An Introduction to Modeling Knowledge and Awareness in Economics and Game Theory

経済学（特にゲーム理論）における知識と気付きのモデリングに関するサーベイ（2015年11月）
・状態空間モデル（情報構造、情報分割）
・共有知識のゲーム理論的定義
・無知（unawareness）モデルへの拡張

Transcript

1. An Introduction to Modeling Knowledge and Awareness in Economics and Game Theory ࠤʑ໦ɹ߁࿕ ๺཮ઌ୺Պֶٕज़େֶӃେֶɹ஌ࣝՊֶݚڀՊ 2015 ೥ 11 ݄
2. ຊࢿྉͷ໨త Կ͔Λ஌͍ͬͯΔʗ஌Βͳ͍ͱ͍͏͜ͱΛɺͲͷΑ͏ʹܗࣜ తʹදݱͰ͖Δ͔ʁ ʢಛʹҙࢥܾఆͱͷؔ࿈Ͱʣ ܦࡁֶ΍ήʔϜཧ࿦Ͱ༻͍ΒΕ͖ͯͨ஌ࣝͷϞσϧͷ঺հ ʢʴ൷൑తݕ౼ʣ ඪ४తͳ஌ࣝͷϞσϧʹ͓͚Δڧ͍ԾఆΛ؇࿨ͨ͠ɺؾ෇͖ ʢawarenessʣͷϞσϧͷ঺հ
3. ܦࡁֶʹ͓͚Δ஌ࣝͷϞσϧ ܦࡁֶʹ͓͚Δඪ४తͳ஌ࣝͷϞσϧͱͯ͠ɺঢ়ଶۭؒϞσ ϧʢಛʹ৘ใؔ਺ͱηοτͰఆٛ͞ΕΔ৘ใߏ଄ϞσϧʣΛ ঺հ͢Δɻ Hintikka (1962) ʹΑΓཧ࿦తجૅཱ͕֬͞ΕɺAumann (1976) ʹΑΓήʔϜཧ࿦ʹ͓͚Δҙࢥܾఆऀͷ஌ࣝͷ෼ੳʹ ༻͍ΒΕΔΑ͏ʹͳͬͨɻ ʢήʔϜཧ࿦ʹΑΔ෼ੳͰ͸ɺϓ ϨΠϠʔͷ࣋ͭ஌͕ࣝॏཁͳ໾ׂΛ࣋ͭʣ ঢ়ଶۭؒϞσϧʹؔ͢ΔࢀߟจݙɿOsborne and Rubinstein (1994, Ch.5)ʀSamuelson (2004)
4. ৘ใߏ଄ ఆٛ (Ω, P) Λʢ͋Δҙࢥܾఆऀͷʣ৘ใߏ଄ʢinformation structureʣ ͱ͍͏ɻΩ ͸ঢ়ଶʢstateʣͷू߹ɺP : Ω → 2Ω \ {ϕ} ͸৘ใؔ਺ ʢinformation functionʣͰ͋Δɻ Ω ͸ҙࢥܾఆʹؔ࿈͢ΔՄೳͳঢ়ଶΛશؚͯΈɺͦͷཁૉ͸ ޓ͍ʹഉଞతɻ P(ω) ͷղऍɿঢ়ଶ ω ∈ Ω ʹ͓͍ͯɺҙࢥܾఆऀ͕Մೳͱߟ ͑Δঢ়ଶͷू߹ ∗ʢඇۭͳ Ω ͷ෦෼ू߹ʣ ௨ৗɺҙࢥܾఆऀ͸৘ใߏ଄Λ஌͍ͬͯΔ͜ͱΛԾఆɻ *௨ৗɺܦࡁֶͷจݙͰ͸ P(ω) Λʢҙࢥܾఆऀ͕࣋ͭʣ ʮ৘ใʯͱݺͿɻͨͩ ͠ɺҰൠʹ͸ҙࢥܾఆऀͷݶఆ߹ཧੑʢهԱɺਪ࿦ɺೝ஌ͷݶք౳ʣ΋ؚΊͨ Ϟσϧͷղऍ΋ՄೳͰ͋Γɺͦͷ৔߹ʹ͸ P(ω) ͸ʮ৴೦ʯ౳ͱݺ͹ΕΔɻP ͸ ՄೳੑରԠʢpossibility correspondenceʣͱ΋ݺ͹ΕΔɻ
5. ৘ใߏ଄ͷྫ ྫ 1 Ω = {s, c, r} ͱ͢Δɻ ʢs = ੖Εɺc = ಶΓɺr = Ӎʣ ԰֎ʹ͍ΔਓɿP(s) = {s}, P(c) = {c}, P(r) = {r} ૭ͷͳ͍ࣨ಺ʹ͍ΔਓɿP(s) = P(c) = P(r) = {s, c, r} ྫ 2 Ω = [0, 1) ͱ͢Δɻ͋Δਓ͕ɺਅͷ਺஋ͷখ਺఺ҎԼୈ 3 Ґ·Ͱ ؍ଌͰ͖Δͱ͢Δɻখ਺఺ҎԼୈ 3 Ґ·Ͱ͕ಉ͡Ͱ͋Δ͍͔ͳΔ ω, ω′ ∈ Ω ʹ͍ͭͯ΋ɺP(ω) = P(ω′)ɻٯʹɺͦ͏Ͱͳ͍ ω, ω′ ʹ ͍ͭͯ͸ɺP(ω) ̸= P(ω′)ɻ
