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ゲーム理論BASIC 演習18 -フォーク定理とトリガー戦略-
- 2. ήʔϜ܁Γฦ͠ήʔϜʹ͓͚Δઓུ ܁Γฦ͠ήʔϜʹ͓͚Δརಘߧۉ ϛχϚοΫεߦಈͱϛχϚοΫεརಘ ϑΥʔΫఆཧͱτϦΨʔઓུ
- 3. ήʔϜ ϓϨΠϠʔͷू߹ ϓϨΠϠʔ ͷߦಈͷू߹ ϓϨΠϠʔ ͷརಘؔ ͱͯ͠ ܁Γฦ͠ήʔϜʹ͓͚Δઓུ ͱͯ͠ ͱ͠ ͱ͢Δ ͜͜Ͱ 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 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 ήʔϜ܁Γฦ͠ήʔϜʹ͓͚Δઓུ ܁Γฦ͞ΕΔήʔϜͷ͜ͱ
- 4. རಘ ׂҾҼࢠ ͱͯ͠ ΛׂҾརಘͱ͍͍ Λਖ਼نԽརಘͱ͍͏ ܁Γฦ͠ήʔϜʹ͓͚Δφογϡߧۉ ઓུ ͕φογϡ͋ͰߧۉΔͱ ͯ͢ͷϓϨΠϠʔ ʹ͍ͭͯ ܁Γฦ͠ήʔϜʹ͓͚Δ෦ήʔϜશߧۉ ઓུ ͕෦ήʔϜશ͋ͰߧۉΔͱ ҙͷ ͱ ճ·Ͱͷҙͷཤྺ ʹରͯ͠ ͱ ͔Βಋ͔ΕΔઓུͷ ͕ ճҎ߱ͷ෦ήʔϜͰφογϡ͋ͰߧۉΔ͜ͱΛ͍͏ δ, 0 δ 1 ∞ ∑ t=1 δt−1 fi (st(x)) f∞i δ (x) = (1 − δ) ∞ ∑ t=1 δt−1 fi (st(x)) x* = (x1 * , ⋯, xn *) i f∞i δ (xi* , x−i* ) ≥ f∞i δ (xi , x−i* ), ∀xi ∈ X∞i x* = (x1 * , ⋯, xn *) t = 1,2,⋯ t − 1 ht−1 x* ht−1 x* ht−1 t ܁Γฦ͠ήʔϜʹ͓͚Δརಘߧۉ ແݶճ܁Γฦ͠Ͱ ศརͳͷͰ͜ΕΛ͏
- 5. ϛχϚοΫεߦಈͱϛχϚοΫεརಘ ήʔϜ ʹ͓͍ͯ Λຬͨ͢ϓϨΠϠʔ Ҏ֎ͷߦಈͷ Λ ʹର͢ΔϛχϚοΫεߦಈͱ͍͏ ͜ͷ ΛϛχϚοΫεརಘΛ͍͍ Ͱද͢ ΛϛχϚοΫεͱ͍͏ ݸਓ߹ཧత ήʔϜ ʹ͓͚Δߦಈͷ ͕ Λຬͨ͢ͱ͖ ݸਓ߹ཧతͰ͋Δͱ͍͍ Λຬͨ͢ͱ͖ ݸ͘ڧਓ߹ཧతͰ͋Δͱ͍͏ 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 ) G s = (s1 , ⋯, sn ) fi (s) ≥ vi , ∀i ∈ N s fi (s) vi , ∀i ∈ N s ϛχϚοΫεߦಈͱϛχϚοΫεརಘ
