The document defines a model for coalition formation with three players (1, 2, 3). It specifies the values of different coalitions and defines the feasible set and payoff functions. It then calculates the payoff functions sij(x) for each pair of players i,j in terms of the values of the different coalitions and the feasible set constraints.

1. ήʔϜཧ࿦#4*$ԋश
ΧʔωϧΛ‫ٻ‬ΊΔ

2. Χʔωϧͷఆٛͷ֬ೝ
ަবू߹ͱΧʔωϧͷؔ܎
લճͷ໰୊

3. Χʔωϧͷఆٛͷ֬ೝ
ఆٛɿఏ‫ܞ‬ ͷෆຬ
ಛੑؔ਺‫ܗ‬ήʔϜ ʹ͓͍ͯ
഑෼ ͱఏ‫ܞ‬ ΛͱΔͱ͖
Λ഑෼ ʹର͢Δఏ‫ܞ‬ ͷෆຬͱ͍͏
ਓͷϓϨΠϠʔ
ΛͱΓ
ͱ͓͘
ఆٛɿ࠷େෆຬ
ਓͷϓϨΠϠʔ
ͱ഑෼ ʹ͍ͭͯ
Λ഑෼ ʹ͓͚ΔϓϨΠϠʔ ͷ ʹର͢Δ࠷େෆຬͱ͍͏
S
(N, v) x ∈
𝒜
(v) S ⊆ N
e(S, x) = v(S) −
∑
i∈S
xi x S
i j ∈ N Tij = {S ⊆ N|i ∈ S, j ∉ S}
i j ∈ N x ∈
𝒜
(v)
sij(x) = max
S∈Tij
e(S, x)
x i j

4. Χʔωϧͷఆٛͷ֬ೝ
ఆٛɿ༏Ґ
ਓͷϓϨΠϠʔ
ͱ഑෼ ʹ͍ͭͯ
ͱͳΔͱ͖
഑෼ ʹ͓͍ͯϓϨΠϠʔ ͸ ΑΓ΋༏ҐͰ͋Δͱ͍͍
ͱॻ͘
Ͱ͋Ε͹
ϓϨΠϠʔ ͸ ʹରͯ͠རಘΛ͘Εͱ͸͍͑ͳ͍
ͳͥͳΒ ͸ఏ‫ܞ‬Λ૊ΉΠϯηϯςΟϒ͕ͳ͘ͳΔͨΊ
i j ∈ N x ∈
𝒜
(v)
sij(x) sji(x) xj v({j})
x i j i ≻x j
xj ≤ v({j}) i j
j
ਓͷϓϨΠϠʔͷ
࠷େෆຬΛൺֱ

5. Χʔωϧͷఆٛͷ֬ೝ
ఆٛɿ‫ߧۉ‬ঢ়ଶ
ਓͷϓϨΠϠʔ
ͱ഑෼ ʹ͍ͭͯ
Ͱ΋ͳ͘ Ͱ΋ͳ͍ͱ͖
഑෼ ʹ͓͍ͯϓϨΠϠʔ ͱ ͸‫ߧۉ‬ঢ়ଶͰ͋Δͱ͍͍
ͱॻ͘
ఆٛɿΧʔωϧ
೚ҙͷਓͷϓϨΠϠʔ͕‫ߧۉ‬ঢ়ଶʹ͋ΔΑ͏ͳ഑෼ͷશମ
ΛΧʔωϧͱ͍͏
i j ∈ N x ∈
𝒜
(v)
i ≻x j j ≻x i
x i j i ∼x j
𝒦
= {x ∈
𝒜
(v)|i ∼x j ∀i, j ∈ N, i ≠ j}

