The document discusses Bayesian games and equilibrium strategies. It first defines a Bayesian game model involving two players, with Player 1 choosing between two strategies and Player 2 choosing one strategy. It then examines the conditions for a strategy profile to represent a Bayesian Nash equilibrium, where no player can increase their expected payoff by switching to another strategy, given the other player's strategy. An example equilibrium strategy profile is presented and verified to satisfy the equilibrium conditions.

1. ήʔϜཧ࿦#4*$ୈճ
ϕΠδΞϯήʔϜʹ͓͚Δઓུͱ‫ߧۉ‬

2. 1. ϕΠδΞϯήʔϜͷઓུΛཧղ
2. ϕΠδΞϯφογϡ‫ ͱߧۉ‬
φογϡ‫ߧۉ‬Λཧղ
ຊಈըͷ໨త

3. 1. ϕΠδΞϯήʔϜʹ͓͚Δઓུͱ‫ߧۉ‬
(1)ઓུͷఆٛ
(2)ϕΠδΞϯφογϡ‫ߧۉ‬
(3)φογϡ‫ߧۉ‬
(4)ϕΠδΞϯφογϡ‫ͱߧۉ‬φογϡ‫ߧۉ‬ͷಉ஋ੑ
໨࣍

4. Strategy and Equilibrium in Bayesian game
ϕΠδΞϯήʔϜʹ͓͚Δઓུͱ‫ߧۉ‬

5. ઓུͷఆٛ
ઓུͷఆٛ
৘ใෆ‫׬‬උήʔϜ ͓ΑͼϕΠδΞϯήʔϜ ʹ͓͍ͯ
֤ϓϨΠϠʔ͸ࣗ෼ͷλΠϓ͕Θ্͔ͬͨͰ
ߦಈΛબͿΏ͑ʹϓϨΠϠʔͷઓུ͸ؔ਺ Ͱ༩͑ΒΕΔ
ྫ
ϓϨΠϠʔ
ͷөըબ୒໰୊
ϓϨΠϠʔͷλΠϓ
ϓϨΠϠʔͷλΠϓ
G* = (N, {Si
, Ti
, fi
, pi
}i∈N) G** = (N, {Si
, Ti
, fi
}i∈N, p)
i xi
: Ti
→ Si
T1
= {SW, HP}
x1
(SW) = S, x1
(HP) = S
x1
(SW) = S, x1
(HP) = H
x1
(SW) = H, x1
(HP) = S
x1
(SW) = H, x1
(HP) = H
T2
= {SW}
x2
(SW) = S
x2
(SW) = W
̍ʘ 4 )
4 ̎
)
̍
̍ʘ 4 )
4
)
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6. ઓུͷఆٛ
ઓུͷఆٛ
৘ใෆ‫׬‬උήʔϜ ͓ΑͼϕΠδΞϯήʔϜ ʹ͓͍ͯ
֤ϓϨΠϠʔ͸ࣗ෼ͷλΠϓ͕Θ্͔ͬͨͰ
ߦಈΛબͿΏ͑ʹϓϨΠϠʔͷઓུ͸ؔ਺ Ͱ༩͑ΒΕΔ
ྫ
ϓϨΠϠʔ
ͷөըબ୒໰୊
ϓϨΠϠʔͷλΠϓ
ϓϨΠϠʔͷλΠϓ
G* = (N, {Si
, Ti
, fi
, pi
}i∈N) G** = (N, {Si
, Ti
, fi
}i∈N, p)
i xi
: Ti
→ Si
T1
= {SW, HP}
x1
(SW) = S, x1
(HP) = S
x1
(SW) = S, x1
(HP) = H
x1
(SW) = H, x1
(HP) = S
x1
(SW) = H, x1
(HP) = H
T2
= {SW}
x2
(SW) = S
x2
(SW) = H
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4 ̎
)
̍
̍ʘ 4 )
4
)
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S − S
S − H
H − S
H − H
S
H