6. “߹ཧతͳ”৘ใߏ଄ ௨ৗɺҎԼ͕Ծఆ͞ΕΔɻ ʢ߹ཧతͳҙࢥܾఆऀͳΒɺ͜ΕΒͷ ੑ࣭Λຬͨͩ͢Ζ͏ʣ P1ɿશͯͷ ω ∈ Ω ʹ͍ͭͯɺω ∈ P(ω)ʢҙࢥܾఆऀ͸ɺਅ ͷঢ়ଶΛՄೳੑ͔Βআ֎͢Δ͜ͱ͸ͳ͍ʣ P2ɿω′ ∈ P(ω) ⇒ P(ω′) ⊆ P(ω)ʢҙࢥܾఆऀͷਪ࿦ͷ੔߹ ੑɿω′′ ∈ P(ω′) \ P(ω) ͷଘࡏ͸ɺω′ ∈ P(ω) ͱໃ६ʣ∗ P3ɿω′ ∈ P(ω) ⇒ P(ω′) ⊇ P(ω)ʢҙࢥܾఆऀͷਪ࿦ͷ੔߹ ੑɿω′′ ∈ P(ω) \ P(ω′) ͷଘࡏ͸ɺω′ ∈ P(ω) ͱໃ६ʣ *ω′′ ∈ P(ω′) \ P(ω) ͕ଘࡏ͢Δͱ͢Δɻঢ়ଶ ω ʹ͓͍ͯɺω′′ / ∈ P(ω) Ͱ͋Δ ͕ɺω′′ ∈ P(ω′) Ͱ͋Δ͜ͱ͔Βɺҙࢥܾఆऀ͸ʮਅͷঢ়ଶ͸ ω′ Ͱ͸ͳ͍ʯͱ ਪ࿦͢ΔͰ͋Ζ͏ɻ͜Ε͸ ω′ / ∈ P(ω) Λҙຯ͢Δ͕ɺω′ ∈ P(ω) ͱໃ६͢Δɻ P3 ΋ಉ༷ͷϩδοΫɻ
7. ৘ใ෼ׂ ໋୊ ৘ใߏ଄͕෼ׂతͰ͋Δ͜ͱ͸ɺP1ʙP3 Λຬͨ͢͜ͱͱಉ஋Ͱ ͋Δɻͨͩ͠ɺ৘ใߏ଄͕෼ׂతʢpartitionalʣͱ͸ɺશͯͷ ω ∈ Ω ʹؔͯ͠ P(ω) ͕ཁૉͰ͋ΔΑ͏ͳ Ω ͷ෼ׂ͕ଘࡏ͠ɺ͔ ͭ P1 ͕ຬͨ͞ΕΔ͜ͱɻ ෼ׂతͳ৘ใߏ଄Λ৘ใ෼ׂʢinformation partitionʣͱ΋͍͏ ∗ɻ ߹ཧతͳҙࢥܾఆऀͷ৘ใߏ଄ͱͯ͠ɺ௨ৗɺ৘ใ෼ׂ͕Ծఆ͞ ΕΔɻ ྫɿల։ܗήʔϜʹ͓͚Δ৘ใू߹ ʢઌͷྫ 1ɺྫ 2 ΋৘ใ෼ׂʣ *ʮP ͕෼ׂతͰ͋Δʯͱ͍͏͜ͱ΋͋Δɻ·ͨɺΩ ͷ෼ׂΛ৘ใ෼ׂͱݺͿ͜ ͱ΋͋Δɻ
8. ෼ׂతͰͳ͍৘ใߏ଄ͷྫ ྫ 3 Ω = Z ͱ͢Δɻ͋Δਓ͕ɺਅͷ਺஋ n ∈ Z Λڭ͑ΒΕͨͱ͖ɺ n − 1, n, n + 1 ͷ͍ͣΕ͔Ͱ͋ͬͨ͜ͱͷΈΛهԱ͍ͯ͠Δͱ͢ Δɻ͢ͳΘͪɺશͯͷ n ∈ Z ʹ͍ͭͯɺP(n) = {n − 1, n, n + 1} Ͱ͋Δɻ͜ͷ৘ใߏ଄͸ɺP1 Λຬ͕ͨ͢ɺP2 ٴͼ P3 Λຬͨ͞ ͣɺ෼ׂతͰͳ͍ɻ ͜ͷΑ͏ʹ෼ׂతͰͳ͍৘ใߏ଄Λߟྀ͢Δ͜ͱͰɺݶఆ߹ཧత ͳҙࢥܾఆऀ΋͋Δఔ౓͸දݱͰ͖Δɻ ʢRubinstein, 1998ʣ ˞͜ͷྫ͸ɺهԱͷෆ׬શੑ͚ͩͰͳ͘ɺਪ࿦ͷෆ੔߹΋ؚΉɻ
9. ࣄ৅ͱ஌ࣝ ఆٛ E ⊆ Ω Λࣄ৅ʢeventʣͱ͍͏ɻω ∈ Ω ʹ͓͍ͯɺҙࢥܾఆऀ͕ࣄ ৅ E Λ஌͍ͬͯΔʢknowʣ͜ͱΛɺP(ω) ⊆ E Ͱ͋Δ͜ͱͱఆٛ ͢Δɻ P(ω) ⊆ E Ͱ͋Δҙࢥܾఆऀ͸ɺω ʹ͓͍ͯɺE ʹؚ·ΕΔ͍ͣ Ε͔ͷঢ়ଶ͕࣮ݱ͢Δ͜ͱΛ஌͍ͬͯΔɻ ˞ʮ৴͍ͯ͡ΔʯͰ͸ͳ͘ʮ஌͍ͬͯΔʯ ʁˠ P1 Λཁ੥͢ΔݶΓ ʮ஌͍ͬͯΔʯͰ͋Δɻ ʢP1 ͔Β௚઀ಋग़͞ΕΔ K3 Λࢀরʣ ྫ 4 ྫ 1 ʹ͓͍ͯɺE = {s, c} ͱ͢ΔʢE ͸ʮ੖Ε·ͨ͸ಶΓʯ ʣ ɻ s ∈ Ω ʹ͓͍ͯɺ԰֎ʹ͍Δਓ͸ E Λ஌͍ͬͯΔ͕ʢP(s) ⊆ Eʣ ɺ ૭ͷͳ͍ࣨ಺ʹ͍Δਓ͸ E Λ஌Βͳ͍ʢP(s) ̸⊆ Eʣ ɻ
10. ஌ࣝؔ਺ ఆٛ K : 2Ω → 2Ω Λ஌ࣝؔ਺ʢknowledge functionʣͱݺͼɺશͯͷ E ⊆ Ω ʹ͍ͭͯɺK(E) = {ω ∈ Ω|P(ω) ⊆ E} ͱఆٛ͢Δɻ K(E) ͷղऍɿҙࢥܾఆऀ͕ࣄ৅ E Λ஌Δঢ়ଶͷू߹ʢK(E) ࣗ਎ ΋Ұͭͷࣄ৅ʣ ྫ 5 ྫ 1 ʹ͓͍ͯɺE = {s, c} ͱ͢ΔʢE ͸ʮ੖Ε·ͨ͸ಶΓʯ ʣ ɻ԰ ֎ʹ͍Δਓʹͱͬͯ͸ K(E) = {s, c} Ͱ͋Γɺ૭ͷͳ͍ࣨ಺ʹ͍ Δਓʹͱͬͯ͸ K(E) = ϕ Ͱ͋Δɻ
