https://youtu.be/4UP2uUN9JqI

1. ήʔϜཧ࿦#4*$ୈճ
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬

2. 1. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬Λཧղ
ຊಈըͷ໨త

3. 1. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬
(1)ల։‫ܗ‬ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬ͷ໰୊఺
(2)෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷఆٛ
(3)෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷ‫ٻ‬Ίํ
໨࣍

4. Subgame Perfect Equilibrium
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬

5. ల։‫ܗ‬ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬ͷ໰୊఺
u1
1
u2
1
P1
P2
P2
a1
1
a1
2
a2
1
a2
2
a2
3
a2
4
(0, 0)
(10, 5)
(5, 10)
(0, 0)
u2
2
ల։‫ܗ‬ήʔϜ ઓུ‫ܗ‬ήʔϜ
̍ʘ
a1
1
a1
2
a2
1 − a2
3
a2
1 − a2
4
a2
2 − a2
3
a2
2 − a2
4
ิ଍
७ઓུ͸ߦಈઓུͷҰ෦Ͱ͋Δɿ
ϓϨΠϠʔͷ७ઓུ ͸
ϓϨΠϠʔͷ७ઓུ ͸
a1
1 b1
= ((b1
(a1
1), b1
(a1
2))) = ((1, 0))
a2
1 − a2
3 b2
= ((b2
(a2
1), b1
(a2
2)), (b2
(a2
3), b1
(a2
4))) = ((1, 0), (1, 0))

6. ల։‫ܗ‬ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬ͷ໰୊఺
u1
1
u2
1
P1
P2
P2
a1
1
a1
2
a2
1
a2
2
̍ʘ
a1
1
a1
2
a2
1 − a2
3
a2
1 − a2
4
a2
2 − a2
3
a2
2 − a2
4
a2
3
a2
4
(0, 0)
(10, 5)
(5, 10)
(0, 0)
u2
2
ల։‫ܗ‬ήʔϜ ઓུ‫ܗ‬ήʔϜ

7. ల։‫ܗ‬ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬ͷ໰୊఺
u1
1
u2
1
P1
P2
P2
a1
1
a1
2
a2
1
a2
2
̍ʘ
a1
1
a1
2
a2
1 − a2
3
a2
1 − a2
4
a2
2 − a2
3
a2
2 − a2
4
a2
3
a2
4
(0, 0)
(10, 5)
(5, 10)
(0, 0)
u2
2
ల։‫ܗ‬ήʔϜ
φογϡ‫ߧۉ‬
߹ཧత͔ʁ
ઓུ‫ܗ‬ήʔϜ

8. ల։‫ܗ‬ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬ͷ໰୊఺
໰୊఺ɿ౸ୡ͠ͳ͍৘ใू߹ʹ͓͍ͯརಘΛ࠷େͱ͢ΔΑ͏ͳߦಈ ʹ࠷దͳߦಈ
Λͱ͍ͬͯͳ͍
u1
1
u2
1
P1
P2
P2
a1
1
a1
2
a2
1
a2
2
̍ʘ
a1
1
a1
2
a2
1 − a2
3
a2
1 − a2
4
a2
2 − a2
3
a2
2 − a2
4
a2
3
a2
4
(0, 0)
(10, 5)
(5, 10)
(0, 0)
u2
2
ల։‫ܗ‬ήʔϜ
φογϡ‫ߧۉ‬
߹ཧత͔ʁ
ઓུ‫ܗ‬ήʔϜ

9. ల։‫ܗ‬ήʔϜʹ͓͚Δφογϡ‫ߧۉ‬ͷ໰୊఺
໰୊఺ɿ౸ୡ͠ͳ͍৘ใू߹ʹ͓͍ͯརಘΛ࠷େͱ͢ΔΑ͏ͳߦಈʹ࠷దͳߦಈΛͱ͍ͬͯͳ͍
ˠ౸ୡ͠ͳ͍৘ใू߹ʹ͓͚Δ߹ཧੑΛߟྀ͢Δ‫ߧۉ‬Λߟ͑Δ
u1
1
u2
1
P1
P2
P2
a1
1
a1
2
a2
1
a2
2
̍ʘ
a1
1
a1
2
a2
1 − a2
3
a2
1 − a2
4
a2
2 − a2
3
a2
2 − a2
4
a2
3
a2
4
(0, 0)
(10, 5)
(5, 10)
(0, 0)
u2
2
ల։‫ܗ‬ήʔϜ
φογϡ‫ߧۉ‬
߹ཧత͔ʁ
ઓུ‫ܗ‬ήʔϜ

10. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷఆٛ
ఆٛ ෦෼ήʔϜ
ల։‫ܗ‬ήʔϜ ʹ͓͍ͯ
ͷ෦෼໦ ʹؔͯ͠
͢΂ͯͷ৘ใू߹͕ ͷ఺ͱ Ҏ֎ͷ఺Λಉ࣌ʹ‫ؚ‬Ή͜ͱ͸ͳ͍ͱ͢Δ
͜ͷͱ͖
ͷ֤ߏ੒ཁૉΛ෦෼໦ ʹ੍‫ͨ͠ݶ‬ల։‫ܗ‬ήʔϜΛ ͷ෦෼ήʔϜͱ͍͏
஫
෦෼໦ͱ͸ ʹؔͯ͠
Ͱ͋Γ
͜ͷ Ͱ໦ͷߏ଄Λ࣋ͭ΋ͷͰ͋Δ
஫
ͦͷ΋ͷ΋෦෼໦ͱͯ͠ߟ͑
શମͷήʔϜ΋෦෼ήʔϜͱ͢Δ
Γ = (K, p, P, U, h) K K′
K′ K′
Γ K′ Γ
K = (V, E) V′ ⊆ V, E′ ⊆ E V′, E′
K

11. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷఆٛ
ఆٛ ෦෼ήʔϜ
ల։‫ܗ‬ήʔϜ ʹ͓͍ͯ
ͷ෦෼໦ ʹؔͯ͠
͢΂ͯͷ৘ใू߹͕ ͷ఺ͱ Ҏ֎ͷ఺Λಉ࣌ʹ‫ؚ‬Ή͜ͱ͸ͳ͍ͱ͢Δ
͜ͷͱ͖
ͷ֤ߏ੒ཁૉΛ෦෼໦ ʹ੍‫ͨ͠ݶ‬ల։‫ܗ‬ήʔϜΛ ͷ෦෼ήʔϜͱ͍͏
஫
෦෼໦ͱ͸ ʹؔͯ͠
Ͱ͋Γ
͜ͷ Ͱ໦ͷߏ଄Λ࣋ͭ΋ͷͰ͋Δ
஫
ͦͷ΋ͷ΋෦෼໦ͱͯ͠ߟ͑
શମͷήʔϜ΋෦෼ήʔϜͱ͢Δ
Γ = (K, p, P, U, h) K K′
K′ K′
Γ K′ Γ
K = (V, E) V′ ⊆ V, E′ ⊆ E V′, E′
K
u1
1
u2
1
x0
x1
a1
1
a1
2
a2
1
a2
2
a2
3
a2
4
w1
x2
w2
w3
w4
u2
2
u2
1
x1
a2
1
a2
2
w1
w2
V = (x0, x1, x2, w1, w2, w3, w4)
E = (a1
1, a1
2, a2
1, a2
2, a3
2, a4
2)
V′ = (x1, w1, w2)
E′ = (a2
1, a2
2,)

12. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷఆٛ
ఆٛ ෦෼ήʔϜ
ల։‫ܗ‬ήʔϜ ʹ͓͍ͯ
ͷ෦෼໦ ʹؔͯ͠
͢΂ͯͷ৘ใू߹͕ ͷ఺ͱ Ҏ֎ͷ఺Λಉ࣌ʹ‫ؚ‬Ή͜ͱ͸ͳ͍ͱ͢Δ
͜ͷͱ͖
ͷ֤ߏ੒ཁૉΛ෦෼໦ ʹ੍‫ͨ͠ݶ‬ల։‫ܗ‬ήʔϜΛ ͷ෦෼ήʔϜͱ͍͏
Γ = (K, p, P, U, h) K K′
K′ K′
Γ K′ Γ
u1
1
u2
1
x0
x1
a1
1
a1
2
a2
1
a2
2
a2
3
a2
4
w1
x2
w2
w3
w4
V = (x0, x1, x2, w1, w2, w3, w4)
E = (a1
1, a1
2, a2
1, a2
2, a3
2, a4
2)
৘ใू߹ ͕
෦෼໦ ͱ෦෼໦ ͷ఺Λಉ࣌ʹ‫ؚ‬Ή
Ώ͑ʹ ΋ ΋෦෼ήʔϜͰ͸ͳ͍
u2
1
K′ K′′
K′ K′′
K′
K′′

13. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷఆٛ
ఆٛ ෦෼ήʔϜ‫׬‬શ‫
ߧۉ‬
ల։‫ܗ‬ήʔϜ ʹ͓͍ͯ
ߦಈઓུ ͕෦෼ήʔϜ‫׬‬શ‫͋Ͱߧۉ‬Δͱ͸
͢΂ͯͷ෦෼ήʔϜʹରͯ͠
͕φογϡ‫ߧۉ‬Λಋ͘͜ͱͰ͋Δ
Γ = (K, p, P, U, h) b* = (b*1
, ⋯, b*n
)
b*
u1
1
u2
1
P1
P2
P2
a1
1
a1
2
a2
1
a2
2
a2
3
a2
4
(0, 0)
(10, 5)
(5, 10)
(0, 0)
u2
2
ϓϨΠϠʔ͕෦෼ήʔϜ Ͱ
࠷దͳߦಈΛͱ͍ͬͯͳ͍
Γ′′
Γ′
Γ′′
ల։‫ܗ‬ήʔϜ

14. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷఆٛ
ఆٛ ෦෼ήʔϜ‫׬‬શ‫
ߧۉ‬
ల։‫ܗ‬ήʔϜ ʹ͓͍ͯ
ߦಈઓུ ͕෦෼ήʔϜ‫׬‬શ‫͋Ͱߧۉ‬Δͱ͸
͢΂ͯͷ෦෼ήʔϜʹରͯ͠
͕φογϡ‫ߧۉ‬Λಋ͘͜ͱͰ͋Δ
Γ = (K, p, P, U, h) b* = (b*1
, ⋯, b*n
)
b*
u1
1
u2
1
P1
P2
P2
a1
1
a1
2
a2
1
a2
2
a2
3
a2
4
(0, 0)
(10, 5)
(5, 10)
(0, 0)
u2
2
Γ′
Γ′′
ϓϨΠϠʔ͕෦෼ήʔϜ Ͱ
࠷దͳߦಈΛͱ͍ͬͯΔ
Γ′
ల։‫ܗ‬ήʔϜ
ϓϨΠϠʔ
ͱ΋ʹશମͷήʔϜͰ
࠷దͳߦಈΛͱ͍ͬͯΔ
ϓϨΠϠʔ͕෦෼ήʔϜ Ͱ
࠷దͳߦಈΛͱ͍ͬͯΔ
Γ′′

15. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷఆٛ
ఆٛ ෦෼ήʔϜ‫׬‬શ‫
ߧۉ‬
ల։‫ܗ‬ήʔϜ ʹ͓͍ͯ
ߦಈઓུ ͕෦෼ήʔϜ‫׬‬શ‫͋Ͱߧۉ‬Δͱ͸
͢΂ͯͷ෦෼ήʔϜʹରͯ͠
͕φογϡ‫ߧۉ‬Λಋ͘͜ͱͰ͋Δ
Γ = (K, p, P, U, h) b* = (b*1
, ⋯, b*n
)
b*
u1
1
u2
1
P1
P2
P2
a1
1
a1
2
a2
1
a2
2
a2
3
a2
4
(0, 0)
(10, 5)
(5, 10)
(0, 0)
u2
2
Γ′
Γ′′
ల։‫ܗ‬ήʔϜ
̍ʘ
a1
1
a1
2
a2
1 − a2
3
a2
1 − a2
4
a2
2 − a2
3
a2
2 − a2
4
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬͸
७ઓུͰද‫͢ݱ‬Ε͹
ߦಈઓུͰද‫͢ݱ‬Ε͹
(a1
1, a2
2 − a2
3)
(b*1
,b*2
) = (((b2
(a1
1), b1
(a1
2)),(((b2
(a2
1), b1
(a2
2)), (b2
(a2
3), b1
(a2
4))))
= (((1, 0)),((0, 1), (1, 0)))

16. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷ‫ٻ‬Ίํ
ӈͷྫͷΑ͏ͳ‫׬‬શ‫ه‬ԱͰ͋Δల։‫ܗ‬ήʔϜΛߟ͑Δ
u2
1
u1
2
u1
1
P1
P2
(8, 8)
(10, 0)
(0, 10)
(1, 1)
(2, 0)
(0, 2)
b1
1
b1
2
b2
1
P2
P1
u2
2
P1
b2
2
b2
3
b2
4
b1
3
b1
4
b1
3
b1
4

17. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷ‫ٻ‬Ίํ
ӈͷྫͷΑ͏ͳ‫׬‬શ‫ه‬ԱͰ͋Δల։‫ܗ‬ήʔϜΛߟ͑Δ
ߦಈઓུΛ ͱ͢Δ
෦෼ήʔϜ ʹ͓͍ͯφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΔ
෦෼ήʔϜʹ͓͚ΔߦಈઓུΛ
ͱͯ͠
͸
͜ͷ෦෼ήʔϜͰͷφογϡ‫͋Ͱߧۉ‬Δ
(b1
, b2
) = (((b1
1, b1
2), (b1
3, b1
4)), ((b2
1, b2
2), (b2
3, b2
4)))
Γ′
(b′1
, b′2
) = (((b1
3, b1
4)), ((b2
1, b2
2)))
(b′1
, b′2
) = (((0, 1)), (((0, 1)))
u2
1
u1
2
u1
1
P1
P2
(8, 8)
(10, 0)
(0, 10)
(1, 1)
(2, 0)
(0, 2)
b1
1
b1
2
b2
1
P2
P1
u2
2
P1
b2
2
b2
3
b2
4
b1
3
b1
4
b1
3
b1
4
Γ′
̍ʘ
b1
3
b1
4
b2
1
b2
2
= b1
(a1
3)

18. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷ‫ٻ‬Ίํ
ӈͷྫͷΑ͏ͳ‫׬‬શ‫ه‬ԱͰ͋Δల։‫ܗ‬ήʔϜΛߟ͑Δ
ߦಈઓུΛ ͱ͢Δ
ఆٛʢॖ໿ήʔϜ
ήʔϜ ʹ͓͍ͯ
෦෼ήʔϜ ͱ ʹ͓͚Δߦಈઓུͷ૊ ʹରͯ͠
Λ
ϓϨΠϠʔͷ‫ظ‬଴རಘϕΫτϧ
ʹஔ͖‫͖Ͱͯ͑׵‬ΔήʔϜΛ
෦෼ήʔϜ ͱߦಈઓུͷ૊ ʹΑΔ ͷॖ໿ήʔϜͱ͍͏
(b1
, b2
) = (((b1
1, b1
2), (b1
3, b1
4)), ((b2
1, b2
2), (b2
3, b2
4)))
Γ
Γ′ Γ′ b′ Γ′
HS(b′) = (H1
S(b′), ⋯, Hn
S(b′))
Γ′ b′ Γ
u1
1
P1
(1, 1)
(2, 0)
(0, 2)
b1
1
b1
2
P2
u2
2
b2
3
b2
4
̍ʘ
b1
3
b1
4
b2
1
b2
2
= b1
(a1
3)
෦෼ήʔϜ ͱ
ߦಈઓུ ʹ͓͚Δ
ॖ໿ήʔϜ
Γ′
(b′1
, b′2
)

19. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷ‫ٻ‬Ίํ
ӈͷྫͷΑ͏ͳ‫׬‬શ‫ه‬ԱͰ͋Δల։‫ܗ‬ήʔϜΛߟ͑Δ
ߦಈઓུΛ ͱ͢Δ
෦෼ήʔϜ ʹ͓͍ͯφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΔ
෦෼ήʔϜʹ͓͚ΔߦಈઓུΛ
ͱͯ͠
͸
͜ͷ෦෼ήʔϜͰͷφογϡ‫͋Ͱߧۉ‬Δ
(b1
, b2
) = (((b1
1, b1
2), (b1
3, b1
4)), ((b2
1, b2
2), (b2
3, b2
4)))
Γ′′
b′′2
= ((b2
3, b2
4)) b′′2
= ((0, 1))
u1
1
P1
(1, 1)
(2, 0)
(0, 2)
b1
1
b1
2
P2
u2
2
b2
3
b2
4
Γ′′

20. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷ‫ٻ‬Ίํ
ӈͷྫͷΑ͏ͳ‫׬‬શ‫ه‬ԱͰ͋Δల։‫ܗ‬ήʔϜΛߟ͑Δ
ߦಈઓུΛ ͱ͢Δ
෦෼ήʔϜ ʹ͓͍ͯφογϡ‫ߧۉ‬Λ‫ٻ‬ΊΔ
෦෼ήʔϜʹ͓͚ΔߦಈઓུΛ
ͱͯ͠
͸
͜ͷ෦෼ήʔϜͰͷφογϡ‫͋Ͱߧۉ‬Δ
(b1
, b2
) = (((b1
1, b1
2), (b1
3, b1
4)), ((b2
1, b2
2), (b2
3, b2
4)))
Γ′′
b′′2
= ((b2
3, b2
4)) b′′2
= ((0, 1))
u1
1
P1
(1, 1)
(0, 2)
b1
1
b1
2

21. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷ‫ٻ‬Ίํ
ӈͷྫͷΑ͏ͳ‫׬‬શ‫ه‬ԱͰ͋Δల։‫ܗ‬ήʔϜΛߟ͑Δ
ߦಈઓུΛ ͱ͢Δ
શମͷήʔϜʹ͓͚ΔϓϨΠϠʔͷ࠷దͳߦಈ͸
ͱͯ͠
Ͱ͋Δ
(b1
, b2
) = (((b1
1, b1
2), (b1
3, b1
4)), ((b2
1, b2
2), (b2
3, b2
4)))
b′′1
= ((b1
1, b1
2)) b′′1
= ((1, 0))
u1
1
P1
(1, 1)
(0, 2)
b1
1
b1
2

22. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷ‫ٻ‬Ίํ
ӈͷྫͷΑ͏ͳ‫׬‬શ‫ه‬ԱͰ͋Δల։‫ܗ‬ήʔϜΛߟ͑Δ
ߦಈઓུΛ ͱ͢Δ
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬͸
෦෼ήʔϜͱॖ໿ήʔϜͷ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬Λ߹੒ͯ͠
શମͷήʔϜͷ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬Λߏ੒Ͱ͖Δ
(b1
, b2
) = (((b1
1, b1
2), (b1
3, b1
4)), ((b2
1, b2
2), (b2
3, b2
4)))
(b1
, b2
) = (((1, 0), (0, 1)), ((0, 1), (1, 0)))
u2
1
u1
2
u1
1
P1
P2
(8, 8)
(10, 0)
(0, 10)
(1, 1)
(2, 0)
(0, 2)
1
0
0
P2
P1
u2
2
P1
1
0
1
Γ′
Γ′′
0
1
0
1

23. ෦෼ήʔϜ‫׬‬શ‫ߧۉ‬ͷ‫ٻ‬Ίํ
ఆཧ
఺ͷ਺͕༗‫׬Ͱݸݶ‬શ‫ه‬ԱήʔϜͰ͋Δల։‫ܗ‬ήʔϜ͸
෦෼ήʔϜ‫׬‬શ‫͕ߧۉ‬ଘࡏ͢Δ
‫׬‬શz৘ใzήʔϜͰΈͨ‫ޙ‬Ζ޲͖‫ؼ‬ೲ๏ͷҰൠԽΛߦ͍ূ໌͢Δʢิ଍
෦෼ήʔϜ͕ͦΕࣗମ͔͠෦෼ήʔϜΛ΋ͨͳ͍৔߹Λ
‫ۃ‬খͳ෦෼ήʔϜͱ͍͏͜ͷͱ͖
෦෼ήʔϜ‫׬‬શ‫ٻߧۉ‬Ίํ͸
͢΂ͯͷ‫ۃ‬খͳ෦෼ήʔϜʹ͓͚Δ‫఺ߧۉ‬Λ‫ٻ‬ΊΔ
‫ۃ‬খͳ෦෼ήʔϜΛ
‫ٻ‬ΊͨߦಈઓུͰಘΒΕΔ‫ظ‬଴རಘͰஔ͖‫͑׵‬Δ ॖ໿ήʔϜΛ࡞Δ
ॖ໿ήʔϜʹ͓͚Δ͢΂ͯͷ‫ۃ‬খͳ෦෼ήʔϜʹ͓͚Δ‫఺ߧۉ‬Λ‫ٻ‬ΊΔ
‫ۃ‬খͳ෦෼ήʔϜΛ
‫ٻ‬ΊͨߦಈઓུͰಘΒΕΔ‫ظ‬଴རಘͰஔ͖‫͑׵‬Δ ॖ໿ήʔϜͷॖ໿ήʔϜΛ࡞Δ
ҎԼಉ͡ૢ࡞Λߦ͍
ॳ‫Ͱ఺ظ‬ͷϓϨΠϠʔͷߦಈ͕֬ఆͨ࣌͠఺ͰऴྃͱͳΔ
͜ͷͱ͖֤ʑͷ෦෼ήʔϜͰͱͬͨߦಈઓུͷ૊͸ɺશମͷ෦෼ήʔϜ‫׬‬શ‫ͳʹߧۉ‬Δ

24. ήʔϜཧ࿦#4*$ୈճ
෦෼ήʔϜ‫׬‬શ‫ ߧۉ‬
‫׬‬
ήʔϜཧ࿦#4*$ୈճʹͭͮ͘
See you next timeʂ