The document provides information about calculating marginal gains values for different orderings of elements in a set. It defines the marginal gain function and describes calculating marginal gains for 3 elements in a set under different orderings of those elements. Equations for the marginal gain function are provided along with numeric examples to illustrate the calculations.

1. ήʔϜཧ࿦#4*$ԋश
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ

2. γϟʔϓϨΠ஋ͷఆٛͷ֬ೝ
લճͷ໰୊
ߟ࡯

3. γϟʔϓϨΠ஋ͷఆٛͷ֬ೝ
ఆٛɿߩ‫౓ݙ‬
ಛੑؔ਺‫ܗ‬ήʔϜ ʹ͓͍ͯ
೚ҙͷఏ‫ܞ‬ ͱ ʹ‫·ؚ‬Εͳ͍ϓϨΠϠʔ ΛͱΔ
͜ͷͱ͖
Λߩ‫͏͍ͱ౓ݙ‬
ͱ͢Ε͹
ߩ‫౓ݙ‬͸
ͱ͢Ε͹
ߩ‫౓ݙ‬͸
(N, v) S S i
v(S ∪ {i}) − v(S)
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
S = {2,3} i = 1
v({2,3} ∪ {1}) − v({2,3}) = v({1,2,3}) − v({2,3}) = 10 − 0 = 10
S = {1,3} i = 2
v({1,3} ∪ {2}) − v({1,3}) = v({1,2,3}) − v({1,3}) = 10 − 10 = 0

4. γϟʔϓϨΠ஋ͷఆٛͷ֬ೝ
ਓͷϓϨΠϠʔΛฒ΂ͨॱྻΛ ͱ͠
‫ݸ‬ͷॱྻ ͷશମΛ
ਓͣͭϓϨΠϠʔ͕ՃΘΔͱ͍͏ఏ‫ܗܞ‬੒Λߟ͓͑ͯΓ
ॱྻ ʹ͓͚ΔϓϨΠϠʔ ͷߩ‫౓ݙ‬͸
Ͱ༩͑ΒΕΔ
͸ ͷઌߦऀͱ͍͏
ϓϨΠϠʔ ͕ Ͱ͋Δͱ͖ͷઌߦऀ ͷू߹Λ Ͱද͢
ͨͩ͠
ͷͱ͖͸ઌߦऀ͕͍ͳ͍ͷͰ
Λ༻͍Ε͹
ͷॱྻ ʹ͓͚Δߩ‫౓ݙ‬͸
n π = (π(1), …, π(n)) n! π Π
π π(k)
v({π(1), ⋯, π(k − 1), π(k)}) − v({π(1), ⋯, π(k − 1)})
π(1), ⋯, π(k − 1) π(k)
i π(k) π(1), ⋯, π(k − 1) Pπ,i
π(1) Pπ,i
= ∅
Pπ,i
i π v(Pπ,i
∪ {i}) − v(Pπ,i
)

5. γϟʔϓϨΠ஋ͷఆٛͷ֬ೝ
ఆٛɿγϟʔϓϨΠ஋
ಛੑؔ਺‫ܗ‬ήʔϜ ʹ͓͍ͯ
ਓͣͭϓϨΠϠʔ͕ՃΘΓશମఏ‫ܗ͕ܞ‬੒͞ΕΔ ‫ݸ‬ͷఏ‫ܗܞ‬੒͕
͢΂ͯಉ֬͡཰Ͱ‫͜ى‬Δͱ͢Δ͜ͷͱ͖ͷ֤ϓϨΠϠʔͷߩ‫౓ݙ‬ͷ‫ظ‬଴஋Λ
ʹ͓͚ΔͦΕͧΕͷϓϨΠϠʔͷγϟʔϓϨΠ஋ͱ͍͏
ϓϨΠϠʔ ͷγϟʔϓϨΠ஋Λ ͱ͢Δͱ
Λ୯ʹγϟʔϓϨΠ஋ͱ͍͏
(N, v)
n!
(N, v)
i ϕi(v)
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ϕ(v) = (ϕ1(v)⋯, ϕn(v))

6.
͜ͷಛੑؔ਺Λ༻͍ͯ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊͯΈΑ͏
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໰୊ 1
2 3
x1 = 4
x3 = 5
(
2,
3
2
,
5
2)
ίΞ
҆ఆू߹
ަবू߹
Χʔωϧ
ਔ

