https://youtu.be/UTR5yTMpooI

2.
۩ମྫ
ΧϥΠʹεϞϩσΟϯεΩʔղ
۩ମྫ
‫ݸ‬ਓ୯ௐੑެཧ
Kalai, Ehud, and Meir Smorodinsky. Other Solutions to Nash's Bargaining
Problem. Econometrica 43, no. 3 (1975): 513-18.

3. ަব໰୊ͷఆٛ
ఆٛɿަব໰୊͸
࣮‫ݱ‬Մೳू߹ɿ
ަবͷෆҰக఺ɿ
ͷ૊ ʹΑͬͯఆٛ͞ΕΔ
Ծఆ
w ͸࣍‫ۭؒ਺࣮ݩ‬ ͷίϯύΫτͰತͳ෦෼ू߹
w
w ͷ఺ ͕ଘࡏͯ͠
U
d = (d1
, d2
)
(U, d)
U ℜ2
d ∈ U
U u = (u1
, u2
) ui
di
(i = 1, 2)
0
u2
u1
(d1
, d2
) = (20, 20)
U
100
100

4. φογϡަবղʹ͓͚Δͭͷެཧ
ύϨʔτ࠷దੑ
͢΂ͯͷ ʹରͯ͠ Ͱ
গͳ͘ͱ΋ͭͷ ʹରͯ͠
Ͱ͋Δ ͷ఺ ͸ଘࡏ͠ͳ͍
ରশੑ
ަব໰୊ ͕ରশͳΒ͹
ˎަব໰୊͕ରশͰ͋Δͱ͸
ͳΒ͹
ਖ਼࣍ม‫͔׵‬Βͷಠཱੑ
ަব໰୊ ͕ަব໰୊ ͔Βޮ༻ͷਖ਼࣍ม‫׵‬ ʹΑͬͯಘΒΕΔͱ͖
ແؔ܎ͳ݁Ռ͔Βͷಠཱੑ
̎ͭͷަব໰୊ ͱ ʹ͓͍ͯ
ͱ͢Δ
΋͠
ͳΒ͹
Ͱ͋Δ
i = 1, 2 ti
≥ fi
(U, d) i
ti
fi
(U, d) U t = (t1
, t2
)
(U, d) f1
(U, d) = f2
(U, d)
d1
= d2
(u1
, u2
) ∈ U (u2
, u1
) ∈ U
(U′

, d′

) (U, d) ui′

= αi
ui
+ βi
, αi
0 (i = 1,2)
fi
(U′

, d′

) = αi
fi
(U, d) + βi
, αi
0 (i = 1,2)
(U, d) (T, d) T ⊂ U
f(U, d) ∈ T f(T, d) = f(U, d)

5. φογϡަবղʹ͓͚Δͭͷެཧ
ύϨʔτ࠷దੑ
͢΂ͯͷ ʹରͯ͠ Ͱ
গͳ͘ͱ΋ͭͷ ʹରͯ͠
Ͱ͋Δ ͷ఺ ͸ଘࡏ͠ͳ͍
ରশੑ
ަব໰୊ ͕ରশͳΒ͹
ˎަব໰୊͕ରশͰ͋Δͱ͸
ͳΒ͹
ਖ਼࣍ม‫͔׵‬Βͷಠཱੑ
ަব໰୊ ͕ަব໰୊ ͔Βޮ༻ͷਖ਼࣍ม‫׵‬ ʹΑͬͯಘΒΕΔͱ͖
ແؔ܎ͳ݁Ռ͔Βͷಠཱੑ
̎ͭͷަব໰୊ ͱ ʹ͓͍ͯ
ͱ͢Δ
΋͠
ͳΒ͹
Ͱ͋Δ
i = 1, 2 ti
≥ fi
(U, d) i
ti
fi
(U, d) U t = (t1
, t2
)
(U, d) f1
(U, d) = f2
(U, d)
d1
= d2
(u1
, u2
) ∈ U (u2
, u1
) ∈ U
(U′