7. ϕΠδΞϯφογϡ‫ߧۉ‬
ϕΠδΞϯφογϡ‫ߧۉ‬
ઓུͷ૊ ʹ͓͍ͯ
֤ϓϨΠϠʔͷ֤λΠϓ ʹ͍ͭͯ
ͷ༩͑Δߦಈ͕ଞͷϓϨΠϠʔͷ֤λΠϓͷߦಈʹର͢Δ࠷ద൓Ԡͱͳ͍ͬͯΔͱ͖
ͭ·Γ
͢΂ͯͷ ͱͷ͢΂ͯͷλΠϓ ʹରͯ͠
ͱͳ͍ͬͯΔͱ͖
ΛϕΠδΞϯφογϡ‫͏͍ͱߧۉ‬
ͷͱ͖
͜ΕΑΓ
ͷ༩͑Δߦಈ ಛʹλΠϓ)1
͸࠷ద൓ԠͰ͸ͳ͍
x* = (x*1
, ⋯, x*n
) i ti
x*i
i ∈ N i ti
∑
t−1
∈T−1
fi
(x*i
(ti
), x*−i
(t−i
), ti
, t−i
)pi
(t−i
|ti
) ≥
∑
t−1
∈T−1
fi
(si
, x*−i
(t−i
), ti
, t−i
)pi
(t−i
|ti
) ∀si
∈ Si
x* = (x*1
, ⋯, x*n
)
x1
(SW) = S, x1
(HP) = H x2
(SW) = S
f1
(x1
(SW), x2
(SW), SW, SW)p1
(SW|SW) = f1
(S, S, SW, SW) = 2
f1
(x1
(HP), x2
(SW), HP, SW)p1
(SW|HP) = f1
(H, S, HP, SW) = 0
f1
(x1
(HP), x2
(SW), HP, SW)p1
(SW|HP) = f1
(H, S, HP, SW) = 0 1 = f1
(S, S, HP, SW) = f1
(S, x2
(SW), HP, SW)p1
(SW|HP)
x1
(SW) = S, x1
(HP) = H
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4 ̎
)
̍
̍ʘ 4 )
4
)
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8. ϕΠδΞϯφογϡ‫ߧۉ‬
ϕΠδΞϯφογϡ‫ߧۉ‬
ઓུͷ૊ ʹ͓͍ͯ
֤ϓϨΠϠʔͷ֤λΠϓ ʹ͍ͭͯ
ͷ༩͑Δߦಈ͕ଞͷϓϨΠϠʔͷ֤λΠϓͷߦಈʹର͢Δ࠷ద൓Ԡͱͳ͍ͬͯΔͱ͖
ͭ·Γ
͢΂ͯͷ ͱͷ͢΂ͯͷλΠϓ ʹରͯ͠
ͱͳ͍ͬͯΔͱ͖
ΛϕΠδΞϯφογϡ‫͏͍ͱߧۉ‬
ͷͱ͖
x* = (x*1
, ⋯, x*n
) i ti
x*i
i ∈ N i ti
∑
t−1
∈T−1
fi
(x*i
(ti
), x*−i
(t−i
), ti
, t−i
)pi
(t−i
|ti
) ≥
∑
t−1
∈T−1
fi
(si
, x*−i
(t−i
), ti
, t−i
)pi
(t−i
|ti
) ∀si
∈ Si
x* = (x*1
, ⋯, x*n
)
x1
(SW) = S, x1
(HP) = H x2
(SW) = S
f1
(x1
(SW), x2
(SW), SW, SW)p1
(SW|SW) = f1
(S, S, SW, SW) = 2
f1
(x1
(HP), x2
(SW), HP, SW)p1
(SW|HP) = f1
(H, S, HP, SW) = 0
f2
(x1
(SW), x2
(SW), SW, SW)p2
(SW|SW)
+ f2
(x1
(HP), x2
(SW), HP, SW)p2
(HP|SW) = f2
(S, S, SW, SW)p2
(SW|SW) + f2
(H, S, HP, SW)p2
(HP|SW)
= 3p2
(SW|SW) + 0p2
(HP|SW)
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4 ̎
)
̍
̍ʘ 4 )
4
)
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9. ϕΠδΞϯφογϡ‫ߧۉ‬
ϕΠδΞϯφογϡ‫ߧۉ‬
ઓུͷ૊ ʹ͓͍ͯ
֤ϓϨΠϠʔͷ֤λΠϓ ʹ͍ͭͯ
ͷ༩͑Δߦಈ͕ଞͷϓϨΠϠʔͷ֤λΠϓͷߦಈʹର͢Δ࠷ద൓Ԡͱͳ͍ͬͯΔͱ͖
ͭ·Γ
͢΂ͯͷ ͱͷ͢΂ͯͷλΠϓ ʹରͯ͠
ͱͳ͍ͬͯΔͱ͖
ΛϕΠδΞϯφογϡ‫͏͍ͱߧۉ‬
ͷͱ͖
ͷ༩͑Δߦಈ͸ͦΕͧΕ࠷ద൓ԠͱͳΔ
x* = (x*1
, ⋯, x*n
) i ti
x*i
i ∈ N i ti
∑
t−1
∈T−1
fi
(x*i
(ti
), x*−i
(t−i
), ti
, t−i
)pi
(t−i
|ti
) ≥
∑
t−1
∈T−1
fi
(si
, x*−i
(t−i
), ti
, t−i
)pi
(t−i
|ti
) ∀si
∈ Si
x* = (x*1
, ⋯, x*n
)
x1
(SW) = S, x1
(HP) = S x2
(SW) = S
f1
(x1
(SW), x2
(SW), SW, SW)p1
(SW|SW) = f1
(S, S, SW, SW) = 2
f1
(x1
(HP), x2
(SW), HP, SW)p1
(SW|HP) = f1
(S, S, HP, SW) = 1
x1
(SW) = S, x1
(HP) = S
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4 ̎
)
̍
̍ʘ 4 )
4
)
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10. ϕΠδΞϯφογϡ‫ߧۉ‬
ͷͱ͖
ϓϨΠϠʔͷߦಈΛ ʹมߋ͢Δͱ
Ώ͑ʹ ͷ༩͑Δߦಈ͸ ʹରͯ͠࠷ద൓ԠͱͳΔ
Ҏ্ΑΓ
͸ϕΠδΞϯφογϡ‫ߧۉ‬
x1
(SW) = S, x1
(HP) = S x2
(SW) = S
f2
(x1
(SW), x2
(SW), SW, SW)p2
(SW|SW)
+ f2
(x1
(HP), x2
(SW), HP, SW)p2
(HP|SW) = f2
(S, S, SW, SW)p2
(SW|SW) + f2
(S, S, HP, SW)p2
(HP|SW)
= 3p2
(SW|SW) + 3p2
(HP|SW) = 3
H
f2
(x1
(SW), H, SW, SW)p2
(SW|SW)
+ f2
(x1
(HP), H, HP, SW)p2
(HP|SW) = f2
(S, H, SW, SW)p2
(SW|SW) + f2
(S, H, HP, SW)p2
(HP|SW)
= 0p2
(SW|SW) + 0p2
(HP|SW) = 0 3
x2
(SW) = S x1
(SW) = S, x1
(HP) = S
x1
(SW) = S, x1
(HP) = S x2
(SW) = S
̍ʘ 4 )
4 ̎
)
̍
̍ʘ 4 )
4
)
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11. ϕΠδΞϯφογϡ‫ߧۉ‬
ͷͱ͖
ϓϨΠϠʔͷߦಈΛ ʹมߋ͢Δͱ
Ώ͑ʹ ͷ༩͑Δߦಈ͸ ʹରͯ͠࠷ద൓ԠͱͳΔ
·ͨ
ϓϨΠϠʔͷ֤λΠϓʹ͓͚Δߦಈ ͸ ʹରͯ͠࠷ద൓ԠͱͳΔ
Ҏ্ΑΓ
͸ϕΠδΞϯφογϡ‫ߧۉ‬
x1
(SW) = H, x1
(HP) = H x2
(SW) = H
f2
(x1
(SW), x2
(SW), SW, SW)p2
(SW|SW)
+ f2
(x1
(HP), x2
(SW), HP, SW)p2
(HP|SW) = f2
(H, H, SW, SW)p2
(SW|SW) + f2
(H, H, HP, SW)p2
(HP|SW)
= 3p2
(SW|SW) + 3p2
(HP|SW) = 1
S
f2
(x1
(SW), S, SW, SW)p2
(SW|SW)
+ f2
(x1
(HP), S, HP, SW)p2
(HP|SW) = f2
(H, S, SW, SW)p2
(SW|SW) + f2
(H, S, HP, SW)p2
(HP|SW)
= 0p2
(SW|SW) + 0p2
(HP|SW) = 0 1
x2
(SW) = H x1
(SW) = H, x1
(HP) = H
H x2
(SW) = H
x1
(SW) = H, x1
(HP) = H x2
(SW) = H
̍ʘ 4 )
4 ̎
)
̍
̍ʘ 4 )
4
)
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)1޷͖
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12. φογϡ‫ߧۉ‬
φογϡ‫ߧۉ‬
ϕΠδΞϯήʔϜʹ͓͚Δઓུͷ૊ ʹ͓͍ͯ
͢΂ͯͷ ʹରͯ͠
ͱͳ͍ͬͯΔͱ͖
Λφογϡ‫͏͍ͱߧۉ‬
֤λΠϓ͝ͱͰ͸ͳ͘
ͭҎ্ͷλΠϓͷߦಈΛಉ࣌ʹมߋͰ͖Δͱߟ͑Δ
֤ϓϨΠϠʔ͕߹ཧతͰ
٬‫؍‬త֬཰͔Βओ‫؍‬త֬཰Λ‫ܗ‬੒Ͱ͖Δ
ͭ·Γ͢΂ͯͷϓϨΠϠʔʹ͍ͭͯ
ɹ
ͱͳΔͷͰ͋Ε͹
ϕΠδΞϯφογϡ‫ͱߧۉ‬φογϡ‫ߧۉ‬͸ಉ஋ͱͳΔ
x* = (x*1
, ⋯, x*n
) i ∈ N
∑
t∈T
fi
(x*i
(ti
), x*−i
(t−i
), ti
, t−i
)p(ti
, t−i
) ≥
∑
t∈T
fi
(xi
(ti
), x*−i
(t−i
), ti
, t−i
)p(ti
, t−i
) ∀xi
∈ Xi
x* = (x*1
, ⋯, x*n
)
pi
(t *−i
|t*i
) =
p(t *i
, t*−i
)
∑t−i∈Ti p(t *i ,t−i)

