https://youtu.be/MoPmU06tAiY

1. ήʔϜཧ࿦#4*$ୈճ
ަবήʔϜɿަব໰୊ͱφογϡަবղ

2. ‫ྗڠ‬ήʔϜཧ࿦ͷલఏ
ਓ‫ྗڠ‬ήʔϜ ަবήʔϜ
ަব໰୊ͷྫ
ަব໰୊ͷఆٛ
ઓུ‫ܗ‬ήʔϜ͔Βަব໰୊΁
ަব໰୊ͷղ
ަবղ͕ຬͨ͢΂͖ͭͷެཧ

3. ‫ྗڠ‬ήʔϜཧ࿦ͷલఏ
͜Ε·Ͱͷඇ‫ྗڠ‬ήʔϜཧ࿦Ͱ͸ɺ
࿩͠߹͍ͷ݁Ռ౸ୡͨ͠߹ҙ͕߆ଋྗΛ΋ͨͳ͍
‫ྗڠ‬ήʔϜཧ࿦Ͱ͸߹ҙʹ߆ଋྗ͕͋Δͱ͍͏લఏ
AʘB ໧ൿ ࣗന
໧ൿ
ࣗന
໧ൿ
͠Α͏ͥ
͍͍ͥʙ

4. ਓ‫ྗڠ‬ήʔϜ ަবήʔϜ
‫݁ͨ͠ྗڠ‬Ռͱͯ͠ͲͷΑ͏ͳঢ়ଶΛ࣮‫͔͖΂͢ݱ‬ʁ
φογϡͷަব໰୊ͱͯ͠ఆࣜԽ͓Αͼφογϡަবղ
ϓϨΠϠʔؒͰ‫ྗڠ‬ͷ߹ҙ͕࣮‫͢ݱ‬Δ͔Ͳ͏͔ʁ
ඍ෼ۙࣅΞϓϩʔν·ͨ͸ఏҊԠ౴ήʔϜ
‫ڞ‬ಉࣄ‫͠ۀ‬Α͏ͥ
‫͢ྗڠ‬Δ͚Ͳ
ͳΜ͔ར఺͋Δͷʁ
རӹ͸
Ͳ͏෼͚Δͷ͞ʁ

5. ަব໰୊ͷྫ
ਓͷਓ෺
#͕‫ڞ‬ಉࣄ‫Ͱۀ‬ສԁͷརӹΛ͑Δ͜ͱ͕Ͱ͖Δͱ͢Δ
͸୯ಠͰࣄ‫ۀ‬Λ͢Ε͹ສԁ
#͸୯ಠͰࣄ‫ۀ‬Λ͢Ε͹ສԁΛ͑Δ͜ͱ͕Ͱ͖Δͱ͢Δ
͜ͷͱ͖
#͕‫ڞͯ͠ྗڠ‬ಉࣄ‫ۀ‬Λߦ͏ͱ͢Δͱ͖ʹಘΒΕΔສԁ
͸ͲͷΑ͏ʹ෼഑͢Ε͹Α͍͔ʁ

6. ަ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

7. ઓུ‫ܗ‬ήʔϜ͔Βަব໰୊΁
૬ؔࠞ߹ઓུΛߟ͑Δ͜ͱͰ࣮‫ݱ‬ՄೳྖҬΛඳ͚Δ
ఆٛɿਓઓུ‫ܗ‬ήʔϜʹؔͯ͠
ઓུͷ૊ Λ૬ؔઓུͱ‫ ͼݺ‬
૬ؔઓུͷू߹ ্ͷ֬཰෼෍Λ૬ؔࠞ߹ઓུͱ‫Ϳݺ‬
͜ΕʹΑΓ‫ظ‬଴རಘ͕ҎԼͰఆ·Δɿ
s = (s1
, s2
)
S1
× S2
Fi
(q) =
∑
s1
∈S1
∑
s2
∈S2
q(s1
, s2
)f1
(s1
, s2
)

8. ઓུ‫ܗ‬ήʔϜ͔Βަব໰୊΁
AʘB ໧ൿ ࣗന
໧ൿ
ࣗന
AʘB αοΧʔ ໺‫ٿ‬
αοΧʔ
໺‫ٿ‬
0
u2
u1
U
5
−4
6
6
5
−4 −3
−3
0
u2
u1
U
1
2
−1
−1
1 2

9. ઓུ‫ܗ‬ήʔϜ͔Βަব໰୊΁
AʘB ໧ൿ ࣗന
໧ൿ
ࣗന
AʘB αοΧʔ ໺‫ٿ‬
αοΧʔ
໺‫ٿ‬
0
u2
u1
U
5
−4
6
6
5
−4 −3
−3
0
u2
u1
U
1
2
−1
−1
1 2
ަবͷෆҰக఺͸ʁ
ϛχϚοΫεརಘ
φογϡ‫ߧۉ‬ͷརಘ
FUDʜ

