ゲーム理論 BASIC 演習71 -3人ゲーム分析：コア-

1. ήʔϜཧ࿦#4*$ԋश
ਓήʔϜ෼ੳɿίΞ

2. ԋशͷਓήʔϜΛಛੑؔ਺‫ܗ‬ήʔϜ΁
ίΞΛ‫ٻ‬ΊΔ

3. φογϡ‫ߧۉ‬͸̏ͭɿ
((p,1 − p), (q,1 − q), (r,1 − r)) = ((0,1), (1,0), (1,0)), ((1,0), (0,1), (0,1))
((
3
4
,
1
4)
, (0, 1),
(
1
2
,
1
2))
ԋशͷਓήʔϜΛಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
p
1 − p
p
1 − p
q 1 − q q 1 − q
r 1 − r
(a2, b1, c1) (a1, b2, c2)

4. ‫ྗڠ‬ήʔϜͷ࿮૊ΈͰߟ͑Δ֤ϓϨΠϠʔ͸֤ઓུΛબͿ͜ͱʹ߆ଋྗͷ͋Δ߹ҙ͕Մೳ
ఏ‫·ؚʹ಺ܞ‬ΕΔརಘͷ࿨Λಛੑؔ਺ͷ஋ͱ͢Δ
ͨͩ͠ ఏ‫֎ܞ‬ͷϓϨΠϠʔ͕࠷ѱͳߦಈΛऔΔͱߟ͑ɺ
ͦͷத͔Βఏ‫࠷ͯͬͱʹܞ‬΋େ͖͘ͳΔ஋Λಛੑؔ਺஋ͱ͢ΔʢϚοΫεϛχʣ
ԋशࢀর
શϓϨΠϠʔ͕‫͢ྗڠ‬ΔͷͰ͋Ε͹ ઓུͷ૊ ΛऔΕ͹ རಘͷ࿨ͱͯ͠
(a2, b1, c1)
v({1,2,3}) = 6
ԋशͷਓήʔϜΛಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

5. ‫ྗڠ‬ήʔϜͷ࿮૊ΈͰߟ͑Δ֤ϓϨΠϠʔ͸֤ઓུΛબͿ͜ͱʹ߆ଋྗͷ͋Δ߹ҙ͕Մೳ
ఏ‫·ؚʹ಺ܞ‬ΕΔརಘͷ࿨Λಛੑؔ਺ͷ஋ͱ͢Δ
ͨͩ͠ ఏ‫֎ܞ‬ͷϓϨΠϠʔ͕࠷ѱͳߦಈΛऔΔͱߟ͑ɺ
ͦͷத͔Βఏ‫࠷ͯͬͱʹܞ‬΋େ͖͘ͳΔ஋Λಛੑؔ਺஋ͱ͢ΔʢϚοΫεϛχʣ
ϓϨΠϠʔ ͕‫͢ྗڠ‬Δ͜ͱΛߟ͑Δઓུͷ૊ ΛͱΔͱ͖ ϓϨΠϠʔ͸
ϓϨΠϠʔ ͷརಘͷ࿨Λ࠷খͱ͢Δͱͯ͠ ΛͱΔͱߟ͑Δ
͜ͷͱ͖ϓϨΠϠʔ ͷརಘͷ࿨͸
(a1, b1)
c2
ԋशͷਓήʔϜΛಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

6. ‫ྗڠ‬ήʔϜͷ࿮૊ΈͰߟ͑Δ֤ϓϨΠϠʔ͸֤ઓུΛબͿ͜ͱʹ߆ଋྗͷ͋Δ߹ҙ͕Մೳ
ఏ‫·ؚʹ಺ܞ‬ΕΔརಘͷ࿨Λಛੑؔ਺ͷ஋ͱ͢Δ
ͨͩ͠ ఏ‫֎ܞ‬ͷϓϨΠϠʔ͕࠷ѱͳߦಈΛऔΔͱߟ͑ɺ
ͦͷத͔Βఏ‫࠷ͯͬͱʹܞ‬΋େ͖͘ͳΔ஋Λಛੑؔ਺஋ͱ͢ΔʢϚοΫεϛχʣ
ϓϨΠϠʔ ͕‫͢ྗڠ‬Δ͜ͱΛߟ͑Δ
ઓུͷ૊ ΛͱΔͱ͖ རಘͷ࿨͸ઓུͷ૊ ΛͱΔͱ͖ རಘͷ࿨͸
ઓུͷ૊ ΛͱΔͱ͖ རಘͷ࿨͸ઓུͷ૊ ΛͱΔͱ͖ རಘͷ࿨͸
Ώ͑ʹ ϓϨΠϠʔ ͕‫ͯ͠ྗڠ‬ಘΒΕΔརಘͷ࿨ͷ࠷େ஋͸
(a1, b1) (a1, b2)
(a2, b1) (a2, b2)
v({1,2}) = 1
ԋशͷਓήʔϜΛಛੑؔ਺‫ܗ‬ήʔϜ΁
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