11. ߹ཧతͳҙࢥܾఆऀͷ஌ࣝͷੑ࣭ K1: KΩ = Ω K2: K(E) ∩ K(F) = K(E ∩ F) K3: K(E) ⊆ E K4: K(E) ⊆ KK(E) K5: ¬K(E) ⊆ K¬K(E) ໋୊ʢBacharach, 1985ʣ K1ʙK5 Λຬͨ͢͜ͱ͸ɺ৘ใߏ଄͕෼ׂతͰ͋Δ͜ͱͱಉ஋Ͱ ͋Δɻ K1 ٴͼ K2 ͸ɺ͋ΒΏΔ৘ใؔ਺͕ຬͨ͢ੑ࣭ɻK3, K4, K5 ͸ɺ ͦΕͧΕ P1, P2, P3 ͔Βಋ͔ΕΔɻ
12. ֤ੑ࣭ͷҙຯͱͦͷ൷൑తݕ౼ K1: KΩ = Ω ɹʢશ஌ ominiscienceʣ ҙຯɿৗʹ Ω Λ஌͍ͬͯΔɻ ʢΩ ͷ͍ͣΕ͔ͷঢ়ଶ͕࣮ݱ͢ Δ͜ͱΛ஌͍ͬͯΔʣ ൷൑ɿզʑ͸͍ͭ΋ɺਅͷঢ়ଶΛՄೳੑʹؚΊ͍ͯΔ͔ʁ K2: K(E) ∩ K(F) = K(E ∩ F) ҙຯɿE ͱ F ͷ྆ํΛ஌͍ͬͯΔ͜ͱͱɺE ͔ͭ F Λ஌ͬͯ ͍Δ͜ͱ͸ಉ஋ɻ ൷൑ɿ໰୊ͳͦ͞͏͕ͩɺK2 ΑΓ K2a ͕ಘΒΕΔɻ K2a: E ⊆ F ⇒ K(E) ⊆ K(F) ɹʢlogical ominiscienceʣ ҙຯɿ ʮE ⇒ FʯͳΒɺ ʮE Λ஌͍ͬͯΔ ⇒ F Λ஌͍ͬͯΔʯ ൷൑ɿզʑ͸ɺԿ͔Λ஌͍ͬͯΔͱ͖ɺͦͷ࿦ཧతؼ݁Λશ ͯ஌͍ͬͯΔ͔ʁ
13. ֤ੑ࣭ͷҙຯͱͦͷ൷൑తݕ౼ K3: K(E) ⊆ E ɹʢnondelusionʣ ҙຯɿԿ͔Λ஌͍ͬͯΔͳΒɺͦΕ͸ৗʹਅɻ ൷൑ɿզʑ͕࣋ͭ஌ࣝ͸ɺৗʹਅ͔ʁ K4: K(E) ⊆ KK(E) ɹʢpositive introspectionʣ ҙຯɿԿ͔Λ஌͍ͬͯΔͳΒɺ஌͍ͬͯΔ͜ͱΛ஌͍ͬͯΔɻ ൷൑ɿզʑ͸͍ͭ΋ɺࣗ਎ͷ஌ࣝΛ͍֮ࣗͯ͠Δ͔ʁ K5: ¬K(E) ⊆ K¬K(E) ɹʢnegative introspectionʣ ҙຯɿԿ͔Λ஌Βͳ͍ͳΒɺ஌Βͳ͍͜ͱΛ஌͍ͬͯΔɻ ൷൑ɿզʑ͸͍ͭ΋ɺࣗ਎ͷແ஌Λ͍֮ࣗͯ͠Δ͔ʁ
14. ڞ༗஌ࣝ ͋Δࣄ৅͕ڞ༗஌ࣝͱ͸ɺ ʮશһ͕ͦΕΛ஌͓ͬͯΓɺશһ͕ͦΕ Λ஌͍ͬͯΔ͜ͱΛશһ͕஌͓ͬͯΓɺશһ͕ͦΕΛ஌͍ͬͯΔ ͜ͱΛશһ͕஌͍ͬͯΔ͜ͱΛશһ͕஌͓ͬͯΓʜʢແݶʹଓ ͘ʣ ʯͱ͍͏ঢ়گʢLewis, 1969ʀAumann, 1976ʣ ఆٛ K1, K2 Λҙࢥܾఆऀ 1, 2 ͷ஌ࣝؔ਺ͱ͢Δɻω ∈ Ω ʹ͓͍ͯࣄ৅ E ͕ڞ༗஌ࣝʢcommon knowledgeʣͰ͋Δͱ͸ɺK1(E), K2(E), K1(K2(E)), K2(K1(E))... ͷ͍ͣΕ΋ ω ΛؚΉ͜ͱͰ͋Δɻ ୯ʹ ω ∈ K1(E) ∩ K2(E) ͷ৔߹͸ɺ૬ޓ஌ࣝʢmutual knowledgeʣ ͱ͍͏ɻ ඪ४తͳήʔϜཧ࿦Ͱ͸ɺήʔϜͷߏ଄ͷڞ༗஌ࣝΛԾఆ͢Δɻ
15. ڞ༗஌ࣝ ྫ 6 Ω = {ω1, ω2, ω3, ω4} ͱ͢Δɻҙࢥܾఆऀ 1, 2 ͸ɺͦΕͧΕҎԼͷ Α͏ͳ෼ׂΛੜ੒͢Δ৘ใ෼ׂΛ࣋ͭͱ͢Δɻ P1 = {{ω1}, {ω2, ω3}, {ω4}} P2 = {{ω1}, {ω2}, {ω3, ω4}} ࣄ৅ E = {ω3, ω4} ͸ɺͲͷঢ়ଶʹ͓͍ͯ΋ڞ༗஌ࣝͰͳ͍ɻࣄ৅ F = {ω2, ω3, ω4} ͸ɺω2, ω3, ω4 ∈ Ω ʹ͓͍ͯڞ༗஌ࣝͰ͋Δɻ Ұൠʹɺঢ়ଶ ω ΛؚΉ P1, P2 ͷ finest common coarsening ͷཁૉ ͕ࣄ৅ E ʹؚ·ΕΔ͜ͱͱɺω ʹ͓͍ͯ E ͕ڞ༗஌ࣝͰ͋Δ͜ͱ ͸ಉ஋ɻ ʢAumann, 1976ʣ