- 7. ӈͷΑ͏ͳήʔϜΛߟ͑Δ ͜ͷήʔϜʹ͓͚Δ७ਮઓུφογϡߧۉΛٻΊΑ ϓϨΠϠʔʹର͢ΔϓϨΠϠʔͷϛχϚοΫεߦಈԿ͔ʁ ·ͨϓϨΠϠʔͷϛχϚοΫεརಘΛٻΊΑ ϓϨΠϠʔʹର͢ΔϓϨΠϠʔͷϛχϚοΫεߦಈԿ͔ʁ ·ͨϓϨΠϠʔͷϛχϚοΫεརಘΛٻΊΑ ݸ͘ڧਓ߹ཧతͳߦಈͷΛٻΊΑ φογϡߧۉ ϓϨΠϠʔ͕ ΛऔΕ ϓϨΠϠʔ ΛऔΓͷརಘΛಘΔ ϓϨΠϠʔ͕ ΛऔΕ ϓϨΠϠʔ ΛऔΓͷརಘΛಘΔ ϓϨΠϠʔ͕ ΛऔΕ ϓϨΠϠʔ ΛऔΓͷརಘΛಘΔ ͜ΕΛߟ্͑ͨͰϓϨΠϠʔϓϨΠϠʔͷརಘΛ࠷খͱ͢Δ͔Β ΛͱΔ͜Ε͕ϛχϚοΫεߦಈͱͳΔ ·ͨ ͜ͷͱ͖ͷརಘ͕ϛχϚοΫεརಘͱͳΔ (a1, b1) b1 a1 b2 a2 b3 a3 b3 ϛχϚοΫεߦಈͱϛχϚοΫεརಘ ̍ʘ C C C B B B
- 8. ӈͷΑ͏ͳήʔϜΛߟ͑Δ ͜ͷήʔϜʹ͓͚Δ७ਮઓུφογϡߧۉΛٻΊΑ ϓϨΠϠʔʹର͢ΔϓϨΠϠʔͷϛχϚοΫεߦಈԿ͔ʁ ·ͨϓϨΠϠʔͷϛχϚοΫεརಘΛٻΊΑ ϓϨΠϠʔʹର͢ΔϓϨΠϠʔͷϛχϚοΫεߦಈԿ͔ʁ ·ͨϓϨΠϠʔͷϛχϚοΫεརಘΛٻΊΑ ݸ͘ڧਓ߹ཧతͳߦಈͷΛٻΊΑ ϓϨΠϠʔ͕ ΛऔΕ ϓϨΠϠʔ ΛऔΓͷརಘΛಘΔ ϓϨΠϠʔ͕ ΛऔΕ ϓϨΠϠʔ ΛऔΓͷརಘΛಘΔ ϓϨΠϠʔ͕ ΛऔΕ ϓϨΠϠʔ ΛऔΓͷརಘΛಘΔ ͜ΕΛߟ্͑ͨͰϓϨΠϠʔϓϨΠϠʔͷརಘΛ࠷খͱ͢Δ͔Β ΛͱΔ͜Ε͕ϛχϚοΫεߦಈͱͳΔ ·ͨ ͜ͷͱ͖ͷརಘ͕ϛχϚοΫεརಘͱͳΔ a1 b1 a2 b3 a3 b2 a2 ϛχϚοΫεߦಈͱϛχϚοΫεརಘ ̍ʘ C C C B B B
- 9. ӈͷΑ͏ͳήʔϜΛߟ͑Δ ͜ͷήʔϜʹ͓͚Δ७ਮઓུφογϡߧۉΛٻΊΑ ϓϨΠϠʔʹର͢ΔϓϨΠϠʔͷϛχϚοΫεߦಈԿ͔ʁ ·ͨϓϨΠϠʔͷϛχϚοΫεརಘΛٻΊΑ ϓϨΠϠʔʹର͢ΔϓϨΠϠʔͷϛχϚοΫεߦಈԿ͔ʁ ·ͨϓϨΠϠʔͷϛχϚοΫεརಘΛٻΊΑ ݸ͘ڧਓ߹ཧతͳߦಈͷΛٻΊΑ ϓϨΠϠʔͷϛχϚοΫεརಘ ϓϨΠϠʔͷϛχϚοΫεརಘͰ͋Δ ʮήʔϜ ʹ͓͚Δߦಈͷ ͕ Λຬͨ͢ͱ͖ ݸ͘ڧਓ߹ཧతͰ͋Δͱ͍͏ʯ Ͱ͋Δ͔Β ͱ ݸ͘ڧਓ߹ཧతͰ͋Δ ͷ߹ ͷ߹ G s = (s1 , ⋯, sn ) fi (s) vi , ∀i ∈ N s (a1, b1) (a3, b2) (a1, b1) fi (a1, b1) = 5 2, i = 1,2 (a3, b2) f1 (a3, b2) = 3 2, f2 (a3, b2) = 4 2 ϛχϚοΫεߦಈͱϛχϚοΫεརಘ ̍ʘ C C C B B B
