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ハンカチ落としの戦略最適化
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Katsuki Ohto
July 07, 2020
Science
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ハンカチ落としの戦略最適化
漫画「嘘喰い」に登場するハンカチ落としゲームを簡略化してナッシュ均衡戦略を計算します。
Katsuki Ohto
July 07, 2020
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Transcript
ϋϯΧνམͱ͠ ͷઓུ࠷దԽ Katsuki Ohto Optimization Night #3 LT
ϋϯΧνམͱ͠ ➤ Ұൠతʹ… தఔͷਓͰߦ͏ ϨΫϦΤʔγϣϯήʔϜ ➤ ອըʮӕ͍ᷰʯಡऀʹ… ೋਓͰ໋ΛṌ͚ͯઓ͏ήʔϜ ϋϯΧνΛམͱ͢ํΛʮυϩοϓଆʯৼΓ͍ͯϋϯΧν͕མ ͪͨͷΛ֬ೝ͢ΔํΛʮνΣοΫଆʯͱ͠ɺަޓʹ୲͍ͯ͠
͘ɻਐߦʹ࣌ใΛར༻͢Δɻ1λʔϯ1ؒͱ͠ɺυϩοϓ ଆඞͣͦͷ࣌ؒʹམͱ͞ͳ͚ΕͳΒͣɺνΣοΫଆඞͣ Ұ͚ͩৼΓ͔ͳ͚ΕͳΒͳ͍ɻυϩοϓଆ͕ϋϯΧνΛ མͱͨ͠ॠ͔ؒΒνΣοΫଆͷʮ࠲ࢹͷࡍʯ͕ελʔτɺ༰ث ͷத֖͕։͖γϦϯμʔʹʮྟࢮༀʯ͕ཷ·͍ͬͯ͘ɻৼΓ͍ ͯϋϯΧν͕མ͍ͪͯΕνΣοΫޭͱͳΓத֖͕ด͡Δ͕ ٯʹམ͍ͪͯͳ͚ΕʮϖφϧςΟʯ͕ൃಈɻʮͦͷ࣌Ͱཷ ·͍ͬͯΔྟࢮༀʴ1ؒͰཷ·ΔྔʯΛͦͷͰࣹʢγϦϯ μʔʹྟࢮༀ͕5ؒཷ·ͬͨ߹ଈࣹʣ͞ΕΔɻͦͯ͠ ຺ഥ͕ࢭ·ͬͨͷΛ֬ೝ͔ͯ͠Βʮྟࢮʯ͕ελʔτɻϖφϧ ςΟඵΛফԽͨ͠ΒྟࢮༀͱରʹͳΔͷʮોੜༀʯΛ ্ࣹͨ͠Ͱ15ճݶఆͷ৺ଁϚοαʔδʢ10ඵؒ΄ͲʣΛߦ ͍ɺͦΕͰોੜͰ͖ͨΒήʔϜଓߦɺͰ͖ͳ͚Εউෛ͋Γɻ Wikipedia ʮӕ͍ᷰʯΑΓ
؆ུԽϧʔϧ ➤ υϩοϓଆ …ϋϯΧνΛམͱ͢ଆ ➤ νΣοΫଆ …ৼΓฦΔଆ ➤ ࣌ࠁ T
ඵؒ ➤ ϖφϧςΟʢ㲈ใुʣ ɾνΣοΫଆ͕མͱ͞Εͯ ɹ t ඵޙʹৼΓฦΔ… t ɾνΣοΫଆ͕ઌʹৼΓฦΔ ɹ… T ➤ ອըͰɺ͜ΕΛ෦ήʔϜͱͯ͠ ܁Γฦ͢ཁૉ͕͋Δ͕ɺ ࠓճ͜ͷҰճͷήʔϜ͚ͩߟ͑Δ
ಉ࣌ ήʔϜ ͷੑ࣭
͡ΌΜ͚Μ ➤ άʔΛग़͢ઓུʹ ˠύʔΛग़͢ઓུ͕༗ޮ ➤ ύʔΛग़͢ઓུʹ ˠνϣΩΛग़͢ઓུ͕༗ޮ ➤ νϣΩΛग़͢ઓུʹ ˠάʔΛग़͢ઓུ͕༗ޮ
➤ ઓུ͕॥͠ɺৗʹ࠷దͳ ઓུ͕ఆ·Βͳ͍ φογϡۉߧɹΛߟ͑Δ
φογϡۉߧઓུ ➤ ยํͷϓϨΠϠ͕͔ࣗΒख Λม͑ΔϝϦοτ͕ͳ͍ઓུ ʢͷΈ߹Θͤʣ ➤ ғޟকعͳΒউརखͲΕ͔ ҰͭΛબͿઓུͰOK ➤ ಉ࣌ήʔϜ
ɾ͡ΌΜ͚Μ ෆશใήʔϜ ɾϙʔΧʔɹɾຑ ͜ΕΒͷήʔϜͰ φογϡۉߧ֬తઓུ
ઓུ࠷ద Խ
తؔ ➤ ࣌ࠁ T+1 ·ͰԆ͠ɺνΣοΫଆɺυϩοϓଆ͕Կ͠ͳ͔ͬͨ߹ ࣌ࠁ T+1 ʹߦಈΛىͨ͜͜͠ͱʹ͢Δʢ݁ՌʹӨڹͳ͍ʣ ➤ ઓུ