16. ৘ใߏ଄ͷڞ༗஌ࣝ ʮ৘ใߏ଄ͷڞ༗஌͕ࣝ҉໧తͳલఏʯ ʢAumann, 1976ʣ ˠڞ༗஌ࣝͷఆٛʹڞ༗஌͕ࣝඞཁʁ Aumannʢ1987, 1999ʣͷओுɿ ৘ใߏ଄͕ڞ༗஌ࣝͰ͋Δ͜ͱ͸ࣗ໌ɻ ʢԾఆͰ͸ͳ͍ʣ ω ∈ Ω ͸ɺੈքͷ׬શͳهड़Ͱ͋Γɺଞऀͷ৘ใߏ଄΋ؚ Ήɻ΋͠ ω ʹ͓͍ͯɺଞऀͷ৘ใߏ଄ʹؔ͢Δෆ࣮֬ੑ͕͋ ΔͳΒɺω ͸ੈքͷهड़ͱͯ͠׬શͰ͸ͳ͍ɻω Λ͞Βʹ෼ ׂ͠ɺΩ Λ֦ு͠ͳ͚Ε͹ͳΒͳ͍ɻ ͜ͷૢ࡞Λे෼ʹ܁Γฦ͠ɺͲͷঢ়ଶ΋৘ใߏ଄ʹؔ͢Δෆ ࣮֬ੑΛؚ·ͳ͍ Ω′ ΛߏஙͰ͖ΔɻΩ′ ্ͷ৘ใؔ਺͸ɺڞ ༗஌ࣝͰ͋Δɻ
17. ৴೦ɿࣄલͱࣄޙ Ω ্ͷ֬཰෼෍ ρ Λߟ͑ɺҙࢥܾఆऀͷࣄલʢʹ৘ใऔಘલʣͷ ৴೦ʢprior beliefʣΛදݱ͍ͯ͠Δͱ͢Δɻ ΋͠ ρ(P(ω)) > 0 ͳΒɺঢ়ଶ ω ʹ͓͚Δҙࢥܾఆऀͷࣄޙͷ৴೦ ʢposterior beliefʣρω ͸ɺҎԼͰ༩͑ΒΕΔɻ ʢϕΠζϧʔϧʹΑ Δ৴೦ߋ৽ʣ ρω(ω′) = { ρ(ω′)/ρ(P(ω)) if ω′ ∈ P(ω) 0 otherwise
18. “Agreeing to disagree”ͷෆՄೳੑ ໋୊ʢAumann, 1976ʣ ༗ݶͷ Ω ্ʹ౳͍͠ࣄલͷ৴೦Λ࣋ͭೋਓͷҙࢥܾఆऀΛߟ͑ Δɻ྆ऀͷ৘ใߏ଄͕෼ׂతͰ͋Γɺ͔ͭ͋Δࣄ৅ E ʹର͢Δ྆ ऀͷࣄޙͷ৴೦͕ڞ༗஌ࣝͳΒ͹ɺ͜ΕΒͷࣄޙͷ৴೦͸౳͍͠ɻ ߹ཧతҙࢥܾఆऀͲ͏͠͸ɺҙݟͷෆҰகʹ߹ҙʢagree to disagreeʣ͠ಘͳ͍ɻ ˞ K5 Λܽ͘ඇ෼ׂతͳ৘ใߏ଄Ͱ΋੒ΓཱͭʢSamet, 1990ʣ ɻ ڞ༗஌ࣝͰ͋Γޓ͍ʹ౳͍͠৴೦͸ common prior ͱݺ͹Εɺ৘ ใෆ׬උήʔϜͷ෼ੳ౳ͰҰൠʹԾఆ͞ΕΔɻ ʢAumann ʹΑΕ ͹ɺࣄલͷ৴೦͕ڞ༗஌ࣝͰ͋Δ͜ͱ΋ࣗ໌ʣ
19. “Agreeing to disagree”ͷෆՄೳੑ ྫ 7 ྫ 6 ͷ৘ใ෼ׂʹ͓͍ͯɺҙࢥܾఆऀ 1, 2 ͷࣄલͷ৴೦͸ͱ΋ʹ Ұ༷෼෍ͱ͢Δɻ ʢશͯͷ ω ʹ͍ͭͯ ρ(ω) = 1/4ʣ E = {ω2, ω3} ʹ͍ͭͯɺω3 ∈ Ω ʹ͓͍ͯɺ྆ऀͷ E ʹର͢ Δࣄޙͷ৴೦͸ͦΕͧΕ 1, 1/2 Ͱ͋Δ͕ɺ͜Ε͸ڞ༗஌ࣝͰ ͸ͳ͍ɻ ʢ͜ͷ৔߹ɺޓ͍ʹ૬खͷࣄޙͷ৴೦Λ஌Βͳ͍ʣ F = {ω2, ω3, ω4} ʹ͍ͭͯɺω3 ∈ Ω ʹ͓͍ͯɺ྆ऀͷ F ʹର ͢Δࣄޙͷ৴೦͸ڞ༗஌ࣝͰ͋Γɺͱ΋ʹ 1ɻ ΋ͬͱ௚ײతͳʢʁʣྫɿࢲͱ͋ͳͨͷڊਓͷ༏উ֬཰ͷݟੵ΋ Γ͕ͦΕͧΕ 2/3, 1/3 Ͱ͋ΔͳΒɺͦΕ͸ڞ༗஌ࣝͰͳ͍͔ɺڞ ༗஌ࣝͰ͋ΔͳΒࣄલͷ৴೦͕ҟͳ͍ͬͯΔɻ
20. E ϝʔϧήʔϜʢRubinstein, 1989ʣ ҙࢥܾఆऀ 1, 2 ͕ɺ֬཰ p > 1/2 Ͱ GAɺ1 − p Ͱ GB ΛϓϨ Π͢Δɻ1 ͸ਅͷήʔϜΛࣄલʹ஌Δ͜ͱ͕Ͱ͖Δ͕ɺ2 ͸ ஌Βͳ͍ɻ ʢҎ্ͷ͜ͱ͸ڞ༗஌ࣝʣ GA ͳΒ (A, A)ɺGB ͳΒ (B, B) ͕๬·͍͠ɻ͔͠͠ɺ2 ͕ೋ ͭͷήʔϜΛ۠ผͰ͖ͳ͍Ҏ্ɺήʔϜʹԠͨ͡ҟͳΔڠௐ ͸ෆՄೳɻ A B A 10,  10 0,  -­‐20 B -­‐20,  0 0,  0 1 2 GA A B A 0,  0 0,  -­‐20 B -­‐20,  0 10,  10 1 2 GB