7.
͜ͷಛੑؔ਺Λ༻͍ͯ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊͯΈΑ͏
શମఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹ͷ֤ϓϨΠϠʔͷߩ‫౓ݙ‬Λߟ͑Δ
ˠˠͷॱͰఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹
͢ͳΘͪ
ˠ ˠ ˠ ͱఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹Ͱ͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
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}
∅ {1} {1,2} {1,2,3}
v({1}) − v(∅) = 0 − 0 = 0
v({1,2}) − v({1}) = 1 − 0 = 1
v({1,2,3}) − v({1,2}) = 6 − 1 = 5
લճͷ໰୊ 1
2 3
x1 = 4
x3 = 5
(
2,
3
2
,
5
2)
ॱྻ
ίΞ
҆ఆू߹
ަবू߹
Χʔωϧ
ਔ

8.
͜ͷಛੑؔ਺Λ༻͍ͯ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊͯΈΑ͏
શମఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹ͷ֤ϓϨΠϠʔͷߩ‫౓ݙ‬Λߟ͑Δ
ˠˠͷॱͰఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹
͢ͳΘͪ
ˠ ˠ ˠ ͱఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹Ͱ͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
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}
∅ {1} {1,3} {1,2,3}
v({1}) − v(∅) = 0 − 0 = 0
v({1,2,3}) − v({1,3}) = 6 − 3 = 3
v({1,3}) − v({1}) = 3 − 0 = 3
લճͷ໰୊ 1
2 3
x1 = 4
x3 = 5
(
2,
3
2
,
5
2)
ॱྻ
ίΞ
҆ఆू߹
ަবू߹
Χʔωϧ
ਔ

9.
͜ͷಛੑؔ਺Λ༻͍ͯ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊͯΈΑ͏
શମఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹ͷ֤ϓϨΠϠʔͷߩ‫౓ݙ‬Λߟ͑Δ
ˠˠͷॱͰఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹
͢ͳΘͪ
ˠ ˠ ˠ ͱఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹Ͱ͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
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}
∅ {2} {1,2} {1,2,3}
v({1,2}) − v({2}) = 1 − 0 = 1
v(∅) − v({2}) = 0 − 0 = 0
v({1,2,3}) − v({1,2}) = 6 − 1 = 5
લճͷ໰୊ 1
2 3
x1 = 4
x3 = 5
(
2,
3
2
,
5
2)
ίΞ
҆ఆू߹
ަবू߹
Χʔωϧ
ਔ
ॱྻ

10.
͜ͷಛੑؔ਺Λ༻͍ͯ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊͯΈΑ͏
શମఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹ͷ֤ϓϨΠϠʔͷߩ‫౓ݙ‬Λߟ͑Δ
ˠˠͷॱͰఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹
͢ͳΘͪ
ˠ ˠ ˠ ͱఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹Ͱ͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
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}
∅ {2} {2,3} {1,2,3}
v({1,2,3}) − v({2,3}) = 6 − 2 = 4
v(∅) − v({2}) = 0 − 0 = 0
v({2,3}) − v({2}) = 2 − 0 = 2
લճͷ໰୊ 1
2 3
x1 = 4
x3 = 5
(
2,
3
2
,
5
2)
ॱྻ
ίΞ
҆ఆू߹
ަবू߹
Χʔωϧ
ਔ

11.
͜ͷಛੑؔ਺Λ༻͍ͯ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊͯΈΑ͏
શମఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹ͷ֤ϓϨΠϠʔͷߩ‫౓ݙ‬Λߟ͑Δ
ˠˠͷॱͰఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹
͢ͳΘͪ
ˠ ˠ ˠ ͱఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹Ͱ͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
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}
∅ {3} {1,3} {1,2,3}
v({1,3}) − v({3}) = 3 − 0 = 3
v({1,2,3}) − v({1,3}) = 6 − 3 = 3
v(∅) − v({3}) = 0 − 0 = 0
લճͷ໰୊ 1
2 3
x1 = 4
x3 = 5
(
2,
3
2
,
5
2)
ॱྻ
ίΞ
҆ఆू߹
ަবू߹
Χʔωϧ
ਔ

12.
͜ͷಛੑؔ਺Λ༻͍ͯ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊͯΈΑ͏
શମఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹ͷ֤ϓϨΠϠʔͷߩ‫౓ݙ‬Λߟ͑Δ
ˠˠͷॱͰఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹
͢ͳΘͪ
ˠ ˠ ˠ ͱఏ‫ܗ͕ܞ‬੒͞ΕΔ৔߹Ͱ͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
ϓϨΠϠʔͷߩ‫౓ݙ‬͸
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}
∅ {3} {2,3} {1,2,3}
v({1,2,3}) − v({2,3}) = 6 − 2 = 4
v({2,3}) − v({3}) = 2 − 0 = 2
v(∅) − v({3}) = 0 − 0 = 0
લճͷ໰୊ 1
2 3
x1 = 4
x3 = 5
(
2,
3
2
,
5
2)
ॱྻ
ίΞ
҆ఆू߹
ަবू߹
Χʔωϧ
ਔ

