https://youtu.be/VXAmzFIi0AM

1. ήʔϜཧ࿦#4*$ԋश
ແ‫ݶ‬ճ‫܁‬Γฦ͠नਓͷδϨϯϚ

2. ੒෼ήʔϜ‫܁‬Γฦ͠ήʔϜʹ͓͚Δઓུ
‫܁‬Γฦ͠ήʔϜʹ͓͚Δརಘ‫ߧۉ‬
ແ‫ݶ‬ճ‫܁‬Γฦ͠नਓͷδϨϯϚ

3. ੒෼ήʔϜ
ϓϨΠϠʔͷू߹
ϓϨΠϠʔ ͷߦಈͷू߹
ϓϨΠϠʔ ͷརಘؔ਺ ͱͯ͠
‫܁‬Γฦ͠ήʔϜʹ͓͚Δઓུ
ͱͯ͠
ͱ͠
ͱ͢Δ
͜͜Ͱ
Λ ճ໨·Ͱͷཤྺͱ͍͏
ϓϨΠϠʔ ͷ ճ໨ͷߦಈ͸
ཤྺ Λ΋ͱʹܾఆ͞ΕΔ
͢ͳΘͪ Ͱ༩͑ΒΕΔ
͜ΕΑΓ‫܁‬Γฦ͠ήʔϜʹ͓͚ΔϓϨΠϠʔ ͷ७ਮઓུ͸ Ͱ༩͑ΒΕΔ
ϓϨΠϠʔ ͷ७ਮઓུͷू߹Λ ͱ͠
ϓϨΠϠʔશମͷઓུͷ૊ͷू߹Λ Ͱද͢
G = (N, {Si
}i∈N, {fi
}i∈N)
N
Si
i
fi
i S = S1
× ⋯ × Sn
fi
: S → ℜ
S = S1
× ⋯ × Sn
St−1 =
t−1
S × ⋯ × S S0 = {∅}
ht−1 = (s1, ⋯, st−1) ∈ St−1 t − 1
i t ht−1 xi
t : St−1 → Si
i xi
= (xi
t)∞
t=1
i X∞i
X∞
= X∞1
× ⋯ × X∞n
੒෼ήʔϜ‫܁‬Γฦ͠ήʔϜʹ͓͚Δઓུ
‫܁‬Γฦ͞ΕΔήʔϜͷ͜ͱ

4. རಘ
ׂҾҼࢠ ͱͯ͠
ΛׂҾརಘͱ͍͍
Λਖ਼‫ن‬Խརಘͱ͍͏
‫܁‬Γฦ͠ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬
ઓུ ͕φογϡ‫͋Ͱߧۉ‬Δͱ͸
͢΂ͯͷϓϨΠϠʔ ʹ͍ͭͯ
‫܁‬Γฦ͠ήʔϜʹ͓͚Δ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬
ઓུ ͕෦෼ήʔϜ‫׬‬શ‫͋Ͱߧۉ‬Δͱ͸
೚ҙͷ ͱ ճ໨·Ͱͷ೚ҙͷཤྺ ʹରͯ͠
ͱ ͔Βಋ͔ΕΔઓུͷ૊ ͕ ճ໨Ҏ߱ͷ෦෼ήʔϜͰφογϡ‫͋Ͱߧۉ‬Δ͜ͱΛ͍͏
δ, 0 δ 1
∞
∑
t=1
δt−1
fi
(st(x))
f∞i
δ (x) = (1 − δ)
∞
∑
t=1
δt−1
fi
(st(x))
x* = (x1
* , ⋯, xn
*) i
f∞i
δ (xi*
, x−i*
) ≥ f∞i
δ (xi
, x−i*
), ∀xi
∈ X∞i
x* = (x1
* , ⋯, xn
*) t = 1,2,⋯ t − 1 ht−1
x* ht−1 x*
ht−1
t
‫܁‬Γฦ͠ήʔϜʹ͓͚Δརಘ‫ߧۉ‬
ແ‫ݶ‬ճ‫܁‬Γฦ͠Ͱ͸
ศརͳͷͰ͜ΕΛ࢖͏

5. ӈͷΑ͏ͳनਓͷδϨϯϚ‫ܕ‬ͷήʔϜΛߟ͑Δ
τϦΨʔઓུΛߟ͑Δ͢ͳΘͪ
ʮճ໨͸$ΛͱΔճ໨Ҏ߱ʹؔͯ͠
લͷ‫ʹ͍ޓͰ·ظ‬$Λબͼଓ͚ͯΔ‫ݶ‬Γ
$ΛબͿ
ͦΕҎ֎Ͱ͋Ε͹
ͦΕҎ߱ͷ‫Ͱظ‬͸%Λબͼଓ͚Δʯ
֤ϓϨΠϠʔ͕τϦΨʔઓུΛͱΔͱͯ͠
͜ͷઓུͷ૊͕φογϡ‫ͳͱߧۉ‬ΔͨΊͷׂҾҼࢠ ͷ৚݅Λ‫ٻ‬ΊΑ
δ
ແ‫ݶ‬ճ‫܁‬Γฦ͠नਓͷδϨϯϚ
̍ʘ % $
%
$