࣌ࠁ i Ͱυϩοϓ͢Δ֬ ͨͩ͠ ࣌ࠁ j ͰνΣοΫ͢Δ֬ ͨͩ͠ ➤ ϖφϧςΟͷظ Di T+1 ∑ i=1 Di = 1 Cj T+1 ∑ j=1 Cj = 1 R(C, D) = T+1 ∑ i=1 Di ( i−1 ∑ j=1 Cj T + T+1 ∑ j=i Cj (j − i)) T=1ͷͱ͖ x = D1, y = C1ͱͯ͠
߆ଋ͖݅ͭͷఀཹ ➤ ϖφϧςΟͷظ ➤ φογϡۉߧͰɺνΣοΫଆɺυϩοϓଆ ͱʹઓུΛมߋ͢ΔΠϯηϯςΟϒ͕ͳ͍ ➤ →֬ͷ͕ 1ͱ͍͏ ɹ߆ଋ͖݅ͭͷఀཹ
➤ →ϥάϥϯδϡͷະఆ๏Λར༻͢Δ R(C, D) = T+1 ∑ i=1 Di ( i−1 ∑ j=1 Cj T + T+1 ∑ j=i Cj (j − i)) T=1ͷͱ͖ x = D1, y = C1ͱͯ͠
ϥάϥϯδϡͷະఆ๏ ➤ ཱࣜ ➤ ల։ ➤ ղऍ ͜ͷࣜ
͔ͭ ͱॻ͚Δ F = R(D, C) − γ( T+1 ∑ i=1 Di − 1) − λ( T+1 ∑ j=1 Cj − 1) ∂F ∂Dt = ∂F ∂Ct = ∂F ∂γ = ∂F ∂λ = 0 ∂F ∂Dt = t−1 ∑ j=1 Cj T + T+1 ∑ j=t Cj (j − t) − γ ∂F ∂Ct = t ∑ i=1 Di (t − i) + T+1 ∑ i=t+1 Di T − λ ∂R ∂Dt = const . ∂R ∂Ct = const .
νΣοΫଆͷԽࣜಋग़ ➤ υϩοϓଆͷ͔ࣜΒελʔτ ➤ ∂F ∂Dt =
0 = ∂F ∂Dt+1 t−1 ∑ j=1 Cj T + T+1 ∑ j=t Cj (j − t) − γ = t ∑ j=1 Cj T + T+1 ∑ j=t+1 Cj (j − t − 1) − γ 0 = Ct T − T+1 ∑ j=t+1 Cj Ct = ∑T+1 j=t+1 Cj T
υϩοϓଆͷԽࣜಋग़ ➤ νΣοΫଆͷ͔ࣜΒελʔτ ➤ ∂F ∂Ct =
0 = ∂F ∂Ct+1 t ∑ i=1 Di (t − i) + T+1 ∑ i=t+1 Di T − λ = t+1 ∑ i=1 Di (t + 1 − i) + T+1 ∑ i=t+2 Di T − λ Dt+1 T = t ∑ i=1 Di Dt+1 = ∑t i=1 Di T
ܭࢉ݁Ռ·ͱΊ ➤ ➤ ➤ ͜Ε͔Βͷ͔ࣜΒφογϡۉߧઓུ͕ఆ·Δ ➤ υϩοϓଆͱνΣοΫଆͰରশʹͳ͍ͬͯΔ ʢ͜͜ϖφϧςΟͷઃఆ࣍ୈʣ Dt+1
= ∑t i=1 Di T T+1 ∑ i=1 Di = 1 Ct = ∑T+1 j=t+1 Cj T T+1 ∑ i=1 Ci = 1
ઓུՄࢹ Խ
φογϡۉߧઓུ ➤ T = 10 ͷ߹Λਤࣔ ➤ υϩοϓଆʮ͙͢མͱ͢ʯ νΣοΫଆʮৼΓ͔ͣʯ ͷ͕֬Ұ൪ߴ͍
➤ ͦΕҎ֎Ͱ֬ΏΔ͔ ʹมԽ
࿈ଓ֦࣌ؒு ➤ T = 100Ͱͷ͚֬ͩԼ͕ Βͣɺ͋ͱ0ʹ͍͍ۙͮͯΔ ➤ ήʔϜΛ࣌ࠁ 0 ~
1 Ͱ࿈ଓత ʹ͢Δ֦ுΛߟ͑Δ ➤ ࿈ଓ֦࣌ؒுͰɺ υϩοϓଆʮ͙͢མͱ͢ʯ νΣοΫଆʮৼΓ͔ͣʯ ͷ֬ͷऩଋ ➤ ͦΕҎ֎Ͱ؇͔ͳ֬ີ ͕ఆ·Δ 1 e
͓ΘΓʹ ➤ ͡ΌΜ͚Μͱಉ͘͡ɺ φογϡۉߧઓུʹରͯ͠ઓུΛมԽͤͯ͞ɺ ϖφϧςΟͷظมԽͤͣɺฏۉҎ্উͭ͜ͱͳ͍ ➤ ͨͩ͠ɺφογϡۉߧΛ಄ʹೖΕ͓ͯ͘͜ͱͰɺ ઌʹࣗͷบΛόϥͣ͞ʹྫྷ੩ʹઓ͏͜ͱͰ͖Δ ➤ ࠓճ͕ࣜ؆୯ͰखܭࢉͰ͖͕ͨɺෳࡶͳ߹ػցͰ࠷దԽ͍ͯ͘͠
߹͋Δ ➤ ໋ΛṌ͚ͯʢṌ͚ͳͯ͘ʣ ϋϯΧνམͱ͠Ͱઓ͏߹ʹ φογϡۉߧͷ͜ͱࢥ͍ग़ͯ͠ཉ͍͠Ͱ͢ɻ