13.
͜ͷಛੑؔ਺Λ༻͍ͯ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊͯΈΑ͏
શମఏ‫ܗ͕ܞ‬੒͞ΕΔͷ౳֬཰Ͱ͋Γ
Ͱ͋Δ
ϓϨΠϠʔͷγϟʔϓϨΠ஋͸
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}
1
3!
=
1
6
ϕ1(v) =
1
6 ∑
π∈Π
(v(Pπ,1
∪ {1}) − v(Pπ,1
))
=
1
6
(0 + 0 + 1 + 4 + 3 + 4) = 2
લճͷ໰୊ 1
2 3
x1 = 4
x3 = 5
(
2,
3
2
,
5
2)
ॱྻ
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ίΞ
҆ఆू߹
ަবू߹
Χʔωϧ
ਔ

14.
͜ͷಛੑؔ਺Λ༻͍ͯ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊͯΈΑ͏
શମఏ‫ܗ͕ܞ‬੒͞ΕΔͷ౳֬཰Ͱ͋Γ
Ͱ͋Δ
ϓϨΠϠʔͷγϟʔϓϨΠ஋͸
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}
1
3!
=
1
6
ϕ2(v) =
1
6 ∑
π∈Π
(v(Pπ,2
∪ {1}) − v(Pπ,2
))
=
1
6
(1 + 3 + 0 + 0 + 3 + 2) =
3
2
લճͷ໰୊ 1
2 3
x1 = 4
x3 = 5
(
2,
3
2
,
5
2)
ॱྻ
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ίΞ
҆ఆू߹
ަবू߹
Χʔωϧ
ਔ

15.
͜ͷಛੑؔ਺Λ༻͍ͯ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊͯΈΑ͏
શମఏ‫ܗ͕ܞ‬੒͞ΕΔͷ౳֬཰Ͱ͋Γ
Ͱ͋Δ
ϓϨΠϠʔͷγϟʔϓϨΠ஋͸
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}
1
3!
=
1
6
ϕ3(v) =
1
6 ∑
π∈Π
(v(Pπ,3
∪ {1}) − v(Pπ,3
))
=
1
6
(5 + 3 + 5 + 2 + 0 + 0) =
5
2
લճͷ໰୊ 1
2 3
x1 = 4
x3 = 5
(
2,
3
2
,
5
2)
ॱྻ
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ίΞ
҆ఆू߹
ަবू߹
Χʔωϧ
ਔ

16.
͜ͷಛੑؔ਺Λ༻͍ͯ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊͯΈΑ͏
Ҏ্ΑΓ
͜ͷήʔϜʹ͓͚ΔγϟʔϓϨΠ஋͸
͜Ε͸Χʔωϧ΍ਔʹҰக
஫ҰൠతʹҰக͢Δͱ͸‫ݶ‬Βͳ͍
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}
(
2,
3
2
,
5
2 )
લճͷ໰୊ 1
2 3
x1 = 4
x3 = 5
(
2,
3
2
,
5
2)
ॱྻ
ίΞ
҆ఆू߹
ަবू߹
Χʔωϧ
ਔ

17.
ίΞɿ
҆ఆू߹ަবू߹͸ίΞͱҰக
Χʔωϧ
ਔ
γϟʔϓϨΠ஋͸
˗ઓུ‫ܗ‬ήʔϜͰͷφογϡ‫͚͓ʹߧۉ‬Δརಘͱൺ΂Δͱ
ʮ ΛͱΔʯͱ͍͏໿ଋ͕ՄೳͰ͋Δ͔Β
རಘͷ࿨͸େ͖͘͢Δ͜ͱ͕Ͱ͖Δ
˗ઓུ‫ܗ‬ήʔϜͰ͸‫ͮ͑ݟ‬Β͍ϓϨΠϠʔͷྗؔ܎͕
഑෼ͷׂΓ౰ͯྔͰ֬ೝͰ͖Δ
FYϓϨΠϠʔͷӨ‫͕ྗڹ‬େ͖͍
FYϓϨΠϠʔ
ͷӨ‫ྗڹ‬͸ಉ͡Ͱ͸ͳ͍
ʢ७ਮઓུφογϡ‫ݟ͚ͩߧۉ‬Δͱಉ͡ʹ‫͑ݟ‬Δ
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}
(
2,
3
2
,
5
2)
(a2, b1, c2)
ߟ࡯
1
2 3
x1 = 4
x3 = 5
(
2,
3
2
,
5
2)
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B
φογϡ‫ߧۉ‬
φογϡ‫ߧۉ‬
ίΞ
҆ఆू߹
ަবू߹
Χʔωϧ
ਔ

18. ήʔϜཧ࿦#4*$ԋश
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
࣍ճɿԋश