1. The document describes a perturbed game Gεk that converges to the original game G as k approaches infinity. Gεk is defined by adding small perturbations εi(si) to the original probabilities in G. 2. As k increases, the perturbations εi(si) approach 0, such that the probabilities in Gεk converge to those in the original game G. This transforms Gεk into G in the limit. 3. The perturbed game framework is used to analyze properties like perfectness in extensive games. Adding small perturbations alleviates issues like mistakes or ties that could occur in the original game formulation.

2.
͜ͷࢿྉ͓Αͼಈը͸ɺҎԼͷಈըΛࢹௌࡁΈͰ͋Δ͜ͱΛલఏʹ࡞ΒΕ͍ͯ·͢ɻ
ˎήʔϜཧ࿦#"4*$γϦʔζ
ˎήʔϜཧ࿦/&95ஞ࣍‫ߧۉ‬
ࢀߟจ‫ݙ‬
(1) Mas-Colell, A., Whinston, M. D., Green, J. R. (1995). “Microeconomic theory”. Oxford university press.
(2) Reinhard Selten (1975). “A reexamination of the perfectness concept for equilibrium points in extensive
games”. International Journal of Game Theory 4: 25 - 55.

3. ໨࣍
ઁಈ‫׬‬શ‫ͱߧۉ‬͸Կ͔ʁ
มಈήʔϜ 1FSUVSCFEHBNF
ͷఆٛ
ઓུ‫ܗ‬ήʔϜͰͷઁಈ‫׬‬શ‫ߧۉ‬ͷఆٛ
༗༻ͳ໋୊
۩ମྫ

4. ઁಈ‫׬‬શ‫ ߧۉ‬USFNCMJOHIBOEQFSGFDUFRVJMJCSJVN
ͱ͸
೥ʹϥΠϯϋϧτɾθϧςϯʹΑͬͯఏএ͞Εͨ΋ͷ
φογϡ‫ߧۉ‬ͷਫ਼៛ԽͷҰͭͰ͋Δ
ϓϨΠϠʔͷz‫ޡ‬ΓzΛ૊ΈࠐΈ
ͦͷ΋ͱͰ
֤ϓϨΠϠʔ͕ͱΔઓུ͕࠷దͰ͋Γଓ͚Δ͔Ͳ͏͔
͜ͷߟ͑ͷ΋ͱͰఆٛ͞Εͨઁಈ‫׬‬શ‫͋Ͱߧۉ‬Ε͹
ऑࢧ഑͞ΕΔઓུΛ‫ؚ‬Ήφογϡ‫ߧۉ‬͸ഉআ͞ΕΔ
ઁಈ‫׬‬શ‫ͱߧۉ‬͸Կ͔ʁ
̍ʘ D E
B
C

5. มಈήʔϜ 1FSUVSCFEHBNF
ઓུ‫ܗ‬ήʔϜ ʹରͯ͠
มಈήʔϜ Λఆٛ͢Δ
͜͜Ͱ͢΂ͯͷϓϨΠϠʔ ͱͦͷ७ઓུ ʹରͯ͠
Λຬͨ͢ ΛબͿ
͜ͷͱ͖
֤ϓϨΠϠʔ͸ࣗ਎ͷ७ઓུ ʹ͍ͭͯ
গͳ͘ͱ΋ ͷ֬཰Ҏ্Ͱ࢖༻͠ͳ͚Ε͹͍͚ͳ͍
ʹ͍ͭͯ
֤ϓϨΠϠʔ͕ Λ࢖ͬͯ͠·͏ NJTUBLF
֬཰Ͱ͋Δ
G = (N, {Pi
}i∈N, {Fi
}i∈N) Gϵ = (N, {Pi
ϵ}i∈N, {Fi
}i∈N)
i si
∈ Si
ϵi
(si
) ∈ (0, 1)
∑
si
∈Si
ϵi
(si
) 1 ϵi
(si
)
Pi
ϵ = {pi
: pi
(si
) ≥ ϵi
(si
), ∀si
∈ Si
and
∑
si
∈Si
pi
(si
) = 1}
si
∈ Si
ϵi
(si
) 0
ϵi
(si
) 0 si
∈ Si
มಈήʔϜ 1FSUVSCFEHBNF
ͷఆٛ

