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<blockquote data-quote="Truename" data-source="post: 4983260" data-attributes="member: 78255">Monte Carlo vs. DPRLet's start by looking at the difference between a stochastic analysis and a DPR analysis. For this one, we'll start very simply. The question we're analyzing: How many rounds does it take for a PC to kill a monster?The PC: a level 1 dwarf great-weapon fighter, using Character Builder's recommended stats, wielding a maul. No feats or powers yet--just basic attacks.PC Attack bonus: 7&nbsp; (4 stat + 2 maul + 1 greatweapon)PC Dmg dice: 2d6 (maul) + 4 (stat)[sblock=Character Builder summary]====== Created Using Wizards of the Coast D&amp;D Character Builder ======Dwarf-Fighter, level 1Dwarf, FighterFINAL ABILITY SCORESStr 18, Con 12, Dex 14, Int 10, Wis 13, Cha 8.STARTING ABILITY SCORESStr 18, Con 10, Dex 14, Int 10, Wis 11, Cha 8.AC: 17 Fort: 16 Reflex: 12 Will: 11HP: 27 Surges: 10 Surge Value: 6TRAINED SKILLSUNTRAINED SKILLSAcrobatics +2, Arcana, Bluff -1, Diplomacy -1, Dungeoneering +3, Endurance +3, Heal +1, History, Insight +1, Intimidate -1, Nature +1, Perception +1, Religion, Stealth +2, Streetwise -1, Thievery +2, Athletics +4FEATSPOWERSITEMSScale Armor, Adventurer's Kit, Maul====== Copy to Clipboard and Press the Import Button on the Summary Tab ======[/sblock]The Monster: a generic level 1 soldier, created using p184 of the DMG.Monster AC: 17&nbsp; (1 level + 16 soldier)Monster HP: 29 (8 soldier + 13 con + (1 level * 8 soldier)A DPR analysis of this fight says the monster lasts 4.6 rounds on average.[sblock=DPR breakdown]Avergae damage for maul: 2d6+4 = ((2 + 12) / 2) + 4 = 11Crit damage for maul: 2d6+4 = 12 + 4 = 16Roll to hit = 17 AC - 7 att bonus = 10Chance to roll miss = (roll to hit - 1) / 20 = 45%Chance to roll crit hit = 1 / 20 = 5%Chance to roll normal hit = 1 - miss chance - crit chance = 1 - 45% - 5% = 50%Average crit damage = crit chance * crit dmg = 5% * 16 = 0.8Average hit damage = normal hit chance * normal dmg = 50% * 11 = 5.5DPR = 0.8 + 4.95 = 5.75Avg rounds = 29 HP / 5.75 DPR = 4.6 rounds[/sblock]The stochastic analysis shows us something completely different. It says the average fight lasts 5.5 rounds, not 4.6. That's a significant difference... big enough that, if correct, pretty much invalidates DPR as a trustworthy approach.[sblock=Source code (in Ruby)][code]REPS = 1000000PRECISION = 1DISPLAY_WIDTH = 75MONSTER_LEVEL = 1MONSTER_CON = 13PC_ATT_BONUS = 7PC_DMG_BONUS = 4class Monster&nbsp; &nbsp; def initialize&nbsp; &nbsp; &nbsp; &nbsp; @ac = MONSTER_LEVEL + 16&nbsp; &nbsp; &nbsp; &nbsp; @hp = 8 + MONSTER_CON + (MONSTER_LEVEL * 8)&nbsp; &nbsp; end&nbsp; &nbsp; &nbsp; &nbsp; def dead?&nbsp; &nbsp; &nbsp; &nbsp; return @hp &lt;= 0&nbsp; &nbsp; end&nbsp; &nbsp; &nbsp; &nbsp; def defend(att, dmg)&nbsp; &nbsp; &nbsp; &nbsp; @hp -= dmg if att &gt;= @ac&nbsp; &nbsp; endendclass Pc&nbsp; &nbsp; def attack(monster)&nbsp; &nbsp; &nbsp; &nbsp; att_roll = die(20)&nbsp; &nbsp; &nbsp; &nbsp; att = (att_roll + PC_ATT_BONUS)&nbsp; &nbsp; &nbsp; &nbsp; dmg = damage(att_roll)&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; monster.defend(att, dmg)&nbsp; &nbsp; end&nbsp; &nbsp; &nbsp; &nbsp; def damage(att_roll)&nbsp; &nbsp; &nbsp; &nbsp; dmg_dice = die(6) + die(6)&nbsp; &nbsp; &nbsp; &nbsp; dmg_dice = 12 if att_roll == 20&nbsp; &nbsp; &nbsp; &nbsp; return dmg_dice + PC_DMG_BONUS&nbsp; &nbsp; endenddef die(size)&nbsp; &nbsp; return 1 + rand(size)enddef fight&nbsp; &nbsp; monster = Monster.new&nbsp; &nbsp; pc = Pc.new&nbsp; &nbsp; round = 0&nbsp; &nbsp; until monster.dead?