The document defines a set N={1,2,3} and a set function v that assigns values to subsets of N. It defines the core C(v) of v as the set of vectors that dominate all other vectors pointwise. The core is characterized as the set of vectors x in the affine space A(v) such that for all subsets S of N excluding the empty set, the sum of the components of x indexed by S is greater than or equal to v(S).

1. ήʔϜཧ࿦#4*$ୈճ
഑෼ͷࢧ഑ͱίΞ

2. ಛੑؔ਺‫ܗ‬ήʔϜͷఆٛ ෮श
഑෼ͷఆٛ ෮श
഑෼ͷ‫ج‬ຊࡾ֯‫ܗ‬
഑෼ͷࢧ഑
ίΞͷఆٛ
༏Ճ๏ੑΛຬͨ͢ήʔϜʹ͓͚ΔίΞ
ਓήʔϜͷίΞͷଘࡏ৚݅
ఆཧͷূ໌ʹؔͯ͠

3. ಛੑؔ਺‫ܗ‬ήʔϜͷఆٛ
ఆٛɿఏ‫ͱܞ‬ಛੑؔ਺
ϓϨΠϠʔͷू߹Λ

ͱ͢Δ
͜ͷͱ͖

ͷ෦෼ू߹Λఏ‫
͍͍ͱܞ‬ఏ‫ܞ‬ͷू߹Λ

Ͱද͢
͜ͷ֤ఏ‫਺࣮ʹܞ‬஋Λ༩͑Δؔ਺Λ
ৡ౉Մೳͳޮ༻Λ΋ͭήʔϜ 56ήʔϜ
ʹ͓͚Δಛੑؔ਺ͱ͍͏

ಛੑؔ਺‫ܗ‬ήʔϜ͸

Ͱ༩͑ΒΕΔ
ྫ

ͷͱ͖









N
N 2N
v : 2N
→ ℜ
(N, v)
N = {1, 2, 3} 2N
= {∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
v(∅) = 0 v({1}) = 10 v({2}) = 20 v({3}) = 10 v({1,2}) = 50 v({1,3}) = 50 v({2,3}) = 70 v({1,2,3}) = 100
શһͷఏ‫ܞ‬Λશମఏ‫Ϳݺͱܞ‬

4. ഑෼ͷఆٛ
ఆٛɿ഑෼
ಛੑؔ਺‫ܗ‬ήʔϜ

ͷརಘϕΫτϧ

͕഑෼Ͱ͋Δͱ͸
‫ݸ‬ਓ߹ཧੑ

શମ߹ཧੑ

͕੒Γཱͭ͜ͱͰ͋Δ
഑෼ͷू߹Λ

Ͱද͢
(N, v) x = (x1, ⋯, xn)
xi ≥ v({i}) ∀i ∈ N
∑
i∈N
xi = v(N)
𝒜
(v)

5. ྫ

ͷͱ͖













ͱͯ͠

ͱઓུతಉ౳ͳಛੑؔ਺

͸








N = {1, 2, 3} 2N
= {∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}}
v(∅) = 0 v({1}) = 10 v({2}) = 20 v({3}) = 10 v({1,2}) = 50 v({1,3}) = 50 v({2,3}) = 70 v({1,2,3}) = 100
v′

(S) = v(S) +
∑
i∈S
βi β1 = − 10 β2 = − 20 β3 = − 10 v v′

v′

(∅) = 0 v′

({1}) = 0 v′

({2}) = 0 v′

({3}) = 0 v′

({1,2}) = 20 v′

({1,3}) = 30 v′

({2,3}) = 40 v′

({1,2,3}) = 60
഑෼ͷ‫ج‬ຊࡾ֯‫ܗ‬
v′

({1,2,3}) = 60
1
2 3
x3
x1
x2
શମ߹ཧੑɿ

x1 + x2 + x3 = 60

1
2
x1l +
1
2
x2l +
1
2
x3l =
1
2
60l
l

6. ഑෼ͷࢧ഑
ఆٛɿ഑෼ͷࢧ഑
̎ͭͷ഑෼

ͱఏ‫ܞ‬

ΛͱΔ࣍ͷ̎৚͕݅ຬͨ͞ΕΔͱ͖

͸

Λ

Λ௨ͯ͠ࢧ഑͢Δͱ͍͍

ͱॻ͘


̎ͭͷ഑෼

ʹରͯ͠
͋Δఏ‫ܞ‬

͕ଘࡏͯ͠

͕੒Γཱͭͱ͖
୯ʹ

͸

Λࢧ഑͢Δͱ͍͍

ͱॻ͘
x, y S ⊆ N
x y S x domS y
v(S) ≥
∑
i∈S
xi
xi yi, ∀i ∈ S
x, y S x domS y
x y x dom y

