ゲーム理論BASIC 第6回 -ナッシュ均衡と旅人のジレンマ-
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ゲーム理論BASIC 第6回 -ナッシュ均衡と旅人のジレンマ-
- 5. φογϡߧۉɿఆٛ ઓུͷ ͕φογϡ͋ͰߧۉΔͱ ͯ͢ͷϓϨΠϠʔʹରͯ͠ Λຬͨ͢ͱ͖Λ͍͏ ͯ͢ͷϓϨΠϠʔ͕࠷ʹ͍ޓదԠઓུΛͱ͍ͬͯΔΑ͏ͳઓུͷ (s*1 , ⋯, s*n ) ∈ S fi (s*i , s*−i ) ≥ fi (si , s*−i ) ∀si ∈ Si ࠷దԠઓུ ϓϨΠϠʔҎ֎ͷઓུΛ ͱ͠ݻఆ͢Δ Λຬͨ͢ͱ͖ ʹର͢Δ࠷దԠઓུͱ͍͏ i s−i ∈ S−i fi (s *i , s−i ) ≥ fi (si , s−i ), ∀si ∈ Si s *i s−i
- 6. φογϡߧۉɿఆٛ ઓུͷ ͕φογϡ͋ͰߧۉΔͱ ͯ͢ͷϓϨΠϠʔʹରͯ͠ Λຬͨ͢ͱ͖Λ͍͏ ͯ͢ͷϓϨΠϠʔ͕࠷ʹ͍ޓదԠઓུΛͱ͍ͬͯΔΑ͏ͳઓུͷ (s*1 , ⋯, s*n ) ∈ S fi (s*i , s*−i ) ≥ fi (si , s*−i ) ∀si ∈ Si ࠷దԠઓུ ϓϨΠϠʔҎ֎ͷઓུΛ ͱ͠ݻఆ͢Δ Λຬͨ͢ͱ͖ ʹର͢Δ࠷దԠઓུͱ͍͏ i s−i ∈ S−i fi (s *i , s−i ) ≥ fi (si , s−i ), ∀si ∈ Si s *i s−i AʘB sA 1 sA 2 sA 3 sB 1 sB 2 ྫ AʘB ൿ ࣗന ൿ ࣗന ྫ
- 7. φογϡߧۉɿఆٛ ઓུͷ ͕φογϡ͋ͰߧۉΔͱ ͯ͢ͷϓϨΠϠʔʹରͯ͠ Λຬͨ͢ͱ͖Λ͍͏ ͯ͢ͷϓϨΠϠʔ͕࠷ʹ͍ޓదԠઓུΛͱ͍ͬͯΔΑ͏ͳઓུͷ (s*1 , ⋯, s*n ) ∈ S fi (s*i , s*−i ) ≥ fi (si , s*−i ) ∀si ∈ Si ࠷దԠઓུ ϓϨΠϠʔҎ֎ͷઓུΛ ͱ͠ݻఆ͢Δ Λຬͨ͢ͱ͖ ʹର͢Δ࠷దԠઓུͱ͍͏ i s−i ∈ S−i fi (s *i , s−i ) ≥ fi (si , s−i ), ∀si ∈ Si s *i s−i AʘB sA 1 sA 2 sA 3 sB 1 sB 2 ྫ AʘB ൿ ࣗന ൿ ࣗന ྫ
- 8. φογϡߧۉɿఆٛ ઓུͷ ͕φογϡ͋ͰߧۉΔͱ ͯ͢ͷϓϨΠϠʔʹରͯ͠ Λຬͨ͢ͱ͖Λ͍͏ ͯ͢ͷϓϨΠϠʔ͕࠷ʹ͍ޓదԠઓུΛͱ͍ͬͯΔΑ͏ͳઓུͷ (s*1 , ⋯, s*n ) ∈ S fi (s*i , s*−i ) ≥ fi (si , s*−i ) ∀si ∈ Si AʘB sA 1 sA 2 sA 3 sB 1 sB 2 ྫ AʘB ൿ ࣗന ൿ ࣗന ྫ ࠷దԠઓུ ϓϨΠϠʔҎ֎ͷઓུΛ ͱ͠ݻఆ͢Δ Λຬͨ͢ͱ͖ ʹର͢Δ࠷దԠઓུͱ͍͏ i s−i ∈ S−i fi (s *i , s−i ) ≥ fi (si , s−i ), ∀si ∈ Si s *i s−i φογϡߧۉ (sA 2 , sB 2 )