13. φογϡ‫ߧۉ‬
φογϡ‫ߧۉ‬
ϕΠδΞϯήʔϜʹ͓͚Δઓུͷ૊ ʹ͓͍ͯ
͢΂ͯͷ ʹରͯ͠
ͱͳ͍ͬͯΔͱ͖
Λφογϡ‫͏͍ͱߧۉ‬
٬‫཰֬؍‬Λ ͱ͢Δ
ͷͱ͖
x* = (x*1
, ⋯, x*n
) i ∈ N
∑
t∈T
fi
(x*i
(ti
), x*−i
(t−i
), ti
, t−i
)p(ti
, t−i
) ≥
∑
t∈T
fi
(xi
(ti
), x*−i
(t−i
), ti
, t−i
)p(ti
, t−i
) ∀xi
∈ Xi
x* = (x*1
, ⋯, x*n
)
p = (1/2, 1/2)
x1
(SW) = S, x1
(HP) = S x2
(SW) = S
f1
(x1
(SW), x2
(SW), SW, SW)p(SW, SW) + f1
(x1
(HP), x2
(SW), HP, SW)p(HP, SW)
= f1
(S, S, SW, SW)
1
2
+ f1
(S, S, HP, SW)
1
2
=
3
2
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4 ̎
)
̍
̍ʘ 4 )
4
)
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14. φογϡ‫ߧۉ‬
φογϡ‫ߧۉ‬
ϕΠδΞϯήʔϜʹ͓͚Δઓུͷ૊ ʹ͓͍ͯ
͢΂ͯͷ ʹରͯ͠
ͱͳ͍ͬͯΔͱ͖
Λφογϡ‫͏͍ͱߧۉ‬
٬‫཰֬؍‬Λ ͱ͢Δ
ͷͱ͖
x* = (x*1
, ⋯, x*n
) i ∈ N
∑
t∈T
fi
(x*i
(ti
), x*−i
(t−i
), ti
, t−i
)p(ti
, t−i
) ≥
∑
t∈T
fi
(xi
(ti
), x*−i
(t−i
), ti
, t−i
)p(ti
, t−i
) ∀xi
∈ Xi
x* = (x*1
, ⋯, x*n
)
p = (1/2, 1/2)
x1
(SW) = S, x1
(HP) = S x2
(SW) = S
f1
(x1
(SW), x2
(SW), SW, SW)p(SW, SW) + f1
(x1
(HP), x2
(SW), HP, SW)p(HP, SW)
= f1
(S, S, SW, SW)
1
2
+ f1
(S, S, HP, SW)
1
2
=
3
2
̍ʘ 4 )
4 ̎
)
̍
̍ʘ 4 )
4
)
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)1޷͖
P1
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44
4)
)4
))