6. ઓུ‫ܗ‬ήʔϜͰͷઁಈ‫׬‬શ‫ߧۉ‬ͷఆٛ
ઓུ‫ܗ‬ήʔϜ ʹ͓͚Δφογϡ‫ߧۉ‬ ͕
ઓུ‫ܗ‬ήʔϜʹ͓͚Δઁಈ‫׬‬શ‫͋Ͱߧۉ‬Δͱ͸
มಈήʔϜͷྻ
Ͱ
ͱͳΔΑ͏ͳ΋ͷ͕ଘࡏ͠
֤มಈήʔϜ ʹ͓͚Δφογϡ‫ߧۉ‬ͷྻ
͕
ͱͳΔͱ͖Λ͍͏
͜͜Ͱ ͸
Λҙຯ͢Δ
͜ͷఆٛͩͱ
֤มಈήʔϜͷφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΔඞཁ͕͋ΔͨΊ
࢖͍ͮΒ͍
G = (N, {Pi
}i∈N, {Fi
}i∈N) p = (p1
, ⋯, pn
)
Gϵk
= (N, {Pi
ϵk
}i∈N, {Fi
}i∈N) k = 1,2,⋯ Gϵk
→ G
Gϵk
= (N, {Pi
ϵk
}i∈N, {Fi
}i∈N) pk = (p1
k , ⋯, pn
k ) k = 1,2,⋯
pk → p
Gϵk
→ G lim
k→∞
ϵi
k(si
) = 0, ∀i ∈ N and ∀si
∈ Si
ઓུ‫ܗ‬ήʔϜͰͷઁಈ‫׬‬શ‫ߧۉ‬ͷఆٛ

7. ઓུ‫ܗ‬ήʔϜͰͷઁಈ‫׬‬શ‫ߧۉ‬ͷඞཁे෼৚݅
ઓུ‫ܗ‬ήʔϜ ʹ͓͚Δφογϡ‫ߧۉ‬ ͕
ઓུ‫ܗ‬ήʔϜʹ͓͚Δઁಈ‫׬‬શ‫͋Ͱߧۉ‬ΔͳΒ͹
‫׬‬શࠞ߹ઓུͷ૊ͷྻ
ͰҎԼͷ৚݅Λຬͨ͢΋ͷ͕ଘࡏ͢Δ
͔ͭ
͢΂ͯͷ ʹ͍ͭͯ
͸
ʹରͯ͠࠷ద൓ԠͱͳΔ
·ͨ
‫ٯ‬΋੒Γཱͭ
‫׬‬શࠞ߹ઓུͱ͸
͢΂ͯͷ७ઓུʹରͯ͠ਖ਼ͷ֬཰Λ༩͑Δ΋ͷ
G = (N, {Pi
}i∈N, {Fi
}i∈N) p = (p1
, ⋯, pn
)
pk = (p1
k , ⋯, pn
k ) k = 1,2,⋯
pk
k→∞
p
i ∈ N pi
p−i
k = (p1
k , ⋯, pi−1
k , pi+1
k ⋯, pn
k ) k = 1,2,⋯
༗༻ͳ໋୊
ఴࣈͳ͠ʹ஫ҙ

8. มಈήʔϜ Λߟ͑Δ
ͱ
ͱͳΔΑ͏ͳ
Λߟ͑Δ
ͨͩ͠
͜ͷͱ͖
ϓϨΠϠʔͷ‫ظ‬଴རಘ͸ Ͱ͋Δ
ಉ༷ʹ
ϓϨΠϠʔͷ‫ظ‬଴རಘ͸ Ͱ͋Δ
ʹ஫ҙ͢Δͱ
มಈήʔϜʹ͓͚Δφογϡ‫ߧۉ‬͸
ͱ͢Δͱ͖
ΑΓ
Gϵk
1 − ϵ1
k (b) ≥ p1
k ≥ ϵ1
k (a) k = 1,2,⋯
1 − ϵ2
k (d) ≥ p2
k ≥ ϵ2
k (c) k = 1,2,⋯ p1
k p2
k
ϵi
(si
) ∈ (0, 1)
∑
si
∈Si
ϵi
(si
) 1, ∀i ∈ {1,2}
F1
(p1
k , p2
k ) = 1 ⋅ p1
k ⋅ p2
k
F2
(p1
k , p2
k ) = 1 ⋅ p1
k ⋅ p2
k
1 1 − ϵ1
k (b) ≥ p1
k ≥ ϵ1
k (a) 0
1 1 − ϵ2
k (d) ≥ p2
k ≥ ϵ2
k (c) 0
((1 − ϵ1
k (b), ϵ1
k (b)) (1 − ϵ2
k (d), ϵ2
k (d)))
Gϵk
→ G lim
k→∞
ϵ1
k (a) = 0, lim
k→∞
ϵ1
k (b) = 0, lim
k→∞
ϵ1
k (c) = 0, lim
k→∞
ϵ1
k (d) = 0
((1 − ϵ1
k (b), ϵ1
k (b)) (1 − ϵ2
k (d), ϵ2
k (d)))
k→∞
((1,0) (1,0))
۩ମྫ
̍ʘ D E
B
C
p1
k
1 − p1
k
p2
k 1 − p2
k
p1
k
0 ϵ1
k (a)
1
1
1 − ϵ1
k (b)
ϵ2
k (c)
1 − ϵ2
k (d)
p2
k

