https://youtu.be/fOnzqOK8Nyc

1. ήʔϜཧ࿦#4*$ୈճ
ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ

2. 1. ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜΛཧղ
ຊಈըͷ໨త

3. 1. ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
(1)ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜͷఆٛ
(2)φογϡ‫ߧۉ‬ͷఆٛ
(3)ϑΥʔΫఆཧ
໨࣍

4. Infinitely Repeated Game
ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ

5. ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜͷఆٛ
੒෼ήʔϜ
ϓϨΠϠʔͷू߹
ϓϨΠϠʔͷߦಈͷू߹
ϓϨΠϠʔͷརಘؔ਺ ͱͯ͠
‫܁‬Γฦ͠ήʔϜʹ͓͚Δઓུ
ͱͯ͠
ͱ͠
ͱ͢Δ
͜͜Ͱ
Λ ճ໨·Ͱͷཤྺͱ͍͏
ϓϨΠϠʔͷճ໨ͷߦಈ͸
ཤྺ Λ΋ͱʹܾఆ͞ΕΔ
͢ͳΘͪ Ͱ༩͑ΒΕΔ
͜ΕΑΓ‫܁‬Γฦ͠ήʔϜʹ͓͚ΔϓϨΠϠʔͷ७ਮઓུ͸ Ͱ༩͑ΒΕΔ
ϓϨΠϠʔͷ७ਮઓུͷू߹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
T → ∞

6. ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜͷఆٛ
‫܁‬Γฦ͠ήʔϜʹ͓͚Δརಘ
ׂҾҼࢠ ͱͯ͠
ΛׂҾརಘͱ͍͏
Λਖ਼‫ن‬Խརಘͱ͍͏
Ҏ্ΑΓ
ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜΛ
Ͱఆٛ͢Δ
δ, 0 δ 1
∞
∑
t=1
δt−1
fi
(st(x))
f∞i
δ (x) = (1 − δ)
∞
∑
t=1
δt−1
fi
(st(x))
G∞
(δ) = (N, {X∞i
}i∈N, {f∞i
δ }i∈N)
ແ‫ݶ‬౳ൺ‫਺ڃ‬
∞
∑
t=1
δt−1
=
1
1 − δ

7. φογϡ‫ߧۉ‬ͷఆٛ
ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬
ʹ͓͍ͯ
ઓུͷ૊ ͕φογϡ‫͋Ͱߧۉ‬Δͱ͸
͢΂ͯͷϓϨΠϠʔʹ͍ͭͯ
G∞
(δ) = (N, {X∞i
}i∈N, {f∞i
δ }i∈N) x* = (x1*
, ⋯, xn*
) i
f∞i
δ (xi*
, x−i*
) ≥ f∞i
δ (xi
, x−i*
), ∀xi
∈ X∞i

8. ϑΥʔΫఆཧ
ϛχϚοΫεߦಈͱϛχϚοΫεརಘ
੒෼ήʔϜ ʹ͓͍ͯ
Λຬͨ͢ϓϨΠϠʔҎ֎ͷߦಈͷ૊ Λʹର͢ΔϛχϚοΫεߦಈͱ͍͏
͜ͷ ΛϛχϚοΫεརಘΛ͍͍
Ͱද͢ ΛϛχϚοΫε఺ͱ͍͏
G
max
si
fi
(si
, ̂
s−i
) = min
s−i
max
si
fi
(si
, s−i
)
i ̂
s−i
i
max
si
fi
(si
, ̂
s−i
) vi
v = (v1
, ⋯, vn
)
̍ʘ E F G
B
C
D
ϓϨΠϠʔʹର͢ΔϛχϚοΫεߦಈG
ϓϨΠϠʔͷϛχϚοΫεརಘ

9. ϑΥʔΫఆཧ
ϛχϚοΫεߦಈͱϛχϚοΫεརಘ
੒෼ήʔϜ ʹ͓͍ͯ
Λຬͨ͢ϓϨΠϠʔҎ֎ͷߦಈͷ૊ Λʹର͢ΔϛχϚοΫεߦಈͱ͍͏
͜ͷ ΛϛχϚοΫεརಘΛ͍͍
Ͱද͢ ΛϛχϚοΫε఺ͱ͍͏
G
max
si
fi
(si
, ̂
s−i
) = min
s−i
max
si
fi
(si
, s−i
)
i ̂
s−i
i
max
si
fi
(si
, ̂
s−i
) vi
v = (v1
, ⋯, vn
)
̍ʘ E F G
B
C
D
ϓϨΠϠʔʹର͢ΔϛχϚοΫεߦಈG
ϓϨΠϠʔʹର͢ΔϛχϚοΫεߦಈC
ϓϨΠϠʔͷϛχϚοΫεརಘ
ϓϨΠϠʔͷϛχϚοΫεརಘ

10. ϑΥʔΫఆཧ
‫ݸ‬ਓ߹ཧత
੒෼ήʔϜ ʹ͓͚Δߦಈͷ૊ ͕
Λຬͨ͢ͱ͖
͸‫ݸ‬ਓ߹ཧతͰ͋Δͱ͍͍
Λຬͨ͢ͱ͖
͸‫ݸ͘ڧ‬ਓ߹ཧతͰ͋Δͱ͍͏
G s = (s1
, ⋯, sn
)
fi
(s) ≥ vi
, ∀i ∈ N s
fi
(s) vi
, ∀i ∈ N s
̍ʘ E F G
B
C
D
ϓϨΠϠʔʹର͢ΔϛχϚοΫεߦಈG
ϓϨΠϠʔʹର͢ΔϛχϚοΫεߦಈC
ϓϨΠϠʔͷϛχϚοΫεརಘ
ϓϨΠϠʔͷϛχϚοΫεརಘ
‫ݸ͘ڧ‬ਓ߹ཧత B
E
ͱ D
F