- 10. ӈͷήʔϜΛήʔϜͱͯ͠ ແݶճ܁Γฦ͢͜ͱΛߟ͑Δ ݸ͘ڧਓ߹ཧతͳߦಈͷ ແݶճ܁Γฦ͠ήʔϜʹ͓͍ͯ φογϡͰߧۉୡͰ͖Δ͜ͱΛ֬ೝ͢ΔͨΊʹ τϦΨʔઓུͱͯ͠ҎԼΛߟ͑Δ ϓϨΠϠʔͷτϦΨʔઓུ ʮճ ΛͱΔظҎ߱ʹ͍ͭͯ ϓϨΠϠʔ͕ Λͱ͍ͬͯΔݶΓ ΛͱΔϓϨΠϠʔ͕ͦΕҎ֎ͷߦಈΛͱͬͨ߹ ϛχϚοΫεߦಈͰ͋Δ Λબͼଓ͚Δʯ ϓϨΠϠʔͷτϦΨʔઓུ ʮճ ΛͱΔظҎ߱ʹ͍ͭͯ ϓϨΠϠʔ͕ Λͱ͍ͬͯΔݶΓ ΛͱΔϓϨΠϠʔ͕ͦΕҎ֎ͷߦಈΛͱͬͨ߹ ϛχϚοΫεߦಈͰ͋Δ Λબͼଓ͚Δʯ ͜ͷઓུͷ͕ແݶճ܁Γฦ͠ήʔϜʹ͓͚ΔφογϡͳͱߧۉΔͨΊͷׂҾҼࢠ ͷ݅ΛٻΊΑ ͜ͷઓུͷແݶճ܁Γฦ͠ήʔϜʹ͓͚Δ෦ήʔϜશͤߟ ͔ߧۉΑ (a3, b2) a3 b2 a3 a2 b2 a3 b2 b3 δ ϑΥʔΫఆཧͱτϦΨʔઓུ ̍ʘ C C C B B B
- 11. ϓϨΠϠʔͷτϦΨʔઓུ ʮճ ΛͱΔظҎ߱ʹ͍ͭͯ ϓϨΠϠʔ͕ Λͱ͍ͬͯΔݶΓ ΛͱΔϓϨΠϠʔ͕ͦΕҎ֎ͷߦಈΛͱͬͨ߹ ϛχϚοΫεߦಈͰ͋Δ Λબͼଓ͚Δʯ ϓϨΠϠʔͷτϦΨʔઓུ ʮճ ΛͱΔظҎ߱ʹ͍ͭͯ ϓϨΠϠʔ͕ Λͱ͍ͬͯΔݶΓ ΛͱΔϓϨΠϠʔ͕ͦΕҎ֎ͷߦಈΛͱͬͨ߹ ϛχϚοΫεߦಈͰ͋Δ Λબͼଓ͚Δʯ ͜ͷઓུͷ͕ແݶճ܁Γฦ͠ήʔϜʹ͓͚ΔφογϡͳͱߧۉΔͨΊͷׂҾҼࢠ ͷ݅ΛٻΊΑ ϓϨΠϠʔʹؔͯ͠ τϦΨʔઓུʹै͑ਖ਼نԽརಘͱͯ͠ΛಘΔ͜ͱ͕Ͱ͖ ϓϨΠϠʔ͕τϦΨʔઓུʹै͍ͬͯΔݶΓ རಘΛߴ͘͢Δ͜ͱͰ͖ͳ͍ ΑͬͯϓϨΠϠʔͷτϦΨʔઓུ࠷దԠ a3 b2 a3 a2 b2 a3 b2 b3 δ ϑΥʔΫఆཧͱτϦΨʔઓུ ̍ʘ C C C B B B
- 12. ϓϨΠϠʔͷτϦΨʔઓུ ʮճ ΛͱΔظҎ߱ʹ͍ͭͯ ϓϨΠϠʔ͕ Λͱ͍ͬͯΔݶΓ ΛͱΔϓϨΠϠʔ͕ͦΕҎ֎ͷߦಈΛͱͬͨ߹ ϛχϚοΫεߦಈͰ͋Δ Λબͼଓ͚Δʯ ϓϨΠϠʔͷτϦΨʔઓུ ʮճ ΛͱΔظҎ߱ʹ͍ͭͯ ϓϨΠϠʔ͕ Λͱ͍ͬͯΔݶΓ ΛͱΔϓϨΠϠʔ͕ͦΕҎ֎ͷߦಈΛͱͬͨ߹ ϛχϚοΫεߦಈͰ͋Δ Λબͼଓ͚Δʯ ͜ͷઓུͷ͕ແݶճ܁Γฦ͠ήʔϜʹ͓͚ΔφογϡͳͱߧۉΔͨΊͷׂҾҼࢠ ͷ݅ΛٻΊΑ Ұํ ϓϨΠϠʔʹؔͯ͠ ͋Δظ ʹ ͔Β ΛͱΔ͜ͱͰҰ࣌తʹརಘΛେ͖͘͢Δ͜ͱ͕Ͱ͖Δ ͦͷΑ͏ͳߦಈઓུ ظҎ߱ ϓϨΠϠʔ͕ Λͱͬͯ͘ΔͷͰ ࠷େͰͷརಘΛ͑Δ͜ͱʹͳΔ Ώ͑ʹτϦΨʔઓུ͕࠷దԠͱͳΔͨΊͷ݅ a3 b2 a3 a2 b2 a3 b2 b3 δ t a3 a2 t + 1 b3 (1 − δ){3 + 3δ1 + ⋯ + 3δt−2 + 