7. ίΞͷఆٛ
ఆٛɿίΞ
ಛੑؔ਺‫ܗ‬ήʔϜ

ͷίΞͱ͸
ଞͷ͍͔ͳΔ഑෼ʹ΋ࢧ഑͞Εͳ͍഑෼ͷू߹Ͱ͋Δ

ͳΔ

͕ଘࡏ͠ͳ͍









഑෼

ʹରͯ͠

‫ͼٴ‬

Λߟ͑Δͱ



ΑΓ

͸

Λ௨ͯ͠

Λࢧ഑͢Δ
(N, v)
C(v) = {x ∈
𝒜
(v)|y dom x y }
v′

(∅) = 0 v′

({1}) = 0 v′

({2}) = 0 v′

({3}) = 0 v′

({1,2}) = 20 v′

({1,3}) = 30 v′

({2,3}) = 40 v′

({1,2,3}) = 60
(y1, y2, y3) = (30,20,10) (x1, x2, x3) = (20,25,15) S = {2,3}
v′

({2,3}) = 40 ≥ x2 + x3 = 40
x2 y2 x3 y3
x S = {2,3} y
1
2 3
(30,20,10)
(20,25,15)

8. ίΞͷఆٛ
ఆٛɿίΞ
ಛੑؔ਺‫ܗ‬ήʔϜ

ͷίΞͱ͸
ଞͷ͍͔ͳΔ഑෼ʹ΋ࢧ഑͞Εͳ͍഑෼ͷू߹Ͱ͋Δ








(N, v)
v′

(∅) = 0 v′

({1}) = 0 v′

({2}) = 0 v′

({3}) = 0 v′

({1,2}) = 20 v′

({1,3}) = 30 v′

({2,3}) = 40 v′

({1,2,3}) = 60
1
2 3
x1 ≤ 20
x2 ≤ 30
x3 ≤ 40
ίΞ

9. ༏Ճ๏ੑΛຬͨ͢ήʔϜʹ͓͚ΔίΞ
ఆཧ
ಛੑؔ਺‫ܗ‬ήʔϜ

ʹ͓͍ͯ

͕༏Ճ๏ੑΛຬͨ͢ͱ͖
ίΞ͸

‫͑׵͍ݴ‬Ε͹
ίΞ͸ఏ‫߹ܞ‬ཧੑΛຬͨ͢഑෼ͱ͍͏͜ͱ
(N, v) v
C(v) = {x ∈
𝒜
|
∑
i∈S
xi ≥ v(S), ∀S ⊂ N, S ≠ ∅, N}
1
2 3
x1 ≤ 20
x2 ≤ 30
x3 ≤ 40
ίΞ
ఏ‫߹ܞ‬ཧੑ