15. φογϡ‫ߧۉ‬
φογϡ‫ߧۉ‬
ϕΠδΞϯήʔϜʹ͓͚Δઓུͷ૊ ʹ͓͍ͯ
͢΂ͯͷ ʹରͯ͠
ͱͳ͍ͬͯΔͱ͖
Λφογϡ‫͏͍ͱߧۉ‬
٬‫཰֬؍‬Λ ͱ͢Δ
ͷͱ͖
x* = (x*1
, ⋯, x*n
) i ∈ N
∑
t∈T
fi
(x*i
(ti
), x*−i
(t−i
), ti
, t−i
)p(ti
, t−i
) ≥
∑
t∈T
fi
(xi
(ti
), x*−i
(t−i
), ti
, t−i
)p(ti
, t−i
) ∀xi
∈ Xi
x* = (x*1
, ⋯, x*n
)
p = (1/2, 1/2)
x1
(SW) = S, x1
(HP) = S x2
(SW) = S
f1
(x1
(SW), x2
(SW), SW, SW)p(SW, SW) + f1
(x1
(HP), x2
(SW), HP, SW)p(HP, SW)
= f1
(S, S, SW, SW)
1
2
+ f1
(S, S, HP, SW)
1
2
=
3
2
f2
(x1
(SW), x2
(SW), SW, SW)p(SW, SW) + f2
(x1
(HP), x2
(SW), HP, SW)p(HP, SW)
= f2
(S, S, SW, SW)
1
2
+ f2
(S, S, HP, SW)
1
2
= 3
̍ʘ 4 )
4 ̎
)
̍
̍ʘ 4 )
4
)
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)1޷͖
P1
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44
4)
)4
))