, d′

) (U, d) ui′

= αi
ui
+ βi
, αi
0 (i = 1,2)
fi
(U′

, d′

) = αi
fi
(U, d) + βi
, αi
0 (i = 1,2)
(U, d) (T, d) T ⊂ U
f(U, d) ∈ T f(T, d) = f(U, d)
0
u2
u1
U
100
100
(d1
, d2
) = (20, 20)
(50, 50)
ແؔ܎ͳ݁Ռ

6. ۩ମྫ
ҎԼͷೋͭͷަব໰୊Λߟ͑Δ͍ͣΕ΋ަবͷෆҰக఺͸
φογϡަবղ͸ʁ ԋशࢀর
d = (0, 0)
u2
u1
1
0
(0.75, 0.75)
1
0.75
u2
u1
1
0 1
0.7
0.75
(1, 0.7)
ަব໰୊1:(S1, d) ަব໰୊2:(S2, d)
S1 S2

7. ۩ମྫ
ҎԼͷೋͭͷަব໰୊Λߟ͑Δ͍ͣΕ΋ަবͷෆҰக఺͸
φογϡަবղ͸ʁ ԋशࢀর
d = (0, 0)
u2
u1
1
0
(0.75, 0.75)
1
0.75
u2
u1
1
0 1
0.7
0.75
(1, 0.7)
ަব໰୊1:(S1, d) ަব໰୊2:(S2, d)
S1 S2

8. ۩ମྫ
ަব໰୊ʹൺ΂
ަব໰୊Ͱ͸ϓϨΠϠʔʹ͍ͭͯಘΒΕΔརಘ ޮ༻
͸େ͖͍
͔͠͠
φογϡަবղͰ͸͔Βͱখ͘͞ͳΔ
ϓϨΠϠʔ͸ೲಘͰ͖Δ͔ʁ
u2
u1
1
0 1
0.7 (1, 0.7)
S1
S2

9. ΧϥΠʹεϞϩσΟϯεΩʔղ
ΧϥΠͱεϞϩσΟϯεΩʔ͸ެཧʹ୅ΘΔެཧΛ௥Ճ͠
ผͷަবղΛఏࣔͨ͠
͜ͷަবղ͸
ʮ֤‫ݸ‬ਓͷ࠷΋๬·͍͠རಘʯͱʮަবղʹ͓͚Δརಘʯͷൺ͕
ϓϨΠϠʔؒͰ౳͘͠ͳΔΑ͏ʹܾΊΒΕΔ
u2
u1
1
0 1
0.7 (1, 0.7)
S1
S2
͜ͷ఺͕ߟྀ͞ΕΔ

10. ࠓճͷ໰୊͸
ެཧͷʮແؔ܎ͳ݁Ռ͔Βͷಠཱੑʯ‫ى‬Ҽ
ΧϥΠͱεϞϩσΟϯεΩʔ͸͜ͷެཧʹ୅ΘΔެཧΛ௥Ճ͠
ผͷަবղΛఏࣔͨ͠
͜ͷަবղ͸
ʮ֤‫ݸ‬ਓͷ࠷΋๬·͍͠རಘʯͱʮަবղʹ͓͚Δརಘʯͷൺ͕
ϓϨΠϠʔؒͰ౳͘͠ͳΔΑ͏ʹܾΊΒΕΔ
۩ମྫ
u2
u1
1
0
(0.75, 0.75)
1
0.75
u2
u1
1
0 1
0.7
0.75
(1, 0.7)
ަব໰୊1:(S1, d) ަব໰୊2:(S2, d)
S1 S2
ཧ૝఺ ཧ૝఺
Nash
KS
(0.77, 0.77)

