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Similar to ゲーム理論BASIC 第45回 -シャープレイ値に関する定理 補足 証明5-
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ゲーム理論BASIC 第45回 -シャープレイ値に関する定理 補足 証明5-
- 3. ఆཧ ެ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
- 4. γϟʔϓϨΠͷެཧ ެ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
- 5. ఆཧͷূ໌ ެཧΛຬͨؔ͢ ͱ͢Δҙͷ ʹ͍ͭͯ ิΑΓ ͱͳΔ ݸͷ࣮ ͕ͨͩҰͭଘࡏ͢Δ ͳΔ ଘࡏ͠͏ΔͷͰ ͱॻ͖͑ΒΕΔ͜Εӈลͷ߲ΛҠ߲ͯ͠ ެཧΑΓ ҙͷ ʹର͠ ρ : V → ℜn v ∈ V v = ∑ R:R⊆N,R≠∅ cRvR 2n − 1 cR, R ⊆ N, R ≠ ∅ cR 0 cR, R ⊆ N, R ≠ ∅ v = ∑ R:R⊆N,R≠∅ cRvR = ∑ R:R⊆N,R≠∅,cR≥0 cRvR − ∑ R:R⊆N,R≠∅,cR0 |cR |vR v + ∑ R:R⊆N,R≠∅,cR0 |cR |vR = ∑ R:R⊆N,R≠∅,cR≥0 cRvR i ∈ N ρi( v + ∑ R:R⊆N,R≠∅,cR0 |cR |vR) = ρi( ∑ R:R⊆N,R≠∅,cR≥0 cRvR) ⇔ ρi(v) + ρi( ∑ R:R⊆N,R≠∅,cR0 |cR |vR) = ρi( ∑ R:R⊆N,R≠∅,cR≥0 cRvR) vR(S) = { 1 R ⊆ S 0 otherwise ิҙͷఏܞ ʹରͯ͠ ಛੑؔ Λ ͱఆٛ͢Δͱ ҙͷಛੑؔ ʹରͯ͠ ͳΔ ݸͷ࣮ ͕ͨͩҰͭଘࡏ͢Δ R ⊆ N, R ≠ ∅ vR ∈ V vR(S) = { 1 R ⊆ S 0 otherwise v ∈ V v = ∑ R:R⊆N,R≠∅ cRvR 2n − 1 cR, R ⊆ N, R ≠ ∅ ެཧՃ๏ੑ ҙͷ ʹରͯ͠ v, u ∈ V ψi(v + u) = ψi(v) + ψi(u) ∀i ∈ N
- 6. ఆཧͷূ໌ ެཧͱิΑΓ Ώ͑ʹ ࠓ Ұҙʹఆ·Δͷ͔ͩΒ Ұҙʹఆ·Δ ূ໌ऴྃ ρi(v) + ρi( ∑ R:R⊆N,R≠∅,cR0 |cR |vR) = ρi( ∑ R:R⊆N,R≠∅,cR≥0 cRvR) ρi( ∑ R:R⊆N,R≠∅,cR0 |cR |vR) = ∑ R:R⊆N,R≠∅,cR0 ρi(|cR |vR) = ∑ R:R⊆N,i∈R,cR0 |cR | |R| ρi( ∑ R:R⊆N,R≠∅,cR≥0 cRvR) = ∑ R:R⊆N,R≠∅,cR≥0 ρi(cRvR) = ∑ R:R⊆N,i∈R,cR≥0 cR |R| ρi(v) + ∑ R:R⊆N,i∈R,cR0 |cR | |R| = ∑ R:R⊆N,i∈R,cR≥0 cR |R| ⇔ ρi(v) = ∑ R:R⊆N,i∈R,cR≥0 cR |R| − ∑ R:R⊆N,i∈R,cR0 |cR | |R| = ∑ R:R⊆N,i∈R cR |R| cR, R ⊆ N, R ≠ ∅ ρi(v) ެཧՃ๏ੑ ҙͷ ʹରͯ͠ v, u ∈ V ψi(v + u) = ψi(v) + ψi(u) ∀i ∈ N ิެཧΛຬͨؔ͢ ͱ͢Δ͜ͷͱ͖֤ ͱҙͷ࣮ ʹ͍ͭͯ ͱͳΔ ψ : V → ℜn vR ∈ V R ⊆ N, R ≠ ∅ c 0 ψi(cvR) = { c |R| i ∈ R 0 i ∉ R
