This document describes a Bayesian game model. It defines the key components of a Bayesian game, including the players (N), the strategy sets (Si and Ti) and payoff functions (fi) for each player i, and the prior beliefs (p). It provides an example with two players (P1 and P2), each having two strategies (S and W). The prior probabilities are p = (1/12, 1/2, 1/12, 1/3). It also shows how to calculate a player's posterior beliefs after observing another player's strategy choice.

1. ήʔϜཧ࿦#4*$ୈճ
৘ใෆ‫׬‬උήʔϜͱϕΠδΞϯήʔϜ

2. 1. ৘ใෆ‫׬‬උήʔϜΛཧղ
2. ϕΠδΞϯήʔϜΛཧղ
ຊಈըͷ໨త

3. 1. ৘ใෆ‫׬‬උήʔϜ
(1)͜Ε·ͰͷήʔϜͱͷҧ͍
(2)৘ใෆ‫׬‬උήʔϜͷఆٛ
(A)λΠϓ
(B)རಘؔ਺
(C)ϕΠδΞϯԾઆ
(D)৘ใෆ‫׬‬උήʔϜͷఆٛ
2. ϕΠδΞϯήʔϜ
(1)ϕΠδΞϯήʔϜͷఆٛ
(2)ϕΠζಉ஋Ծઆ
(3)໦ͷද‫ݱ‬
໨࣍

4. Game with incomplete information
৘ใෆ‫׬‬උήʔϜ

5. ͜Ε·ͰͷήʔϜͱͷҧ͍
৘ใ‫׬‬උήʔϜͱ৘ใෆ‫׬‬උήʔϜ
͜Ε·Ͱߟ͖͑ͯͨήʔϜͰ͸
֤ϓϨΠϠʔ͕ʮઓུͷू߹ʯʮརಘʯʮ໦ͷߏ଄ʯΛ೺Ѳͯ͠ҙࢥܾఆΛߦ͏
͜ͷΑ͏ʹ৘ใ͕‫ڞ‬༗஌ࣝͱͳ͍ͬͯΔήʔϜΛ৘ใ‫׬‬උͳήʔϜͱ͍͏
ྫ
ক‫
ع‬ғ‫Ͳͳޟ‬
Ұํ
‫࣮ࣾݱ‬ձʹ͓͍ͯ
ήʔϜͷঢ়‫گ‬Λෆ‫׬‬શʹ͔͠೺ѲͰ͖͍ͯͳ͍ঢ়‫͕͋گ‬Δ
ϓϨΠϠʔ͕ήʔϜͷ৘ใΛ೺Ѳ͍ͯ͠ͳ͍Α͏ͳήʔϜΛ৘ใෆ‫׬‬උͳήʔϜͱ͍͏
ྫ
ϙʔΧʔͳͲ
ྫ
ফඅऀͷ޷ΈͳͲ
ࠓճ͸֤ϓϨΠϠʔ͕ଞͷϓϨΠϠʔͷརಘؔ਺Λ‫׬‬શʹ͸஌Βͳ͍ঢ়‫گ‬Λߟ͑Δ
Ͳ͏ϞσϧԽ͢Δ͔ʁ
̍ʘ 'JHIU TD
'JHIU
TD
̍ʘ 'JHIU TD
'JHIU
TD
̍ʘ 'JHIU TD
'JHIU
TD
4USPOH
P1
8FBL
P1

6. ৘ใෆ‫׬‬උήʔϜͷఆٛ
λΠϓ
རಘؔ਺͕͋Δม਺Ͱܾఆ͞ΕΔͱ͢Δ͢΂ͯͷϓϨΠϠʔ͸͜ͷͷਅͷ஋Λ஌Βͳ͍
ɹ͞Βʹ͸
ͱ ‫ݸ‬ͷม਺ʹ෼ղͰ͖Δͱ͠
֤ϓϨΠϠʔ͸ ͷਅͷ஋Λ஌Δ͜ͱ͸Ͱ͖Δ͕
ɹଞͷϓϨΠϠʔͷ Λ஌Βͳ͍΋ͷͱ͢Δ
ͷΑ͏ʹෳ਺ͷ஋ͷ৔߹Λ࣋ͭ͜ͱ΋͋Γ
ΛϓϨΠϠʔͷ৘ใϕΫτϧͱ‫ Ϳݺ‬
͕औΓ͏Δ஋ͷू߹Λ Ͱද͢
͜ͷ৘ใϕΫτϧ͸‫ݸ‬ਓͷଐੑ ֶྺ
ੜ࢈ٕज़
Λද͠
·ͨ৘ใϕΫτϧ͸ࣗ෼ͷ΋ͷ͔͠஌Βͳ͍ͷͰ
‫ݸ‬ਓ৘ใͰ΋͋Δ
͜͜Ͱ͸৘ใϕΫτϧΛλΠϓͱ‫Ϳݺ‬
t t
t t = (t1
, ⋯, tn
) n i ti
tj
(j ≠ i)
ti
= (ti
1, ⋯, ti
ki) ti
i
ti
Ti

7. ৘ใෆ‫׬‬උήʔϜͷఆٛ
རಘؔ਺
རಘؔ਺͸λΠϓʹΑܾͬͯ·ΔͷͰ
Ͱද͞ΕΔ
fi
(s1
, ⋯, sn
, t1
, ⋯, tn
)