6. Χʔωϧͷఆٛͷ֬ೝ
ఆٛɿ‫ߧۉ‬ঢ়ଶ
ਓͷϓϨΠϠʔ
ͱ഑෼ ʹ͍ͭͯ
Ͱ΋ͳ͘ Ͱ΋ͳ͍ͱ͖
഑෼ ʹ͓͍ͯϓϨΠϠʔ ͱ ͸‫ߧۉ‬ঢ়ଶͰ͋Δͱ͍͍
ͱॻ͘
ͱͳΔͨΊʹ͸
͸഑෼ͷ‫ݸ‬ਓ߹ཧੑΑΓ
ʹ஫ҙͯ͠
͔ͭ
͢ͳΘͪ
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i j ∈ N x ∈
𝒜
(v)
i ≻x j j ≻x i
x i j i ∼x j
i ∼x j x ∈
𝒜
(v) xi ≥ v({i}) xj ≥ v({j})
sij(x) ≥ sji(x) sij(x) ≤ sji(x) sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
ఆٛɿ༏Ґ
ਓͷϓϨΠϠʔ
ͱ഑෼ ʹ͍ͭͯ
ͱͳΔͱ͖
഑෼ ʹ͓͍ͯϓϨΠϠʔ ͸ ΑΓ΋༏ҐͰ͋Δͱ͍͍
ͱॻ͘
i j ∈ N x ∈
𝒜
(v)
sij(x) sji(x) xj v({j})
x i j i ≻x j
͔ͭ
xj ≥ v({j}) xj ≤ v({j})
͔ͭ
xi ≥ v({i}) xi ≤ v({i})

7. ަবू߹ͱΧʔωϧͷؔ܎
ఆཧɿަবू߹ͱΧʔωϧ
ήʔϜ ͕༏Ճ๏ੑΛຬͨ͢ͱ͢Δ͜ͷͱ͖ΧʔωϧΛ
ަবू߹Λ ͱ͢Δͱ
(N, v)
𝒦
(v) ℬ(v)
𝒦
(v) ⊆ ℬ(v)
1
2 3
x1 = 10
x2 = 3
x3 = 0
ަবू߹
Χʔωϧ

8.
͜ͷಛੑؔ਺Λ༻͍ͯ
ΧʔωϧΛ‫ٻ‬ΊͯΈΑ͏
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2
v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|5 ≥ x3, 3 ≥ x2, 4 ≥ x1}
લճͷ໰୊ ͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
φογϡ‫ߧۉ‬
φογϡ‫ߧۉ‬
1
2 3
x1 ≤ 4
x2 ≤ 3
x3 ≤ 5
ίΞ
҆ఆू߹
ަবू߹

9.
͜ͷಛੑؔ਺Λ༻͍ͯ
ΧʔωϧΛ‫ٻ‬ΊͯΈΑ͏
ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|5 ≥ x3, 3 ≥ x2, 4 ≥ x1}
x = (x1, x2, x3) ∈
𝒜
(v)
s12(x) = max(v({1,3}) − x1 − x3), v({1}) − x1) )
= max(3 − x1 − x3, 0 − x1 )
=
{
3 − x1 − x3 (0 ≤ x3 ≤ 3)
−x1 (3 x3 ≤ 6)
s13(x) = max(v({1,2}) − x1 − x2, v({1}) − x1 )
=
{
1 − x1 − x2 (0 ≤ x2 ≤ 1)
−x1 (1 x2 ≤ 6)
લճͷ໰୊
1
2 3
x1 ≤ 4
x2 ≤ 3
x3 ≤ 5
ίΞ
҆ఆू߹
ަবू߹

10.
͜ͷಛੑؔ਺Λ༻͍ͯ
ΧʔωϧΛ‫ٻ‬ΊͯΈΑ͏
ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|5 ≥ x3, 3 ≥ x2, 4 ≥ x1}
x = (x1, x2, x3) ∈
𝒜
(v)
s21(x) = max(v({2,3}) − x2 − x3), v({2}) − x2) )
= max(2 − x2 − x3, 0 − x2 )
=
{
2 − x2 − x3 (0 ≤ x3 ≤ 2)
−x2 (2 x3 ≤ 6)
s23(x) = max(v({1,2}) − x1 − x2), v({2}) − x2) )
= max(1 − x1 − x2, 0 − x2 )
=
{
1 − x1 − x2 (0 ≤ x1 ≤ 1)
−x2 (1 x1 ≤ 6)
લճͷ໰୊
1
2 3
x1 ≤ 4
x2 ≤ 3
x3 ≤ 5
ίΞ
҆ఆू߹
ަবू߹

