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ゲーム理論BASIC 第20回 -無限回繰り返しゲーム-
This document discusses infinitely repeated games. It defines infinitely repeated games and their properties. It also covers trigger strategies, where players adopt punishment strategies if another player deviates from cooperation. Discount factors are introduced to model players valuing future payoffs less than current payoffs. The document discusses how trigger strategies can support cooperative outcomes as subgame perfect equilibria in infinitely repeated games using discount factors.
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ゲーム理論BASIC 第20回 -無限回繰り返しゲーム-
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- 5. ແݶճ܁Γฦ͠ήʔϜͷఆٛ ήʔϜ ϓϨΠϠʔͷू߹ ϓϨΠϠʔͷߦಈͷू߹ ϓϨΠϠʔͷརಘؔ ͱͯ͠ ܁Γฦ͠ήʔϜʹ͓͚Δઓུ ͱͯ͠ ͱ͠ ͱ͢Δ ͜͜Ͱ Λ ճ·Ͱͷཤྺͱ͍͏ ϓϨΠϠʔͷճͷߦಈ ཤྺ Λͱʹܾఆ͞ΕΔ ͢ͳΘͪ Ͱ༩͑ΒΕΔ ͜ΕΑΓ܁Γฦ͠ήʔϜʹ͓͚ΔϓϨΠϠʔͷ७ਮઓུ Ͱ༩͑ΒΕΔ ϓϨΠϠʔͷ७ਮઓུͷू߹Λ ͱ͠ ϓϨΠϠʔશମͷઓུͷͷू߹Λ Ͱද͢ G = (N, {Si }i∈N, {fi }i∈N) N Si i fi i S = S1 × ⋯ × Sn fi : S → ℜ S = S1 × ⋯ × Sn St−1 = t−1 S × ⋯ × S S0 = {∅} ht−1 = (s1, ⋯, st−1) ∈ St−1 t − 1 i t ht−1 xi t : St−1 → Si i xi = (xi t)∞ t=1 i X∞i X∞ = X∞1 × ⋯ × X∞n T → ∞
- 6. ແݶճ܁Γฦ͠ήʔϜͷఆٛ ܁Γฦ͠ήʔϜʹ͓͚Δརಘ ׂҾҼࢠ ͱͯ͠ ΛׂҾརಘͱ͍͏ Λਖ਼نԽརಘͱ͍͏ Ҏ্ΑΓ ແݶճ܁Γฦ͠ήʔϜΛ Ͱఆٛ͢Δ δ, 0 δ 1 ∞ ∑ t=1 δt−1 fi (st(x)) f∞i δ (x) = (1 − δ) ∞ ∑ t=1 δt−1 fi (st(x)) G∞ (δ) = (N, {X∞i }i∈N, {f∞i δ }i∈N) ແݶൺڃ ∞ ∑ t=1 δt−1 = 1 1 − δ
