湖水汚水ゲーム：ナッシュ均衡, 混合戦略まで考えた支配戦略

1. ήʔϜཧ࿦#4*$ԋश
‫ބ‬ਫԚਫήʔϜɿφογϡ‫ߧۉ‬
ࠞ߹ઓུ·Ͱߟ͑ͨࢧ഑ઓུ

2. ‫ބ‬ਫԚਫήʔϜ
φογϡ‫ߧۉ‬Λ‫ٻ‬ΊΔ
ࠞ߹ઓུ·Ͱߟ͑ͨࢧ഑ઓུ

3. ͱ͋Δ‫ބ‬ͷपΓʹ ͭͷ޻৔͕͋Δɽ
͜ΕΒͷ޻৔͸੡඼ੜ࢈͢Δʹ͋ͨͬͯ
Ԛਫ͕ੜ͡ΔͨΊ
ԚਫΛॲཧ͢Δඞཁ͕͋Δ͕
͜Ε͸ٛ຿Ͱ͸ͳ͍ͱ͢Δ
ͯ͞
֤޻৔͕‫͢ྲྀʹބ‬ԚਫΛড়Խॲཧ͢Δʹ͸ ͚ͩඅ༻͕͔͔Δ
‫ݸ‬ͷ޻৔͕ԚਫΛॲཧ͠ͳ͍Ͱ‫ͯ΂͢
ͱ͢ྲྀʹބ‬ͷ‫ʹۀا‬ରͯ͠ඃ֐͕ੜ͡Δ
ͦΕ͸ֹۚ‫Ͱࢉ׵‬ ͷඅ༻ͱͳΔ
͜ͷͱ͖
֤޻৔͸ԚਫΛॲཧ͢ΔͩΖ͏͔ʁॲཧ͢Δʁ͠ͳ͍ʁ
ઓུ‫ܗ‬ήʔϜΛߟ͑
φογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ಛੑؔ਺‫ܗ‬ήʔϜͱͯ͠ද‫
͠ݱ‬ίΞΛ‫ٻ‬ΊΑʢ࣍ճʣ
͋Δఏ‫ܞ‬Λߟ͑ͨ৔߹ʹ
ఏ‫·ؚʹܞ‬Εͳ͍ϓϨΠϠʔ͸
ͦͷఏ‫࠷ͯͬͱʹܞ‬ѱͳߦಈΛͱΔͱͯ͠ߟ͑ͯΑ͍ͦͷ΋ͱͰఏ‫ܞ‬͸࠷ળͷߦಈΛͱΔͱ͢Δ
3
100
u
60u
‫ބ‬ਫԚਫήʔϜ

4. ͯ͞
֤޻৔͕‫͢ྲྀʹބ‬ԚਫΛড়Խॲཧ͢Δʹ͸ ͚ͩඅ༻͕͔͔Δ
‫ݸ‬ͷ޻৔͕ԚਫΛॲཧ͠ͳ͍Ͱ‫ͯ΂͢
ͱ͢ྲྀʹބ‬ͷ‫ʹۀا‬ରͯ͠ඃ֐͕ੜ͡ΔͦΕ͸ֹۚ‫Ͱࢉ׵‬ ͷඅ༻ͱͳΔ
ઓུ‫ܗ‬ήʔϜΛߟ͑
φογϡ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ
΋ಉ༷ҎԼརಘදࢀর
100
u 60u
N = {1,2,3} Si
= (clean, no action) = (c, n) i ∈ N
f1
(c, c, c) = − 100
f1
(c, n, c) = f1
(c, c, n) = − 100 − 1 ⋅ 60 = − 160
f1
(c, n, n) = − 100 − 2 ⋅ 60 = − 220
φογϡ‫ߧۉ‬Λ‫ٻ‬ΊΔ
͕D
1ʘ2 D O
D
O
͕O
1ʘ2 D O
D
O
f1
(n, c, c) = 0 − 1 ⋅ 60 = − 60
f1
(n, n, c) = f1
(n, c, n) = 0 − 2 ⋅ 60 = − 120
f1
(n, n, n) = 0 − 3 ⋅ 60 = − 180
φογϡ‫ ߧۉ‬O
O
O