- 9. φογϡߧۉɿఆٛ ઓུͷ ͕φογϡ͋ͰߧۉΔͱ ͯ͢ͷϓϨΠϠʔʹରͯ͠ Λຬͨ͢ͱ͖Λ͍͏ ͯ͢ͷϓϨΠϠʔ͕࠷ʹ͍ޓదԠઓུΛͱ͍ͬͯΔΑ͏ͳઓུͷ (s*1 , ⋯, s*n ) ∈ S fi (s*i , s*−i ) ≥ fi (si , s*−i ) ∀si ∈ Si AʘB sA 1 sA 2 sA 3 sB 1 sB 2 ྫ AʘB ൿ ࣗന ൿ ࣗന ྫ ࠷దԠઓུ ϓϨΠϠʔҎ֎ͷઓུΛ ͱ͠ݻఆ͢Δ Λຬͨ͢ͱ͖ ʹର͢Δ࠷దԠઓུͱ͍͏ i s−i ∈ S−i fi (s *i , s−i ) ≥ fi (si , s−i ), ∀si ∈ Si s *i s−i φογϡߧۉ (sA 2 , sB 2 ) φογϡߧۉ ࣗന ࣗന (
- 15. ཱྀਓͷδϨϯϚ ϧʔϧ ਓ͕ಉֹۚ͡Λॻ͚ ͦΕΛࠎ౹ͷՁͱ͠ ਓͷཱྀߦऀʹͦͷֹۚΛ͍͢ ϧʔϧ Ұํ͕ଞํΑΓֹ͍ۚΛॻ͍ͨ߹ ͜ͷֹ͍ۚΛࠎ౹ͷՁ֨ͱ͢Δ ֹ͍ۚΛॻཱྀ͍ͨߦऀʹࠎ౹ͷՁ֨ʹυϧͷֹ͕ͨۚ͠ࢧΘΕ ߴֹ͍ۚΛॻཱྀ͍ͨߦऀ͔Βࠎ౹ͷՁ͔֨ΒυϧͷҾֹ͍͕ͨࢧΘΕΔ AʘB ɾɾɾ ɾɾɾ ɾɾɾ 99 + 2 99 − 2
- 16. ཱྀਓͷδϨϯϚ ϧʔϧ ਓ͕ಉֹۚ͡Λॻ͚ ͦΕΛࠎ౹ͷՁͱ͠ ਓͷཱྀߦऀʹͦͷֹۚΛ͍͢ ϧʔϧ Ұํ͕ଞํΑΓֹ͍ۚΛॻ͍ͨ߹ ͜ͷֹ͍ۚΛࠎ౹ͷՁ֨ͱ͢Δ ֹ͍ۚΛॻཱྀ͍ͨߦऀʹࠎ౹ͷՁ֨ʹυϧͷֹ͕ͨۚ͠ࢧΘΕ ߴֹ͍ۚΛॻཱྀ͍ͨߦऀ͔Βࠎ౹ͷՁ͔֨ΒυϧͷҾֹ͍͕ͨࢧΘΕΔ AʘB ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ
- 17. ཱྀਓͷδϨϯϚ ϧʔϧ ਓ͕ಉֹۚ͡Λॻ͚ ͦΕΛࠎ౹ͷՁͱ͠ ਓͷཱྀߦऀʹͦͷֹۚΛ͍͢ ϧʔϧ Ұํ͕ଞํΑΓֹ͍ۚΛॻ͍ͨ߹ ͜ͷֹ͍ۚΛࠎ౹ͷՁ֨ͱ͢Δ ֹ͍ۚΛॻཱྀ͍ͨߦऀʹࠎ౹ͷՁ֨ʹυϧͷֹ͕ͨۚ͠ࢧΘΕ ߴֹ͍ۚΛॻཱྀ͍ͨߦऀ͔Βࠎ౹ͷՁ͔֨ΒυϧͷҾֹ͍͕ͨࢧΘΕΔ AʘB ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ
- 18. ཱྀਓͷδϨϯϚ ϧʔϧ ਓ͕ಉֹۚ͡Λॻ͚ ͦΕΛࠎ౹ͷՁͱ͠ ਓͷཱྀߦऀʹͦͷֹۚΛ͍͢ ϧʔϧ Ұํ͕ଞํΑΓֹ͍ۚΛॻ͍ͨ߹ ͜ͷֹ͍ۚΛࠎ౹ͷՁ֨ͱ͢Δ ֹ͍ۚΛॻཱྀ͍ͨߦऀʹࠎ౹ͷՁ֨ʹυϧͷֹ͕ͨۚ͠ࢧΘΕ ߴֹ͍ۚΛॻཱྀ͍ͨߦऀ͔Βࠎ౹ͷՁ͔֨ΒυϧͷҾֹ͍͕ͨࢧΘΕΔ AʘB ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ
- 19. ཱྀਓͷδϨϯϚ ϧʔϧ ਓ͕ಉֹۚ͡Λॻ͚ ͦΕΛࠎ౹ͷՁͱ͠ ਓͷཱྀߦऀʹͦͷֹۚΛ͍͢ ϧʔϧ Ұํ͕ଞํΑΓֹ͍ۚΛॻ͍ͨ߹ ͜ͷֹ͍ۚΛࠎ౹ͷՁ֨ͱ͢Δ ֹ͍ۚΛॻཱྀ͍ͨߦऀʹࠎ౹ͷՁ֨ʹυϧͷֹ͕ͨۚ͠ࢧΘΕ ߴֹ͍ۚΛॻཱྀ͍ͨߦऀ͔Βࠎ౹ͷՁ͔֨ΒυϧͷҾֹ͍͕ͨࢧΘΕΔ AʘB ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ
- 20. ཱྀਓͷδϨϯϚ ϧʔϧ ਓ͕ಉֹۚ͡Λॻ͚ ͦΕΛࠎ౹ͷՁͱ͠ ਓͷཱྀߦऀʹͦͷֹۚΛ͍͢ ϧʔϧ Ұํ͕ଞํΑΓֹ͍ۚΛॻ͍ͨ߹ ͜ͷֹ͍ۚΛࠎ౹ͷՁ֨ͱ͢Δ ֹ͍ۚΛॻཱྀ͍ͨߦऀʹࠎ౹ͷՁ֨ʹυϧͷֹ͕ͨۚ͠ࢧΘΕ ߴֹ͍ۚΛॻཱྀ͍ͨߦऀ͔Βࠎ౹ͷՁ͔֨ΒυϧͷҾֹ͍͕ͨࢧΘΕΔ AʘB ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ
- 21. ཱྀਓͷδϨϯϚ ϧʔϧ ਓ͕ಉֹۚ͡Λॻ͚ ͦΕΛࠎ౹ͷՁͱ͠ ਓͷཱྀߦऀʹͦͷֹۚΛ͍͢ ϧʔϧ Ұํ͕ଞํΑΓֹ͍ۚΛॻ͍ͨ߹ ͜ͷֹ͍ۚΛࠎ౹ͷՁ֨ͱ͢Δ ֹ͍ۚΛॻཱྀ͍ͨߦऀʹࠎ౹ͷՁ֨ʹυϧͷֹ͕ͨۚ͠ࢧΘΕ ߴֹ͍ۚΛॻཱྀ͍ͨߦऀ͔Βࠎ౹ͷՁ͔֨ΒυϧͷҾֹ͍͕ͨࢧΘΕΔ AʘB ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ
- 22. ཱྀਓͷδϨϯϚ ϧʔϧ ਓ͕ಉֹۚ͡Λॻ͚ ͦΕΛࠎ౹ͷՁͱ͠ ਓͷཱྀߦऀʹͦͷֹۚΛ͍͢ ϧʔϧ Ұํ͕ଞํΑΓֹ͍ۚΛॻ͍ͨ߹ ͜ͷֹ͍ۚΛࠎ౹ͷՁ֨ͱ͢Δ ֹ͍ۚΛॻཱྀ͍ͨߦऀʹࠎ౹ͷՁ֨ʹυϧͷֹ͕ͨۚ͠ࢧΘΕ ߴֹ͍ۚΛॻཱྀ͍ͨߦऀ͔Βࠎ౹ͷՁ͔֨ΒυϧͷҾֹ͍͕ͨࢧΘΕΔ AʘB ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ
- 23. ཱྀਓͷδϨϯϚ ϧʔϧ ਓ͕ಉֹۚ͡Λॻ͚ ͦΕΛࠎ౹ͷՁͱ͠ ਓͷཱྀߦऀʹͦͷֹۚΛ͍͢ ϧʔϧ Ұํ͕ଞํΑΓֹ͍ۚΛॻ͍ͨ߹ ͜ͷֹ͍ۚΛࠎ౹ͷՁ֨ͱ͢Δ ֹ͍ۚΛॻཱྀ͍ͨߦऀʹࠎ౹ͷՁ֨ʹυϧͷֹ͕ͨۚ͠ࢧΘΕ ߴֹ͍ۚΛॻཱྀ͍ͨߦऀ͔Βࠎ౹ͷՁ͔֨ΒυϧͷҾֹ͍͕ͨࢧΘΕΔ AʘB ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ ɾɾɾ φογϡ ߧۉ ʢυϧυϧʣ ύϨʔτ࠷ద υϧ υϧ