8. ৘ใෆ‫׬‬උήʔϜͷఆٛ
རಘؔ਺
རಘؔ਺͸λΠϓʹΑܾͬͯ·ΔͷͰ
Ͱද͞ΕΔ
ϕΠδΞϯԾઆ
ϓϨΠϠʔ͸ଞͷϓϨΠϠʔͷλΠϓ ͷਅͷ஋Λ༧૝͢Δ
ϓϨΠϠʔͷओ‫؍‬త༧૝͸ ‫ݸ‬ͷม਺ ͷಉ࣌֬཰෼෍ Ͱද‫͞ݱ‬Ε
ϓϨΠϠʔ͸֬཰෼෍ ʹΑΔརಘͷ‫ظ‬଴஋Λ࠷େԽ͢Δ
͸ࣗ෼ͷλΠϓ ʹґଘ͠
ͱද‫͖Ͱݱ‬Δ
fi
(s1
, ⋯, sn
, t1
, ⋯, tn
)
i t1
, ⋯, ti−1
, ti+1
, ⋯, tn
i n − 1 t1
, ⋯, ti−1
, ti+1
, ⋯, tn
pi
i pi
pi
ti
pi
(t1
, ⋯, ti−1
, ti+1
, ⋯, tn
|ti
)

9. ৘ใෆ‫׬‬උήʔϜͷఆٛ
৘ใෆ‫׬‬උήʔϜͷఆٛ
৘ใෆ‫׬‬උήʔϜ͸ Ͱද͞ΕΔ
B
͸ϓϨΠϠʔͷू߹
C
͸ϓϨΠϠʔͷߦಈͷू߹
D
͸ϓϨΠϠʔͷλΠϓͷू߹
E
͸ϓϨΠϠʔͷརಘؔ਺Ͱ͋Γ
F
͸ϓϨΠϠʔͷओ‫؍‬త༧૝Ͱ͋Γ
֤ ʹରͯ͠
௚ੵू߹ ্ͷಉ࣌֬཰෼෍
G * = (N, {Si
, Ti
, fi
, pi
}i∈N)
N
Si
i
Ti
i
fi
i fi
: S1
× ⋯ × Sn
× T1
× ⋯ × Tn
→ ℜ
pi
i ti
∈ Ti
T1
× ⋯Ti−1
× Ti+1
⋯ × Tn
pi
(t1
, ⋯, ti−1
, ti+1
, ⋯, tn
|ti
)

10. Bayesian game
ϕΠδΞϯήʔϜ

11. ϕΠδΞϯήʔϜͷఆٛ
ϕΠδΞϯήʔϜ
৘ใෆ‫׬‬උͳήʔϜΛ‫׬‬උͳήʔϜʹม‫࡯ߟͯ͠׵‬
֤ϓϨΠϠʔ͸λΠϓͷ૊શମ ্ʹ
‫ڞ‬௨ͷࣄલ֬཰෼෍ Λ‫ڞ‬༗͍ͯ͠Δͱ͢Δ
͞Βʹ֤ϓϨΠϠʔ͸͜ͷࣄલ֬཰෼෍ Λ࢖ͬͯ
ϕΠδΞϯήʔϜ͸ Ͱද͞ΕΔ
T = T1
× ⋯ × Tn
p
p
pi
(t *−i
|t*i
) =
p(t *i
, t*−i
)
∑t−i∈Ti p(t *i ,t−i)
G** = (N, {Si
, Ti
, fi
}i∈N, p)
̍ʘ 'JHIU TD
'JHIU
TD
̍ʘ 'JHIU TD
'JHIU
TD
4USPOH
4USPOH
P1
P2
T1
= (S, W) T2
= (S, W)
T = ((S, S), (S, W), (W, S), (W, W))
p = (1/12, 1/2, 1/12, 1/3)
p1
(S|S) =
p(S, S)
p(S, S) + p(S, W)
=
1/12
1/12 + 1/2
=
1
7
4USPOH
8FBL
P1
P2
̍ʘ 'JHIU TD
'JHIU
TD
8FBL
4USPOH
P1
P2
̍ʘ 'JHIU TD
'JHIU
TD
8FBL
8FBL
P1
P2

12. ϕΠζಉ஋Ծઆ
ϕΠζಉ஋Ծઆ
৘ใෆ‫׬‬උήʔϜ ͱ
ϕΠδΞϯήʔϜ ͸
ϓϨΠϠʔʹͱͬͯͷઓ্ུͷ‫͔఺؍‬Β͸ಉ஋Ͱ͋Γ
ϓϨΠϠʔͷઓུͷબ୒͸ͲͪΒͷήʔϜͰ΋
ಉܾ͡ఆϧʔϧʹै͏
Ҏ߱
͜ͷԾઆΛ࠾༻ͯ͠
৘ใෆ‫׬‬උήʔϜΛϕΠδΞϯήʔϜʹม‫ͯ͠׵‬෼ੳ͢Δ
G * = (N, {Si
, Ti
, fi
, pi
}i∈N) G** = (N, {Si
, Ti
, fi
}i∈N, p)

13. ໦ͷද‫ݱ‬
ϕΠδΞϯήʔϜ͸ҎԼͷΑ͏ͳήʔϜͷ໦Ͱ΋ද‫͖Ͱݱ‬Δɿ
P1
P2
P2
'
N
P1
P1
P1
ࣗવͱ‫ݺ‬͹ΕΔ
Ծ૝తͳϓϨΠϠʔΛஔ͘
P2
P2
P2
P2
P2
P2
4USPOH
4USPOH
8FBL
8FBL
'
'
'
T1
= (S, W) T2
= (S, W)
T = ((S, S), (S, W), (W, S), (W, W))
p = (1/12, 1/2, 1/12, 1/3)

14. ήʔϜཧ࿦#4*$ୈճ
৘ใෆ‫׬‬උήʔϜͱϕΠδΞϯήʔϜ
‫׬‬
ήʔϜཧ࿦#4*$ୈճʹͭͮ͘
See you next timeʂ