5. ࠓճ
७ઓུ·Ͱߟ͕͑ͨ
ࠞ߹ઓུ·Ͱߟ͑Δඞཁ͸ͳ͍͔ʁ
֤ϓϨΠϠʔ͕ࢧ഑ઓུΛ࣋ͭ৔߹
ࠞ߹ઓུ·Ͱߟ͑ͯ΋
ͦͷࢧ഑ઓུͷ૊ ࢧ഑ઓུ‫ͱߧۉ‬΋͍͏
ͷΈ͕φογϡ‫ͳʹߧۉ‬Δ
΋͠
ࢧ഑ઓུ͕͋Δͱ͢Ε͹
ͦΕ͸७ઓུͰ͋Δ
‫͑׵͍ݴ‬Ε͹
ͭҎ্ͷ७ઓུʹର͠ਖ਼ͷ֬཰Λ༩͑ΔΑ͏ͳ
ࠞ߹ઓུ͸
ࢧ഑ઓུʹͳΒͳ͍
w w
ࠞ߹ઓུ·Ͱߟ͑ͨࢧ഑ઓུ
͕D
1ʘ2 D O
D
O
͕O
1ʘ2 D O
D
O
φογϡ‫ ߧۉ‬O
O
O
ϓϨΠϠʔ ͷ̎ͭͷࠞ߹ઓུ ʹ͍ͭͯ
Λຬͨ͢ͱ͖ ͸ Λࢧ഑͢Δͱ͍͏
ŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠ
ϓϨΠϠʔ ͷઓུ ʹ͍ͭͯ ʹରͯ͠
Λຬͨ͢ͱ͖ Λࢧ഑ઓུͱ͍͏
i pi
, p′

i
∈ Pi
Fi
(pi
, p−i
) Fi
(p′

i
, p−i
), ∀p−i
∈ P−i
pi
p′

i
i pi
∈ Pi
∀p′

i
∈ Pi
{pi
}
Fi
(pi
, p−i
) Fi
(p′

i
, p−i
), ∀p−i
∈ P−i
pi
ࠞ߹ઓུͷࢧ഑ઓུ

6. ֤ϓϨΠϠʔ͕ࢧ഑ઓུΛ࣋ͭ৔߹
ࠞ߹ઓུ·Ͱߟ͑ͯ΋
ͦͷࢧ഑ઓུͷ૊ ࢧ഑ઓུ‫ͱߧۉ‬΋͍͏
ͷΈ͕φογϡ‫ͳʹߧۉ‬Δ
͕ࢧ഑ઓུ‫͢ͱߧۉ‬Δ
φογϡ‫͋Ͱߧۉ‬Δ͜ͱͷূ໌ɿ
φογϡ‫͢ͱ͍ͳͰߧۉ‬Δͱ
͋ΔϓϨΠϠʔ ʹ͍ͭͯ
ଞͷϓϨΠϠʔͷઓུͷ૊ ʹର͠
࠷ద൓ԠͱͳΔ ͕ଘࡏ͢Δͭ·Γ࠷ద൓Ԡͷఆ͔ٛΒ
ઓུͷ૊ ʹର͠
౰વ
ͱͳΔ͕
ࠓ ͸ࢧ഑ઓུ͔ͩΒ Ͱ͋Δ͸ͣͳͷͰໃ६
Ώ͑ʹ
ࢧ഑ઓུ‫ߧۉ‬ ͷ΋ͱͰ
͢΂ͯͷϓϨΠϠʔʹ͍ͭͯ
ଞͷϓϨΠϠʔͷઓུͷ૊ʹର͠࠷ద൓Ԡͱͳ͍ͬͯΔͭ·Γφογϡ‫͋Ͱߧۉ‬Δ
p ∈ P1
× ⋯ × Pn
i
p−i
∈ P−i
p′

i
∈ Pi
p−i
∈ P−i
Fi
(p′

i
, p−i
) ≥ Fi
(p′

′

i
, p−i
), ∀p′

′

i
∈ Pi
Fi
(p′

i
, p−i
) ≥ Fi
(pi
, p−i
) pi
Fi
(pi
, p−i
) Fi
(p′

i
, p−i
)
p ∈ P1
× ⋯ × Pn
ࠞ߹ઓུ·Ͱߟ͑ͨࢧ഑ઓུ
ϓϨΠϠʔ ͷ̎ͭͷࠞ߹ઓུ ʹ͍ͭͯ
Λຬͨ͢ͱ͖ ͸ Λࢧ഑͢Δͱ͍͏
ŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠ
ϓϨΠϠʔ ͷઓུ ʹ͍ͭͯ ʹରͯ͠
Λຬͨ͢ͱ͖ Λࢧ഑ઓུͱ͍͏
i pi
, p′