7. ਓήʔϜʹ͓͍ͯ ઓུ‫ܗ‬ήʔϜ͔Βಛੑؔ਺‫ܗ‬ήʔϜΛ࡞Γ ҎԼͷ໰͍ʹ౴͑Α
༏Ճ๏ੑΛຬͨ͢ήʔϜ͔ʁ
ίΞΛਤղͤΑ
ίΞΛ‫ٻ‬ΊΔ
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

8. v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4
ίΞΛ‫ٻ‬ΊΔ
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

9. v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2
ίΞΛ‫ٻ‬ΊΔ
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

10. v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2
v({1}) = 0 v({2}) = 0 v({3}) = 0
ίΞΛ‫ٻ‬ΊΔ
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

11. ಛੑؔ਺ͷੑ࣭
ఆٛɿ༏Ճ๏ੑ
ಛੑؔ਺ ͕༏Ճ๏ੑΛຬͨ͢ͱ͸
‫ަʹ͍ޓ‬ΘΒͳ͍೚ҙͷఏ‫ܞ‬ ͱ ʹରͯ͠
͕੒Γཱͭ͜ͱͰ͋Δ
v
S T S ∩ T = ∅
v(S ∪ T) ≥ v(S) + v(T)

12. ༏Ճ๏ੑΛຬͨ͢ήʔϜ͔ʁ͸͍
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2
v({1}) = 0 v({2}) = 0 v({3}) = 0
v({1,2,3}) ≥ v({1,2}) + v({3})
ίΞΛ‫ٻ‬ΊΔ
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

13. ༏Ճ๏ੑΛຬͨ͢ήʔϜʹ͓͚ΔίΞ
ఆཧ
ಛੑؔ਺‫ܗ‬ήʔϜ ʹ͓͍ͯ ͕༏Ճ๏ੑΛຬͨ͢ͱ͖
ίΞ͸
ఆٛɿఏ‫ܞ‬ ͷෆຬ
ಛੑؔ਺‫ܗ‬ήʔϜ ʹ͓͍ͯ ഑෼ ͱఏ‫ܞ‬ ΛͱΔͱ͖
Λ഑෼ ʹର͢Δఏ‫ܞ‬ ͷෆຬͱ͍͏
ίΞʹ‫·ؚ‬ΕΔ഑෼͸ ͱͳΔͨΊ
͢΂ͯͷఏ‫ܞ‬ ʹ͍ͭͯෆຬ͕ͳ͍ঢ়ଶͱ΋‫͑ݴ‬Δ
(N, v) v
C(v) = {x ∈
𝒜
(v)|
∑
i∈S
xi ≥ v(S), ∀S ⊂ N, S ≠ ∅, N}
S
(N, v) x ∈
𝒜
(v) S ⊆ N
e(S, x) = v(S) −
∑
i∈S
xi x S
e(S, x) ≤ 0
S ⊆ N
ఏ‫߹ܞ‬ཧੑ

14. ίΞΛਤղͤΑ
഑෼Λ ͱ͢Δͱ
શମ߹ཧੑ ‫ݸ‬ਓ߹ཧੑ ఏ‫߹ܞ‬ཧੑΛຬͨ͢ͷͰ
શମ߹ཧੑ
‫ݸ‬ਓ߹ཧੑ
ఏ‫߹ܞ‬ཧੑ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2
v({1}) = 0 v({2}) = 0 v({3}) = 0
x = (x1, x2, x3)
x1 + x2 + x3 = 6
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
x1 + x2 ≥ 1, x1 + x3 ≥ 4, x2 + x3 ≥ 2
ίΞΛ‫ٻ‬ΊΔ
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

15. ίΞΛਤղͤΑ
഑෼Λ ͱ͢Δͱ
શମ߹ཧੑ ‫ݸ‬ਓ߹ཧੑ ఏ‫߹ܞ‬ཧੑΛຬͨ͢ͷͰ
શମ߹ཧੑ
‫ݸ‬ਓ߹ཧੑ
ఏ‫߹ܞ‬ཧੑ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2
v({1}) = 0 v({2}) = 0 v({3}) = 0
x = (x1, x2, x3)
x1 + x2 + x3 = 6
x1 ≥ 0, x2 ≥ 0, x3 ≥ 0
x1 + x2 ≥ 1, x1 + x3 ≥ 4, x2 + x3 ≥ 2 ⇒ x3 ≤ 5, x2 ≤ 2, x1 ≤ 4
ίΞΛ‫ٻ‬ΊΔ
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
1
2 3
x1 ≤ 4
x2 ≤ 2
x3 ≤ 5
ίΞ

16. ήʔϜཧ࿦#4*$ԋश
ਓήʔϜ෼ੳɿίΞ