&nbsp; &nbsp; &nbsp; &nbsp; round += 1&nbsp; &nbsp; &nbsp; &nbsp; pc.attack(monster)&nbsp; &nbsp; &nbsp;&nbsp; end&nbsp; &nbsp; &nbsp; return roundenddef analyze&nbsp; &nbsp; results = {}&nbsp; &nbsp; total_rounds = 0&nbsp; &nbsp; max_value = 0&nbsp; &nbsp; REPS.times do&nbsp; &nbsp; &nbsp; rounds = fight&nbsp; &nbsp; &nbsp; results[rounds] = 0 unless results[rounds]&nbsp; &nbsp; &nbsp; new_value = results[rounds] += 1&nbsp; &nbsp; &nbsp; max_value = new_value if new_value &gt; max_value&nbsp; &nbsp; &nbsp; total_rounds += rounds&nbsp; &nbsp; end&nbsp; &nbsp; &nbsp; &nbsp; results.keys.sort.each do |key|&nbsp; &nbsp; &nbsp; value = &quot;&quot;&nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; ticks = (results[key].to_f / max_value.to_f * DISPLAY_WIDTH).to_i&nbsp; &nbsp; &nbsp; ticks.times do&nbsp; &nbsp; &nbsp; &nbsp; value += &quot;=&quot;&nbsp; &nbsp; &nbsp; end&nbsp; &nbsp; &nbsp; puts &quot;#{key.to_s.rjust(2)}: #{value}&quot; if value != &quot;&quot;&nbsp; &nbsp; end&nbsp; &nbsp; &nbsp; &nbsp; avg_rounds = total_rounds.to_f / REPS&nbsp; &nbsp; format = &quot;%.#{PRECISION}f&quot;&nbsp; &nbsp; puts &quot;Fights simulated: #{REPS}&quot;&nbsp; &nbsp; puts &quot;Average # of rounds per fight: #{format % avg_rounds}&quot;end&nbsp; &nbsp; analyze[/code][/sblock]Why the difference? I'm not entirely sure. (Hopefully it's not a bug! That's why I included the source code.) I think it's partly because, in a real fight, some damage is &quot;wasted&quot; after the monster hits zero HP.The Monte Carlo analysis also gives us a histogram that summarizes all of the fights. For this simulation, I ran a million fights. 'Cause I could.[code]LEVEL 1 SOLDIER VS. DWARF GREATWEAPON FIGHTER (no feats, no powers)&nbsp; 2&nbsp; (2.3%): =======&nbsp; 3 (18.0%): ==================================================&nbsp; 4 (38.7%): ===================================================================&nbsp; 5 (57.8%): =============================================================&nbsp; 6 (72.6%): ===============================================&nbsp; 7 (83.0%): =================================&nbsp; 8 (89.8%): =====================&nbsp; 9 (94.0%): =============&nbsp;10 (96.6%): ========&nbsp;11 (98.1%): ====&nbsp;12 (99.0%): ==&nbsp;13 (99.4%): =Fights simulated: 1000000Average # of rounds per fight: 5.5[/code]So, although the average # of rounds is 5.5, the majority of fights take 4 rounds. Over 80% of fights take 7 rounds or less to complete, although a very small fraction of this simulation's fights dragged on and on, presumably due to lots of bad rolls.You can see how the Monte Carlo simulation gives us a much richer, more accurate analysis than the DPR approach. And we're just barely getting started. Next up: feats, powers, and monsters that actually fight back!</blockquote>

[QUOTE="Truename, post: 4983260, member: 78255"] [b]Monte Carlo vs. DPR[/b] Let's start by looking at the difference between a stochastic analysis and a DPR analysis. For this one, we'll start very simply. The question we're analyzing: How many rounds does it take for a PC to kill a monster? [B]The PC[/B]: a level 1 dwarf great-weapon fighter, using Character Builder's recommended stats, wielding a maul. No feats or powers yet--just basic attacks. PC Attack bonus: 7 (4 stat + 2 maul + 1 greatweapon) PC Dmg dice: 2d6 (maul) + 4 (stat) [sblock=Character Builder summary] ====== Created Using Wizards of the Coast D&D Character Builder ====== Dwarf-Fighter, level 1 Dwarf, Fighter FINAL ABILITY SCORES Str 18, Con 12, Dex 14, Int 10, Wis 13, Cha 8. STARTING ABILITY SCORES Str 18, Con 10, Dex 14, Int 10, Wis 11, Cha 8. AC: 17 Fort: 16 Reflex: 12 Will: 11 HP: 27 Surges: 10 Surge Value: 6 TRAINED SKILLS UNTRAINED SKILLS Acrobatics +2, Arcana, Bluff -1, Diplomacy -1, Dungeoneering +3, Endurance +3, Heal +1, History, Insight +1, Intimidate -1, Nature +1, Perception +1, Religion, Stealth +2, Streetwise -1, Thievery +2, Athletics +4 FEATS POWERS ITEMS Scale Armor, Adventurer's Kit, Maul ====== Copy to Clipboard and Press the Import Button on the Summary Tab ====== [/sblock] [B]The Monster[/B]: a generic level 1 