This document describes a finite repeated game model. It defines the game as consisting of a finite number of players N, strategy sets for each player Si, and payoff functions fi for each player. The game is played repeatedly for a finite number of periods T. It also describes how player strategies can change over time based on the history of past play.

1. ήʔϜཧ࿦#4*$ୈճ
༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ

2. 1. ‫܁‬Γฦ͠ήʔϜΛཧղ
2. ‫܁‬Γฦ͠ήʔϜͷઓུ
རಘΛཧղ
ຊಈըͷ໨త

3. 1. ༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
(1)༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
(2)‫܁‬Γฦ͠ήʔϜʹ͓͚Δઓུ
(3)‫܁‬Γฦ͠ήʔϜʹ͓͚Δརಘ
(4)‫܁‬Γฦ͠ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬
(5)ఆཧͷ঺հ
(6)۩ମྫ
໨࣍

4. Finitely Repeated Game
༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ

5. ༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
˒ಉҰͷήʔϜ͕‫܁‬Γฦ͞ΕΔঢ়‫܁ʹگ‬Γฦ͠ήʔϜ
˒ա‫ڈ‬ͷߦಈ ཤྺ
Λ΋ͱʹߦಈΛબ୒͢Δ
P1
P2
P2
ࣗന
໧ൿ
(−2, − 2)
(6, − 4)
(−4, 6)
(5, 5)
ࣗന
໧ൿ
ࣗന
໧ൿ
ల։‫ܗ‬ήʔϜ
ઓུ‫ܗ‬ήʔϜ
̍ʘ ࣗന ໧ൿ
ࣗന
໧ൿ

6. ༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
˒ಉҰͷήʔϜ͕‫܁‬Γฦ͞ΕΔঢ়‫܁ʹگ‬Γฦ͠ήʔϜ
˒ա‫ڈ‬ͷߦಈ ཤྺ
Λ΋ͱʹߦಈΛબ୒͢Δ
P1
P2
P2
ࣗന
໧ൿ
(−2 + (−2)δ, − 2 + (−2)δ)
(−2 + 6δ, − 2 + (−4)δ)
(−2 + (−4)δ, − 2 + 6δ)
(−2 + 5δ, − 2 + 5δ)
ࣗന
໧ൿ
ࣗന
໧ൿ
̍ʘ ࣗന ໧ൿ
ࣗന
໧ൿ
ల։‫ܗ‬ήʔϜ
ઓུ‫ܗ‬ήʔϜ
P1
P2
P2
ࣗന
໧ൿ
ࣗന
໧ൿ
ࣗന
໧ൿ
(−2, − 2)
͸ׂҾҼࢠͰ͋Γ
͜ΕʹΑΓকདྷརಘΛ‫ࡏݱ‬Ձ஋ʹ‫ࢉ׵‬
δ 0 δ 1

7. ༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
˒ಉҰͷήʔϜ͕‫܁‬Γฦ͞ΕΔঢ়‫܁ʹگ‬Γฦ͠ήʔϜ
˒ա‫ڈ‬ͷߦಈ ཤྺ
Λ΋ͱʹߦಈΛબ୒͢Δ
P1
P2
P2
ࣗന
໧ൿ
(−2 + (−2)δ, − 2 + (−2)δ)
(−2 + 6δ, − 2 + (−4)δ)
(−2 + (−4)δ, − 2 + 6δ)
(−2 + 5δ, − 2 + 5δ)
ࣗന
໧ൿ
ࣗന
໧ൿ
̍ʘ ࣗന ໧ൿ
ࣗന
໧ൿ
ల։‫ܗ‬ήʔϜ
ઓུ‫ܗ‬ήʔϜ
P1
P2
P2
ࣗന
໧ൿ
ࣗന
໧ൿ
ࣗന
໧ൿ
(−2, − 2)
੒෼ήʔϜ
͸ׂҾҼࢠͰ͋Γ
͜ΕʹΑΓকདྷརಘΛ‫ࡏݱ‬Ձ஋ʹ‫ࢉ׵‬
δ 0 δ 1

8. ༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
˒ಉҰͷήʔϜ͕‫܁‬Γฦ͞ΕΔঢ়‫܁ʹگ‬Γฦ͠ήʔϜ
˒ա‫ڈ‬ͷߦಈ ཤྺ
Λ΋ͱʹߦಈΛબ୒͢Δ
P1
P2
P2
ࣗന
໧ൿ
ࣗന
໧ൿ
ࣗന
໧ൿ
̍ʘ ࣗന ໧ൿ
ࣗന
໧ൿ
ల։‫ܗ‬ήʔϜ
ઓུ‫ܗ‬ήʔϜ
P1
P2
P2
ࣗന
໧ൿ
ࣗന
໧ൿ
ࣗന
໧ൿ
(−2, − 2)
੒෼ήʔϜ
P1
P2
P2
ࣗന
໧ൿ
ࣗന
໧ൿ
ࣗന
໧ൿ
(−2 + 6δ + (−2)δ2
, − 2 + (−4)δ + (−2)δ2
)
(−2 + 6δ + 6δ2
, − 2 + (−4)δ + (−4)δ2
)
(−2 + 6δ + (−4)δ2
, − 2 + (−4)δ + 6δ2
)
(−2 + 6δ + 5δ2
, − 2 + (−4)δ + 5δ2
)

9. ༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
˒ಉҰͷήʔϜ͕‫܁‬Γฦ͞ΕΔঢ়‫܁ʹگ‬Γฦ͠ήʔϜ
˒ա‫ڈ‬ͷߦಈ ཤྺ
Λ΋ͱʹߦಈΛબ୒͢Δ
੒෼ήʔϜ
ϓϨΠϠʔͷू߹
ϓϨΠϠʔͷߦಈͷू߹
ϓϨΠϠʔͷརಘؔ਺ ͱͯ͠
੒෼ήʔϜΛ ճ
‫܁‬Γฦ͢ήʔϜΛߟ͑Δ
ઓུ͓ΑͼརಘΛఆٛ͢Δ
G = (N, {Si
}i∈N, {fi
}i∈N)
N
Si
i
fi
i S = S1
× ⋯ × Sn
fi
: S → ℜ
T T ∞

10. ‫܁‬Γฦ͠ήʔϜʹ͓͚Δઓུ
ճ໨ʹ͓͍ͯ
֤ϓϨΠϠʔ͸ ճ໨·Ͱʹબ͹ΕͨߦಈΛ͢΂ͯ஌্ͬͨͰߦಈΛબ୒͢Δ
‫܁‬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
t t − 1
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
t=1
i XTi
XT
= XT1
× ⋯ × XTn

11. ‫܁‬Γฦ͠ήʔϜʹ͓͚Δઓུ
ճ໨ʹ͓͍ͯ
֤ϓϨΠϠʔ͸ ճ໨·Ͱʹબ͹ΕͨߦಈΛ͢΂ͯ஌্ͬͨͰߦಈΛબ୒͢Δ
‫܁‬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
ʹରͯ͠
ճ໨
ճ໨
ճ໨
ʜ
ճ໨
t t − 1
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
t=1
i XTi
XT
= XT1
× ⋯ × XTn
x = (x1
, ⋯, xn
) ∈ XT
s1(x) = (x1
1(∅), ⋯, xn
1(∅)) s2(x) = (x1
2(s1(x)), ⋯, xn
2(s1(x))) s3(x) = (x1
3(s1(x), s2(x)), ⋯, xn
3(s1(x), s2(x)))
t st(x) = (x1
t (s1(x)⋯, st−1(x)), ⋯, xn
t (s1(x)⋯, st−1(x)))

12. ‫܁‬Γฦ͠ήʔϜʹ͓͚Δརಘ
ϓϨΠϠʔͷརಘ͸
୅දతͳ΋ͷͰ
ฏ‫ۉ‬རಘ·ͨ͸ׂҾརಘ͕͋Δ
‫܁‬Γฦ͠ήʔϜʹ͓͚Δརಘ
બ୒ͷ૊ͷྻ ͱ͢Δ
Λฏ‫ۉ‬རಘͱ͍͏
·ͨ
ׂҾҼࢠ ͱͯ͠
ΛׂҾརಘͱ͍͏
ฏ‫ۉ‬རಘͰརಘΛ༩͑ΔήʔϜΛ
ׂҾརಘͰརಘΛ༩͑ΔήʔϜΛ
s(x) = (st(x))T
t=1
¯
fTi
(x) =
1
T
T
∑
t=1
fi
(st(x))
δ, 0 δ 1
fTi
δ (x) =
T
∑
t=1
δt−1
fi
(st(x))
ḠT
= (N, {XTi
}i∈N, {¯
fTi
}i∈N)
GT
(δ) = (N, {XTi
}i∈N, {fTi
δ }i∈N)