i
∈ Pi
Fi
(pi
, p−i
) Fi
(p′

i
, p−i
), ∀p−i
∈ P−i
pi
p′

i
i pi
∈ Pi
∀p′

i
∈ Pi
{pi
}
Fi
(pi
, p−i
) Fi
(p′

i
, p−i
), ∀p−i
∈ P−i
pi

7. ֤ϓϨΠϠʔ͕ࢧ഑ઓུΛ࣋ͭ৔߹
ࠞ߹ઓུ·Ͱߟ͑ͯ΋
ͦͷࢧ഑ઓུͷ૊ ࢧ഑ઓུ‫ͱߧۉ‬΋͍͏
ͷΈ͕φογϡ‫ͳʹߧۉ‬Δ
͕ࢧ഑ઓུ‫͢ͱߧۉ‬Δ
།ҰͰ͋Δ͜ͱͷূ໌ɿ
ଞʹφογϡ‫͕ߧۉ‬ଘࡏ͢Δͱ͠
ͦΕΛ
ͱ͢Δ
͢Δͱ
͋ΔϓϨΠϠʔ ʹ͍ͭͯ͸
Ͱ͋Δ
ଞͷϓϨΠϠʔͷઓུͷ૊ ʹର͠
͸࠷ద൓Ԡ͔ͩΒ
͕੒Γཱͭ
౰વ
ͱͳΔ͕
ࠓ ͸ࢧ഑ઓུ͔ͩΒ Ͱ͋Δ͸ͣͳͷͰໃ६
Ώ͑ʹ
ࢧ഑ઓུ‫ߧۉ‬Ҏ֎ͷφογϡ‫ߧۉ‬͸ଘࡏ͠ͳ͍
p ∈ P1
× ⋯ × Pn
p′

∈ P1
× ⋯ × Pn
i p′

i
≠ pi
p′

−i
∈ P−i
p′

i
∈ Pi
Fi
(p′

i
, p′

−i
) ≥ Fi
(p′

′

i
, p′

−i
), ∀p′

′

i
∈ Pi
Fi
(p′

i
, p′

−i
) ≥ Fi
(pi
, p′

−i
) pi
Fi
(pi
, p′

−i
) Fi
(p′

i
, p′

−i
)
ࠞ߹ઓུ·Ͱߟ͑ͨࢧ഑ઓུ
ϓϨΠϠʔ ͷ̎ͭͷࠞ߹ઓུ ʹ͍ͭͯ
Λຬͨ͢ͱ͖ ͸ Λࢧ഑͢Δͱ͍͏
ŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠ
ϓϨΠϠʔ ͷઓུ ʹ͍ͭͯ ʹରͯ͠
Λຬͨ͢ͱ͖ Λࢧ഑ઓུͱ͍͏
i pi
, p′

i
∈ Pi
Fi
(pi
, p−i
) Fi
(p′

i
, p−i
), ∀p−i
∈ P−i
pi
p′

i
i pi
∈ Pi
∀p′

i
∈ Pi
{pi
}
Fi
(pi
, p−i
) Fi
(p′

i
, p−i
), ∀p−i
∈ P−i
pi

8. ϓϨΠϠʔ ͷ̎ͭͷࠞ߹ઓུ ʹ͍ͭͯ
ɾɾɾ
Λຬͨ͢ͱ͖ ͸ Λࢧ഑͢Δͱ͍͏
ͷ෦෼͸
ҎԼͰஔ͖‫͑׵‬Δ͜ͱ͕Ͱ͖Δɿ
ɾɾɾ
ͭ·Γ
ଞͷϓϨΠϠʔͷઓུ͸ࠞ߹ઓུͰͳ͘
७ઓུΛߟ͑Ε͹े෼
ɾɾɾ
ͲΜͳ Λͱͬͨͱͯ͠΋
ࣜࠨล͕ਖ਼ʹͳΔ
΋͠
͋Δ Ͱ ͱͳΔͱ͢Δͱ
Λߏ੒͢Δઓུʹର͠
֤ϓϨΠϠʔ ͕
֬཰ Λ༩͑Δࠞ߹ઓུΛߟ͑Δͱ
ࣜӈล͸ ͱͳΓ
ʹໃ६
Ώ͑ʹ
‫͢ͱͩ
ʹٯ‬Ε͹
ΑΓ໌Β͔ʹ
ࠨล͸ͲΜͳ ʹ͍ͭͯ΋ਖ਼ʹͳΔ
i pi
, p′

