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ゲーム理論 BASIC 演習72 -3人ゲーム分析:仁-
- 3. ਔͷఆٛͷ֬ೝ
ఆٛɿఏܞ ͷෆຬ
ಛੑؔܗήʔϜ ʹ͓͍ͯ ͱఏܞ ΛͱΔͱ͖
Λ ʹର͢Δఏܞ ͷෆຬͱ͍͏
ʹର֤ͯ͠ఏܞ ͷෆຬ Λ
େ͖ͳͷ͔ΒॱʹฒͨϕΫτϧΛ ͱද͢
ͨͩ͠ ͯ͢ͷʹରͯ͠ ͱ ʹؔͯ͠ෆຬͳͷͰআ֎ͯ͠ߟ͑Δ
Ώ͑ʹ ݸͷෆຬΛฒͨϕΫτϧʹͳΔ
S
(N, v) x ∈
𝒜
(v) S ⊆ N
e(S, x) = v(S) −
∑
i∈S
xi x S
x ∈
𝒜
(v) S ⊆ N e(S, x)
θ(x)
N ∅
θ(x) 2n
− 2
#4*$
·ͨԋशࢀর
- 4. ਔͷఆٛͷ֬ೝ
ఆٛɿड༰త
̎ͭͷ ʹରͯ͠ ͋Δ ͕ଘࡏͯ͠
ͱͳΔͱ͖ ΑΓड༰తͰ͋Δͱ͍͍ ͱද͢
ͷҙຯɿ ͩͱ ͱͳΔͨΊ
Ұఆ
x, y ∈
𝒜
(v) k, 1 ≤ k ≤ 2n
− 4
θl(x) = θl(y) ∀l = 1,⋯, k, θk+1(x) θk+1(y)
x y x ≫ y
2n
− 4 θl(x) = θl(y) ∀l = 1,⋯,2n
− 3 θ2n−2(x) = θ2n−2(y)
n−2
∑
l=1
θl(x) =
∑
S⊂N,S≠N,∅
(
v(S) −
∑
i∈S
xi)
=
∑
S⊂N,S≠N,∅
v(S) −
∑
S⊂N,S≠N,∅
∑
i∈S
xi
=
∑
S⊂N,∅
v(S) −
n−1
∑
|S|=1
n−1C|S|−1 ∑
i∈N
xi =
∑
S⊂N,∅
v(S) −
n−1
∑
|S|=1
n−1C|S|−1v(N)
#4*$
·ͨԋशࢀর
- 7. Λ ͱ͢ΔͳͷͰ
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
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
ਔΛٻΊΔ
- 8. Λ ͱ͢ΔͳͷͰ
࠷େෆຬΛ ͱͯ͠ ͦΕΛ࠷খԽ͢ΔΛߟ͑Δ ͢ͳΘͪҎԼͷઢܭܗըΛղ͘ɿ
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
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
M
min M
s . t . v({1,2}) − (x1 + x2) ≤ M
v({1,3}) − (x1 + x3) ≤ M
v({2,3}) − (x2 + x3) ≤ M
v({1}) − x1 ≤ M v({2}) − x2 ≤ M v({3}) − x3 ≤ M
x1 + x2 + x3 = 6, x1, x2, x3 ≥ 0
ਔΛٻΊΔ
ఏܞ ͷෆຬ
{1,2}
- 9. Λ ͱ͢ΔͳͷͰ
࠷େෆຬΛ ͱͯ͠ ͦΕΛ࠷খԽ͢ΔΛߟ͑Δ ͢ͳΘͪҎԼͷઢܭܗըΛղ͘ɿ
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
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
M
min M
s . t . 1 − (x1 + x2) ≤ M
4 − (x1 + x3) ≤ M
2 − (x2 + x3) ≤ M
−x1 ≤ M −x2 ≤ M −x3 ≤ M
x1 + x2 + x3 = 6, x1, x2, x3 ≥ 0
ਔΛٻΊΔ
min M
s . t . x3 ≤ M + 5 x2 ≤ M + 2 x1 ≤ M + 4
−x1 ≤ M −x2 ≤ M −x3 ≤ M
−3M ≤ x1 + x2 + x3 = 6 ≤ 11 + 3M, x1, x2, x3 ≥ 0
- 10. Λ ͱ͢ΔͳͷͰ
࠷େෆຬΛ ͱͯ͠ ͦΕΛ࠷খԽ͢ΔΛߟ͑Δ ͢ͳΘͪҎԼͷઢܭܗըΛղ͘ɿ
͜ͷ੍݅ʹ͓͍ͯ ͕ଘࡏ͢ΔͨΊͷ݅
Ҏ্ͷ݅Λຬͨ͢தͰ ࠷খͱͳΔ ͜ͷͱ͖
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
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
M
min M
s . t . −M ≤ x1 ≤ M + 4 −M ≤ x2 ≤ M + 2 −M ≤ x3 ≤ M + 5 −3M ≤ x1 + x2 + x3 = 6 ≤ 11 + 3M, x1, x2, x3 ≥ 0
x = (x1, x2, x3)
−M ≤ M + 4 ⇔ − 2 ≤ M −M ≤ M + 2 ⇔ − 1 ≤ M −M ≤ M + 5 ⇔ −
5
2
= −
15
6
≤ M
−3M ≤ 6 ⇔ − 2 ≤ M 6 ≤ 11 + 3M ⇔ −
5
3
= −
10
6
≤ M
M −1 x2 = 1
ਔΛٻΊΔ
−2
−
5
2
−1
M
−
5
3
- 11. Λ ͱ͢ΔͳͷͰ
࠷େෆຬ ͳͷͰ ֤ఏܞͷෆຬ
ෆຬΛฒΔͱ
Ώ͑ʹ൪ʹେ͖͍ෆຬ Λߟ͑ ࠷খԽΛߟ͑Δ
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
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
M = − 1 x2 = 1
1 − (x1 + x2) = − x1 = ?
4 − (x1 + x3) = − 2 + x2 = − 1
2 − (x2 + x3) = 1 − x3 = ?
−x1 = ? −x2 = − 1 −x3 = ?
(−1, − 1, θ3(x), θ4(x), θ5(x), θ6(x))
M′

