前半 https://youtu.be/n5YQ-AH_ALo 後半 https://youtu.be/z_OPzm86_18

1. ήʔϜཧ࿦#4*$ୈճ
ิ଍ɿγϟʔϓϨΠ஋ʹؔ͢Δఆཧূ໌

2. ߩ‫౓ݙ‬
ॱྻͱઌߦऀ
γϟʔϓϨΠ஋ͷఆٛ
γϟʔϓϨΠ஋ͷผද‫ݱ‬
γϟʔϓϨΠ஋ͷެཧ‫͔ܥ‬Βͷಋग़
ิ୊ͷূ໌

3. 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

4. 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
)

5. 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))

6. γϟʔϓϨΠ஋ͷผද‫ݱ‬
ϓϨΠϠʔ ͷγϟʔϓϨΠ஋Λ ͱ͢Δ
͸
ͱಉ஋Ͱ͋Δ͜͜Ͱ
ʹରͯ͠
ͱͳΔ৔߹ͷ਺͸
‫͋ݸ‬Δ
·ͨ͜ͷͱ͖ߩ‫౓ݙ‬͸ Ώ͑ʹผද‫ݱ‬Λ͑Δ
i ϕi(v)
ϕi(v) =
1
n! ∑
π∈Π
(v(Pπ,i
∪ {i}) − v(Pπ,i
))
ϕi(v) =
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(v(S ∪ {i}) − v(S)) s = |S| n = |N|
π ∈ Π Pπ,i
= S s! × (n − s − 1)!
v(S ∪ {i}) − v(S)
Π = {{1,2,3} {1,3,2} {2,1,3} {2,3,1} {3,1,2} {3,2,1}}
P{1,2,3},1
= ∅ P{1,3,2},1
= ∅ P{2,1,3},1
= {2} P{2,3,1},1
= {2,3} P{3,1,2},1
= {3} P{3,2,1},1
= {2,3}
S {i} N(S ∪ {i})
s! (n − s − 1)!
௨Γ
0!2! = 2 ௨Γ
1!1! = 1 ௨Γ
1!1! = 1 ௨Γ
2!0! = 2

7. γϟʔϓϨΠ஋ͷެཧ‫͔ܥ‬Βͷಋग़
ϓϨΠϠʔͷू߹ ͱ͢Δ
༏Ճ๏ੑΛຬͨ͢ಛੑؔ਺ ͷશମΛ Ͱද͢
֤ήʔϜ ʹରͯ͠
࣍‫ݩ‬ͷ࣮਺ϕΫτϧΛ༩͑Δؔ਺Λ ͱ͢Δ
ϓϨΠϠʔ ʹ͍ͭͯ
ͱͳΔͱ͖
ϓϨΠϠʔ ͸ ʹ͓͍ͯφϧϓϨΠϠʔͰ͋Δͱ͍͏
ϓϨΠϠʔ ʹ͍ͭͯ
ͱͳΔͱ͖
ϓϨΠϠʔ ͸ ʹ͓͍ͯରশͰ͋Δͱ͍͏
̎ͭͷಛੑؔ਺ ʹରͯ͠
Λ
ʹΑͬͯఆٛ͠
ͱ ͷ࿨ͱ͍͏
N
v : 2N
→ ℜ V
(N, v), v ∈ V n ψ : V → ℜn
ψ(v) = (ψ1(v), ⋯, ψn(v))
i ∈ N v(S ∪ {i}) − v(S) = 0 ∀S ⊆ N{i}
i (N, v)
i, j ∈ N, i ≠ j v(S ∪ {i}) = v(S ∪ {j}) ∀S ⊆ N({i} ∪ {j})
i, j (N, v)
v, u ∈ V v + u ∈ V
(v + u)(S) = v(S) + u(S) ∀S ⊆ N v u

8. γϟʔϓϨΠ஋ͷެཧ‫͔ܥ‬Βͷಋग़
ެཧશମ߹ཧੑ
೚ҙͷ ʹରͯ͠
ެཧφϧϓϨΠϠʔʹؔ͢Δੑ࣭
೚ҙͷ ʹରͯ͠
ϓϨΠϠʔ ͕φϧϓϨΠϠʔͰ͋Ε͹
ެཧରশੑ
೚ҙͷ ʹରͯ͠
ϓϨΠϠʔ ͕ ʹ͓͍ͯରশͰ͋Ε͹
ެཧՃ๏ੑ
೚ҙͷ ʹରͯ͠
v ∈ V
∑
i∈N
ψi(v) = v(N)
v ∈ V i ∈ N ψi(v) = 0
v ∈ V i, j (N, v) ψi(v) = ψj(v)
v, u ∈ V ψi(v + u) = ψi(v) + ψi(u) ∀i ∈ N
༨͢͜ͱͳ͘
ߩ‫͚ͳݙ‬Ε͹
ಉ͡ߩ‫ͳݙ‬Βಉ͡ධՁ
ٞ࿦ͷ༨஍͋Γ ‫ݶ‬քߩ‫౓ݙ‬ґଘੑ