21. E ϝʔϧήʔϜ ҎԼͷΑ͏ͳίϛϡχέʔγϣϯγεςϜΛಋೖͨ͠ɻ 1 ͸ɺήʔϜ͕ GB Ͱ͋Δ৔߹ͷΈɺ2 ʹϝʔϧΛૹ৴ 2 ͸ɺϝʔϧΛड৴ͨ͠৔߹ͷΈɺ1 ʹ֬ೝϝʔϧΛૹ৴ 1 ͸ɺϝʔϧΛड৴ͨ͠৔߹ͷΈɺ2 ʹ֬ೝϝʔϧΛૹ৴ 2 ͸ɺ· · · ʢҎԼಉ༷ʣ ϝʔϧ͸֬཰ ϵ < 1/2 Ͱಧ͔ͳ͍ɻͲͪΒ͔ͷϝʔϧ͕ಧ͔ͳ ͔ͬͨ࣌఺Ͱ΍ΓͱΓ͸ऴྃ͠ɺήʔϜͰͷબ୒Λߦ͏ɻ Ω = {GA, (GB, 0), (GB, 1), (GB, 2), (GB, 3), . . . } ͨͩ͠ (GB, n) ͸ɺήʔϜ͕ GB Ͱɺૹ৴ʹܭ n ճ੒ޭͨ͠ঢ়ଶ P1 = {{GA}, {(GB, 0), (GB, 1)}, {(GB, 2), (GB, 3)}, . . . } P2 = {{GA, (GB, 0)}, {(GB, 1), (GB, 2)}, {(GB, 3), (GB, 4)}, . . . } ҙࢥܾఆऀ i ͸ɺPi (ω) ʹԠͯ͡ A ·ͨ͸ B ͷબ୒Λߦ͏ɻ
22. E ϝʔϧήʔϜ ήʔϜ͕ GB ͷ৔߹ɺϝʔϧͷૹड৴Λ௨ͯ͡ɺ ʢϵ ͕খ͍͞΄Ͳʣ GB ͸ڞ༗஌ࣝʹۙͮ͘ɻ ʢͨͩ͠ɺ׬શͳڞ༗஌ࣝʹ͸ͳΒͳ͍ʣ ˠ͜ͷγεςϜ͸ɺήʔϜʹԠͨ͡ڠௐΛՄೳʹ͢Δ͔ʁ ʢ௚ײ తʹ͸ɺՄೳͳΑ͏ʹࢥ͑Δʣ ໋୊ʢRubinstein, 1989ʣ E ϝʔϧήʔϜʹ͓͚Δ།ҰͷφογϡۉߧͰ͸ɺҙࢥܾఆऀ 1, 2 ͱ΋ɺৗʹ A Λબ୒͢Δɻ ͨͱ͑ਅͷήʔϜ͕ GB Ͱɺ྆ऀ͕ͦͷ͜ͱΛ஌͍ͬͯͯ΋ʢ“΄ ΅”ڞ༗஌ࣝͰ΋ʣ ɺ(B, B) Ͱڠௐ͢Δ͜ͱ͸ͳ͍ɻ
23. ஌ࣝͷϞσϧ͔Βؾ෇͖ͷϞσϧ΁ ͜Ε·Ͱݟ͖ͯͨඪ४తͳϞσϧͰ͸ɺҙࢥܾఆऀ͕ৗʹࣗ ෼͕ԿΛ஌Βͳ͍͔Λ஌͍ͬͯΔ͜ͱ͕Ծఆ͞ΕΔɻ ʢK5ʣ ͜ͷͨΊɺݱ࣮తͳਓؒͷೝࣝͱͦΕʹجͮ͘ߦಈΛ͏·͘ આ໌Ͱ͖ͳ͍ɻ ඪ४తϞσϧͷݶքΛࢦఠ͠ɺࠀ෰͢ΔࢼΈ ͕ 1990 ೥୅ΑΓ੝ΜʹߦΘΕΔΑ͏ʹͳͬͨɻ Կ͔ʹؾ෇͍͍ͯΔ͜ͱͱɺͦͷ൓ରͷؾ෇͍͍ͯͳ͍͜ͱ ʢunawareness, ຊࢿྉͰ͸ʮແ஌ʯͱݺͿʣΛϞσϧԽ͢Δ ۙ೥ͷࢼΈΛ঺հ͢Δɻ
24. ஌Βͳ͍͜ͱΛ஌͍ͬͯΔʁ K5ɿҙࢥܾఆऀ͕ৗʹࣗ෼͕ԿΛ஌Βͳ͍͔Λ֮ࣗɻ ˠ৽ͨͳ৘ใͷऔಘ͸ɺ ʮڻ͖ʯʹͳΓಘͳ͍ʢͦͷΑ͏ͳ৘ใ ͷଘࡏՄೳੑ͸ॳΊ͔Β஌͍ͬͯͨʣ ɻϕΠζϧʔϧʹΑΔ৴೦ ͷߋ৽Λಋ͚ͩ͘ɻ ݱ࣮ͷਓؒ͸ɺ͍ͭ΋ͦ͏͔ʁ χϡʔτϯ͸ɺ૬ରੑཧ࿦Λ஌Βͳ͍͜ͱΛ஌͍ͬͯͨʁ ʮੈͷதʹ͸ɺ஌Βͳ͍͜ͱΛ஌Βͳ͍͜ͱʢunknown unknownʣ͕͋Δʯ ʢϥϜζϑΣϧυݩถࠃ๷௕׭ʣ Πϊϕʔγϣϯͷछ͸ɺ͍ͭ΋ॳΊ͔Β෼͔͍ͬͯΔʁ ஌͍ͬͯΔʗ஌Βͳ͍Λ൑அͰ͖ͳ͍͚ͩͰͳ͘ɺͦ΋ͦ΋ଘࡏ ࣗମΛೝ͍ࣝͯ͠ͳ͍Α͏ͳࣄ৅͕͋ΔͩΖ͏ɻ ˠͲͷΑ͏ʹϞσϧԽͰ͖Δ͔ʁ
25. ඇ෼ׂతͳ৘ใߏ଄ K5 Λຬͨ͞ͳ͍ඇ෼ׂతͳ৘ใߏ଄Λߟ͑Ε͹Α͍ʁ ྫ 8 Ω = {a, b} ʹؔͯ͠ɺP(a) = {a}, P(b) = {a, b} ͱ͢Δɻ E = {a} ͱ͢Δͱɺ¬K(E) = {b} ̸⊆ K¬K(E) = ϕ Ͱ͋Γɺ K5 ͸ຬͨ͞Εͳ͍ɻ ʢঢ়ଶ b ʹ͓͍ͯɺࣄ৅ E Λ஌Βͳ͍͜ ͱΛ஌Βͳ͍ʣ ͔͠͠ɺa ∈ P(b) Ͱ͋Δɻ ʢঢ়ଶ b ʹ͓͍ͯɺࣄ৅ E ͷՄೳ ੑ͸ೝ͍ࣝͯ͠Δʣ ˠઌͷ໰୊Λ͏·͘ѻ͑ͳ͍ɻ
