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ゲーム理論BASIC 演習59 -有限回繰り返しゲームにおける部分ゲーム完全均衡2-
- 3. ༗ݶճ܁Γฦ͠ήʔϜ ήʔϜ ϓϨΠϠʔͷू߹ ϓϨΠϠʔ ͷߦಈͷू߹ ϓϨΠϠʔ ͷརಘؔ ͱͯ͠ ήʔϜΛ ճ ܁Γฦ͢ήʔϜΛߟ͑Δ ճʹ͓͍ͯ ֤ϓϨΠϠʔ ճ·ͰʹબΕͨߦಈΛ্ͯͬͨ͢ͰߦಈΛબ͢Δ ܁Γฦ͠ήʔϜʹ͓͚Δઓུ ͱͯ͠ ͱ͠ ͱ͢Δ͜͜Ͱ Λ ճ·Ͱͷཤྺͱ͍͏ ϓϨΠϠʔ ͷ ճͷߦಈ ཤྺ Λͱʹܾఆ͞ΕΔ ͢ͳΘͪ Ͱ༩͑ΒΕΔ ͜ΕΑΓ܁Γฦ͠ήʔϜʹ͓͚ΔϓϨΠϠʔ ͷ७ਮઓུ Ͱ༩͑ΒΕΔ ϓϨΠϠʔ ͷ७ਮઓུͷू߹Λ ͱ͠ ϓϨΠϠʔશମͷઓུͷͷू߹Λ Ͱද͢ G = (N, {Si }i∈N, {fi }i∈N) N Si i fi i S = S1 × ⋯ × Sn fi : S → ℜ T T ∞ t t − 1 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 t=1 i XTi XT = XT1 × ⋯ × XTn
- 4. ༗ݶճ܁Γฦ͠ήʔϜ ܁Γฦ͠ήʔϜʹ͓͚Δརಘ ׂҾҼࢠ ͱͯ͠ ΛׂҾརಘͱ͍͏ ܁Γฦ͠ήʔϜʹ͓͚Δφογϡߧۉ ઓུ ͕φογϡ͋ͰߧۉΔͱ ͯ͢ͷϓϨΠϠʔ ʹ͍ͭͯ ׂҾརಘ δ, 0 δ 1 fTi δ (x) = T ∑ t=1 δt−1 fi (st(x)) x* = (x1 * , ⋯, xn *) i fTi δ (xi* , x−i* ) ≥ fTi δ (xi , x−i* ), ∀xi ∈ XTi
- 5. ༗ݶճ܁Γฦ͠ήʔϜʹ͓͚Δ෦ήʔϜશߧۉ Ҏޙͷ෦ήʔϜΛ Ͱද͢ ·ͨ ҙͷϓϨΠϠʔʹ͍ͭͯ ઓུ ͕ ʹಋ͘ઓུΛ Ͱද͢ ۩ମతʹ ΛҰͭߟ͑ͯΈΔ͜ͷཤྺͰୡ͞Εͨ෦ήʔϜ ʹ͓͍ͯ ճͷߦಈ ճ·Ͱͷཤྺ ʹରͯ͠ ༗ݶճ܁Γฦ͠ήʔϜʹ͓͚Δ෦ήʔϜશߧۉ ʹ͓͍ͯ ઓུ ͕෦ήʔϜશ͋ͰߧۉΔͱ ҙͷճ ͱ ճ·Ͱͷҙͷཤྺ ʹରͯ͠ ֤ϓϨΠϠʔ ͷઓུ ͕ಋ͘ઓུ ͷ ͕ ͷφογϡͳͱߧۉΔ͜ͱͰ͋Δ ht−1 GT (δ)(ht−1) xi ∈ XTi GT (δ)(ht−1) xi |ht−1 ht−1 = (s1, s2, ⋯, st−1) GT (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) GT (δ) = (N, {XTi }i∈N, {fTi δ }i∈N) x* = (x1* , ⋯, xn* ) t = 1,2,⋯, T t − 1 ht−1 i xi* xi* |ht−1 x*|ht−1 = (x1* |ht−1, ⋯, xn* |ht−1) GT (δ)(ht−1)