6. ӈͷΑ͏ͳनਓͷδϨϯϚ‫ܕ‬ͷήʔϜΛߟ͑Δ
τϦΨʔઓུΛߟ͑Δ͢ͳΘͪ
ʮճ໨͸$ΛͱΔճ໨Ҏ߱ʹؔͯ͠
લͷ‫ʹ͍ޓͰ·ظ‬$Λબͼଓ͚ͯΔ‫ݶ‬Γ
$ΛબͿ
ͦΕҎ֎Ͱ͋Ε͹
ͦΕҎ߱ͷ‫Ͱظ‬͸%Λબͼଓ͚Δʯ
֤ϓϨΠϠʔ͕τϦΨʔઓུΛͱΔͱͯ͠
͜ͷઓུͷ૊͕φογϡ‫ͳͱߧۉ‬ΔͨΊͷׂҾҼࢠ ͷ৚݅Λ‫ٻ‬ΊΑ
‫ʹ͍ޓ‬τϦΨʔઓུΛͱ͍ͬͯΔͱ͖ͷརಘ͸ ͕બͼଓ͚ΒΕΔͨΊ
ͷརಘΛಘΔ
δ
(C, C)
f∞i
δ (x) = (1 − δ)
∞
∑
t=1
δt−1
5 = (1 − δ){5 + 5δ1
+ ⋯ + 5δt−2
+ 5δt−1
+ 5δt
+ 5δt+1
+ ⋯}( = 5)
ແ‫ݶ‬ճ‫܁‬Γฦ͠नਓͷδϨϯϚ
̍ʘ % $
%
$

7. ӈͷΑ͏ͳनਓͷδϨϯϚ‫ܕ‬ͷήʔϜΛߟ͑Δ
τϦΨʔઓུΛߟ͑Δ͢ͳΘͪ
ʮճ໨͸$ΛͱΔճ໨Ҏ߱ʹؔͯ͠
લͷ‫ʹ͍ޓͰ·ظ‬$Λબͼଓ͚ͯΔ‫ݶ‬Γ
$ΛબͿ
ͦΕҎ֎Ͱ͋Ε͹
ͦΕҎ߱ͷ‫Ͱظ‬͸%Λબͼଓ͚Δʯ
֤ϓϨΠϠʔ͕τϦΨʔઓུΛͱΔͱͯ͠
͜ͷઓུͷ૊͕φογϡ‫ͳͱߧۉ‬ΔͨΊͷׂҾҼࢠ ͷ৚݅Λ‫ٻ‬ΊΑ
‫ʹ͍ޓ‬τϦΨʔઓུΛͱ͍ͬͯΔͱ͖ͷརಘ͸ ͕બͼଓ͚ΒΕΔͨΊ
ͷརಘΛಘΔ
ϓϨΠϠʔʹؔͯ͠
‫ʹظ‬τϦΨʔઓུ͔Βҳ୤ͨ͠৔߹ ͭ·Γ ‫Ͱظ‬%ΛબͿઓུͷ৔߹
ͷརಘ͸ଟͯ͘΋
δ
(C, C)
f∞i
δ (x) = (1 − δ)
∞
∑
t=1
δt−1
5 = (1 − δ){5 + 5δ1
+ ⋯ + 5δt−2
+ 5δt−1
+ 5δt
+ 5δt+1
+ ⋯}( = 5)
t t
f∞1
δ (x) = (1 − δ){5 + 5δ1
+ ⋯ + 5δt−1
+ 6δt−1
+ (−2)δt
+ (−2)δt+1
+ ⋯}
ແ‫ݶ‬ճ‫܁‬Γฦ͠नਓͷδϨϯϚ
̍ʘ % $
%
$
ϓϨΠϠʔ͕%ΛબͿͷͰ
ଟͯ͘΋ͷརಘ