13. ‫܁‬Γฦ͠ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬
ϓϨΠϠʔͷརಘ͸
୅දతͳ΋ͷͰ
ฏ‫ۉ‬རಘ·ͨ͸ׂҾརಘ͕͋Δ
‫܁‬Γฦ͠ήʔϜʹ͓͚Δརಘ
બ୒ͷ૊ͷྻ ͱ͢Δ
Λฏ‫ۉ‬རಘͱ͍͏
·ͨ
ׂҾҼࢠ ͱͯ͠
ΛׂҾརಘͱ͍͏
ฏ‫ۉ‬རಘͰརಘΛ༩͑ΔήʔϜΛ
ׂҾརಘͰརಘΛ༩͑ΔήʔϜΛ
s(x) = (st(x))T
t=1
¯
fTi
(x) =
1
T
T
∑
t=1
fi
(st(x))
δ, 0 δ 1
fTi
δ (x) =
T
∑
t=1
δt−1
fi
(st(x))
ḠT
= (N, {XTi
}i∈N, {¯
fTi
}i∈N)
GT
(δ) = (N, {XTi
}i∈N, {fTi
δ }i∈N)
‫܁‬Γฦ͠ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬
ઓུ ͕φογϡ‫͋Ͱߧۉ‬Δͱ͸
͢΂ͯͷϓϨΠϠʔʹ͍ͭͯ
ฏ‫ۉ‬རಘ
ׂҾརಘ
x* = (x1
* , ⋯, xn
*)
i
¯
fTi
(xi*
, x−i*
) ≥ ¯
fTi
(xi
, x−i*
), ∀xi
∈ XTi
fTi
δ (xi*
, x−i*
) ≥ fTi
δ (xi
, x−i*
), ∀xi
∈ XTi

14. ఆཧͷ঺հ
ఆཧ
੒෼ήʔϜ͕ͨͩҰͭͷφογϡ‫ߧۉ‬ Λ΋ͭͱ͢Δ͜ͷͱ͖ ճ‫܁‬Γฦ͠ήʔϜʹ͓͚Δ
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ ͸ͨͩҰͭଘࡏ͠
ߦಈͷ૊ Λ༩͑Δ
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷఆ͓ٛΑͼূ໌͸ิ଍
s* T
x* s(x*) = (
T
s*, s*, ⋯, s*)

15. ۩ମྫ
ϓϨΠϠʔͷ੒෼ήʔϜͷߦಈͷू߹
ཤྺͷू߹
S1
= {C, D}
S0 = {∅} S1 = {CC, CD, DC, DD}
S2 = {CC − CC, CD − CC, DC − CC, DD − CC, CC − CD, CD − CD, DC − CD, DD − CD,
CC − DC, CD − DC, DC − DC, DD − DC, CC − DD, CD − DD, DC − DD, DD − DD}
P1
P2
P2
%
$
%
$
%
$
̍ʘ % $
%
$
ճ‫܁‬ฦήʔϜ
P1
P2
P2
%
$
%
$
%
$
੒෼ήʔϜ
P1
P2
P2
(−2 + 6δ + (−2)δ2
, − 2 + (−4)δ + (−2)δ2
)
(−2 + 6δ + 6δ2
, − 2 + (−4)δ + (−4)δ2
)
(−2 + 6δ + (−4)δ2
, − 2 + (−4)δ + 6δ2
)
(−2 + 6δ + 5δ2
, − 2 + (−4)δ + 5δ2
)
%
$
%
$
%
$
ϓϨΠϠʔͷઓུ
x1
1 : S0 → {C, D}
x1
2 : S1 → {C, D}
x1
3 : S2 → {C, D}
x1
= (x1
t )3
t=1

16. ήʔϜཧ࿦#4*$ୈճ
༗‫ݶ‬ճ‫܁‬Γฦ͠ήʔϜ
‫׬‬
ήʔϜཧ࿦#4*$ୈճʹͭͮ͘
See you next timeʂ