The document describes the computation of a permutation-based Shapley value for coalition games. It provides an example calculation for a 3-player game with characteristic function values defined for all possible coalitions. All 6 possible orderings of the players are considered and the marginal contribution of each player under each ordering is calculated to obtain the Shapley value distribution.

1. ήʔϜཧ࿦#4*$ԋश
ਓήʔϜ෼ੳɿγϟʔϓϨΠ஋

2. γϟʔϓϨΠ஋ͷఆٛ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ

3. γϟʔϓϨΠ஋ͷఆٛ
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
)
#4*$ࢀর

4. γϟʔϓϨΠ஋ͷఆٛ
ఆٛɿγϟʔϓϨΠ஋
ಛੑؔ਺‫ܗ‬ήʔϜ ʹ͓͍ͯ ਓͣͭϓϨΠϠʔ͕ՃΘΓશମఏ‫ܗ͕ܞ‬੒͞ΕΔ ‫ݸ‬ͷఏ‫ܗܞ‬
੒͕͢΂ͯಉ֬͡཰Ͱ‫͜ى‬Δͱ͢Δ͜ͷͱ͖ͷ֤ϓϨΠϠʔͷߩ‫౓ݙ‬ͷ‫ظ‬଴஋Λ ʹ͓͚Δ
֤ϓϨΠϠʔͷγϟʔϓϨΠ஋ͱ͍͏ϓϨΠϠʔ ͷγϟʔϓϨΠ஋Λ ͱ͢Δͱ
Λ୯ʹγϟʔϓϨΠ஋ͱ͍͏
ผද‫ݱ‬
(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))
ϕi(v) =
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(v(S ∪ {i}) − v(S)), ∀i ∈ N
#4*$ࢀর

5. ਓήʔϜʹ͓͍ͯ ઓུ‫ܗ‬ήʔϜ͔Β‫ٻ‬Ίͨಛੑؔ਺‫ܗ‬ήʔϜʹ͓͍ͯ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΑ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2
v({1}) = 0 v({2}) = 0 v({3}) = 0
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
͕७ઓུD
1ʘ2 C C
B
B
͕७ઓུD
1ʘ2 C C
B
B

6. ֤ϓϨΠϠʔͷߩ‫౓ݙ‬Λߟ͑Ε͹ྑ͍
ͷॱʹఏ‫ܗܞ‬੒͞ΕΔͱ͢Δͱ
ͷߩ‫౓ݙ‬͸
ͷߩ‫౓ݙ‬͸
ͷߩ‫౓ݙ‬͸
ӈදʹ֤஋Λೖྗ͢Δ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
1 → 2 → 3
1 v({1}) − v(∅) = 0 − 0 = 0
2 v({1,2}) − v({1}) = 1 − 0 = 1
3 v({1,2,3}) − v({1,2}) = 6 − 1 = 5
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ॱྻ

7. ֤ϓϨΠϠʔͷߩ‫౓ݙ‬Λߟ͑Ε͹ྑ͍
ͷॱʹఏ‫ܗܞ‬੒͞ΕΔͱ͢Δͱ
ͷߩ‫౓ݙ‬͸
ͷߩ‫౓ݙ‬͸
ͷߩ‫౓ݙ‬͸
ӈදʹ֤஋Λೖྗ͢Δ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
1 → 3 → 2
1 v({1}) − v(∅) = 0 − 0 = 0
2 v({1,2,3}) − v({1,3}) = 6 − 4 = 2
3 v({1,3}) − v({1}) = 4 − 0 = 4
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ॱྻ

8. ֤ϓϨΠϠʔͷߩ‫౓ݙ‬Λߟ͑Ε͹ྑ͍
ͷॱʹఏ‫ܗܞ‬੒͞ΕΔͱ͢Δͱ
ͷߩ‫౓ݙ‬͸
ͷߩ‫౓ݙ‬͸
ͷߩ‫౓ݙ‬͸
ӈදʹ֤஋Λೖྗ͢Δ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
2 → 1 → 3
1 v({1,2}) − v({1}) = 1 − 0 = 1
2 v({2}) − v(∅) = 0 − 0 = 0
3 v({1,2,3}) − v({1,2}) = 6 − 1 = 5
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ॱྻ

