ChatGPTが作成したゲーム理論の問題を解く1

$IBU(15͕࡞੒ͨ͠
ήʔϜཧ࿦ͷ໰୊Λղ͘

໰୊ɿ؆୯ͳೋਓήʔϜͷදΛߟ͑·͢ɻ
ϓϨΠϠʔ͸ߦΛɺϓϨΠϠʔ͸ྻΛબͼ·͢ɻϓϨΠϠʔͷརಘ͕දͷཁૉͷ্ɺϓϨΠϠʔͷརಘ͕දͷ
ཁૉͷԼʹॻ͔Ε͍ͯ·͢ɻϓϨΠϠʔ͸ͭͷߦ͔ΒબͼɺϓϨΠϠʔ͸ͭͷྻ͔Βબͼ·͢ɻ
͜ͷήʔϜʹ͓͍ͯɺࢧ഑ઓུ͕ଘࡏ͢Δ͔Ͳ͏͔Λ൑அ͍ͯͩ͘͠͞ɻ
७ਮઓུφογϡ‫ߧۉ‬Λ‫ٻ‬Ί͍ͯͩ͘͞ɻ
͜ͷήʔϜʹ͓͍ͯɺࠞ߹ઓུφογϡ‫͕ߧۉ‬ଘࡏ͢Δ͔Ͳ͏͔Λ൑அ͍ͯͩ͘͠͞ɻ
$IBU(15͕࡞੒ͨ͠རಘද
ʘ ྻ ྻ
ߦ
ߦ
ߦ

ࢧ഑ઓུ
ϓϨΠϠʔ ͷ̎ͭͷઓུ ʹ͍ͭͯ

Λຬͨ͢ͱ͖ ͸ Λࢧ഑͢Δͱ͍͏
ŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠŠ
ϓϨΠϠʔ ͷઓུ ʹ͍ͭͯ ʹରͯ͠

Λຬͨ͢ͱ͖ Λࢧ഑ઓུͱ͍͏
˒ϓϨΠϠʔ͸ࢧ഑ઓུ͕͋Ε͹
ͦͷઓུΛબͿͱߟ͑ΒΕΔ
i si
, s′

i
∈ Si
fi
(si
, s−i
) fi
(s′

i
, s−i
), ∀s−i
∈ S−i
si
s′

i
i si
∈ Si
∀s′

i
∈ Si
{si
}
fi
(si
, s−i
) fi
(s′

i
, s−i
), ∀s−i
∈ S−i
si
AʘB

sA
1
sA
2
sA
3
sB
1 sB
2
AʘB

sA
1
sA
2
sA
3
sB
1 sB
2
ήʔϜཧ࿦#4*$ΑΓ

φογϡ‫ߧۉ‬ɿఆٛ
ઓུͷ૊ ͕φογϡ‫͋Ͱߧۉ‬Δͱ͸
͢΂ͯͷϓϨΠϠʔʹରͯ͠

Λຬͨ͢ͱ͖Λ͍͏
͢΂ͯͷϓϨΠϠʔ͕‫࠷ʹ͍ޓ‬ద൓ԠઓུΛͱ͍ͬͯΔΑ͏ͳઓུͷ૊
(s*1
, ⋯, s*n
) ∈ S
fi
(s*i
, s*−i
) ≥ fi
(si
, s*−i
) ∀si
∈ Si
AʘB

sA
1
sA
2
sA
3
sB
1 sB
2
ྫ
AʘB ໧ൿ ࣗന
໧ൿ
ࣗന
ྫ
࠷ద൓Ԡઓུ
ϓϨΠϠʔ Ҏ֎ͷઓུΛ ͱ͠‫ݻ‬ఆ͢Δ

Λຬͨ͢ͱ͖ ͸ ʹର͢Δ࠷ద൓Ԡઓུͱ͍͏
i s−i
∈ S−i
fi
(s *i
, s−i
) ≥ fi
(si
, s−i
), ∀si
∈ Si
s *i
s−i
φογϡ‫ߧۉ‬
(sA
2 , sB
2 )
φογϡ‫ߧۉ‬
ࣗന ࣗന
(
ήʔϜཧ࿦#4*$ΑΓ

ʘ ྻ ྻ
ߦ
ߦ
ߦ
ࢧ഑ઓུͳ͠
ࢧ഑ઓུͳ͠
φογϡ‫ߧۉ‬ ߦ ྻ

ʘ ྻ ྻ
ߦ
ߦ
ߦ
ଘࡏ͢Δ
ϓϨΠϠʔ͸ߦ̏ʹ֬཰
ϓϨΠϠʔ͸ྻʹ֬཰ΛৼΔ
ࠞ߹ઓུΛߟ͑Ε͹ྑ͍

ʘ ྻ ྻ
ߦ
ߦ
ߦ
Ҏ֎ͷࠞ߹ઓུ͸͋Δ͔ʁ

ʘ ྻ ྻ
ߦ
ߦ
ߦ
ߦ·ͨ͸ߦʹࢧ഑͞ΕΔઓུ
͜ͷઓུʹਖ਼ͷ֬཰ΛৼΔ͘Β
͍ͳߦ·ͨ͸ߦʹ֬཰Λৼͬ
ͨํ͕Α͍
p
1 − p
r 1 − r
Ҏ֎ͷࠞ߹ઓུ͸͋Δ͔ʁ

ߦ·ͨ͸ߦʹࢧ഑͞ΕΔઓུ
͜ͷઓུʹਖ਼ͷ֬཰ΛৼΔ͘Β͍ͳߦ·ͨ͸ߦʹ֬཰Λৼͬͨํ͕Α͍
೚ҙͷ Ͱ্‫ظه‬଴རಘΛ࠷େʹ͢Δ͜ͱΛߟ͑Ε͹ Λେ͖͘͠ ͱ͓͘ͷ͕ྑ͍
2 ⋅ pr + 1 ⋅ p(1 − r) + {(−1) ⋅ qr + 0 ⋅ (1 − r)} = 2pr + p − pr − qr = (1 + r)p − rq
r (0 ≤ r ≤ 1) p q = 0
ʘ ྻ ྻ
ߦ
ߦ
ߦ
p
r 1 − r
q

ߦΛআ‫݁ͨ͠ڈ‬Ռ ྻ͸ྻʹࢧ഑͞Ε͍ͯΔͷͰ ྻʹ֬཰ৼΔ͘Β͍ͳΒྻʹৼͬͨํ͕ྑ͍
Ώ͑ʹ
ࠞ߹ઓུφογϡ‫ߧۉ‬͸໰୊Ͱ‫ٻ‬Ίͨ७ઓུφογϡ‫ߧۉ‬Ҏ֎ʹ͸ଘࡏ͠ͳ͍

r = 0
((p, q,1 − p − q), (r,1 − r)) = ((0,0,1), (0,1))
ʘ ྻ ྻ
ߦ
ߦ
ߦ
p
r 1 − r
1 − p
0

ChatGPTが作成したゲーム理論の問題を解く1

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