9. γϟʔϓϨΠ஋ͷެཧ‫͔ܥ‬Βͷಋग़
ެཧΛຬͨؔ͢਺ ͸ͨͩ̍ͭʹఆ·Γ
֤ήʔϜ ʹରͯ͠
Ͱ༩͑ΒΕΔ
ψ : V → ℜn
(N, v)
ψi(v) =
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(v(S ∪ {i}) − v(S)), ∀i ∈ N
ิ୊
Ͱ༩͑ΒΕΔؔ਺ ͸ެཧΛຬͨ͢
શମ߹ཧੑΛຬͨ͢ͷ͸Ҏલূ໌͍ͯ͠Δ͕ผূ໌Λ͢Δ
ψ(v) = (ψ1(v), ⋯, ψn(v))
ψi(v) =
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(v(S ∪ {i}) − v(S)), ∀i ∈ N
ψ : V → ℜn

10. γϟʔϓϨΠ஋ͷެཧ‫͔ܥ‬Βͷಋग़
ิ୊
Ͱ༩͑ΒΕΔؔ਺ ͸ެཧΛຬͨ͢
ূ໌
ެཧશମ߹ཧੑʹ͍ͭͯ೚ҙͷ ΛͱΔ͜ͷͱ͖
ψ(v) = (ψ1(v), ⋯, ψn(v))
ψi(v) =
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(v(S ∪ {i}) − v(S)), ∀i ∈ N ψ : V → ℜn
v ∈ V
∑
i∈N
ψi(v) =
∑
i∈N
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(v(S ∪ {i}) − v(S))
=
∑
i∈N
1
n! ( ∑
S:S⊆N,i∉S
s!(n − s − 1)!v(S ∪ {i}) −
∑
S:S⊆N,i∉S
s!(n − s − 1)!v(S)
)
=
∑
i∈N
1
n! ( ∑
S:S⊆N,i∈S
(s − 1)!(n − s)!v(S) −
∑
S:S⊂N,i∉S
s!(n − s − 1)!v(S)
)
=
1
n!(∑
i∈N
∑
S:S⊆N,i∈S
(s − 1)!(n − s)!v(S) −
∑
i∈N
∑
S:S⊂N,i∉S
s!(n − s − 1)!v(S)
)
=
1
n!( ∑
S:S⊆N,S≠∅
∑
i∈S
(s − 1)!(n − s)!v(S) −
∑
S:S⊂N
∑
i∈NS
s!(n − s − 1)!v(S)
)
=
1
n!( ∑
S:S⊆N,S≠∅
s!(n − s)!v(S) −
∑
S:S⊂N
s!(n − s)!v(S)
)
=
1
n!
n!0!v(N) = v(N)
S {i} N(S ∪ {i})
s! (n − s − 1)!
S {i} NS
(s − 1)! (n − s)!
S ⊆ N, i ∉ S ⇒ S ⊂ N, i ∉ S
ͷ͚ࠩͩ
S = N
ۭू߹ ^ ^ ^
^
^
^
^
˔ ˔ ˔ ˔
˔ ˔ ˔ ˔
˔ ˔ ˔ ˔
ۭू߹ ^ ^ ^
^
^
^
^
˔ ˔ ˔ ˔
˔ ˔ ˔ ˔
˔ ˔ ˔ ˔

11. γϟʔϓϨΠ஋ͷެཧ‫͔ܥ‬Βͷಋग़
ิ୊
Ͱ༩͑ΒΕΔؔ਺ ͸ެཧΛຬͨ͢
ެཧφϧϓϨΠϠʔ೚ҙͷ ʹରͯ͠
ϓϨΠϠʔ ͕φϧϓϨΠϠʔͰ͋Ε͹
ূ໌
ϓϨΠϠʔ ʹ͍ͭͯ
φϧϓϨΠϠʔͰ͋Ε͹
Ώ͑ʹ
Ͱ͋Δ
ψ(v) = (ψ1(v), ⋯, ψn(v))
ψi(v) =
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(v(S ∪ {i}) − v(S)), ∀i ∈ N ψ : V → ℜn
v ∈ V i ∈ N ψi(v) = 0
i ∈ N v(S ∪ {i}) − v(S) = 0 ∀S ⊆ N{i}
ψi(v) = 0

