strong>Step 2: including crits</strong></p><p></p><p>A standard critical strike at level 1, taking maximal damage, would be 1d4 + dex mod + 2d6 = 4 + 4 + 12 = 20. With Backstabber, that increases to 24.</p><p></p><p>A standard swing crits 5% of the time, and the rest of the hits do standard damage. I.e. damage is no longer dh, but 0.05*(crit damage) + (h-0.05)d.</p><p></p><p>With Nimble Blade and no Backstabber, this becomes 0.05*20 + ((h + 0.05) - 0.05)*13.5 = 1 + 13.5*h. With Backstabber and no Nimble Blade, this becomes 0.05*24 + (h - 0.05)*15.5 = 1.2 + 15.5h - 0.775 = 0.45 + 15.5h.</p><p></p><p>If 0.45 + 15.5h > 1 + 13.5h, then Backstabber beats Nimble Blade on average, including crits. This is equivalent to saying 2h > 0.55. I certainly hope a rogue with advantage, or any PC in 4e, hits more often than 0.55/2 = 27.5% of the time.</p><p></p><p>The reason the cutoff is lower than before is that accuracy does nothing to improve critical strike damage, but increasing damage dice does.</p><p></p><p><strong>Conclusion</strong></p><p></p><p>This should answer some basic questions and set a template for Sly Flourish vs. Piercing Strike type questions, although it's hard to include debuffing in the mix. Draji Palatial gives an automatic -2 attack debuff regardless of whether it hits, but it's hard to compute the relative merit of the party having +2 defenses against single opponent vs. the expected damage tradeoff of, say, a more accurate Piercing Strike.</p><p></p><p>Thoughts? Corrections? Disagreements?</p></blockquote><p></p>
[QUOTE="crunchy4e, post: 8813739, member: 7038414"] Here are some thoughts, but I'm interested if anyone disagrees or can find an issue with the line of thinking: As a preliminary note, an n-sided die with an even number of sides averages (n+1)/2. So d6 averages 3.5, d12 averages 6.5, and so on. [B]Step 1: ignoring crits[/B] Expected attack damage is E(damage | hit)P(hit). E.g., a 1d4 + 4 attack averages 6.5 on a hit, so the expected DPR is 6.5*(hit chance). Abbreviate this as dh -- d for expected damage, h for hit probability (0 to 1). Ignore crits for now. (I'll include them later.) Increasing accuracy by 1 increases hit probability by 1/20, or 0.05. This changes dh into d(h+0.05) = dh + 0.05d, increasing DPR by d/20. Increasing expected damage by 1 (e.g. increasing a die size) changes dh into (d+1)h = dh + h, increasing DPR by h. Therefore, if h > d/20, then given the choice, it would be better to increase damage by 1. If d/20 > h, it would be better to increase accuracy. Example: a level 1 rogue with 18 dexterity does 1d4 + 4 on several at-will powers. With advantage, a base rogue does an additional 2d6 damage (or 7 on average). Thus, the expected at-will damage with CA is 1d4 + 11, or 13.5. Backstabber increases this expected damage to 15.5, while Nimble Blade increases the hit rate by 5%. Thus: (d+2)h = dh + 2h, and d(h+0.05) = dh + d/20 = dh + 13.5/20 = dh + 0.675. So as a possible answer to my own question, if crits are ignored, Backstabber beats Nimble blade as long as 0.675 > 2h. I certainly expect most rogues (or even any non-optimal PC) to have a hit rate better than 0.675/2 = 33.75%. [B]Step 2: including crits[/B] A standard critical strike at level 1, taking maximal damage, would be 1d4 + dex mod + 2d6 = 4 + 4 + 12 = 20. With Backstabber, that increases to 24. A standard swing crits 5% of the time, and the rest of the hits do standard damage. I.e. damage is no longer dh, but 0.05*(crit damage) + (h-0.05)d. With Nimble Blade and no Backstabber, this becomes 0.05*20 + ((h + 0.05) - 0.05)*13.5 = 1 + 13.5*h. With Backstabber and no Nimble Blade, this becomes 0.05*24 + (h - 0.05)*15.5 = 1.2 + 15.5h - 0.775 = 0.45 + 15.5h. If 0.45 + 15.5h > 1 + 13.5h, then Backstabber beats Nimble Blade on average, including crits. This is equivalent to saying 2h > 0.55. I certainly hope a rogue with advantage, or any PC in 4e, hits more often than 0.55/2 = 27.5% of the time. The reason the cutoff is lower than before is that accuracy does nothing to improve critical strike damage, but increasing damage dice does. [B]Conclusion[/B] This should answer some basic questions and set a template for Sly Flourish vs. Piercing Strike type questions, although it's hard to include debuffing in the mix. Draji Palatial gives an automatic -2 attack debuff regardless of whether it hits, but it's hard to compute the relative merit of the party having +2 defenses against single opponent vs. the expected damage tradeoff of, say, a more accurate Piercing Strike. Thoughts? Corrections? Disagreements? [/QUOTE]
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Charop DPR and nova: accuracy vs. damage
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