More Related Content Similar to ゲーム理論BASIC 演習3 -安定集合を求める- (20) More from ssusere0a682 (20) ゲーム理論BASIC 演習3 -安定集合を求める-3. ҆ఆू߹ͷఆٛͷ֬ೝ
ఆٛɿͷࢧ
̎ͭͷ ͱఏܞ ΛͱΔ࣍ͷ͕̎݅ຬͨ͞ΕΔͱ͖
Λ Λ௨ͯ͠ࢧ͢Δͱ͍͍
ͱॻ͘
̎ͭͷ ʹରͯ͠
͋Δఏܞ ͕ଘࡏͯ͠ ͕Γཱͭͱ͖
୯ʹ Λࢧ͢Δͱ͍͍
ͱॻ͘
x, y S ⊆ N
x y S x domS y
v(S) ≥
∑
i∈S
xi
xi yi, ∀i ∈ S
x, y S x domS y
x y x dom y
͕ఏܞ Ͱ͑Δ
x S
ఏܞ ʹ·ؚΕΔϓϨΠϠʔʹͱͬͯ
͕ΑΓ·͍͠
S
x
7.
͜ͷಛੑؔΛ༻͍ͯ
҆ఆू߹ΛٻΊͯΈΑ͏
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2
v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|5 ≥ x3, 3 ≥ x2, 4 ≥ x1}
લճͷ ͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
φογϡߧۉ
φογϡߧۉ
1
2 3
x1 ≤ 4
x2 ≤ 3
x3 ≤ 5
ίΞ
9.
͜ͷಛੑؔΛ༻͍ͯ
҆ఆू߹ΛٻΊͯΈΑ͏
͕
ίΞʹଐ͢Δʹࢧ͞ΕΔ͜ͱΛࣔ͢
͓ΑͼίΞʹ·ؚΕΔ
ͨͩ͠
Λߟ͑Ε
ͱͳΓ
ʹࢧ͞ΕΔ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2
v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|5 ≥ x3, 3 ≥ x2, 4 ≥ x1}
x = (x1, x2, x3), x2 + x3 2 4 x1 ≤ 6
S = {2,3}
y = (y1, y2, y3) = (4, x2 + ϵ, x3 + ϵ′

)
ϵ 0, ϵ′

0, x2 + ϵ + x3 + ϵ′

= 2
v({2,3}) = 2 ≥ 2 = y2 + y3
yi xi, ∀i ∈ {2,3}
x y
લճͷ ͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
φογϡߧۉ
φογϡߧۉ
1
2 3
x1 ≤ 4
x2 ≤ 3
x3 ≤ 5
ίΞ
͕ఏܞ Ͱ͑Δ
y {2,3}
ఏܞ ʹ·ؚΕΔϓϨΠϠʔ
͕·͍͠
{2,3}
y
10.
͜ͷಛੑؔΛ༻͍ͯ
҆ఆू߹ΛٻΊͯΈΑ͏
͕
ίΞʹଐ͢Δʹࢧ͞ΕΔ͜ͱΛࣔ͢
͓ΑͼίΞʹ·ؚΕΔ
ͨͩ͠
Λߟ͑Ε
ͱͳΓ
ʹࢧ͞ΕΔ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2
v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|5 ≥ x3, 3 ≥ x2, 4 ≥ x1}
x′

= (x′

1, x′

2, x′

3), x′

1 + x′

3 3 3 x′

2 ≤ 6
S = {1,3}
y = (y1, y2, y3) = (x′

1 + ϵ, 3, x′

3 + ϵ′

)
ϵ 0, ϵ′

0, x′

1 + ϵ + x′

3 + ϵ′

= 3
v({1,3}) = 3 ≥ 3 = y1 + y3
yi x′

i, ∀i ∈ {1,3}
x′

y
લճͷ ͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
φογϡߧۉ
φογϡߧۉ
1
2 3
x1 ≤ 4
x2 ≤ 3
x3 ≤ 5
ίΞ
͕ఏܞ Ͱ͑Δ
y {1,3}
ఏܞ ʹ·ؚΕΔϓϨΠϠʔ
͕·͍͠
{1,3}
y
11.
͜ͷಛੑؔΛ༻͍ͯ
҆ఆू߹ΛٻΊͯΈΑ͏
͕
ίΞʹଐ͢Δʹࢧ͞ΕΔ͜ͱΛࣔ͢
͓ΑͼίΞʹ·ؚΕΔ
ͨͩ͠
Λߟ͑Ε
ͱͳΓ
ʹࢧ͞ΕΔ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2
v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|5 ≥ x3, 3 ≥ x2, 4 ≥ x1}
x′

′

= (x′

′

1, x′

′

2, x′

′

3), x′

′

1 + x′

′

2 1 5 x′

′

3 ≤ 6
S = {1,2}
y = (y1, y2, y3) = (x′

′

1 + ϵ, x′

′

2 + ϵ′

,5)
ϵ 0, ϵ′

0, x′

′

1 + ϵ + x′

′

2 + ϵ′

= 1
v({1,2}) = 1 ≥ 1 = y1 + y2
yi x′

′

i , ∀i ∈ {1,2}
x′

′

y
લճͷ ͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
φογϡߧۉ
φογϡߧۉ
1
2 3
x1 ≤ 4
x2 ≤ 3
x3 ≤ 5
ίΞ
͕ఏܞ Ͱ͑Δ
y {1,2}
ఏܞ ʹ·ؚΕΔϓϨΠϠʔ
͕·͍͠
{1,2}
y