12. γϟʔϓϨΠ஋ͷެཧ‫͔ܥ‬Βͷಋग़
ิ୊
Ͱ༩͑ΒΕΔؔ਺ ͸ެཧΛຬͨ͢
ެཧରশੑ೚ҙͷ ʹରͯ͠
ϓϨΠϠʔ ͕ ʹ͓͍ͯରশͰ͋Ε͹
ূ໌
ϓϨΠϠʔ
ʹ͍ͭͯ
ରশͰ͋Ε͹
ψ(v) = (ψ1(v), ⋯, ψn(v))
ψi(v) =
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(v(S ∪ {i}) − v(S)), ∀i ∈ N ψ : V → ℜn
v ∈ V i, j (N, v) ψi(v) = ψj(v)
i, j ∈ N i ≠ j v(S ∪ {i}) = v(S ∪ {j}) ∀S ⊆ N({i} ∪ {j})
ψi(v) − ψj(v) =
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(v(S ∪ {i}) − v(S)) −
1
n! ∑
S:S⊆N,j∉S
s!(n − s − 1)!(v(S ∪ {j}) − v(S))
=
1
n! ∑
S:S⊆N,i∉S,j∈S
s!(n − s − 1)!(v(S ∪ {i}) − v(S)) −
1
n! ∑
S:S⊆N,i∈S,j∉S
s!(n − s − 1)!(v(S ∪ {j}) − v(S))
=
1
n!( ∑
S:S⊆N,i∉S,j∈S
s!(n − s − 1)!v(S ∪ {i}) −
∑
S:S⊆N,i∉S,j∈S
s!(n − s − 1)!v(S)) −
∑
S:S⊆N,i∈S,j∉S
s!(n − s − 1)!v(S ∪ {j}) +
∑
S:S⊆N,i∈S,j∉S
s!(n − s − 1)!v(S)
)
=
1
n!( ∑
S:S⊆N,i,j∉S
(s + 1)!(n − s − 2)!v(S ∪ {i} ∪ {j}) −
∑
S:S⊆N,i,j∉S
(s + 1)!(n − s − 2)!v(S ∪ {j}))
−
∑
S:S⊆N,i,j∉S
(s + 1)!(n − s − 2)!v(S ∪ {j} ∪ {i}) +
∑
S:S⊆N,i,j∉S
(s + 1)!(n − s − 2)!v(S ∪ {i})
)
= 0
ରশੑ
ରশੑ
ରশੑ

13. γϟʔϓϨΠ஋ͷެཧ‫͔ܥ‬Βͷಋग़
ิ୊
Ͱ༩͑ΒΕΔؔ਺ ͸ެཧΛຬͨ͢
ެཧՃ๏ੑ೚ҙͷ ʹରͯ͠
ূ໌
೚ҙͷ ʹ͍ͭͯ
͸ Ͱ͋Δ͜ͷͱ͖
ψ(v) = (ψ1(v), ⋯, ψn(v))
ψi(v) =
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(v(S ∪ {i}) − v(S)), ∀i ∈ N ψ : V → ℜn
v, u ∈ V ψi(v + u) = ψi(v) + ψi(u) ∀i ∈ N
v, u ∈ V v + u ∈ V (v + u)(S) = v(S) + u(S) ∀S ⊆ N
ψi(v + u) =
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!((v + u)(S ∪ {i}) − (v + u)(S))
=
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!{(v(S ∪ {i}) + u(S ∪ {i})) − (v(S) + u(S))}
=
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!{(v(S ∪ {i}) − v(S)) + (u(S ∪ {i}) − u(S))}
=
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(v(S ∪ {i}) − v(S)) +
1
n! ∑
S:S⊆N,i∉S
s!(n − s − 1)!(u(S ∪ {i}) − u(S))
= ψi(v) + ψi(u)

14. ήʔϜཧ࿦#4*$ୈճ
ิ଍ɿγϟʔϓϨΠ஋ʹؔ͢Δఆཧূ໌
࣍ճɿิ଍ɿγϟʔϓϨΠ஋ʹؔ͢Δఆཧূ໌