11.
͜ͷಛੑؔ਺Λ༻͍ͯ
ΧʔωϧΛ‫ٻ‬ΊͯΈΑ͏
ΛͱΔͱ
֤ϓϨΠϠʔͷ࠷େෆຬ͸
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
C(v) = {x ∈
𝒜
(v)|5 ≥ x3, 3 ≥ x2, 4 ≥ x1}
x = (x1, x2, x3) ∈
𝒜
(v)
s31(x) = max(v({2,3}) − x2 − x3), v({3}) − x3) )
= max(2 − x2 − x3, 0 − x3 )
=
{
2 − x2 − x3 (0 ≤ x2 ≤ 2)
−x3 (2 x2 ≤ 6)
s32(x) = max(v({1,3}) − x1 − x3), v({3}) − x3) )
= max(3 − x1 − x3, 0 − x3 )
=
{
3 − x1 − x3 (0 ≤ x1 ≤ 3)
−x3 (3 x1 ≤ 6)
લճͷ໰୊
1
2 3
x1 ≤ 4
x2 ≤ 3
x3 ≤ 5
ίΞ
҆ఆू߹
ަবू߹

12.
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s12(x) =
{
3 − x1 − x3 (0 ≤ x3 ≤ 3)
−x1 (3 x3 ≤ 6)
s13(x) =
{
1 − x1 − x2 (0 ≤ x2 ≤ 1)
−x1 (1 x2 ≤ 6)
s21(x) =
{
2 − x2 − x3 (0 ≤ x3 ≤ 2)
−x2 (2 x3 ≤ 6)
s23(x) =
{
1 − x1 − x2 (0 ≤ x1 ≤ 1)
−x2 (1 x1 ≤ 6)
s31(x) =
{
2 − x2 − x3 (0 ≤ x2 ≤ 2)
−x3 (2 x2 ≤ 6)
s32(x) =
{
3 − x1 − x3 (0 ≤ x1 ≤ 3)
−x3 (3 x1 ≤ 6)
લճͷ໰୊
1
2 3
x1 ≤ 4
x2 ≤ 3
x3 ≤ 5
ίΞ
҆ఆू߹
ަবू߹
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

13.
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s12(x) =
{
3 − x1 − x3 (0 ≤ x3 ≤ 3)
−x1 (3 x3 ≤ 6)
s13(x) =
{
1 − x1 − x2 (0 ≤ x2 ≤ 1)
−x1 (1 x2 ≤ 6)
s21(x) =
{
2 − x2 − x3 (0 ≤ x3 ≤ 2)
−x2 (2 x3 ≤ 6)
s23(x) =
{
1 − x1 − x2 (0 ≤ x1 ≤ 1)
−x2 (1 x1 ≤ 6)
s31(x) =
{
2 − x2 − x3 (0 ≤ x2 ≤ 2)
−x3 (2 x2 ≤ 6)
s32(x) =
{
3 − x1 − x3 (0 ≤ x1 ≤ 3)
−x3 (3 x1 ≤ 6)
લճͷ໰୊
1
2 3
x1 ≤ 4
x2 ≤ 3
x3 ≤ 5
ίΞ
҆ఆू߹
ަবू߹
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})