9. ֤ϓϨΠϠʔͷߩ‫౓ݙ‬Λߟ͑Ε͹ྑ͍
ͷॱʹఏ‫ܗܞ‬੒͞ΕΔͱ͢Δͱ
ͷߩ‫౓ݙ‬͸
ͷߩ‫౓ݙ‬͸
ͷߩ‫౓ݙ‬͸
ӈදʹ֤஋Λೖྗ͢Δ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
2 → 3 → 1
1 v({1,2,3}) − v({2,3}) = 6 − 2 = 4
2 v({2}) − v(∅) = 0 − 0 = 0
3 v({2,3}) − v({2}) = 2 − 0 = 2
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ॱྻ

10. ֤ϓϨΠϠʔͷߩ‫౓ݙ‬Λߟ͑Ε͹ྑ͍
ͷॱʹఏ‫ܗܞ‬੒͞ΕΔͱ͢Δͱ
ͷߩ‫౓ݙ‬͸
ͷߩ‫౓ݙ‬͸
ͷߩ‫౓ݙ‬͸
ӈදʹ֤஋Λೖྗ͢Δ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
3 → 1 → 2
1 v({1,3}) − v({3}) = 4 − 0 = 4
2 v({1,2,3}) − v({1,3}) = 6 − 4 = 2
3 v({3}) − v(∅) = 0 − 0 = 0
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ॱྻ

11. ֤ϓϨΠϠʔͷߩ‫౓ݙ‬Λߟ͑Ε͹ྑ͍
ͷॱʹఏ‫ܗܞ‬੒͞ΕΔͱ͢Δͱ
ͷߩ‫౓ݙ‬͸
ͷߩ‫౓ݙ‬͸
ͷߩ‫౓ݙ‬͸
ӈදʹ֤஋Λೖྗ͢Δ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
3 → 2 → 1
1 v({1,2,3}) − v({2,3}) = 6 − 2 = 4
2 v({2,3}) − v({3}) = 2 − 0 = 2
3 v({3}) − v(∅) = 0 − 0 = 0
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ॱྻ

12. ϓϨΠϠʔͷߩ‫౓ݙ‬ͷ‫ظ‬଴஋ ͢ͳΘͪγϟʔϓϨΠ஋͸
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
ϕ1(v) =
1
3! ∑
π∈Π
(v(Pπ,1
∪ {i}) − v(Pπ,1
))
=
0 + 0 + 1 + 4 + 4 + 4
6
=
13
6
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ॱྻ

13. ϓϨΠϠʔͷߩ‫౓ݙ‬ͷ‫ظ‬଴஋ ͢ͳΘͪγϟʔϓϨΠ஋͸
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
ϕ2(v) =
1
3! ∑
π∈Π
(v(Pπ,2
∪ {i}) − v(Pπ,2
))
=
1 + 2 + 0 + 0 + 2 + 2
6
=
7
6
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ॱྻ

14. ϓϨΠϠʔͷߩ‫౓ݙ‬ͷ‫ظ‬଴஋ ͢ͳΘͪγϟʔϓϨΠ஋͸
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
ϕ3(v) =
1
3! ∑
π∈Π
(v(Pπ,3
∪ {i}) − v(Pπ,3
))
=
5 + 4 + 5 + 2 + 0 + 0
6
=
16
6 (
=
8
3)
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ॱྻ

15. Ҏ্ΑΓ γϟʔϓϨΠ஋͸
ิ଍ɿ഑෼ͱͳ͍ͬͯΔ͜ͱͷ֬ೝ
ิ଍ɿਔ ͱൺ΂Δͱ
ϓϨΠϠʔͷ഑෼͸ए‫ׯ‬Լ͕Δ͕ ߩ‫౓ݙ‬ͷ‫͔఺؍‬Β΋ϓϨΠϠʔ͕࠷΋ήʔϜʹӨ‫ڹ‬
ͷ͋ΔϓϨΠϠʔͱߟ͑Δ͜ͱ͕Ͱ͖Δ
v({1,2,3}) = 6 v({1,2}) = 1 v({1,3}) = 4 v({2,3}) = 2 v({1}) = 0 v({2}) = 0 v({3}) = 0
ϕ(v) = (ϕ1(v), ϕ2(v), ϕ3(v)) =
(
13
6
,
7
6
,
16
6 )
ϕ1(v) + ϕ2(v) + ϕ3(v) =
13
6
+
7
6
+
16
6
=
36
6
= 6
(x1, x2, x3) = (2, 1, 3)
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ॱྻ

16. ήʔϜཧ࿦#4*$ԋश
ਓήʔϜ෼ੳɿγϟʔϓϨΠ஋