26. ແ஌ؔ਺ ࣄ৅ͷଘࡏࣗମΛೝ͍ࣝͯ͠ͳ͍͜ͱΛແ஌ʢunawarenessʣͱݺ Ϳ ∗ɻ ʢ͜Ε·Ͱͷจ຺Ͱͷʮ஌Βͳ͍ʯͱ͸ҟͳΔʣ ఆٛ U : 2Ω → 2Ω Λແ஌ؔ਺ʢunawareness functionʣͱݺͿɻ U(E) ͷղऍɿҙࢥܾఆऀ͕ࣄ৅ E ʹແ஌Ͱ͋Δঢ়ଶͷू߹ ແ஌ؔ਺Λద੾ʹ༻͍Δ͜ͱͰɺҙࢥܾఆऀͷແ஌ΛදݱͰ͖Δ ͷͰ͸ͳ͍͔ʁ *unawareness ͸ʮ஌Βͳ͍ʯͱ͍͏ΑΓʮؾ෇͍͍ͯͳ͍ʯͰ͋Γɺ ʮແ஌ʯͱ ͍͏༁ޠ͕࠷ద͔͸ఆ͔Ͱ͸ͳ͍ɻ ʢগͳ͘ͱ΋ܦࡁֶͰ͜ͷ༁ޠ͕ఆண͍ͯ͠ ΔΘ͚Ͱ͸ͳ͍ʣ
27. ແ஌ͷੑ࣭ ແ஌ؔ਺ͷຬͨ͢΂͖ੑ࣭ͱͯ͠ɺҎԼͷ 3 ͭΛߟ͑Δɻ U1: U(E) ⊆ ¬K(E) ∩ ¬K¬K(E) ɹʢplausibilityʣ Կ͔ʹແ஌Ͱ͋ΔͳΒɺͦͷԿ͔Λ஌Βͳ͍ɺ͔ͭ஌Βͳ͍ ͜ͱΛ஌Βͳ͍ɻ U2: KU(E) = ϕ ɹʢKU introspectionʣ Կ͔ʹແ஌Ͱ͋ΔͳΒɺͦͷແ஌Λ஌Δ͜ͱ͸ͳ͍ɻ U3: U(E) ⊆ UU(E) ɹʢAU introspectionʣ Կ͔ʹແ஌Ͱ͋ΔͳΒɺͦͷແ஌ʹ͍ͭͯແ஌Ͱ͋Δɻ ˞ U2 ͱ U3 Ͱ͸ɺࣄ৅ E Λಛఆ͍ͯ͠Δ͜ͱʹ஫ҙɻ͍ΘΏΔ ʮແ஌ͷ஌ʯ ʢʹର৅͸ಛఆͰ͖ͳ͍͕ɺԿΒ͔ͷࣄฑʹ͍ͭͯແ ஌Ͱ͋Δ͜ͱΛ֮ࣗʣ͕ෆՄೳͩͱݴ͍ͬͯΔΘ͚Ͱ͸ͳ͍ɻ
28. ඪ४తͳঢ়ଶۭؒϞσϧʹΑΔແ஌දݱͷෆՄೳੑ ஌ࣝؔ਺ K Λ৘ใؔ਺ P ͔Βఆٛ͢ΔͷͰ͸ͳ͘ɺΑΓҰൠʹ K(E) ͕ࣄ৅ E Λ஌Δঢ়ଶͷू߹Ͱ͋Δؔ਺ͱͯ͠ѻ͏ɻ ໋୊ʢDekel et al., 1998ʣ (Ω, K, U) ʹ͓͍ͯɺK ͕ K1 Λຬͨ͠ɺU ͕ U1ʙU3 Λຬͨ͢ͳ Βɺશͯͷࣄ৅ E ʹ͍ͭͯ U(E) = ϕɻ K1 Λཁ੥͢ΔݶΓɺ͍͔ͳΔແ஌΋ѻ͑ͳ͍ɻ ˠ͜Ε͸ɺ֓೦తʹ͸ࣗ໌ɿҙࢥܾఆऀ͕ Ω Λهड़Ͱ͖Δ͜ͱͱɺ ແ஌Ͱ͋Δ ω ∈ Ω ͕ଘࡏ͢Δ͜ͱ͸ɺͦ΋ͦ΋ཱ྆͠ಘͳ͍ɻ ҙࢥܾఆऀͷແ஌Λදݱ͢Δʹ͸ɺҙࢥܾఆऀ͕ Ω Λ஌͍ͬͯΔ ͱ͍͏ɺඪ४తͳঢ়ଶۭؒϞσϧͷࠜຊతͳԾఆΛ์غ͢Δඞཁ ͕͋Δɻ
29. ແ஌දݱΛՄೳʹ͢Δঢ়ଶۭؒϞσϧͷҰൠԽ Heifetz et al.ʢ2006ʣͷແ஌ߏ଄ʢunawareness structureʣ ఆٛ ׬උଋ (S, ⪰) ΛҰൠԽঢ়ଶۭؒϞσϧʢgeneralized state-space modelʣͱݺͿɻ S = {Ωα}α∈A ͸ঢ়ଶۭؒͷू߹ɻΣ = ∪ α∈A Ωα ͱ͢Δɻ ⪰ ͸ S ্ͷ൒ॱংɻΩ′ ⪰ Ω ͸ɺΩ′ ͕ Ω ΑΓ΋දݱྗ͕ߴ͍ ʢexpressiveʣ͜ͱΛࣔ͢ɻ Ω′ ⪰ Ω Ͱ͋Δ৔߹ɺrΩ′ Ω : Ω′ → Ω ͳΔࣹӨΛఆٛ͢Δɻ͜Ε ͸ɺΩ′ ͷ֤ঢ়ଶ͕ Ω ͷͲͷঢ়ଶʹରԠ͢Δ͔Λࣔ͢ɻ P : Σ → 2Σ \ {ϕ} ΛՄೳੑରԠʢpossibility correspondenceʣ ͱݺͿɻ ʢP(ω) ͷղऍ͸৘ใؔ਺ͱಉ͡ʣ ˞Ϟσϧͱͯ͠ҙຯΛ੒ͨ͢Ίʹ͍͔ͭ͘ͷ৚͕݅ඞཁ͕ͩɺ͜ ͜Ͱ͸লུ
30. ౤ػతऔҾʢspeculative tradeʣ ͋Δاۀͷॴ༗ऀʢOʣͱɺͦͷങऩΛݕ౼͍ͯ͠Δങ͍ख ʢBʣ͕͍Δɻ ॴ༗ऀ͸͜ͷاۀͷૌুϦεΫʹؾ෇͍͍ͯΔ͕ɺ࣮ࡍʹૌ ু͕ى͜Δ͔ʢlʣى͜Βͳ͍͔ʢ¬lʣ͸஌Βͳ͍ɻҰํɺങ ͍ख͸ૌুϦεΫʹແ஌Ͱ͋Γɺॴ༗ऀ͸ͦΕΛ஌͍ͬͯΔɻ ങ͍ख͸͜ͷاۀͷΠϊϕʔγϣϯͷՄೳੑʹؾ෇͍͍ͯΔ ͕ɺ࣮ࡍʹΠϊϕʔγϣϯ͕ى͜Δ͔ʢnʣى͜Βͳ͍͔ ʢ¬nʣ͸஌Βͳ͍ɻҰํɺॴ༗ऀ͸ΠϊϕʔγϣϯͷՄೳੑ ʹແ஌Ͱ͋Γɺങ͍ख͸ͦΕΛ஌͍ͬͯΔɻ