10. ަব໰୊ͷղ
ަব໰୊ͷղ
ަব໰୊ ͷ଒Λ ͱ͢Δަব໰୊ͷ଒ ʹର͢Δަবղͱ͸
ؔ਺ Ͱ
Λຬͨ͢΋ͷΛ͍͏
(U, d) B B
f : B → ℜ2
f(U, d) = (f1
(U, d), f2
(U, d)) ∈ U

11. ަবղ͕ຬͨ͢΂͖ͭͷެཧ
ަব໰୊ͷφογϡަবղ
߹ҙ͕࣮‫͢ݱ‬ΔͷͰ͋Ε͹ɺ
ͦͷ߹ҙ఺͕ຬͨ͢΂͖ੑ࣭Λͭͷެཧͱͯ͠ఏࣔͨ͠
·ͨ
ͦͷͭͷެཧΛຬͨ͢߹ҙ఺͸ͨͩҰͭఆ·Δ͜ͱΛࣔͨ͠
φογϡ͕ͭͷެཧ͔Βಋ͍ͨަব໰୊ͷղΛφογϡަবղͱ‫ Ϳݺ‬
ύϨʔτ࠷దੑ
ରশੑ
ਖ਼࣍ม‫͔׵‬Βͷಠཱੑ
ແؔ܎ͳ݁Ռ͔Βͷಠཱੑ

12. ަবղ͕ຬͨ͢΂͖ͭͷެཧ
ύϨʔτ࠷దੑ
͢΂ͯͷ ʹରͯ͠ Ͱ
গͳ͘ͱ΋ͭͷʹରͯ͠
Ͱ͋Δ ͷ఺ ͸ଘࡏ͠ͳ͍
i = 1, 2 ti
≥ fi
(U, d) i
ti
fi
(U, d) U t = (t1
, t2
)
0
u2
u1
(d1
, d2
) = (20, 20)
U
100
100
ύϨʔτ࠷ద

13. ަবղ͕ຬͨ͢΂͖ͭͷެཧ
ରশੑ
ަব໰୊ ͕ରশͳΒ͹
ˎަব໰୊͕ରশͰ͋Δͱ͸
ͳΒ͹
(U, d) f1
(U, d) = f2
(U, d)
d1
= d2
(u1
, u2
) ∈ U (u2
, u1
) ∈ U
0
u2
u1
(d1
, d2
) = (20, 20)
U
100
100
ରশੑ

14. ަবղ͕ຬͨ͢΂͖ͭͷެཧ
ਖ਼࣍ม‫͔׵‬Βͷಠཱੑ
ަব໰୊ ͕ަব໰୊ ͔Βޮ༻ͷਖ਼࣍ม‫׵‬
ʹΑͬͯಘΒΕΔͱ͖
(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)
0
u2′
u1′
(d1′
, d2′
) = (20, 20)
U′
100
100
0
u2
u1
(d1
, d2
) = (20, 10)
U
50
100
u2′
= 2u2
( f1
, f2
) = (50, 25)
( f1′
, f2′
) = (50, 50)
ਖ਼࣍ม‫׵‬

15. ަবղ͕ຬͨ͢΂͖ͭͷެཧ
ແؔ܎ͳ݁Ռ͔Βͷಠཱੑ
̎ͭͷަব໰୊ ͱ ʹ͓͍ͯ
ͱ͢Δ
΋͠
ͳΒ͹
Ͱ͋Δ
(U, d) (T, d) T ⊂ U
f(U, d) ∈ T f(T, d) = f(U, d)
0
u2
u1
(d1
, d2
) = (20, 20)
U
100
100
( f1
, f2
) = (50, 50)
T
಺Ͱ࣮‫ݱ‬Մೳ
T

16. ަবղ͕ຬͨ͢΂͖ͭͷެཧ
ఆཧ
ަব໰୊ ͷ଒ ͱ͢Δ
ͭͷެཧΛຬͨ͢ަব໰୊ͷղ ͕Ұҙʹଘࡏ͠
͢΂ͯͷަব໰୊ ʹରͯ͠
͸
ͷղͰ͋Δ
͜ͷղΛφογϡަবղͱ͍͏
·ͨ
Λφογϡੵͱ͍͏
(U, d) B
f : B → ℜ2
(U, d) f(U, d)
max
u∈U:u≥d
(u1
− d1
)(u2
− d2
)
(u1
− d1
)(u2
− d2
)

17. ήʔϜཧ࿦#4*$ୈճ
ަবήʔϜɿަব໰୊ͱφογϡަবղ
࣍ճɿަবήʔϜɿఆཧͷূ໌