9. ࠞ߹ઓུͰද‫͢ݱ‬Ε͹
͕ઁಈ‫׬‬શ‫͋Ͱߧۉ‬Δ͜ͱΛࣔ͢
ͱ
ΛͱΔͨͩ͠
Λຬͨ͢ͱ͢Δ
͜ͷͱ͖
·ͨ
͕
ʹରͯ͠
ͱͳΓ
࠷దͰ͋Δ
ಉ༷ʹ
͕
ʹରͯ͠
ͱͳΓ
࠷దͰ͋Δ
Ώ͑ʹ
͸ઁಈ‫׬‬શ‫͋Ͱߧۉ‬Δ
(a, d) p = (p1
, p2
) = ((1, 0), (1, 0))
p1
k ∈ (0,1) k = 1,2,⋯ p2
k ∈ (0,1) k = 1,2,⋯ p1
k
k→∞
1 p2
k
k→∞
1
((p1
k , 1 − p1
k ), (p2
k , 1 − p2
k ))
k→∞
((1, 0), (1, 0))
p1
p2
k ∈ (0,1) k = 1,2,⋯ F1
(p1
, p2
k ) = 1 ⋅ 1 ⋅ p2
k ≥ 1 ⋅ p′

1
⋅ p2
k , ∀p′

1
∈ P1
p2
p1
k ∈ (0,1) k = 1,2,⋯ F2
(p1
k , p2
) = 1 ⋅ p1
k ⋅ 1 ≥ 1 ⋅ p1
k ⋅ p′

2
, ∀p′

2
∈ P2
p = (p1
, p2
) = ((1, 0), (1, 0))
۩ମྫ
̍ʘ E F
B
C
p1
k
1 − p1
k
p2
k 1 − p2
k

10. ࠞ߹ઓུͰද‫͢ݱ‬Ε͹
͕ઁಈ‫׬‬શ‫ͱ͍͜ͳͰߧۉ‬Λࣔ͢
ͱ
Ͱ
Λຬͨ͢΋ͷΛ೚ҙʹͱΔ
͜ͷͱ͖
ͱͳΔ͕
͕
ʹରͯ͠
ͱͳΓ
࠷దͰͳ͍
Ώ͑ʹ
͸ઁಈ‫׬‬શ‫Ͱߧۉ‬͸ͳ͍
ϓϨΠϠʔ͸
ϓϨΠϠʔ͕‫
ͯͬޡ‬EΛબͿ͜ͱΛߟྀ͢ΔͨΊ
ϓϨΠϠʔʹͱͬͯCΛͱΔ͜ͱ͕࠷దʹͳΒͳ͘ͳΔ
(b, e) p = (p1
, p2
) = ((0, 1), (0, 1))
p1
k ∈ (0,1) k = 1,2,⋯ p2
k ∈ (0,1) k = 1,2,⋯ p1
k
k→∞
0 p2
k
k→∞
0
((p1
k , 1 − p1
k ), (p2
k , 1 − p2
k ))
k→∞
((0, 1), (0, 1))
p1
p2
k ∈ (0,1) k = 1,2,⋯ F1
(p1
, p2
k ) = 1 ⋅ 0 ⋅ p2
k 1 ⋅ p′

′

1
⋅ p2
k , ∃p′

′

1
∈ (0,1)
p = (p1
, p2
) = ((0, 1), (0, 1))
۩ମྫ
̍ʘ E F
B
C
p1
k
1 − p1
k
p2
k 1 − p2
k