https://youtu.be/HvyiOsMdxdE

1. ήʔϜཧ࿦#4*$ୈճ
γϟʔϓϨΠ஋

2. γϟʔϓϨΠ஋֓ཁ
ߩ‫౓ݙ‬ͷఆٛ
ॱྻͱઌߦऀ
γϟʔϓϨΠ஋ͷఆٛ
γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ

3. γϟʔϓϨΠ஋֓ཁ
γϟʔϓϨΠʹΑͬͯ༩͑ΒΕͨղ
Shapley, L. S. (1953) “A Value for n-Person Games,”Annals of Mathematics Studies
28, pp.305-317.
ެཧ‫͔ܥ‬Βಋग़͞ΕͨղͰ͋Δ
͔͠͠
#4*$Ͱ͸ެཧ‫͔ܥ‬Βͷಋग़͸ѻΘͳ͍
ఏ‫ʹܞ‬ର͢Δߩ‫͖ͮجʹ౓ݙ‬ఆٛ͞ΕΔղͰ͋Δ
͜ͷ఺Λઆ໌͢Δ
·ͨ
γϟʔϓϨΠ஋ͷԠ༻ͱͯ͠
౤ථྗࢦ਺ͷઆ໌Λ͢Δ

4. ߩ‫౓ݙ‬
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

5. ॱྻͱઌߦऀ
γϟʔϓϨΠ஋͸શମఏ‫ܞ‬ͷಛੑؔ਺஋ΛͲ͏෼͚Δ͔ɺͱ͍͏ղͰ͋Δ
ͨͩ͠
ਓͣͭϓϨΠϠʔ͕ՃΘΔͱ͍͏ఏ‫ܗܞ‬੒Λߟ͑Δ

6. ॱྻͱઌߦऀ

ਓͷϓϨΠϠʔΛฒ΂ͨॱྻΛ

ͱ͠

‫ݸ‬ͷॱྻ

ͷશମΛ

ਓͷ৔߹
ฒ΂ํ͸

௨Γ







n π = (π(1), …, π(n)) n! π Π
3! = 6 Π = {π1
, π2
, π3
, π4
, π5
, π6
}
π1
= (π1
(1), π1
(2), π1
(3)) = (1,2,3)
π2
= (π2
(1), π2
(2), π2
(3)) = (1,3,2)
π3
= (π3
(1), π3
(2), π3
(3)) = (2,1,3)
π4
= (π4
(1), π4
(2), π4
(3)) = (2,3,1)
π5
= (π3
(1), π5
(2), π5
(3)) = (3,1,2)
π6
= (π6
(1), π6
(2), π6
(3)) = (3,2,1)
ॱྻ

7. ॱྻͱઌߦऀ

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
)

8. γϟʔϓϨΠ஋ͷఆٛ
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))

9. γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ϓϨΠϠʔ

ͷγϟʔϓϨΠ஋Λ

ͱ͢Δͱ


Λ୯ʹγϟʔϓϨΠ஋ͱ͍͏

ॱྻʹରͯ͠͸
ϓϨΠϠʔ͸ઌߦऀ͕͍ͳ͍ͷͰߩ‫౓ݙ‬

ϓϨΠϠʔ͸ઌߦऀͷू߹

Ͱߩ‫౓ݙ‬

ϓϨΠϠʔ͸ઌߦऀͷू߹

Ͱߩ‫౓ݙ‬

i ϕi(v)
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ϕ(v) = (ϕ1(v)⋯, ϕn(v))
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
v({1}) − v(∅) = 0
{1} v({1,2}) − v({1}) = 10
{1,2} v({1,2,3}) − v({1,2}) = 0
ॱྻ

10. γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ϓϨΠϠʔ

ͷγϟʔϓϨΠ஋Λ

ͱ͢Δͱ


Λ୯ʹγϟʔϓϨΠ஋ͱ͍͏

ॱྻʹରͯ͠͸
ϓϨΠϠʔ͸ઌߦऀ͕͍ͳ͍ͷͰߩ‫౓ݙ‬

ϓϨΠϠʔ͸ઌߦऀͷू߹

Ͱߩ‫౓ݙ‬

ϓϨΠϠʔ͸ઌߦऀͷू߹

Ͱߩ‫౓ݙ‬

i ϕi(v)
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ϕ(v) = (ϕ1(v)⋯, ϕn(v))
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
v({1}) − v(∅) = 0
{1,3} v({1,2,3}) − v({1,3}) = 0
{1} v({1,3}) − v({1}) = 10
ॱྻ

11. γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ϓϨΠϠʔ

ͷγϟʔϓϨΠ஋Λ

ͱ͢Δͱ


Λ୯ʹγϟʔϓϨΠ஋ͱ͍͏

ॱྻʹରͯ͠͸
ϓϨΠϠʔ͸ઌߦऀͷू߹

Ͱߩ‫౓ݙ‬

ϓϨΠϠʔ͸ઌߦऀ͕͍ͳ͍ͷͰߩ‫౓ݙ‬

ϓϨΠϠʔ͸ઌߦऀͷू߹

Ͱߩ‫౓ݙ‬

i ϕi(v)
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ϕ(v) = (ϕ1(v)⋯, ϕn(v))
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
{2} v({1,2}) − v({2}) = 10
v({2}) − v(∅) = 0
{1,2} v({1,2,3}) − v({1,2}) = 0
ॱྻ

12. γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ϓϨΠϠʔ

ͷγϟʔϓϨΠ஋Λ

ͱ͢Δͱ


Λ୯ʹγϟʔϓϨΠ஋ͱ͍͏

ॱྻʹରͯ͠͸
ϓϨΠϠʔ͸ઌߦऀͷू߹

Ͱߩ‫౓ݙ‬

ϓϨΠϠʔ͸ઌߦऀ͕͍ͳ͍ͷͰߩ‫౓ݙ‬

ϓϨΠϠʔ͸ઌߦऀͷू߹

Ͱߩ‫౓ݙ‬

i ϕi(v)
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ϕ(v) = (ϕ1(v)⋯, ϕn(v))
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
{2,3} v({1,2,3}) − v({2,3}) = 10
v({2}) − v(∅) = 0
{2} v({2,3}) − v({2}) = 0
ॱྻ

13. γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ϓϨΠϠʔ

ͷγϟʔϓϨΠ஋Λ

ͱ͢Δͱ


Λ୯ʹγϟʔϓϨΠ஋ͱ͍͏

ॱྻʹରͯ͠͸
ϓϨΠϠʔ͸ઌߦऀͷू߹

Ͱߩ‫౓ݙ‬

ϓϨΠϠʔ͸ઌߦऀͷू߹

Ͱߩ‫౓ݙ‬

ϓϨΠϠʔ͸ઌߦऀ͕͍ͳ͍ͷͰߩ‫౓ݙ‬

i ϕi(v)
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ϕ(v) = (ϕ1(v)⋯, ϕn(v))
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
{3} v({1,3}) − v({3}) = 10
{1,3} v({1,2,3}) − v({1,3}) = 0
v({3}) − v(∅) = 0
ॱྻ

14. γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ϓϨΠϠʔ

ͷγϟʔϓϨΠ஋Λ

ͱ͢Δͱ


Λ୯ʹγϟʔϓϨΠ஋ͱ͍͏

ॱྻʹରͯ͠͸
ϓϨΠϠʔ͸ઌߦऀͷू߹

Ͱߩ‫౓ݙ‬

ϓϨΠϠʔ͸ઌߦऀͷू߹

Ͱߩ‫౓ݙ‬

ϓϨΠϠʔ͸ઌߦऀ͕͍ͳ͍ͷͰߩ‫౓ݙ‬

i ϕi(v)
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ϕ(v) = (ϕ1(v)⋯, ϕn(v))
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
{2,3} v({1,2,3}) − v({2,3}) = 10
{3} v({2,3}) − v({3}) = 0
v({3}) − v(∅) = 0
ॱྻ

15. γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ϓϨΠϠʔ

ͷγϟʔϓϨΠ஋Λ

ͱ͢Δͱ


Λ୯ʹγϟʔϓϨΠ஋ͱ͍͏

ϓϨΠϠʔͷγϟʔϓϨΠ஋͸

i ϕi(v)
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ϕ(v) = (ϕ1(v)⋯, ϕn(v))
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
ϕ1(v) =
1
6
(0 + 0 + 10 + 10 + 10 + 10) =
40
6
ॱྻ

16. γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ϓϨΠϠʔ

ͷγϟʔϓϨΠ஋Λ

ͱ͢Δͱ


Λ୯ʹγϟʔϓϨΠ஋ͱ͍͏

ϓϨΠϠʔͷγϟʔϓϨΠ஋͸

i ϕi(v)
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ϕ(v) = (ϕ1(v)⋯, ϕn(v))
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
ϕ2(v) =
1
6
(10 + 0 + 0 + 0 + 0 + 0) =
10
6
ॱྻ

17. γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ϓϨΠϠʔ

ͷγϟʔϓϨΠ஋Λ

ͱ͢Δͱ


Λ୯ʹγϟʔϓϨΠ஋ͱ͍͏

ϓϨΠϠʔͷγϟʔϓϨΠ஋͸

i ϕi(v)
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ϕ(v) = (ϕ1(v)⋯, ϕn(v))
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
ϕ3(v) =
1
6
(0 + 10 + 0 + 0 + 0 + 0) =
10
6
ॱྻ

18. γϟʔϓϨΠ஋Λ‫ٻ‬ΊΔ
ϓϨΠϠʔ

ͷγϟʔϓϨΠ஋Λ

ͱ͢Δͱ


Λ୯ʹγϟʔϓϨΠ஋ͱ͍͏


˗γϟʔϓϨΠ஋ͷఆٛͰ͸഑෼ʹ͍ͭͯ͸৮Ε͍ͯͳ͍͕
༏Ճ๏ੑΛຬͨ͢ήʔϜͰ͋Ε͹഑෼ͱͳΔ ࣍ճҎ߱ূ໌
ࢀߟɿίΞ
ަবू߹
Χʔωϧ͓Αͼਔ͸

i ϕi(v)
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ϕ(v) = (ϕ1(v)⋯, ϕn(v))
v({1,2,3}) = v({1,2}) = v({1,3}) = 10, v({2,3}) = 0, v({1}) = v({2}) = v({3}) = 0
ϕ(v) = (ϕ1(v), ϕ2(v), ϕ3(v)) =
(
40
6
,
10
6
,
10
6 )
(10,0,0)
ॱྻ

19. ήʔϜཧ࿦#4*$ୈճ
γϟʔϓϨΠ஋
࣍ճɿଓɾγϟʔϓϨΠ஋