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ハンカチ落としの戦略最適化

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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。

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Katsuki Ohto

July 07, 2020
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  1. ϋϯΧνམͱ͠ ➤ Ұൠతʹ͸…
 தఔ౓ͷਓ਺Ͱߦ͏
 ϨΫϦΤʔγϣϯήʔϜ ➤ ອըʮӕ͍ᷰʯಡऀʹ͸…
 ೋਓͰ໋ΛṌ͚ͯઓ͏ήʔϜ ϋϯΧνΛམͱ͢ํΛʮυϩοϓଆʯৼΓ޲͍ͯϋϯΧν͕མ ͪͨͷΛ֬ೝ͢ΔํΛʮνΣοΫଆʯͱ͠ɺަޓʹ୲౰͍ͯ͠

    ͘ɻਐߦʹ͸࣌ใΛར༻͢Δɻ1λʔϯ͸1෼ؒͱ͠ɺυϩοϓ ଆ͸ඞͣͦͷ࣌ؒ಺ʹམͱ͞ͳ͚Ε͹ͳΒͣɺνΣοΫଆ͸ඞͣ Ұ౓͚ͩৼΓ޲͔ͳ͚Ε͹ͳΒͳ͍ɻυϩοϓଆ͕ϋϯΧνΛ མͱͨ͠ॠ͔ؒΒνΣοΫଆͷʮ࠲ࢹͷࡍʯ͕ελʔτɺ༰ث ͷத֖͕։͖γϦϯμʔʹʮྟࢮༀʯ͕ཷ·͍ͬͯ͘ɻৼΓ޲͍ ͯϋϯΧν͕མ͍ͪͯΕ͹νΣοΫ੒ޭͱͳΓத֖͕ด͡Δ͕ ٯʹམ͍ͪͯͳ͚Ε͹ʮϖφϧςΟʯ͕ൃಈɻʮͦͷ࣌఺Ͱཷ ·͍ͬͯΔྟࢮༀʴ1෼ؒͰཷ·ΔྔʯΛͦͷ৔Ͱ஫ࣹʢγϦϯ μʔʹྟࢮༀ͕5෼ؒ෼ཷ·ͬͨ৔߹΋ଈ஫ࣹʣ͞ΕΔɻͦͯ͠ ຺ഥ͕ࢭ·ͬͨͷΛ֬ೝ͔ͯ͠Βʮྟࢮʯ͕ελʔτɻϖφϧ ςΟඵ਺ΛফԽͨ͠ΒྟࢮༀͱରʹͳΔ੒෼ͷʮોੜༀʯΛ஫ ্ࣹͨ͠Ͱ15ճݶఆͷ৺ଁϚοαʔδʢ໿10ඵؒ΄ͲʣΛߦ ͍ɺͦΕͰોੜͰ͖ͨΒήʔϜଓߦɺͰ͖ͳ͚Ε͹উෛ͋Γɻ Wikipedia ʮӕ͍ᷰʯΑΓ
  2. ؆ུԽϧʔϧ ➤ υϩοϓଆ
 …ϋϯΧνΛམͱ͢ଆ ➤ νΣοΫଆ
 …ৼΓฦΔଆ ➤ ࣌ࠁ T

    ඵؒ ➤ ϖφϧςΟʢ㲈ใुʣ
 ɾνΣοΫଆ͕མͱ͞Εͯ
 ɹ t ඵޙʹৼΓฦΔ… t
 ɾνΣοΫଆ͕ઌʹৼΓฦΔ
 ɹ… T ➤ ອըͰ͸ɺ͜ΕΛ෦෼ήʔϜͱͯ͠
 ܁Γฦ͢ཁૉ͕͋Δ͕ɺ
 ࠓճ͸͜ͷҰճͷήʔϜ͚ͩߟ͑Δ
  3. ໨తؔ਺ ➤ ࣌ࠁ T+1 ·ͰԆ௕͠ɺνΣοΫଆɺυϩοϓଆ͕Կ΋͠ͳ͔ͬͨ৔߹ ͸࣌ࠁ T+1 ʹߦಈΛىͨ͜͜͠ͱʹ͢Δʢ݁ՌʹӨڹ͸ͳ͍ʣ ➤ 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
  4. ߆ଋ৚͖݅ͭͷఀཹ఺໰୊ ➤ ϖφϧςΟͷظ଴஋
 ➤ φογϡۉߧͰ͸ɺνΣοΫଆɺυϩοϓଆ
 ͱ΋ʹઓུΛมߋ͢ΔΠϯηϯςΟϒ͕ͳ͍ ➤ →֬཰ͷ࿨͕ 1ͱ͍͏
 ɹ߆ଋ৚͖݅ͭͷఀཹ఺໰୊

    ➤ →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
  5. ϥάϥϯδϡͷະఆ৐਺๏ ➤ ཱࣜ
 
 ➤ ల։
 
 ➤ ղऍ
 ͜ͷࣜ͸

    ͔ͭ ͱ΋ॻ͚Δ 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 .
  6. νΣοΫଆͷ઴Խࣜಋग़ ➤ 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
  7. υϩοϓଆͷ઴Խࣜಋग़ ➤ 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
  8. ࿈ଓ֦࣌ؒு ➤ T = 100Ͱ͸୺ͷ֬཰͚ͩԼ͕ Βͣɺ͋ͱ͸0ʹ͍͍ۙͮͯΔ ➤ ήʔϜΛ࣌ࠁ 0 ~

    1 Ͱ࿈ଓత ʹ͢Δ֦ுΛߟ͑Δ ➤ ࿈ଓ֦࣌ؒுͰ͸ɺ
 υϩοϓଆʮ͙͢མͱ͢ʯ
 νΣοΫଆʮৼΓ޲͔ͣʯ
 ͷ֬཰ͷऩଋ஋͸ ➤ ͦΕҎ֎Ͱ͸؇΍͔ͳ֬཰ີ ౓͕ఆ·Δ 1 e