14.
˗ ͷ৔߹
Ͱ͋Γ
ͱͳΔͨΊʹ͸
Ͱ͋Ε͹
Ͱ͋Ε͹
ͳΒ͹
͔͠͠
͜ͷ৔߹ ͱͳΓ
ͱͳΔͨΊෆద
ͳΒ͹
͔͠͠
͜ͷ৔߹ ͱͳΓ
ෆద
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s12(x) =
{
3 − x1 − x3 (0 ≤ x3 ≤ 3)
−x1 (3 x3 ≤ 6)
s21(x) =
{
2 − x2 − x3 (0 ≤ x3 ≤ 2)
−x2 (2 x3 ≤ 6)
0 ≤ x3 ≤ 2 s12(x) = 3 − x1 − x3 s21(x) = 2 − x2 − x3 1 ∼x 2
s12(x) = s21(x) ⇔ 3 − x1 − x3 = 2 − x2 − x3 ⇔ x1 = x2 + 1
x3 = 0 x1 + x2 + x3 = 6 ⇔ x2 + 1 + x2 + 0 = 6 ⇔ x2 =
5
2
x3 = 2 x1 + x2 + x3 = 6 ⇔ x2 + 1 + x2 + 2 = 6 ⇔ x2 =
3
2
s12(x) s21(x) ⇔ 3 − x1 − x3 2 − x2 − x3 ⇔ x1 x2 + 1 x2 = v({2}) = 0
x1 x2 + 1 = 0 + 1
x1 + x2 + x3 1 + 0 + 2 = 3 6
s12(x) s21(x) ⇔ 3 − x1 − x3 2 − x2 − x3 ⇔ x1 x2 + 1 x1 = v({1}) = 0
0 = x1 x2 + 1 ≥ 1
લճͷ໰୊
1
2 3
x1 = 4
x3 = 5
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
(
7
2
,
5
2
,0
)
x3 = 2
(
5
2
,
3
2
,2
)

15.
˗ ͷ৔߹
Ͱ͋Γ
ͱͳΔͨΊʹ͸
഑෼ͷશମ߹ཧੑΛར༻
ͳΒ͹
ෆద
ͳΒ͹
͔͠͠
͜ͷ৔߹ ͱͳΔͨΊෆద
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s12(x) =
{
3 − x1 − x3 (0 ≤ x3 ≤ 3)
−x1 (3 x3 ≤ 6)
s21(x) =
{
2 − x2 − x3 (0 ≤ x3 ≤ 2)
−x2 (2 x3 ≤ 6)
2 x3 ≤ 3 s12(x) = 3 − x1 − x3 s21(x) = − x2 1 ∼x 2
s12(x) = s21(x) ⇔ 3 − x1 − x3 = − x2 ⇔ x2 =
3
2
s12(x) s21(x) ⇔ 3 − x1 − x3 − x2 ⇔ x2
3
2
x2 = v({2}) = 0
s12(x) s21(x) ⇔ 3 − x1 − x3 − x2 ⇔ x2
3
2
x1 = v({1}) = 0
x1 + x2 + x3 0 +
3
2
+ 3 6
લճͷ໰୊
1
2 3
x1 = 4
x2 = 3
x3 = 5
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
x3 = 2
(
7
2
,
5
2
,0
)
(
5
2
,
3
2
,2
)
x3 = 3
(
3
2
,
3
2
,3
)

16.
˗ ͷ৔߹
Ͱ͋Γ
ͱͳΔͨΊʹ͸
ͳΒ͹
ͱͳΔͨΊෆద
ͳΒ͹
ͱͳΔͨΊෆద
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s12(x) =
{
3 − x1 − x3 (0 ≤ x3 ≤ 3)
−x1 (3 x3 ≤ 6)
s21(x) =
{
2 − x2 − x3 (0 ≤ x3 ≤ 2)
−x2 (2 x3 ≤ 6)
3 x3 ≤ 6 s12(x) = − x1 s21(x) = − x2 1 ∼x 2
s12(x) = s21(x) ⇔ − x1 = − x2 ⇔ x1 = x2
s12(x) s21(x) ⇔ − x1 − x2 ⇔ x1 x2 x2 = v({2}) = 0 x1 0
s12(x) s21(x) ⇔ − x1 − x2 ⇔ x1 x2 x1 = v({1}) = 0 x2 0
લճͷ໰୊
1
2 3
x1 = 4
x2 = 3
x3 = 5
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
x3 = 2
(
7
2
,
5
2
,0
)
(
5
2
,
3
2
,2
)
x3 = 3
(
3
2
,
3
2
,3
)