31. ౤ػతऔҾͷϞσϧԽ Schipper (2014) (based on Heifetz et al., 2006)
32. ౤ػతऔҾͷϞσϧԽ S = {S{n,l} , S{n} , S{l} , Sϕ} S{n,l} Ͱ͸ૌুϦεΫͱΠϊϕʔγϣϯͷ྆ํͷෆ࣮֬ੑ͕ هड़͞Ε͍ͯΔɻS{l} Ͱ͸લऀ͔͠هड़͞Ε͍ͯͳ͍ɻ S{n,l} ⪰ S{n} ⪰ Sϕ ͔ͭ S{n,l} ⪰ S{l} ⪰ Sϕ ਤͷάϨʔ఺ઢ͸ɺࣹӨΛද͢ɻ rS{n,l} S{n} (n, l) = rS{n,l} S{n} (¬n, l) = nʢS{n,l} ͷ n, l ٴͼ ¬n, l ͸ɺ S{n} Ͱ͸ͱ΋ʹ n ͱೝࣝ͞ΕΔʣ ਤͷ੨ͱ੺ͷ໼ҹٴͼ࿮͸ɺͦΕͧΕॴ༗ऀͱങ͍खͷՄೳ ੑରԠΛද͢ɻ શͯͷ ω ∈ S{n,l} ʹ͍ͭͯɺPO (ω) = S{l} ʢॴ༗ऀͷೝࣝʣ શͯͷ ω ∈ S{l} ʹ͍ͭͯɺPB (ω) = Sϕ ʢങ͍खͷೝࣝʹؔ͢ Δॴ༗ऀͷೝࣝʣ
33. ౤ػతऔҾ ∗ ͸ى͜Δ͔ʁ ݱঢ়ɺاۀͷגՁ͸Ұג 100 ԁͰ͋Δɻૌু͕ى͖ͨ৔߹ʹ͸ 10 ԁԼ͕ΓɺΠϊϕʔγϣϯ͕ى͖ͨ৔߹ʹ͸ 10 ԁ্͕Δ͢Δɻ Q. ങ͍खͷҰג 100 ԁͰͷങऩҊʹɺॴ༗ऀ͸Ԡ͡Δ͔ʁ A. Yes. ॴ༗ऀʹͱͬͯגՁͷظ଴஋͸ 90ʙ100 ԁɻങ͍खʹͱͬͯ ͸ 100ʙ110 ԁɻ ങऩҊ͸ɺ૒ํͷࢹ఺͔Βݟͯ߹ཧతɻ ʢޓ͍ʹɺ૬खͷג ՁͷධՁ஋͸ 100 ԁͩͱࢥ͍ͬͯΔʣ *౤ػతऔҾʜ৘ใͷࠩҟͷΈʹΑΓੜ͡ΔऔҾɻඪ४తͳ৘ใ෼ׂϞσϧͰ ͸ɺ͋Δࡒʹؔ͢Δങ͍खͷධՁ஋͕ചΓखͷධՁ஋Λ্ճΔ͜ͱ͕ڞ༗஌ࣝ ʹͳΔ͜ͱ͸ͳ͘ɺ౤ػతऔҾ͸ੜ͡ͳ͍ɻ ʢMilgrom and Stokey, 1982ʣ
34. ؾ෇͖ͷϞσϧԽ ແ஌ߏ଄ʹ͓͍֦ͯுతʹఆٛ͞ΕΔແ஌ؔ਺ U ͸ɺU1ʙU3 ʢʴͦͷଞͷؾ෇͖ʹؔ͢Δ͍͔ͭ͘ͷެཧʣΛຬͨ͢ɻ ˠैདྷϞσϧͷʮ஌͍ͬͯΔʗ஌Βͳ͍ʯʹՃ͑ͯɺ ʮؾ෇͍ͯ ͍Δʗؾ෇͍͍ͯͳ͍ʯͷදݱ͕Մೳʹɻ ҙࢥܾఆऀͷແ஌Λߟྀͨ͠ήʔϜཧ࿦Ϟσϧʢgames with unawarenessʣ΁ͷల։ e.g. Heifetz et al. (2013), Halpern and Rego (2014ʣ ແ஌ߏ଄Ͱ͸ɺ ʮແ஌ͷ஌ʯ ʢawareness of unawarenessʣΛදݱͰ ͖ͳ͍ɻˠࠓޙͷݚڀ՝୊
35. ࢀߟจݙ R. J. Aumann. Agreeing to disagree. The Annals of Statistics, 4(6):1236–1239, 1976. R. J. Aumann. Correlated equilibrium as an expansion of Bayesian rationality. Econometrica, 55(1):1–18, 1987. R. J. Aumann. Interactive epistemology I: Knowledge. International Journal of Game Theory, 28:263–300, 