11. ϑΥʔΫఆཧ
ϑΥʔΫఆཧ
੒෼ήʔϜ ͷ‫ݸ͘ڧ‬ਓ߹ཧతͳ೚ҙͷߦಈͷ૊ ʹରͯ͠
কདྷརಘͷׂҾҼࢠ͕
͕੒ΓཱͭͳΒ͹
ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ ͷφογϡ‫ߧۉ‬ ͕ଘࡏͯ͠
͕੒Γཱͭ
˒ϑΥʔΫ͸ຽؒ఻ঝͷҙຯ
͜ͷఆཧͷҙຯ͢Δͱ͜Ζ͸
ׂҾҼࢠ͕େ͖͚Ε͹
‫ݸ‬ਓ߹ཧతͳߦಈͷ૊͕φογϡ‫ʹߧۉ‬Αͬͯୡ੒Ͱ͖Δ
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, ⋯)
δ
̍ʘ % $
%
$
‫ ͏߹͠ྗڠʹ͍ޓ‬$
$
Λ
औΓଓ͚ΔΑ͏ͳ‫͕ߧۉ‬ଘࡏ

12. ήʔϜཧ࿦#4*$ୈճ
ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
‫׬‬
ήʔϜཧ࿦#4*$ୈճʹͭͮ͘
See you next timeʂ

13. φογϡ‫ߧۉ‬෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷఆٛ
ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬
ʹ͓͍ͯ
ઓུͷ૊ ͕φογϡ‫͋Ͱߧۉ‬Δͱ͸
͢΂ͯͷϓϨΠϠʔʹ͍ͭͯ
Ҏ‫ޙ‬ͷ෦෼ήʔϜΛ Ͱද͢
·ͨ
೚ҙͷϓϨΠϠʔʹ͍ͭͯ
ઓུ ͕ ʹಋ͘ઓུΛ Ͱද͢
۩ମతʹ
ΛҰͭߟ͑ͯΈΔ͜ͷཤྺͰୡ੒͞Εͨ෦෼ήʔϜ ʹ͓͍ͯ
ճ໨ͷߦಈ͸
ճ·Ͱͷཤྺ ʹରͯ͠
ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜʹ͓͚Δ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬
ʹ͓͍ͯ
ઓུ ͕෦෼ήʔϜ‫׬‬શ‫͋Ͱߧۉ‬Δͱ͸
೚ҙͷճ਺ ͱ ճ·Ͱͷ೚ҙͷཤྺ ʹରͯ͠
֤ϓϨΠϠʔͷઓུ ͕ಋ͘ઓུ ͷ૊
͕
ͷφογϡ‫ͳͱߧۉ‬Δ͜ͱͰ͋Δ
G∞
(δ) = (N, {X∞i
}i∈N, {f∞i
δ }i∈N) x* = (x1*
, ⋯, xn*
) i
f∞i
δ (xi*
, x−i*
) ≥ f∞i
δ (xi
, x−i*
), ∀xi
∈ X∞i
ht−1 G∞
(δ)(ht−1)
xi
∈ X∞i
G∞
(δ)(ht−1) xi
|ht−1
ht−1 = (s1, s2, ⋯, st−1) G∞
(δ)(ht−1)
t′ t′− 1 ht′−1 = (s′
1, s′
2, ⋯, s′
t′−1) (xi
|ht−1)t′(ht′−1) = xi
(s1, s2, ⋯, st−1, s′
1, s′
2, ⋯, s′
t′−1)
G∞
(δ) = (N, {X∞i
}i∈N, {f∞i
δ }i∈N) x* = (x1*
, ⋯, xn*
)
t = 1,2,⋯ t − 1 ht−1 i xi*
xi*
|ht−1
x*|ht−1 = (x1*
|ht−1, ⋯, xn*
|ht−1) G∞
(δ)(ht−1)

14. ‫׬‬શϑΥʔΫఆཧ
ఆཧ
੒෼ήʔϜ ͷߦಈͷ૊ ͕φογϡ‫͋Ͱ఺ߧۉ‬Γ
Λຬͨ͢೚ҙͷߦಈͷ૊ ΛͱΔ͜ͷͱ͖
কདྷརಘͷׂҾҼࢠ͕
͕੒ΓཱͭͳΒ͹
ແ‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ ͷ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ ͕ଘࡏͯ͠
͕੒Γཱͭ
‫׬‬શϑΥʔΫఆཧ
੒෼ήʔϜ ͷ‫ݸ͘ڧ‬ਓ߹ཧతͳརಘϕΫτϧΛୡ੒͢ΔΑ͏ͳ
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ ͕ଘࡏ͠
͕੒Γཱͭ
ϑΥʔΫఆཧ͕
͋Δ৚݅ͷ΋ͱͰ͸෦෼ήʔϜ‫׬‬શ‫ͯؔ͠ʹߧۉ‬΋੒ཱ͢Δ ূ໌͸#4*$Ͱ͸͠ͳ͍
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, ⋯)
G
x* = (x1*
, ⋯, xn*
) s(x*) = (s, s, ⋯)

15. φογϡ‫ߧۉ‬෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷఆٛ
P1
P2
P2
%
$
%
$
%
$
̍ʘ % $
%
$
ແ‫ݶ‬ճ‫܁‬ฦήʔϜ
P1
P2
P2
%
$
%
$
%
$
੒෼ήʔϜ
P1
P2
P2
%
$
%
$
%
$
G∞
(δ)(h1) = G∞
(δ)({DD})