i
∈ Pi
Fi
(pi
, p−i
) Fi
(p′

i
, p−i
), ∀p−i
∈ P−i
pi
p′

i
Fi
(pi
, s−i
) Fi
(p′

i
, s−i
), ∀s−i
∈ S−i
Fi
(pi
, p−i
) − Fi
(p′

i
, p−i
) =
∑
s−i
∈S−i
[∏
k≠i
pk
(sk
)
](
Fi
(pi
, s−i
) − Fi
(p′

i
, s−i
)
)
p−i
∈ P−i
s′

−i
∈ S−i
Fi
(pi
, s′

−i
) ≤ Fi
(p′

i
, s′

−i
) s′

−i
∈ S−i
k ≠ i
1 Fi
(pi
, s′

−i
) − Fi
(p′

i
, s′

−i
) ≤ 0
Fi
(pi
, s−i
) Fi
(p′

i
, s−i
), ∀s−i
∈ S−i
p−i
∈ P−i
ࠞ߹ઓུ·Ͱߟ͑ͨࢧ഑ઓུ
ϓϨΠϠʔ ͷ̎ͭͷࠞ߹ઓུ ʹ͍ͭͯ
Λຬͨ͢ͱ͖ ͸ Λࢧ഑͢Δͱ͍͏
ŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠ
ϓϨΠϠʔ ͷઓུ ʹ͍ͭͯ ʹରͯ͠
Λຬͨ͢ͱ͖ Λࢧ഑ઓུͱ͍͏
i pi
, p′

i
∈ Pi
Fi
(pi
, p−i
) Fi
(p′

i
, p−i
), ∀p−i
∈ P−i
pi
p′

i
i pi
∈ Pi
∀p′

i
∈ Pi
{pi
}
Fi
(pi
, p−i
) Fi
(p′

i
, p−i
), ∀p−i
∈ P−i
pi

9. ΋͠
ࢧ഑ઓུ͕͋Δͱ͢Ε͹
ͦΕ͸७ઓུͰ͋Δ
ϓϨΠϠʔ ͷઓུ ͕ࢧ഑ઓུͰ͋Δͱ͢Δ
ͭҎ্ͷ७ઓུʹରͯ͠ਖ਼ͷ֬཰Λ༩͍͑ͯΔͱ͢Δ
·ͨ
ͦͷਖ਼ͷ֬཰Λ༩͑ΒΕͯΔ७ઓུΛ ͱ͢Δ ˎ
͕ࢧ഑ઓུͰ͋Δ͔Β
ఆٛΑΓ ʹରͯ͠
͕੒Γཱͭ౰વࣗ਎ͷ७ઓུʹରͯ͠΋੒Γཱ͔ͭΒ
͕੒Γཱͭࠓ
্‫ͨͬ࠾Ͱه‬ ʹ͍ͭͯ΋੒ΓཱͭΘ͚͔ͩΒ
ʹର͠
྆ล ʹ͓͚Δ֬཰Λ͔͚ͯ࿨ΛͱΕ͹
͜͜Ͱ ˎ
͕ޮ͍ͯ
େখؔ܎͕อଘ͞ΕΔ͜ͱʹ஫ҙ
ͱͳΓ
ໃ६
Ώ͑ʹࢧ഑ઓུ͕͋ΔͷͳΒ
ͦΕ͸ͻͱͭͷ७ઓུʹͷΈਖ਼ͷ֬཰Λ༩͑Δ
i pi
∈ Pi
si
1, si
2, ⋯, si
N
pi
∀p′

i
∈ Pi
{pi
}
Fi
(pi
, s−i
) Fi
(p′

i
, s−i
), ∀s−i
∈ S−i
Fi
(pi
, s−i
) Fi
(si
, s−i
), ∀s−i
∈ S−i
si
1, si
2, ⋯, si
N k = 1,2,⋯, N
Fi
(pi
, s−i
) Fi
(si
k, s−i
), ∀s−i
∈ S−i
pi
N
∑
k=1
pi
k(si
k)Fi
(pi
, s−i
)
N
∑
k=1
pi
kFi
(si
k, s−i
), ∀s−i
∈ S−i
⇔ Fi
(pi
, s−i
) Fi
(pi
, s−i
), ∀s−i
∈ S−i
ࠞ߹ઓུ·Ͱߟ͑ͨࢧ഑ઓུ
ϓϨΠϠʔ ͷ̎ͭͷࠞ߹ઓུ ʹ͍ͭͯ
Λຬͨ͢ͱ͖ ͸ Λࢧ഑͢Δͱ͍͏
ŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠ
ϓϨΠϠʔ ͷઓུ ʹ͍ͭͯ ʹରͯ͠
Λຬͨ͢ͱ͖ Λࢧ഑ઓུͱ͍͏
i pi
, p′

i
∈ Pi
Fi
(pi
, s−i
) Fi
(p′

i
, s−i
), ∀s−i
∈ S−i
pi
p′

i
i pi
∈ Pi
∀p′

i
∈ Pi
{pi
}
Fi
(pi
, s−i
) Fi
(p′

i
, s−i
), ∀s−i
∈ S−i
pi
N
∑
k=1
pi
k(si
k) = 1

10. ήʔϜཧ࿦#4*$ԋश
‫ބ‬ਫԚਫήʔϜɿφογϡ‫ߧۉ‬
ࠞ߹ઓུ·Ͱߟ͑ͨࢧ഑ઓུ