The document discusses a set function v defined on subsets of a ground set N. It defines v on certain singleton and multiple element subsets. It then calculates the Shapley value φ1(v) which equals 1, indicating the marginal contribution of element 1 to any set it joins. It also shows the order of inclusion of elements into sets.

1. ήʔϜཧ࿦#4*$ԋश
౤ථήʔϜʹ͓͚ΔγϟʔϓϨΠwγϡʔϏοΫࢦ਺

2. ౤ථήʔϜʹ͓͚Δ‫ڋ‬൱‫ݖ‬ϓϨΠϠʔಠࡋऀ
γϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺
໰୊
ղ౴

3. ౤ථήʔϜʹ͓͚Δ‫ڋ‬൱‫ݖ‬ϓϨΠϠʔಠࡋऀ
ಛੑؔ਺͕ ͷ஋ΛͱΓ ͱͳΔಛੑؔ਺‫ܗ‬ήʔϜ Λ୯७ήʔϜͱ͍͏
୯७ήʔϜͷ͏ͪ
ͳΒ͹
ͳΒ͹
ͷ̎ͭͷੑ࣭Λຬͨ͢΋ͷΛ ౤ථήʔϜͱ͍͏
ͱͳΔఏ‫ܞ‬Λউརఏ‫͍͍ͱܞ‬ ͱͳΔఏ‫ܞ‬Λഊ๺ఏ‫͏͍ͱܞ‬
উརఏ‫ܞ‬ͷશମΛ Ͱද͢
͋ΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬ϓϨΠϠʔͱ͸ ͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔͷ͜ͱͰ͋Δ
͋ΔϓϨΠϠʔ͕ಠࡋऀͰ͋Δͱ͸ ͜ͷϓϨΠϠʔΛ‫ؚ‬Ήશͯͷఏ‫͕ܞ‬উརఏ‫ͳͱܞ‬Δͱ͖Λ͍͏
ˎϓϨΠϠʔ ͕ಠࡋऀͩͱ͢Δͱ ਓఏ‫͍͓ͯʹܞ‬ ͕੒ΓཱͭͨΊ ͱͳΔ
Ώ͑ʹ౤ථήʔϜʹ͓͍ͯ ಠࡋऀ͸ߴʑਓ͔͠ଘࡏ͠ͳ͍
౤ථήʔϜʹ͍ͭͯ͸zήʔϜཧ࿦#4*$ୈճγϟʔϓϨΠ஋Ԡ༻ɿ౤ථྗࢦ਺l΋ࢀর
v(N) = 1 (N, v)
S ⊆ T v(S) ≤ v(T)
v(S) = 1 v(NS) = 0
v(S) = 1 v(S) = 0
𝒲
i v({i}) = 1 v(N{i}) = 0

4. γϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺
౤ථήʔϜʹ͓͍ͯ͸ ಛੑؔ਺ͷ஋͸ ͷ͍ͣΕ͔Ͱ
ʹͳΕ͹ ʹͳΔఏ‫ܞ‬ ͷಛੑؔ਺஋͸
Αͬͯ ߩ‫͕౓ݙ‬ ͱͳΔ৔߹͸
͕উརఏ‫Ͱܞ‬ ͕উརఏ‫Ͱܞ‬͸ͳ͍৔߹͚ͩʹͳΔ
Ώ͑ʹ
ͨͩ͠
౤ථήʔϜʹ͓͚ΔγϟʔϓϨΠ஋Λ
γϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺ 44ࢦ਺ ͱ͍͏
v(S) = 1 S ⊆ T T v(T) = 1
v(S ∪ {i}) − v(S) = 1
S ∪ {i} S
ϕi(v) =
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(v(S ∪ {i}) − v(S))
=
∑
S∪{i}∈
𝒲
,S∉
𝒲
s!(n − s − 1)!
n!
s = |S|
i
n − s − 1
n
s
͕ՃΘΓউརఏ‫ܞ‬
i

5. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ܾٞ‫ݖ‬Λ΋ͭϓϨΠϠʔ͕ ͋ΔձࣾͷܾٞҊʹର͠ ౤ථ͢Δঢ়‫گ‬Λߟ͑Δ
͜ͷͱ͖ ա൒਺ϧʔϧΛ࠾༻͢Δ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
໰୊

6. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Γ
͔ͭϓϨΠϠʔΛ‫ؚ‬Ήఏ‫͕ܞ‬ඞͣউརఏ‫ܞ‬ͷͨΊ ϓϨΠϠʔ͸ಠࡋऀͰ΋͋Δ
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 0 v({1}) = 1 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3} {1}
𝒲
= {{1,2,3} {1,2} {1,3} {1}}
ϕ1(v) =
∑
S∪{1}∈
𝒲
,S∉
𝒲
s!(3 − s − 1)!
3!
=
0!(3 − 0 − 1)!
3!
+
1!(3 − 1 − 1)!
3!
+
1!(3 − 1 − 1)!
3!
+
2!(3 − 2 − 1)!
3!
=
1 ⋅ 2
6
+
1 ⋅ 1
6
+
1 ⋅ 1
6
+
2 ⋅ 1
6
= 1
ղ౴
∅ → {1} {3}→{1,3} {2}→{1,2} {2,3}→{1,2,3}

7. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Γ
͔ͭϓϨΠϠʔΛ‫ؚ‬Ήఏ‫͕ܞ‬ඞͣউརఏ‫ܞ‬ͷͨΊ ϓϨΠϠʔ͸ಠࡋऀͰ΋͋Δ
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 0 v({1}) = 1 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3} {1}
𝒲
= {{1,2,3} {1,2} {1,3} {1}}
ϕ1(v) =
∑
S∪{1}∈
𝒲
,S∉
𝒲
s!(3 − s − 1)!
3!
=
0!(3 − 0 − 1)!
3!
+
1!(3 − 1 − 1)!
3!
+
1!(3 − 1 − 1)!
3!
+
2!(3 − 2 − 1)!
3!
=
1 ⋅ 2
6
+
1 ⋅ 1
6
+
1 ⋅ 1
6
+
2 ⋅ 1
6
= 1
ղ౴
∅ → {1} {3}→{1,3} {2}→{1,2} {2,3}→{1,2,3}
ಠࡋऀͰ͋Ε͹44ࢦ਺͸

8. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Γ
͔ͭϓϨΠϠʔΛ‫ؚ‬Ήఏ‫͕ܞ‬ඞͣউརఏ‫ܞ‬ͷͨΊ ϓϨΠϠʔ͸ಠࡋऀͰ΋͋Δ
Ͱ͋Δ͔Β ղ౴
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 0 v({1}) = 1 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3} {1}
𝒲
= {{1,2,3} {1,2} {1,3} {1}}
ϕ1(v) + ϕ2(v) + ϕ3(v) = v({1,2,3}) = 1 (ϕ1(v), ϕ2(v), ϕ3(v)) = (1, 0, 0)
ղ౴

9. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Δ
ಠࡋऀ͸ଘࡏ͠ͳ͍
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 0 v({1}) = 0 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3}
𝒲
= {{1,2,3} {1,2} {1,3}}
ϕ1(v) =
∑
S∪{1}∈
𝒲
,S∉
𝒲
s!(3 − s − 1)!
3!
=
1!(3 − 1 − 1)!
3!
+
1!(3 − 1 − 1)!
3!
+
2!(3 − 2 − 1)!
3!
=
1 ⋅ 1
6
+
1 ⋅ 1
6
+
2 ⋅ 1
6
=
4
6
ղ౴
{3}→{1,3} {2}→{1,2} {2,3}→{1,2,3}

10. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Δ
ಠࡋऀ͸ଘࡏ͠ͳ͍
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 0 v({1}) = 0 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3}
𝒲
= {{1,2,3} {1,2} {1,3}}
ϕ2(v) =
∑
S∪{2}∈
𝒲
,S∉
𝒲
s!(3 − s − 1)!
3!
=
1!(3 − 1 − 1)!
3!
=
1 ⋅ 1
6
=
1
6
ղ౴
{1}→{1,2}

11. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Δ
ಠࡋऀ͸ଘࡏ͠ͳ͍
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 0 v({1}) = 0 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3}
𝒲
= {{1,2,3} {1,2} {1,3}}
ϕ3(v) =
∑
S∪{3}∈
𝒲
,S∉
𝒲
s!(3 − s − 1)!
3!
=
1!(3 − 1 − 1)!
3!
=
1 ⋅ 1
6
=
1
6
ղ౴
{1}→{1,3}

12. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Δ
ಠࡋऀ͸ଘࡏ͠ͳ͍
ղ౴
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 0 v({1}) = 0 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3}
𝒲
= {{1,2,3} {1,2} {1,3}}
(ϕ1(v), ϕ2(v), ϕ3(v)) =
(
4
6
,
1
6
,
1
6)
=
(
2
3
,
1
6
,
1
6)
ղ౴

13. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͸͍ͳ͍ͷͰ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ͸ଘࡏ͠ͳ͍
·ͨ ಠࡋऀ΋͍ͳ͍
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 1 v({1}) = 0 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3} {2,3}
𝒲
= {{1,2,3} {1,2} {1,3} {2,3}}
ϕ1(v) =
∑
S∪{1}∈
𝒲
,S∉
𝒲
s!(3 − s − 1)!
3!
=
1!(3 − 1 − 1)!
3!
+
1!(3 − 1 − 1)!
3!
+
2!(3 − 2 − 1)!
3!
=
1 ⋅ 1
6
+
1 ⋅ 1
6
=
2
6
ղ౴
{3}→{1,3} {2}→{1,2} {2,3}→{1,2,3}

14. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͸͍ͳ͍ͷͰ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ͸ଘࡏ͠ͳ͍
·ͨ ಠࡋऀ΋͍ͳ͍
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 1 v({1}) = 0 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3} {2,3}
𝒲
= {{1,2,3} {1,2} {1,3} {2,3}}
ϕ2(v) =
∑
S∪{2}∈
𝒲
,S∉
𝒲
s!(3 − s − 1)!
3!
=
1!(3 − 1 − 1)!
3!
+
1!(3 − 1 − 1)!
3!
=
1 ⋅ 1
6
+
1 ⋅ 1
6
=
2
6
ղ౴
{1}→{1,2} {3}→{2,3}

15. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͸͍ͳ͍ͷͰ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ͸ଘࡏ͠ͳ͍
·ͨ ಠࡋऀ΋͍ͳ͍
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 1 v({1}) = 0 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3} {2,3}
𝒲
= {{1,2,3} {1,2} {1,3} {2,3}}
ϕ3(v) =
∑
S∪{3}∈
𝒲
,S∉
𝒲
s!(3 − s − 1)!
3!
=
1!(3 − 1 − 1)!
3!
+
1!(3 − 1 − 1)!
3!
=
1 ⋅ 1
6
+
1 ⋅ 1
6
=
2
6
ղ౴
{1}→{1,2} {2}→{2,3}

16. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
ಛੑؔ਺͸ҎԼͷΑ͏ʹͳΔ
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͸͍ͳ͍ͷͰ ‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔ͸ଘࡏ͠ͳ͍
·ͨ ಠࡋऀ΋͍ͳ͍
ղ౴
v({1,2,3}) = 1 v({1,2}) = 1 v({1,3}) = 1 v({2,3}) = 1 v({1}) = 0 v({2}) = 0 v({3}) = 0
{1,2,3} {1,2} {1,3} {2,3}
𝒲
= {{1,2,3} {1,2} {1,3} {2,3}}
(ϕ1(v), ϕ2(v), ϕ3(v)) =
(
2
6
,
2
6
,
2
6)
=
(
1
3
,
1
3
,
1
3)
ղ౴

17. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Γ
͔ͭϓϨΠϠʔΛ‫ؚ‬Ήఏ‫͕ܞ‬ඞͣউརఏ‫ܞ‬ͷͨΊ ϓϨΠϠʔ͸ಠࡋऀͰ΋͋Δ
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 0
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 1 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4} {1}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4} {1}}
ϕ1(v) =
∑
S∪{1}∈
𝒲
,S∉
𝒲
s!(4 − s − 1)!
4!
=
0!(4 − 0 − 1)!
4!
+
1!(4 − 1 − 1)!
4!
+
1!(4 − 1 − 1)!
4!
+
1!(4 − 1 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
3!(4 − 3 − 1)!
4!
=
1 ⋅ 6
4!
+
1 ⋅ 2
4!
+
1 ⋅ 2
4!
+
1 ⋅ 2
4!
+
2 ⋅ 1
4!
+
2 ⋅ 1
4!
+
2 ⋅ 1
4!
+
6 ⋅ 1
4!
=
24
4 ⋅ 3 ⋅ 2 ⋅ 1
= 1
ղ౴
∅ → {1} {3}→{1,3}
{2}→{1,2} {2,3}→{1,2,3}
{4}→{1,4} {2,4}→{1,2,4} {3,4}→{1,3,4} {2,3,4}→{1,2,3,4}

18. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Γ
͔ͭϓϨΠϠʔΛ‫ؚ‬Ήఏ‫͕ܞ‬ඞͣউརఏ‫ܞ‬ͷͨΊ ϓϨΠϠʔ͸ಠࡋऀͰ΋͋Δ
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 0
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 1 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4} {1}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4} {1}}
ϕ1(v) =
∑
S∪{1}∈
𝒲
,S∉
𝒲
s!(4 − s − 1)!
4!
=
0!(4 − 0 − 1)!
4!
+
1!(4 − 1 − 1)!
4!
+
1!(4 − 1 − 1)!
4!
+
1!(4 − 1 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
3!(4 − 3 − 1)!
4!
=
1 ⋅ 6
4!
+
1 ⋅ 2
4!
+
1 ⋅ 2
4!
+
1 ⋅ 2
4!
+
2 ⋅ 1
4!
+
2 ⋅ 1
4!
+
2 ⋅ 1
4!
+
6 ⋅ 1
4!
=
24
4 ⋅ 3 ⋅ 2 ⋅ 1
= 1
ղ౴
∅ → {1} {3}→{1,3}
{2}→{1,2} {2,3}→{1,2,3}
{4}→{1,4} {2,4}→{1,2,4} {3,4}→{1,3,4} {2,3,4}→{1,2,3,4}
ಠࡋऀͰ͋Ε͹44ࢦ਺͸

19. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Γ
͔ͭϓϨΠϠʔΛ‫ؚ‬Ήఏ‫͕ܞ‬ඞͣউརఏ‫ܞ‬ͷͨΊ ϓϨΠϠʔ͸ಠࡋऀͰ΋͋Δ
Ͱ͋Δ͔Β ղ౴
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 0
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 1 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4} {1}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4} {1}}
ϕ1(v) + ϕ2(v) + ϕ3(v) + ϕ4(v) = v({1,2,3,4}) = 1 (ϕ1(v), ϕ2(v), ϕ3(v), ϕ4(v)) = (1, 0, 0, 0)
ղ౴

20. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Δ
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 0
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4}}
ϕ1(v) =
∑
S∪{1}∈
𝒲
,S∉
𝒲
s!(4 − s − 1)!
4!
=
1!(4 − 1 − 1)!
4!
+
1!(4 − 1 − 1)!
4!
+
1!(4 − 1 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
3!(4 − 3 − 1)!
4!
=
1 ⋅ 2
4!
+
1 ⋅ 2
4!
+
1 ⋅ 2
4!
+
2 ⋅ 1
4!
+
2 ⋅ 1
4!
+
2 ⋅ 1
4!
+
6 ⋅ 1
4!
=
18
4 ⋅ 3 ⋅ 2 ⋅ 1
=
3
4
ղ౴
{3}→{1,3}
{2}→{1,2} {2,3}→{1,2,3}
{4}→{1,4} {2,4}→{1,2,4} {3,4}→{1,3,4} {2,3,4}→{1,2,3,4}

21. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Δ
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 0
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4}}
ϕ2(v) =
∑
S∪{2}∈
𝒲
,S∉
𝒲
s!(4 − s − 1)!
4!
=
1!(4 − 1 − 1)!
4!
=
1 ⋅ 2
4!
=
2
4 ⋅ 3 ⋅ 2 ⋅ 1
=
1
12
ղ౴
{1}→{1,2}

22. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Δ
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 0
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4}}
ϕ3(v) =
∑
S∪{3}∈
𝒲
,S∉
𝒲
s!(4 − s − 1)!
4!
=
1!(4 − 1 − 1)!
4!
=
1 ⋅ 2
4!
=
2
4 ⋅ 3 ⋅ 2 ⋅ 1
=
1
12
ղ౴
{1}→{1,3}

23. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Δ
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 0
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4}}
ϕ4(v) =
∑
S∪{4}∈
𝒲
,S∉
𝒲
s!(4 − s − 1)!
4!
=
1!(4 − 1 − 1)!
4!
=
1 ⋅ 2
4!
=
2
4 ⋅ 3 ⋅ 2 ⋅ 1
=
1
12
ղ౴
{1}→{1,4}

24. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
͢΂ͯͷউརఏ‫·ؚʹܞ‬ΕΔϓϨΠϠʔ͕‫ڋ‬൱‫ݖ‬Λ΋ͭϓϨΠϠʔͰ͋Δ
ղ౴
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 0
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 1 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {1,2} {1,3} {1,4}}
(ϕ1(v), ϕ2(v), ϕ3(v), ϕ4(v)) =
(
3
4
,
1
12
,
1
12
,
1
12 )
ղ౴

25. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ͜ͷͱ͖ ա
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 1
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 0 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2} {1,3}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2} {1,3}}
ϕ1(v) =
∑
S∪{1}∈
𝒲
,S∉
𝒲
s!(4 − s − 1)!
4!
=
1!(4 − 1 − 1)!
4!
+
1!(4 − 1 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
=
1 ⋅ 2
4!
+
1 ⋅ 2
4!
+
2 ⋅ 1
4!
+
2 ⋅ 1
4!
+
2 ⋅ 1
4!
=
10
4 ⋅ 3 ⋅ 2 ⋅ 1
=
5
12
ղ౴
{3}→{1,3}
{2}→{1,2} {2,3}→{1,2,3} {2,4}→{1,2,4} {3,4}→{1,3,4}

26. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ͜ͷͱ͖ ա
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 1
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 0 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2} {1,3}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2} {1,3}}
ϕ2(v) =
∑
S∪{2}∈
𝒲
,S∉
𝒲
s!(4 − s − 1)!
4!
=
1!(4 − 1 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
=
1 ⋅ 2
4!
+
2 ⋅ 1
4!
+
2 ⋅ 1
4!
=
6
4 ⋅ 3 ⋅ 2 ⋅ 1
=
1
4
ղ౴
{1}→{1,2} {3,4}→{2,3,4} {1,4}→{1,2,4}

27. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ͜ͷͱ͖ ա
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 1
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 0 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2} {1,3}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2} {1,3}}
ϕ3(v) =
∑
S∪{3}∈
𝒲
,S∉
𝒲
s!(4 − s − 1)!
4!
=
1!(4 − 1 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
+
2!(4 − 2 − 1)!
4!
=
1 ⋅ 2
4!
+
2 ⋅ 1
4!
+
2 ⋅ 1
4!
=
6
4 ⋅ 3 ⋅ 2 ⋅ 1
=
1
4
ղ౴
{1}→{1,3} {2,4}→{2,3,4} {1,4}→{1,3,4}

28. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ͜ͷͱ͖ ա
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 1
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 0 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2} {1,3}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2} {1,3}}
ϕ4(v) =
∑
S∪{4}∈
𝒲
,S∉
𝒲
s!(4 − s − 1)!
4!
=
2!(4 − 2 − 1)!
4!
=
2 ⋅ 1
4!
=
2
4 ⋅ 3 ⋅ 2 ⋅ 1
=
1
12
ղ౴
{2,3}→{2,3,4}

29. ԋशͰ‫ݕ‬౼ͨ͠ಛ੡ؔ਺‫ܗ‬ήʔϜʹ͓͚ΔγϟʔϓϨΠɾγϡʔϏοΫ౤ථྗࢦ਺Λ‫ٻ‬ΊΑ͜ͷͱ͖ ա
ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ ϓϨΠϠʔ͸ͷܾٞ‫ݖ‬Λ΋ͭ৔߹
͜ͷήʔϜʹ͓͍ͯ উརఏ‫ܞ‬͸ Ͱ͋Γ
ղ౴
v({1,2,3,4}) = 1 v({1,2,3}) = 1 v({1,2,4}) = 1 v({1,3,4}) = 1 v({2,3,4}) = 1
v({1,2}) = 1 v({1,3}) = 1 v({1,4}) = 0 v({2,3}) = 0 v({2,4}) = 0 v({3,4}) = 0
v({1}) = 0 v({2}) = 0 v({3}) = 0 v({4}) = 0
{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2} {1,3}
𝒲
= {{1,2,3,4} {1,2,3} {1,2,4} {1,3,4} {2,3,4} {1,2} {1,3}}
(ϕ1(v), ϕ2(v), ϕ3(v), ϕ4(v)) =
(
5
12
,
3
12
,
3
12
,
1
12 )
=
(
5
12
,
1
4
,
1
4
,
1
12 )
ղ౴

30. ήʔϜཧ࿦#4*$ԋश
౤ථήʔϜʹ͓͚ΔγϟʔϓϨΠwγϡʔϏοΫࢦ਺