17.
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s12(x) =
{
3 − x1 − x3 (0 ≤ x3 ≤ 3)
−x1 (3 x3 ≤ 6)
s13(x) =
{
1 − x1 − x2 (0 ≤ x2 ≤ 1)
−x1 (1 x2 ≤ 6)
s21(x) =
{
2 − x2 − x3 (0 ≤ x3 ≤ 2)
−x2 (2 x3 ≤ 6)
s23(x) =
{
1 − x1 − x2 (0 ≤ x1 ≤ 1)
−x2 (1 x1 ≤ 6)
s31(x) =
{
2 − x2 − x3 (0 ≤ x2 ≤ 2)
−x3 (2 x2 ≤ 6)
s32(x) =
{
3 − x1 − x3 (0 ≤ x1 ≤ 3)
−x3 (3 x1 ≤ 6)
લճͷ໰୊
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
1
2 3
x1 = 4
x2 = 3
x3 = 5
(
7
2
,
5
2
,0
)
(
5
2
,
3
2
,2
)
(
3
2
,
3
2
,3
)

18.
˗ ͷ৔߹
Ͱ͋Γ
ͱͳΔͨΊʹ͸
ͳΒ͹
͔͠͠
͜ͷ৔߹ ͱͳΔͨΊෆద
ͳΒ͹
͔͠͠
͜ͷ৔߹ ͱͳΓ
ͱͳΔͨΊෆద
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s13(x) =
{
1 − x1 − x2 (0 ≤ x2 ≤ 1)
−x1 (1 x2 ≤ 6)
s31(x) =
{
2 − x2 − x3 (0 ≤ x2 ≤ 2)
−x3 (2 x2 ≤ 6)
0 ≤ x2 ≤ 1 s13(x) = 1 − x1 − x2 s31(x) = 2 − x2 − x3 1 ∼x 3
s13(x) = s31(x) ⇔ 1 − x1 − x2 = 2 − x2 − x3 ⇔ x3 = x1 + 1
s13(x) s31(x) ⇔ 1 − x1 − x2 2 − x2 − x3 ⇔ x3 x1 + 1 x3 = v({3}) = 0
0 = x3 x1 + 1 ≥ 1
s13(x) s31(x) ⇔ 1 − x1 − x2 2 − x2 − x3 ⇔ x3 x1 + 1 x1 = v({1}) = 0
x3 x1 + 1 = 0 + 1
x1 + x2 + x3 0 + 1 + 1 = 2 6
લճͷ໰୊
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
1
2 3
x1 = 4
x2 = 3
x3 = 5
(
7
2
,
5
2
,0
)
(
5
2
,0,
7
2 )
x2 = 1
(2,1,3)

19.
˗ ͷ৔߹
Ͱ͋Γ
ͱͳΔͨΊʹ͸
ͳΒ͹
͔͠͠
͜ͷ৔߹ ͱͳΔͨΊෆద
ͳΒ͹
ෆద
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s13(x) =
{
1 − x1 − x2 (0 ≤ x2 ≤ 1)
−x1 (1 x2 ≤ 6)
s31(x) =
{
2 − x2 − x3 (0 ≤ x2 ≤ 2)
−x3 (2 x2 ≤ 6)
1 x2 ≤ 2 s13(x) = − x1 s31(x) = 2 − x2 − x3 1 ∼x 3
s13(x) = s31(x) ⇔ − x1 = 2 − x2 − x3 ⇔ x1 = 2
s13(x) s31(x) ⇔ − x1 2 − x2 − x3 ⇔ x1 2 x3 = v({3}) = 0
x1 + x2 + x3 2 + 2 + 0 = 4 6
s13(x) s31(x) ⇔ − x1 2 − x2 − x3 ⇔ x1 2 x1 = v({1}) = 0
લճͷ໰୊
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
1
2 3
x1 = 4
x2 = 3
x3 = 5
(
7
2
,
5
2
,0
)
(
5
2
,0,
7
2 )
x2 = 1
(2,1,3)
(
5
2
,
3
2
,2
)
x2 = 2
(2,2,2)