10. ༏Ճ๏ੑΛຬͨ͢ήʔϜʹ͓͚ΔίΞ
ఆཧ
ಛੑؔ਺‫ܗ‬ήʔϜ

ʹ͓͍ͯ

͕༏Ճ๏ੑΛຬͨ͢ͱ͖
ίΞ͸









ίΞͷ഑෼Λ

ͱ͢Δͱఏ‫߹ܞ‬ཧੑͱ
‫ݸ‬ਓ߹ཧੑ

ͱશମ߹ཧੑ

ΑΓ



(N, v) v
C(v) = {x ∈
𝒜
|
∑
i∈S
xi ≥ v(S), ∀S ⊂ N, S ≠ ∅, N}
v′

(∅) = 0 v′

({1}) = 0 v′

({2}) = 0 v′

({3}) = 0 v′

({1,2}) = 20 v′

({1,3}) = 30 v′

({2,3}) = 40 v′

({1,2,3}) = 60
(x1, x2, x3)
xi ≥ 0, ∀i ∈ N x1 + x2 + x3 = 60
x1 + x2 ≥ v′

({1,2}) = 20 ⇒ 0 ≤ x3 ≤ 40
x1 + x3 ≥ v′

({1,3}) = 30 ⇒ 0 ≤ x2 ≤ 30
x2 + x3 ≥ v′

({2,3}) = 40 ⇒ 0 ≤ x1 ≤ 20
1
2 3
x1 ≤ 20
x2 ≤ 30
x3 ≤ 40
ίΞ

11. ਓήʔϜͷίΞͷଘࡏ৚݅
ఆཧ
ਓͷήʔϜ

ʹ͓͍ͯ

͕༏Ճ๏ੑΛຬͨ͢ͱ͖
ίΞͷଘࡏ৚݅͸









ίΞͷ഑෼Λ

ͱ͢Δͱఏ‫߹ܞ‬ཧੑͱ
‫ݸ‬ਓ߹ཧੑ

ͱશମ߹ཧੑ

ΑΓ



(N, v) v
v({1,2}) + v({1,3}) + v({2,3}) ≤ 2v({1,2,3})
v′

(∅) = 0 v′

({1}) = 0 v′

({2}) = 0 v′

({3}) = 0 v′

({1,2}) = 20 v′

({1,3}) = 30 v′

({2,3}) = 40 v′

({1,2,3}) = 60
(x1, x2, x3)
xi ≥ 0, ∀i ∈ N x1 + x2 + x3 = 60
x1 + x2 ≥ v′

({1,2}) = 20 ⇒ 0 ≤ x3 ≤ 40
x1 + x3 ≥ v′

({1,3}) = 30 ⇒ 0 ≤ x2 ≤ 30
x2 + x3 ≥ v′

({2,3}) = 40 ⇒ 0 ≤ x1 ≤ 20
1
2 3
x1 ≤ 20
x2 ≤ 30
x3 ≤ 40
ίΞ

12. ਓήʔϜͷίΞͷଘࡏ৚݅
ఆཧ
ਓͷήʔϜ

ʹ͓͍ͯ

͕༏Ճ๏ੑΛຬͨ͢ͱ͖
ίΞͷଘࡏ৚݅͸









ίΞͷ഑෼Λ

ͱ͢Δͱఏ‫߹ܞ‬ཧੑͱ
‫ݸ‬ਓ߹ཧੑ

ͱશମ߹ཧੑ

ΑΓ



(N, v) v
v({1,2}) + v({1,3}) + v({2,3}) ≤ 2v({1,2,3})
v(∅) = 0 v({1}) = 0 v({2}) = 0 v({3}) = 0 v({1,2}) = 30 v({1,3}) = 40 v({2,3}) = 40 v({1,2,3}) = 50
(x1, x2, x3)
xi ≥ 0, ∀i ∈ N x1 + x2 + x3 = 50
x1 + x2 ≥ v({1,2}) = 30 ⇒ 0 ≤ x3 ≤ 20
x1 + x3 ≥ v({1,3}) = 40 ⇒ 0 ≤ x2 ≤ 10
x2 + x3 ≥ v({2,3}) = 40 ⇒ 0 ≤ x1 ≤ 10
1
2 3
x1 ≤ 10
x2 ≤ 10
x3 ≤ 20

13. ఆཧͷূ໌ʹؔͯ͠
ఆཧ
ಛੑؔ਺‫ܗ‬ήʔϜ

ʹ͓͍ͯ

͕༏Ճ๏ੑΛຬͨ͢ͱ͖
ίΞ͸

ఆཧ
ਓͷήʔϜ

ʹ͓͍ͯ

͕༏Ճ๏ੑΛຬͨ͢ͱ͖
ίΞͷଘࡏ৚݅͸

/ਓήʔϜͰͷଘࡏূ໌ͳͲ͸/95ࢀরͱ͠ɺ͜͜Ͱ͸লུ
(N, v) v
C(v) = {x ∈
𝒜
|
∑
i∈S
xi ≥ v(S), ∀S ⊂ N, S ≠ ∅, N}
(N, v) v
v({1,2}) + v({1,3}) + v({2,3}) ≤ 2v({1,2,3})

14. ήʔϜཧ࿦#4*$ୈճ
഑෼ͷࢧ഑ͱίΞ
࣍ճɿࢢ৔ήʔϜͱίΞ