11. ‫ݸ‬ਓ୯ௐੑެཧ
ަব໰୊ ʹରͯ͠
ΛҎԼͰఆٛ͢Δɿ
ަব໰୊ͷఆٛʹ͓͚ΔԾఆΑΓ
ͷίϯύΫτੑ͔Β Ͱ͋Δ·ͨ
·ͨ
ʹ͍ͭͯ
ΛҎԼͰఆٛ͢Δɿ
(S, d) b(S) = (b1
(S), b2
(S))
b1
(S) = sup{u1
∈ ℜ : ∃u2
∈ ℜ, (u1
, u2
) ∈ S}
b2
(S) = sup{u2
∈ ℜ : ∃u1
∈ ℜ, (u1
, u2
) ∈ S}
S b(S) = (b1
(S), b2
(S)) ∈ S (d1
. d2
) (b1
(S), b2
(S))
u1
≤ b1
(S) gS(u1
)
gS(u1
) =
{
u2
if (u1
, u2
) is pareto of (S, d)
b2
(S) otherwise
u2
u1
0
S
b1
(S)
b2
(S)
ʹ͍ͭͯ
u1
≤ b1
(S) gS(u1
)

12. ‫ݸ‬ਓ୯ௐੑެཧ
ަব໰୊ ʹରͯ͠
ΛҎԼͰఆٛ͢Δɿ
·ͨ
ʹ͍ͭͯ
ΛҎԼͰఆٛ͢Δɿ
‫ݸ‬ਓ୯ௐੑ
̎ͭͷަব໰୊ ͱ ʹ͓͍ͯ
΋͠
Ͱ͋Γ
͔ͭ೚ҙͷ ʹ͍ͭͯ
ͳΒ͹
Λຬͨ͢
্‫ه‬͸ϓϨΠϠʔʹ͍ͭͯड़΂͍ͯΔ͕
ϓϨΠϠʔ΋ಉ༷
ͭͷަব໰୊ʹ͓͍ͯ
ϓϨΠϠʔ͕ཁ‫͏͠ٻ‬Δ͢΂ͯͷརಘʹ͍ͭͯ
ϓϨΠϠʔͷୡ੒Մೳͳརಘ͕େ͖͘Ͱ͖ΔͷͰ͋Ε͹
ަব݁Ռʹ͓͍ͯ΋
ϓϨΠϠʔͷಘΔརಘ΋େ͖͘͢Δඞཁ͕͋Δ
(S, d) b(S) = (b1
(S), b2
(S))
b1
(S) = sup{u1
∈ ℜ : ∃u2
∈ ℜ, (u1
, u2
) ∈ S} b2
(S) = sup{u2
∈ ℜ : ∃u1
∈ ℜ, (u1
, u2
) ∈ S}
u1
≤ b1
(S) gS(u1
) gS(u1
) =
{
u2
if (u1
, u2
) is pareto of (S, d)
b2
(S) otherwise
(S1, d) (S2, d)
b1
(S1) = b1
(S2) u1
gS2
(u1
) ≤ gS1
(u1
)
f2
(S2, d) ≤ f2
(S1, d)
u2
u1
0
S1
b1
(S1) = b1
(S2)
b2
(S1) = b2
(S2)
೚ҙͷ ʹ͍ͭͯ
u1
gS2
(u1
) ≤ gS1
(u1
)
S2

13. ఆཧ
ύϨʔτ࠷దੑ
ରশੑ
ਖ਼࣍ม‫͔׵‬Βͷಠཱੑ͓Αͼ‫ݸ‬ਓ୯ௐੑΛຬͨ͢
ަবղ ͸ͨͩҰͭଘࡏ͢Δ
·ͨ
͜ͷlؔ਺z ͸ҎԼͷੑ࣭Λຬͨ͢ɿ
ަব໰୊ ʹ͍ͭͯ
ͱ Λ݁Ϳઢ Λߟ͑Δ
͜ͷͱ͖ ্ͷ ͷ࠷େ఺͕ΧϥΠʹεϞϩσΟϯεΩʔղ Ͱ͋Δ
μ
μ
(S, d) d b(S) L(d, b(S))
S L(d, b(S)) μ(S, d)
u2
u1
1
0 1
0.7 (1, 0.7)
ަব໰୊2:(S2, d)
S2
ཧ૝఺
Nash
KS
(0.77, 0.77)