20.
˗ ͷ৔߹
Ͱ͋Γ
ͱͳΔͨΊʹ͸
ͳΒ͹
͔͠͠
͜ͷ৔߹ ͱͳΔͨΊෆద
ͳΒ͹
͔͠͠
͜ͷ৔߹ ͱͳΔͨΊෆద
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s13(x) =
{
1 − x1 − x2 (0 ≤ x2 ≤ 1)
−x1 (1 x2 ≤ 6)
s31(x) =
{
2 − x2 − x3 (0 ≤ x2 ≤ 2)
−x3 (2 x2 ≤ 6)
2 x2 ≤ 6 s13(x) = − x1 s31(x) = − x3 1 ∼x 3
s13(x) = s31(x) ⇔ − x1 = − x3 ⇔ x1 = x3
s13(x) s31(x) ⇔ − x1 − x3 ⇔ x1 x3 x3 = v({3}) = 0
x1 x3 = 0
s13(x) s31(x) ⇔ − x1 − x3 ⇔ x1 x3 x1 = v({1}) = 0
x1 = 0 x3
લճͷ໰୊
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
1
2 3
x1 = 4
x2 = 3
x3 = 5
(
7
2
,
5
2
,0
)
(
5
2
,0,
7
2 )
x2 = 1
(2,1,3)
(
5
2
,
3
2
,2
)
x2 = 2
(2,2,2)

21.
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s12(x) =
{
3 − x1 − x3 (0 ≤ x3 ≤ 3)
−x1 (3 x3 ≤ 6)
s13(x) =
{
1 − x1 − x2 (0 ≤ x2 ≤ 1)
−x1 (1 x2 ≤ 6)
s21(x) =
{
2 − x2 − x3 (0 ≤ x3 ≤ 2)
−x2 (2 x3 ≤ 6)
s23(x) =
{
1 − x1 − x2 (0 ≤ x1 ≤ 1)
−x2 (1 x1 ≤ 6)
s31(x) =
{
2 − x2 − x3 (0 ≤ x2 ≤ 2)
−x3 (2 x2 ≤ 6)
s32(x) =
{
3 − x1 − x3 (0 ≤ x1 ≤ 3)
−x3 (3 x1 ≤ 6)
લճͷ໰୊
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
1
2 3
x1 = 4
x2 = 3
x3 = 5
(
7
2
,
5
2
,0
)
(
5
2
,0,
7
2 )
(2,1,3)
(
5
2
,
3
2
,2
)
(2,2,2)
(
3
2
,
3
2
,3
)

22.
˗ ͷ৔߹
Ͱ͋Γ
ͱͳΔͨΊʹ͸
ͳΒ͹
͔͠͠
͜ͷ৔߹ ͱͳΔͨΊෆద
ͳΒ͹
͔͠͠
͜ͷ৔߹ ͱͳΓ
ͱͳΔͨΊෆద
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s23(x) =
{
1 − x1 − x2 (0 ≤ x1 ≤ 1)
−x2 (1 x1 ≤ 6)
s32(x) =
{
3 − x1 − x3 (0 ≤ x1 ≤ 3)
−x3 (3 x1 ≤ 6)
0 ≤ x1 ≤ 1 s23(x) = 1 − x1 − x2 s32(x) = 3 − x1 − x3 2 ∼x 3
s23(x) = s32(x) ⇔ 1 − x1 − x2 = 3 − x1 − x3 ⇔ x3 = x2 + 2
s23(x) s32(x) ⇔ 1 − x1 − x2 3 − x1 − x3 ⇔ x3 x2 + 2 x3 = v({3}) = 0
0 = x3 x2 + 1
s23(x) s32(x) ⇔ 1 − x1 − x2 3 − x1 − x3 ⇔ x3 x2 + 2 x2 = v({2}) = 0
x3 x2 + 2 = 0 + 2
x1 + x2 + x3 1 + 0 + 2 = 3 6
લճͷ໰୊
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
1
2 3
x1 = 4
x2 = 3
x3 = 5
(
7
2
,
5
2
,0
)
(
5
2
,0,
7
2 )
x1 = 1
(2,1,3)
(0,2,4)
(
1,
3
2
,
7
2)