16. φογϡ‫ߧۉ‬
φογϡ‫ߧۉ‬
ϕΠδΞϯήʔϜʹ͓͚Δઓུͷ૊ ʹ͓͍ͯ
͢΂ͯͷ ʹରͯ͠
ͱͳ͍ͬͯΔͱ͖
Λφογϡ‫͏͍ͱߧۉ‬
٬‫཰֬؍‬Λ ͱ͢Δ
ͷͱ͖
x* = (x*1
, ⋯, x*n
) i ∈ N
∑
t∈T
fi
(x*i
(ti
), x*−i
(t−i
), ti
, t−i
)p(ti
, t−i
) ≥
∑
t∈T
fi
(xi
(ti
), x*−i
(t−i
), ti
, t−i
)p(ti
, t−i
) ∀xi
∈ Xi
x* = (x*1
, ⋯, x*n
)
p = (1/2, 1/2)
x1
(SW) = S, x1
(HP) = H x2
(SW) = S
f1
(x1
(SW), x2
(SW), SW, SW)p(SW, SW) + f1
(x1
(HP), x2
(SW), HP, SW)p(HP, SW)
= f1
(S, S, SW, SW)
1
2
+ f1
(H, S, HP, SW)
1
2
= 1
f2
(x1
(SW), x2
(SW), SW, SW)p(SW, SW) + f2
(x1
(HP), x2
(SW), HP, SW)p(HP, SW)
= f2
(S, S, SW, SW)
1
2
+ f2
(H, S, HP, SW)
1
2
=
3
2
̍ʘ 4 )
4 ̎
)
̍
̍ʘ 4 )
4
)
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)1޷͖
P1
̍ʘ 4 )
44
4)
)4
))