8. ӈͷΑ͏ͳनਓͷδϨϯϚ‫ܕ‬ͷήʔϜΛߟ͑Δ
τϦΨʔઓུΛߟ͑Δ͢ͳΘͪ
ʮճ໨͸$ΛͱΔճ໨Ҏ߱ʹؔͯ͠
લͷ‫ʹ͍ޓͰ·ظ‬$Λબͼଓ͚ͯΔ‫ݶ‬Γ
$ΛબͿ
ͦΕҎ֎Ͱ͋Ε͹
ͦΕҎ߱ͷ‫Ͱظ‬͸%Λબͼଓ͚Δʯ
֤ϓϨΠϠʔ͕τϦΨʔઓུΛͱΔͱͯ͠
͜ͷઓུͷ૊͕φογϡ‫ͳͱߧۉ‬ΔͨΊͷׂҾҼࢠ ͷ৚݅Λ‫ٻ‬ΊΑ
‫ʹ͍ޓ‬τϦΨʔઓུΛͱ͍ͬͯΔͱ͖ͷརಘ͸ ͕બͼଓ͚ΒΕΔͨΊ
ͷརಘΛಘΔ
ϓϨΠϠʔʹؔͯ͠
‫ʹظ‬τϦΨʔઓུ͔Βҳ୤ͨ͠৔߹ͷརಘ͸ଟͯ͘΋
τϦΨʔઓུ͕φογϡ‫ͳͱߧۉ‬Δ৚݅͸
ϓϨΠϠʔ΋ಉ༷ͳͷͰ͜Εׂ͕ҾҼࢠ ͷ৚݅Ͱ͋Δ
δ
(C, C)
f∞i
δ (x) = (1 − δ)
∞
∑
t=1
δt−1
5 = (1 − δ){5 + 5δ1
+ ⋯ + 5δt−2
+ 5δt−1
+ 5δt
+ 5δt+1
+ ⋯}( = 5)
t
f∞1
δ (x) = (1 − δ){5 + 5δ1
+ ⋯ + 5δt−1
+ 6δt−1
+ (−2)δt
+ (−2)δt+1
+ ⋯}
(1 − δ){5 + 5δ1
+ ⋯ + 5δt−1
+ 6δt−1
+ (−2)δt
+ (−2)δt+1
+ ⋯} ≤ (1 − δ){5 + 5δ1
+ ⋯ + 5δt−2
+ 5δt−1
+ 5δt
+ 5δt+1
+ ⋯}
⇔ 6δt−1
+ (−2)δt
+ (−2)δt+1
+ ⋯ ≤ 5δt−1
+ 5δt
+ 5δt+1
+ ⋯
⇔ δt−1
≤ δt
{7 + 7δ1
+ ⋯} =
7δt
1 − δ
⇔
1
8
≤ δ δ
ແ‫ݶ‬ճ‫܁‬Γฦ͠नਓͷδϨϯϚ
̍ʘ % $
%
$

9. ϑΥʔΫఆཧ
੒෼ήʔϜ ͷ‫ݸ͘ڧ‬ਓ߹ཧతͳ೚ҙͷߦಈͷ૊ ʹରͯ͠
কདྷརಘͷׂҾҼࢠ ͕
͕੒ΓཱͭͳΒ͹
ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ ͷφογϡ‫ߧۉ‬ ͕ଘࡏͯ͠
͕੒Γཱͭ
ࠓճͷ໰୊Ͱ͋Ε͹
͸‫ݸ͘ڧ‬ਓ߹ཧతͳߦಈͷ૊Ͱ͋Γ
G s = (s1
, ⋯, sn
) ∈ S δ
δ ≥
maxti fi
(ti
, s−i
) − fi
(s)
maxti fi(ti, s−i) − vi
, ∀i ∈ N
G∞
(δ) x* = (x1*
, ⋯, xn*
) s(x*) = (s, s, ⋯)
s = (C, C)
max
ti
∈{C,D}
fi
(ti
, s−i
) = max
ti
∈{C,D}
fi
(ti
, C) = fi
(D, C) = 6
fi
(s) = fi
(C, C) = 5
vi
= − 2
δ ≥
maxti fi
(ti
, s−i
) − fi
(s)
maxti fi(ti, s−i) − vi
=
6 − 5
6 − (−2)
=
1
8
ఆཧ
̍ʘ % $
%
$
ϓϨΠϠʔʹର͢Δ
ϛχϚοΫεߦಈ

10. ఆཧ
੒෼ήʔϜ ͷߦಈͷ૊ ͕φογϡ‫͋Ͱ఺ߧۉ‬Γ
Λຬͨ͢೚ҙͷߦಈͷ૊ ΛͱΔ͜ͷͱ͖
কདྷརಘͷׂҾҼࢠ ͕
͕੒ΓཱͭͳΒ͹
ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ ͷ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ ͕ଘࡏͯ͠
͕੒Γཱͭ
ࠓճͷ໰୊Ͱ͸
Ͱ͋Γ
Ͱ͋Ε͹
τϦΨʔઓུͷ૊͸෦෼ήʔϜ‫׬‬શ‫ͳʹߧۉ‬Δ
ͭ·Γ
೚ҙͷཤྺʹରͯ͠औΒΕ͍ͯΔߦಈ͕φογϡ‫ͳͱߧۉ‬Δ
ͬ͘͟Γ‫͑ݴ‬͹
‫ߧۉ‬ύεҎ֎ͷ
$
$
͔Βҳ୤ͨ͠‫ޙ‬ͷήʔϜͰ͸
%
%
͕औΒΕଓ͚
͜Ε͕φογϡ‫ͳͱߧۉ‬Δ͜ͱ͕ࣔͤΔ লུ
G s* = (s1*
, ⋯, sn*
) ∈ S
fi
(s) fi
(s*), ∀i ∈ N
s = (s1
, ⋯, sn
) ∈ S δ
δ ≥
maxti fi
(ti
, s−i
) − fi
(s)
maxti fi(ti, s−i) − fi(s*)
, ∀i ∈ N
G∞
(δ) x* = (x1*
, ⋯, xn*
) s(x*) = (s, s, ⋯)
fi
(s*) = vi 1
8
≤ δ
ఆཧ
̍ʘ % $
%
$

11. ήʔϜཧ࿦#4*$ԋश
ແ‫ݶ‬ճ‫܁‬Γฦ͠नਓͷδϨϯϚ
࣍ճɿԋश