23.
˗ ͷ৔߹
Ͱ͋Γ
ͱͳΔͨΊʹ͸
ͳΒ͹
͔͠͠
ͱͳΔͨΊෆద
ͳΒ͹
ෆద
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s23(x) =
{
1 − x1 − x2 (0 ≤ x1 ≤ 1)
−x2 (1 x1 ≤ 6)
s32(x) =
{
3 − x1 − x3 (0 ≤ x1 ≤ 3)
−x3 (3 x1 ≤ 6)
1 x1 ≤ 3 s23(x) = − x2 s32(x) = 3 − x1 − x3 2 ∼x 3
s23(x) = s32(x) ⇔ − x2 = 3 − x1 − x3 ⇔ x2 =
3
2
s23(x) s32(x) ⇔ − x2 3 − x1 − x3 ⇔ x2
3
2
x3 = v({3}) = 0
x1 + x2 + x3 3 +
3
2
+ 0 6
s23(x) s32(x) ⇔ − x2 3 − x1 − x3 ⇔ x2
3
2
x2 = v({2}) = 0
લճͷ໰୊
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
1
2 3
x1 = 4
x3 = 5
(
7
2
,
5
2
,0
)
(
5
2
,0,
7
2 )
x1 = 1
(0,2,4)
x1 = 3
(
1,
3
2
,
7
2)
(
3,
3
2
,
3
2)

24.
˗ ͷ৔߹
Ͱ͋Γ
ͱͳΔͨΊʹ͸
ͳΒ͹
ͱͳΓෆద
ͳΒ͹
ͱͳΓෆద
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s23(x) =
{
1 − x1 − x2 (0 ≤ x1 ≤ 1)
−x2 (1 x1 ≤ 6)
s32(x) =
{
3 − x1 − x3 (0 ≤ x1 ≤ 3)
−x3 (3 x1 ≤ 6)
3 x1 ≤ 6 s23(x) = − x2 s32(x) = − x3 2 ∼x 3
s23(x) = s32(x) ⇔ − x2 = − x3 ⇔ x2 = x3
s23(x) s32(x) ⇔ − x2 − x3 ⇔ x2 x3 x3 = v({3}) = 0
x2 x3 = 0
s23(x) s32(x) ⇔ − x2 − x3 ⇔ x2 x3 x2 = v({2}) = 0
x3 x2 = 0
લճͷ໰୊
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
1
2 3
x1 = 4
x3 = 5
(
7
2
,
5
2
,0
)
(
5
2
,0,
7
2 )
x1 = 1
(0,2,4)
(
1,
3
2
,
7
2)
x1 = 3
(
3,
3
2
,
3
2)

25.
͔ͭ ͔ͭ Λຬͨ͢഑෼͸
͜ͷ഑෼ͷΈ͕Χʔωϧʹ‫·ؚ‬ΕΔ഑෼Ͱ͋Δ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 3 v({2,3}) = 2 v({1}) = v({2}) = v({3}) = 0
s12(x) =
{
3 − x1 − x3 (0 ≤ x3 ≤ 3)
−x1 (3 x3 ≤ 6)
s13(x) =
{
1 − x1 − x2 (0 ≤ x2 ≤ 1)
−x1 (1 x2 ≤ 6)
s21(x) =
{
2 − x2 − x3 (0 ≤ x3 ≤ 2)
−x2 (2 x3 ≤ 6)
s23(x) =
{
1 − x1 − x2 (0 ≤ x1 ≤ 1)
−x2 (1 x1 ≤ 6)
s31(x) =
{
2 − x2 − x3 (0 ≤ x2 ≤ 2)
−x3 (2 x2 ≤ 6)
s32(x) =
{
3 − x1 − x3 (0 ≤ x1 ≤ 3)
−x3 (3 x1 ≤ 6)
1 ∼x 2 1 ∼x 3 2 ∼x 3
(
2,
3
2
,
5
2)
લճͷ໰୊
ͱͳΔͨΊʹ͸
·ͨ͸
ͳΒ͹
·ͨ͸
ͳΒ͹
i ∼x j
sij(x) = sji(x)
sij(x) sji(x) xj = v({j})
sij(x) sji(x) xi = v({i})
1
2 3
x1 = 4
x3 = 5
(
7
2
,
5
2
,0
)
(
5
2
,0,
7
2 )
(
2,
3
2
,
5
2)
(0,2,4)
x2 = 3

26. ήʔϜཧ࿦#4*$ԋश
ΧʔωϧΛ‫ٻ‬ΊΔ
࣍ճɿԋश