- 7. φογϡߧۉͷఆٛ ແݶճ܁Γฦ͠ήʔϜʹ͓͚Δφογϡߧۉ ʹ͓͍ͯ ઓུͷ ͕φογϡ͋ͰߧۉΔͱ ͯ͢ͷϓϨΠϠʔʹ͍ͭͯ G∞ (δ) = (N, {X∞i }i∈N, {f∞i δ }i∈N) x* = (x1* , ⋯, xn* ) i f∞i δ (xi* , x−i* ) ≥ f∞i δ (xi , x−i* ), ∀xi ∈ X∞i
- 8. ϑΥʔΫఆཧ ϛχϚοΫεߦಈͱϛχϚοΫεརಘ ήʔϜ ʹ͓͍ͯ Λຬͨ͢ϓϨΠϠʔҎ֎ͷߦಈͷΛʹର͢ΔϛχϚοΫεߦಈͱ͍͏ ͜ͷ ΛϛχϚοΫεརಘΛ͍͍ Ͱද͢ ΛϛχϚοΫεͱ͍͏ G max si fi (si , ̂ s−i ) = min s−i max si fi (si , s−i ) i ̂ s−i i max si fi (si , ̂ s−i ) vi v = (v1 , ⋯, vn ) ̍ʘ E F G B C D ϓϨΠϠʔʹର͢ΔϛχϚοΫεߦಈG ϓϨΠϠʔͷϛχϚοΫεརಘ
- 9. ϑΥʔΫఆཧ ϛχϚοΫεߦಈͱϛχϚοΫεརಘ ήʔϜ ʹ͓͍ͯ Λຬͨ͢ϓϨΠϠʔҎ֎ͷߦಈͷΛʹର͢ΔϛχϚοΫεߦಈͱ͍͏ ͜ͷ ΛϛχϚοΫεརಘΛ͍͍ Ͱද͢ ΛϛχϚοΫεͱ͍͏ G max si fi (si , ̂ s−i ) = min s−i max si fi (si , s−i ) i ̂ s−i i max si fi (si , ̂ s−i ) vi v = (v1 , ⋯, vn ) ̍ʘ E F G B C D ϓϨΠϠʔʹର͢ΔϛχϚοΫεߦಈG ϓϨΠϠʔʹର͢ΔϛχϚοΫεߦಈC ϓϨΠϠʔͷϛχϚοΫεརಘ ϓϨΠϠʔͷϛχϚοΫεརಘ
- 10. ϑΥʔΫఆཧ ݸਓ߹ཧత ήʔϜ ʹ͓͚Δߦಈͷ ͕ Λຬͨ͢ͱ͖ ݸਓ߹ཧతͰ͋Δͱ͍͍ Λຬͨ͢ͱ͖ ݸ͘ڧਓ߹ཧతͰ͋Δͱ͍͏ G s = (s1 , ⋯, sn ) fi (s) ≥ vi , ∀i ∈ N s fi (s) vi , ∀i ∈ N s ̍ʘ E F G B C D ϓϨΠϠʔʹର͢ΔϛχϚοΫεߦಈG ϓϨΠϠʔʹର͢ΔϛχϚοΫεߦಈC ϓϨΠϠʔͷϛχϚοΫεརಘ ϓϨΠϠʔͷϛχϚοΫεརಘ ݸ͘ڧਓ߹ཧత B E ͱ D F
- 11. ϑΥʔΫఆཧ ϑΥʔΫఆཧ ήʔϜ ͷݸ͘ڧਓ߹ཧతͳҙͷߦಈͷ ʹରͯ͠ কདྷརಘͷׂҾҼࢠ͕ ͕ΓཱͭͳΒ ແݶճ܁Γฦ͠ήʔϜ ͷφογϡߧۉ ͕ଘࡏͯ͠ ͕Γཱͭ ˒ϑΥʔΫຽؒঝͷҙຯ ͜ͷఆཧͷҙຯ͢Δͱ͜Ζ ׂҾҼࢠ͕େ͖͚Ε ݸਓ߹ཧతͳߦಈͷ͕φογϡʹߧۉΑͬͯୡͰ͖Δ G s = (s1 , ⋯, sn ) ∈ S δ δ ≥ maxti fi (ti , s−i ) − fi (s) maxti fi(ti, s−i) − vi , ∀i ∈ N G∞ (δ) x* = (x1* , ⋯, xn* ) s(x*) = (s, s, ⋯) δ ̍ʘ % $ % $ ͏߹͠ྗڠʹ͍ޓ$ $ Λ औΓଓ͚ΔΑ͏ͳ͕ߧۉଘࡏ