- 6. ӈͷήʔϜΛճ܁Γฦ͢ήʔϜΛߟ͑Δ ҎԼͷઓུ͕φογϡ͋ͰߧۉΔͨΊͷׂҾҼࢠ ͷ݅ΛٻΊΑ ϓϨΠϠʔ ճͷߦಈ ճͷߦಈ ཤྺ͕ ͳΒ ͦΕҎ֎ͷཤྺͷ߹ ϓϨΠϠʔ ճͷߦಈ ճͷߦಈ ཤྺ͕ ͳΒ ͦΕҎ֎ͷཤྺͷ߹ ͷઓུͷ෦ήʔϜશͰߧۉͳ͍͜ͱΛઆ໌ͤΑ ෦ήʔϜશߧۉΛٻΊΑ δ a (a, d) c b d (a, d) f e ̍ʘ E F G B C D
- 7. ղ ӈͷήʔϜΛճ܁Γฦ͢ήʔϜΛߟ͑Δ ҎԼͷઓུ͕φογϡ͋ͰߧۉΔͨΊͷׂҾҼࢠ ͷ݅ΛٻΊΑ ϓϨΠϠʔ ճͷߦಈ ճͷߦಈ ཤྺ͕ ͳΒ ͦΕҎ֎ͷཤྺͷ߹ ϓϨΠϠʔ ճͷߦಈ ճͷߦಈ ཤྺ͕ ͳΒ ͦΕҎ֎ͷཤྺͷ߹ ϓϨΠϠʔʹ͍ͭͯ ϓϨΠϠʔͷઓུʹରͯ͠࠷దԠʹͳΔ݅ΛٻΊΔ ্هઓུͷͱͰಘΒΕΔརಘ ճͷߦಈͰ ΛऔΔΑ͏ͳઓུΛͱΔͱ ϓϨΠϠʔ͕̎GΛऔΔͨΊ ΑΓେ͖͍རಘΛಘΔ͜ͱͳ͍ ճͷߦಈ ΛऔΔΑ͏ͳઓུΛͱΔͱ ϓϨΠϠʔ͕̎FΛऔΔͨΊ ࠷େͰ ͷརಘΛಘΔ ࠷దԠʹͳΔͨΊʹ δ a (a, d) c b d (a, d) f e 10 + 5δ a 10 + 5δ c 13 + δ 10 + 5δ ≥ 13 + δ ⇔ δ ≥ 3 4 ̍ʘ E F G B C D
- 8. ղ ӈͷήʔϜΛճ܁Γฦ͢ήʔϜΛߟ͑Δ ҎԼͷઓུ͕φογϡ͋ͰߧۉΔͨΊͷׂҾҼࢠ ͷ݅ΛٻΊΑ ϓϨΠϠʔ ճͷߦಈ ճͷߦಈ ཤྺ͕ ͳΒ ͦΕҎ֎ͷཤྺͷ߹ ϓϨΠϠʔ ճͷߦಈ ճͷߦಈ ཤྺ͕ ͳΒ ͦΕҎ֎ͷཤྺͷ߹ ͷઓུͷ෦ήʔϜશͰߧۉͳ͍͜ͱΛઆ໌ͤΑ Ҏ֎ͷཤྺͷ෦ήʔϜʹ͓͍ͯ ͷઓུͷ͕ͱΒΕΔ͕ ͜ΕφογϡͰߧۉͳ͍ͨΊ δ a (a, d) c b d (a, d) f e (a, d) (b, e) ̍ʘ E F G B C D
- 9. ղ ӈͷήʔϜΛճ܁Γฦ͢ήʔϜΛߟ͑Δ ෦ήʔϜશߧۉΛٻΊΑ ҙͷཤྺʹ͍ͭͯ ઓུͷ͕ಋ͘ճͷߦಈͷ͕φογϡͯͬͳʹߧۉΔඞཁ͕͋Δ Ώ͑ʹ ճͷઓུͷ͕ ͱͳΔઓུͷΈ͕෦ήʔϜશͳʹߧۉΔ ͜ΕΛલఏͱͯ͠ རಘදΛॻ͚ ͜ΕΑΓ ճͷઓུͷ ͱͳΔͷ͕ ෦ήʔϜશͳͱߧۉΔΏ͑ʹ ϓϨΠϠʔ ճͷߦಈ ճͷߦಈ ҙͷཤྺʹର͠ ϓϨΠϠʔ ճͷߦಈ ճͷߦಈ ҙͷཤྺʹର͠ (c, f ) (c, f ) c c f f ̍ʘ E F G B C D ̍ʘ E F G B Ў Ў Ў Ў Ў Ў C Ў Ў Ў Ў Ў Ў D Ў Ў Ў Ў Ў Ў
- 10. ఆཧ ఆཧ ήʔϜ͕ͨͩҰͭͷφογϡߧۉ Λͭͱ͢Δ͜ͷͱ͖ ճ܁Γฦ͠ήʔϜʹ͓͚Δ ෦ήʔϜશߧۉ ͨͩҰͭଘࡏ͠ ߦಈͷ Λ༩͑Δ ήʔϜཧ#4*$ୈճิࢀর s* T x* s(x*) = ( T s*, s*, ⋯, s*) ̍ʘ E F G B C D