1999. M. Bacharach. Some extensions of a claim of Aumann in an axiomatic model of knowledge. Journal of Economic Theory, 37(1):167–190, 1985. E. Dekel, B. L. Lipman, and A. Rustichini. Standard state-space models preclude unawareness. Econometrica, 66(1):159–173, 1998. J. Y. Halpern and L. C. Rego. Extensive games with possibly unaware players. Mathematical Social Sciences, 70:42-58, 2014. A. Heifetz, M. Meier, and B. C. Schipper. Interactive unawareness. Journal of Economic Theory, 130(1):78–94, 2006. A. Heifetz, M. Meier, and B. C. Schipper. Dynamic unawareness and rationalizable behavior. Games and Economic Behavior, 81:50-68, 2013. J. Hintikka. Knowledge and Belief. Cornell University Press, Ithaca, 1962.
36. ࢀߟจݙ D. Lewis. Conventions: A Philosophical Study. Harvard University Press, Cambridge, 1969. P. Milgrom and N. Stokey. Information, trade and common knowledge. Journal of Economic Theory, 26(1):17-27, 1982 M. J. Osborne and A. Rubinstein. A Course in Game Theory. MIT Press, Cambridge, 1994. A. Rubinstein. The electronic mail games, American Economic Review, 79: 385–391, 1989. A. Rubinstein. Modeling Bounded Rationality. MIT Press, Cambridge, 1998. D. Samet. Ignoring ignorance and agreeing to disagree. Journal of Economic Theory, 52:190–207, 1990. L. Samuelson. Modeling knowledge in economic analysis. Journal of Economic Literature, 42(2):367–403, 2004. B. C. Schipper. Unawareness–a gentle introduction to both the literature and the special issue. Mathematical Social Sciences, 70:1-9, 2014