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- 13. φογϡߧۉ෦ήʔϜશߧۉͷఆٛ ແݶճ܁Γฦ͠ήʔϜʹ͓͚Δφογϡߧۉ ʹ͓͍ͯ ઓུͷ ͕φογϡ͋ͰߧۉΔͱ ͯ͢ͷϓϨΠϠʔʹ͍ͭͯ Ҏޙͷ෦ήʔϜΛ Ͱද͢ ·ͨ ҙͷϓϨΠϠʔʹ͍ͭͯ ઓུ ͕ ʹಋ͘ઓུΛ Ͱද͢ ۩ମతʹ ΛҰͭߟ͑ͯΈΔ͜ͷཤྺͰୡ͞Εͨ෦ήʔϜ ʹ͓͍ͯ ճͷߦಈ ճ·Ͱͷཤྺ ʹରͯ͠ ແݶճ܁Γฦ͠ήʔϜʹ͓͚Δ෦ήʔϜશߧۉ ʹ͓͍ͯ ઓུ ͕෦ήʔϜશ͋ͰߧۉΔͱ ҙͷճ ͱ ճ·Ͱͷҙͷཤྺ ʹରͯ͠ ֤ϓϨΠϠʔͷઓུ ͕ಋ͘ઓུ ͷ ͕ ͷφογϡͳͱߧۉΔ͜ͱͰ͋Δ G∞ (δ) = (N, {X∞i }i∈N, {f∞i δ }i∈N) x* = (x1* , ⋯, xn* ) i f∞i δ (xi* , x−i* ) ≥ f∞i δ (xi , x−i* ), ∀xi ∈ X∞i ht−1 G∞ (δ)(ht−1) xi ∈ X∞i G∞ (δ)(ht−1) xi |ht−1 ht−1 = (s1, s2, ⋯, st−1) G∞ (δ)(ht−1) t′ t′− 1 ht′−1 = (s′ 1, s′ 2, ⋯, s′ t′−1) (xi |ht−1)t′(ht′−1) = xi (s1, s2, ⋯, st−1, s′ 1, s′ 2, ⋯, s′ t′−1) G∞ (δ) = (N, {X∞i }i∈N, {f∞i δ }i∈N) x* = (x1* , ⋯, xn* ) t = 1,2,⋯ t − 1 ht−1 i xi* xi* |ht−1 x*|ht−1 = (x1* |ht−1, ⋯, xn* |ht−1) G∞ (δ)(ht−1)
- 14. શϑΥʔΫఆཧ ఆཧ ήʔϜ ͷߦಈͷ ͕φογϡ͋ͰߧۉΓ Λຬͨ͢ҙͷߦಈͷ ΛͱΔ͜ͷͱ͖ কདྷརಘͷׂҾҼࢠ͕ ͕ΓཱͭͳΒ ແݶճ܁Γฦ͠ήʔϜ ͷ෦ήʔϜશߧۉ ͕ଘࡏͯ͠ ͕Γཱͭ શϑΥʔΫఆཧ ήʔϜ ͷݸ͘ڧਓ߹ཧతͳརಘϕΫτϧΛୡ͢ΔΑ͏ͳ ෦ήʔϜશߧۉ ͕ଘࡏ͠ ͕Γཱͭ ϑΥʔΫఆཧ͕ ͋Δ݅ͷͱͰ෦ήʔϜશͯؔ͠ʹߧۉཱ͢Δ ূ໌#4*$Ͱ͠ͳ͍ G s* = (s1* , ⋯, sn* ) ∈ S fi (s) fi (s*), ∀i ∈ N s = (s1 , ⋯, sn ) ∈ S δ δ ≥ maxti fi (ti , s−i ) − fi (s) maxti fi(ti, s−i) − fi(s*) , ∀i ∈ N G∞ (δ) x* = (x1* , ⋯, xn* ) s(x*) = (s, s, ⋯) G x* = (x1* , ⋯, xn* ) s(x*) = (s, s, ⋯)
- 15. φογϡߧۉ෦ήʔϜશߧۉͷఆٛ P1 P2 P2 % $ % $ % $ ̍ʘ % $ % $ ແݶճ܁ฦήʔϜ P1 P2 P2 % $ % $ % $ ήʔϜ P1 P2 P2 % $ % $ % $ G∞ (δ)(h1) = G∞ (δ)({DD})