3δt−1 + 3δt + 3δt+1 + ⋯} ≥ (1 − δ){3 + 3δ1 + ⋯ + 3δt−2 + 4δt−1 + 2δt + 2δt+1 + ⋯} ⇔ 3δt−1 + 3δt + 3δt+1 + ⋯ ≥ 4δt−1 + 2δt + 2δt+1 + ⋯ ⇔ δt + δt+1 + ⋯ ≥ δt−1 ⇔ δ1 + δ2 + ⋯ ≥ 1 ⇔ δ 1 − δ ≥ 1 ⇔ δ ≥ 1 2 ϑΥʔΫఆཧͱτϦΨʔઓུ ̍ʘ C C C B B B
- 13. ϑΥʔΫఆཧ ήʔϜ ͷݸ͘ڧਓ߹ཧతͳҙͷߦಈͷ ʹରͯ͠ কདྷརಘͷׂҾҼࢠ ͕ ͕ΓཱͭͳΒ ແݶճ܁Γฦ͠ήʔϜ ͷφογϡߧۉ ͕ଘࡏͯ͠ ͕Γཱͭ ࠓճͷͰ͋Ε ݸ͘ڧਓ߹ཧతͳߦಈͷͰ͋Γ ϓϨΠϠʔʹ͍ͭͯ ϓϨΠϠʔʹ͍ͭͯ ͷ݅ͱͳΔΏ͑ʹ݅ͷ͍͠ݫ 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, ⋯) s = (a3, b2) max t1 ∈{a1,a2,a3} f1 (t1 , b2) = max t1 ∈{a1,a2,a3} f1 (t1 , b2) = fi (a2, b2) = 4 f1 (s) = f1 ((a3, b2)) = 3 v1 = 2 δ ≥ maxt1 f1 (t1 , b2) − f1 (s) maxt1 f1(t1, b2) − v1 = 4 − 3 4 − 2 = 1 2 δ ≥ 0 δ ≥ 1 2 ఆཧ ̍ʘ C C C B B B
- 14. ϓϨΠϠʔͷτϦΨʔઓུ ʮճ ΛͱΔظҎ߱ʹ͍ͭͯ ϓϨΠϠʔ͕ Λͱ͍ͬͯΔݶΓ ΛͱΔϓϨΠϠʔ͕ͦΕҎ֎ͷߦಈΛͱͬͨ߹ ϛχϚοΫεߦಈͰ͋Δ Λબͼଓ͚Δʯ ϓϨΠϠʔͷτϦΨʔઓུ ʮճ ΛͱΔظҎ߱ʹ͍ͭͯ ϓϨΠϠʔ͕ Λͱ͍ͬͯΔݶΓ ΛͱΔϓϨΠϠʔ͕ͦΕҎ֎ͷߦಈΛͱͬͨ߹ ϛχϚοΫεߦಈͰ͋Δ Λબͼଓ͚Δʯ ͜ͷઓུͷແݶճ܁Γฦ͠ήʔϜʹ͓͚Δ෦ήʔϜશͤߟ ͔ߧۉΑ ϓϨΠϠʔ͕ҳͨ͠߹ϓϨΠϠʔͲͷͯؔ͠ʹظ ͱΓଓ͚Δ ͜ͷͱ͖ ֤ͯʹظઓུͷ ͱͳΔ͜ΕΒήʔϜʹ͓͚ΔφογϡͰߧۉͳ͍ ͱ͘ʹ ΛͱΓଓ͚Δ෦ήʔϜʹ͓͍ͯ ϓϨΠϠʔͱͯ͠ ΛબͿΑ͏ͳߦಈઓུʹม͑Δ͜ͱͰརಘΛେ͖͘Ͱ͖Δ Ώ͑ʹҙͷཤྺʹରͯ͠φογϡͰߧۉͳ͍ʢϓϨΠϠʔ͕࠷దͳߦಈͱͳ͍ͬͯͳ͍ ͨΊ ෦ήʔϜશͰߧۉͳ͍ a3 b2 a3 a2 b2 a3 b2 b3 b3 (a1, b3) (a2, b3) (a3, b3) (a2, b3) a3 ϑΥʔΫఆཧͱτϦΨʔઓུ ̍ʘ C C C B B B