soldier, created using p184 of the DMG. Monster AC: 17 (1 level + 16 soldier) Monster HP: 29 (8 soldier + 13 con + (1 level * 8 soldier) A DPR analysis of this fight says the monster lasts [B]4.6 rounds[/B] on average. [sblock=DPR breakdown] Avergae damage for maul: 2d6+4 = ((2 + 12) / 2) + 4 = 11 Crit damage for maul: 2d6+4 = 12 + 4 = 16 Roll to hit = 17 AC - 7 att bonus = 10 Chance to roll miss = (roll to hit - 1) / 20 = 45% Chance to roll crit hit = 1 / 20 = 5% Chance to roll normal hit = 1 - miss chance - crit chance = 1 - 45% - 5% = 50% Average crit damage = crit chance * crit dmg = 5% * 16 = 0.8 Average hit damage = normal hit chance * normal dmg = 50% * 11 = 5.5 DPR = 0.8 + 4.95 = 5.75 Avg rounds = 29 HP / 5.75 DPR = 4.6 rounds [/sblock] The stochastic analysis shows us something [I]completely[/I] different. It says the average fight lasts [B]5.5 rounds[/B], not 4.6. That's a significant difference... big enough that, if correct, pretty much invalidates DPR as a trustworthy approach. [sblock=Source code (in Ruby)] [code] REPS = 1000000 PRECISION = 1 DISPLAY_WIDTH = 75 MONSTER_LEVEL = 1 MONSTER_CON = 13 PC_ATT_BONUS = 7 PC_DMG_BONUS = 4 class Monster def initialize @ac = MONSTER_LEVEL + 16 @hp = 8 + MONSTER_CON + (MONSTER_LEVEL * 8) end def dead? return @hp <= 0 end def defend(att, dmg) @hp -= dmg if att >= @ac end end class Pc def attack(monster) att_roll = die(20) att = (att_roll + PC_ATT_BONUS) dmg = damage(att_roll) monster.defend(att, dmg) end def damage(att_roll) dmg_dice = die(6) + die(6) dmg_dice = 12 if att_roll == 20 return dmg_dice + PC_DMG_BONUS end end def die(size) return 1 + rand(size) end def fight monster = Monster.new pc = Pc.new round = 0 until monster.dead? round += 1 pc.attack(monster) end return round end def analyze results = {} total_rounds = 0 max_value = 0 REPS.times do rounds = fight results[rounds] = 0 unless results[rounds] new_value = results[rounds] += 1 max_value = new_value if new_value > max_value total_rounds += rounds end results.keys.sort.each do |key| value = "" ticks = (results[key].to_f / max_value.to_f * DISPLAY_WIDTH).to_i ticks.times do value += "=" end puts "#{key.to_s.rjust(2)}: #{value}" if value != "" end avg_rounds = total_rounds.to_f / REPS format = "%.#{PRECISION}f" puts "Fights simulated: #{REPS}" puts "Average # of rounds per fight: #{format % avg_rounds}" end analyze [/code][/sblock] Why the difference? I'm not entirely sure. (Hopefully it's not a bug! That's why I included the source code.) I think it's partly because, in a real fight, some damage is "wasted" after the monster hits zero HP. The Monte Carlo analysis also gives us a histogram that summarizes all of the fights. For this simulation, I ran a million fights. 'Cause I could. [code] LEVEL 1 SOLDIER VS. DWARF GREATWEAPON FIGHTER (no feats, no powers) 2 (2.3%): ======= 3 (18.0%): ================================================== 4 (38.7%): =================================================================== 5 (57.8%): ============================================================= 6 (72.6%): =============================================== 7 (83.0%): ================================= 8 (89.8%): ===================== 9 (94.0%): ============= 10 (96.6%): ======== 11 (98.1%): ==== 12 (99.0%): == 13 (99.4%): = Fights simulated: 1000000 Average # of rounds per fight: 5.5 [/code]So, although the average # of rounds is 5.5, the majority of fights take 4 rounds. Over 80% of fights take 7 rounds or less to complete, although a very small fraction of this simulation's fights dragged on and on, presumably due to lots of bad rolls. You can see how the Monte Carlo simulation gives us a much richer, more accurate analysis than the DPR approach. And we're just barely getting started. Next up: feats, powers, and monsters that actually fight back! [/QUOTE]

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