17. φογϡ‫ߧۉ‬
φογϡ‫ߧۉ‬
ϕΠδΞϯήʔϜʹ͓͚Δઓུͷ૊ ʹ͓͍ͯ
͢΂ͯͷ ʹରͯ͠
ͱͳ͍ͬͯΔͱ͖
Λφογϡ‫͏͍ͱߧۉ‬
٬‫཰֬؍‬Λ ͱ͢Δ
φογϡ‫ߧۉ‬͸ 44
4
͓Αͼ ))
)
͢ͳΘͪ
x* = (x*1
, ⋯, x*n
) i ∈ N
∑
t∈T
fi
(x*i
(ti
), x*−i
(t−i
), ti
, t−i
)p(ti
, t−i
) ≥
∑
t∈T
fi
(xi
(ti
), x*−i
(t−i
), ti
, t−i
)p(ti
, t−i
) ∀xi
∈ Xi
x* = (x*1
, ⋯, x*n
)
p = (1/2, 1/2)
x1
(SW) = S, x1
(HP) = S x2
(SW) = S
x1
(SW) = H, x1
(HP) = H x2
(SW) = H
̍ʘ 4 )
4 ̎
)
̍
̍ʘ 4 )
4
)
48޷͖
P1
)1޷͖
P1
̍ʘ 4 )
44
4)
)4
))

18. ϕΠδΞϯφογϡ‫ͱߧۉ‬φογϡ‫ߧۉ‬ͷಉ஋ੑ
ϕΠδΞϯφογϡ‫ͱߧۉ‬φογϡ‫ߧۉ‬ͷಉ஋ੑ
ϕΠδΞϯφογϡ‫͋Ͱߧۉ‬Δ͜ͱ͸
φογϡ‫͋Ͱߧۉ‬ΔͨΊͷඞཁे෼৚݅Ͱ͋Δ
͜ͷఆཧΑΓ
Ҏ߱
ϕΠδΞϯήʔϜͰͷφογϡ‫ߧۉ‬Λ‫ٻ‬Ίͯ
ͦΕΛϕΠδΞϯφογϡ‫͢ͱͱ͜Ϳݺͱߧۉ‬Δ

19. ήʔϜཧ࿦#4*$ୈճ
ϕΠδΞϯήʔϜʹ͓͚Δઓུͱ‫ ߧۉ‬
‫׬‬
ήʔϜཧ࿦#4*$ୈճʹͭͮ͘
See you next timeʂ