ਔΛٻΊΔ
−2
−
5
2
−1
M
−
5
3
ఏܞ ͷෆຬ֬ఆ
{1,3}
ఏܞ ͷෆຬ֬ఆ
{2}
- 12. Λ ͱ͢ΔͳͷͰ
Λલఏͱ͠ ൪ʹେ͖͍ෆຬΛ ͱͯ͠ ͦΕΛ࠷খԽ͢ΔΛߟ͑Δ
͢ͳΘͪҎԼͷઢܭܗըΛղ͘ɿ
݅Λཧ͢Δͱ
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
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
x2 = 1 M′

( − 1)
min M′

s . t . 1 − (x1 + x2) = − x1 ≤ M′

⇔ x3 − 5 ≤ M′

2 − (x2 + x3) = 1 − x3 ≤ M′

⇔ x1 − 4 ≤ M′

−x1 ≤ M′

−x3 ≤ M′

x1 + 1 + x3 = 6, x1, x3 ≥ 0
−M′

≤ x1 ≤ M′

+ 4 1 − M′

≤ x3 ≤ M′

+ 5 1 − 2M′

≤ x1 + x3 = 5 ≤ 2M′

+ 9
ਔΛٻΊΔ
- 13. Λ ͱ͢ΔͳͷͰ
Λલఏͱ͠ ൪ʹେ͖͍ෆຬΛ ͱͯ͠ ͦΕΛ࠷খԽ͢ΔΛߟ͑Δ
͜ΕΒ݅ʹ͓͍ͯ ͕ଘࡏ͢ΔͨΊʹ
Ҏ্ͷ݅Λຬͨ͢தͰ ࠷খͱͳΔ ͜ͷͱ͖ ͱͳΔ
Αͬͯ ਔ
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
(x1, x2, x3) x1 + x2 + x3 = v({1,2,3}) = 6 xi ≥ v({i}) = 0, ∀i ∈ {1,2,3}
x2 = 1 M′

( − 1)
−M′

≤ x1 ≤ M′

+ 4 1 − M′

≤ x3 ≤ M′

+ 5 1 − 2M′

≤ x1 + x3 = 5 ≤ 2M′

+ 9
x1, x3
−M′

≤ M′

+ 4 ⇔ − 2 ≤ M′

1 − M′

≤ M′

+ 5 ⇔ − 2 ≤ M′

1 − 2M′

≤ 5 ⇔ − 2 ≤ M′

5 ≤ 2M′

+ 9 ⇔ − 2 ≤ M′

M′

−2 x1 = 2 x3 = 3
(x1, x2, x3) = (2, 1, 3)
ਔΛٻΊΔ