- 7. γϟʔϓϨΠͷಋग़ ެ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 ઌ΄ͲͷٞͰ ΛٻΊ͕ͨ ্هͷࣜΛಋग़͢Δʹ ࣮ࡍʹ Λ͢ࢉܭΔඞཁ͕͋Δ ψi(v) = ∑ R:R⊆N,i∈R cR |R| cR, R ⊆ N, R ≠ ∅
- 8. γϟʔϓϨΠͷಋग़ ͷ Ͱ༩͑ΒΕΔͨͩ͠ ࣮ࡍ ҙͷ ʹ͍ͭͯ v = ∑ R:R⊆N,R≠∅ cRvR cR cR = ∑ T⊆R (−1)|R|−|T| v(T) = ∑ T⊆R (−1)r−t v(T) r = |R|, t = |T| S ⊆ N ∑ R:R⊆N,R≠∅ cRvR(S) = ∑ R:R⊆S,R≠∅ cR = ∑ R:R⊆S,R≠∅ ∑ T⊆R (−1)r−t v(T) = ∑ T ⊆ S ∑ R ⊆ S, R ⊇ T (−1)r−t v(T) = ∑ T⊆S s ∑ r=t (−1)r−t ( s − t r − t) v(T) vR(S) = { 1 R ⊆ S 0 otherwise ۭू߹ ^ ^ ^ ^ ^ ^ ^ ^ ˔ ˔ ^ ˔ ˔ ^ ˔ ˔ ^ ˔ ˔ ˔ ˔ ^ ˔ ˔ ˔ ˔ ^ ˔ ˔ ˔ ˔ ^ ˔ ˔ ˔ ˔ ˔ ˔ ˔ ˔ R T Λݻఆ͔ͯ͠ΒΛͱΔ ҰఆΈͳͤΔ ू߹ ͷେ͖͞Ͱ͋Γ ΛؚΉΑ͏ͳ ͷ߹ͷ ӈදྫɿ ͷ߹ Ҏ֎ͷΓͷਓͷબͼํ ୭બͳ͍ͷؚΉ ͔ͩΒ ɹ ͔֬ʹ T v(T) r R T R s ∑ r=t ( s − t r − t) T = {1} 3 ∑ r=1 ( 3 − 1 r − 1) = ( 3 − 1 1 − 1) + ( 3 − 1 2 − 1) + ( 3 − 1 3 − 1) = 1 + 2 + 1 = 4 R = {1}, {1,2}, {1,3}, {1,2,3}
- 9. γϟʔϓϨΠͷಋग़ ͷ Ͱ༩͑ΒΕΔͨͩ͠ ࣮ࡍ ҙͷ ʹ͍ͭͯ ೋ߲ఆཧɿ Λ༻͍Δ ͨͩ͠ v = ∑ R:R⊆N,R≠∅ cRvR cR cR = ∑ T⊆R (−1)|R|−|T| v(T) = ∑ T⊆R (−1)r−t v(T) r = |R|, t = |T| S ⊆ N ∑ R:R⊆N,R≠∅ cRvR(S) = ∑ R:R⊆S,R≠∅ cR = ∑ R:R⊆S,R≠∅ ∑ T⊆R (−1)r−t v(T) = ∑ T ⊆ S ∑ R ⊆ S, R ⊇ T (−1)r−t v(T) = ∑ T⊆S s ∑ r=t (−1)r−t ( s − t r − t) v(T) = ∑ T⊆S (1 − 1)s−t v(T) = v(S) (1 + x)n = n ∑ r=0 ( n r) xr s ∑ r=t ( s − t r − t) (−1)r−t = s−t ∑ r′  =0 ( s − t r′  ) (−1)r′  = (1 − 1)s−t r′  = r − t ͷͱ͖ ͷͱ͖ T = S (1 − 1)s−s v(S) = 00 v(S) = v(S) T ⊂ S (1 − 1)s−t v(T) = 0s−t v(S) = 0
- 10. γϟʔϓϨΠͷಋग़ ͱ͓͘ͱ ͷͱ͖ ͱ͢Ε ʹҙͯ͠ ψi(v) = ∑ R:R⊆N,i∈R cR |R| = ∑ R:R⊆N,i∈R 1 r ∑ S⊆R (−1)r−s v(S) = ∑ S⊆N ∑ R⊇S∪{i} 1 r (−1)r−s v(S) αi(S) = ∑ R⊇S∪{i} 1 r (−1)r−s i ∈ S S = S′  ∪ {i}, i ∉ S′  S ∪ {i} = S′  ∪ {i} αi(S) = ∑ R⊇S∪{i} 1 r (−1)r−s = ∑ R⊇S′  ∪{i} 1 r (−1)r−(s−1)−1 = ∑ R⊇S′  ∪{i} 1 r (−1)r−s′  −1 = − αi(S′  ) ψi(v) = ∑ S⊆N αi(S)v(S) = ∑ S ⊆ N i ∈ S αi(S)v(S) + ∑ S ⊆ N i ∉ S αi(S)v(S) = ∑ S′  ⊆ N i ∉ S′  αi(S′  ∪ {i})v(S′  ∪ {i}) + ∑ S ⊆ N i ∉ S αi(S)v(S) = ∑ S′  ⊆ N i ∉ S′  αi(S)v(S′  ∪ {i}) + ∑ S ⊆ N i ∉ S αi(S)v(S) = ∑ S′  ⊆ N i ∉ S′  (−αi(S′  ))v(S′  ∪ {i}) + ∑ S ⊆ N i ∉ S αi(S)v(S) = ∑ S ⊆ N i ∉ S αi(S)(−v(S ∪ {i}) + v(S)) cR = ∑ S⊆R (−1)|R|−|S| v(S) = ∑ S⊆R (−1)r−s v(S) ۭू ߹ ^ ^ ^ ^ ^ ^ ^ ^ ˔ ˔ ^ ˔ ˔ ˔ ˔ ^ ˔ ˔ ˔ ˔ ^ ˔ ˔ ˔ ˔ ˔ ˔ ˔ ˔ R S i = 1
- 11. γϟʔϓϨΠͷಋग़ ͱ͓͘ͱ ͷͱ͖ ͱ͢Ε ʹҙͯ͠ ψi(v) = ∑ R:R⊆N,i∈R cR |R| = ∑ R:R⊆N,i∈R 1 r ∑ S⊆R (−1)r−s v(S) = ∑ S⊆N ∑ R⊇S∪{i} 1 r (−1)r−s v(S) αi(S) = ∑ R⊇S∪{i} 1 r (−1)r−s i ∈ S S = S′  ∪ {i}, i ∉ S′  S ∪ {i} = S′  ∪ {i} αi(S) = ∑ R⊇S∪{i} 1 r (−1)r−s = ∑ R⊇S′  ∪{i} 1 r (−1)r−(s−1)−1 = ∑ R⊇S′  ∪{i} 1 r (−1)r−s′  −1 = − αi(S′  ) ψi(v) = ∑ S⊆N αi(S)v(S) = ∑ S ⊆ N i ∈ S αi(S)v(S) + ∑ S ⊆ N i ∉ S αi(S)v(S) = ∑ S′  ⊆ N i ∉ S′  αi(S′  ∪ {i})v(S′  ∪ {i}) + ∑ S ⊆ N i ∉ S αi(S)v(S) = ∑ S′  ⊆ N i ∉ S′  αi(S)v(S′  ∪ {i}) + ∑ S ⊆ N i ∉ S αi(S)v(S) = ∑ S′  ⊆ N i ∉ S′  (−αi(S′  ))v(S′  ∪ {i}) + ∑ S ⊆ N i ∉ S αi(S)v(S) = ∑ S ⊆ N i ∉ S αi(S)(−v(S ∪ {i}) + v(S)) cR = ∑ S⊆R (−1)|R|−|S| v(S) = ∑ S⊆R (−1)r−s v(S) ۭू ߹ ^ ^ ^ ^ ^ ^ ^ ^ ˔ ˔ ^ ˔ ˔ ˔ ˔ ^ ˔ ˔ ˔ ˔ ^ ˔ ˔ ˔ ˔ ˔ ˔ ˔ ˔ R S i = 1 ψi(v) = 1 n! ∑ S:S⊆N,i∉S s!(n − s − 1)!(v(S ∪ {i}) − v(S)), ∀i ∈ N ψi(v) = ∑ S:S⊆N,i∉S s!(n − s − 1)! n! (v(S ∪ {i}) − v(S))
- 12. γϟʔϓϨΠͷಋग़ ʹ͍ͭͯ i ∉ S αi(S) = ∑ R ⊇ S ∪ {i} R ⊆ N 1 r (−1)r−s = n ∑ r=s+1 1 r (−1)r−s ( n − s − 1 r − s − 1) = n ∑ r=s+1 (−1)r−s ( n − s − 1 r − s − 1) 1 r = n ∑ r=s+1 (−1)r−s ( n − s − 1 r − s − 1)∫ 1 0 xr−1 dx = n ∑ r=s+1 ∫ 1 0 ( n − s − 1 r − s − 1) (−1)r−s xr−1 dx S {i} N(S ∪ {i}) ʹඞͣ·ؚΕΔ R ʹ ͓Αͼ ·ؚΕ͍ͯΔ͔Β ͱͳΔͱ͖ͷ߹ͷ Γͷ ਓ͔Β બͿ߹ͷͰ͋Δ Γͷਓͱͯ͠ ਓ͔Β ਓ·ͰͰ ͱͯ͠ ਓ͔Β ਓ·Ͱ R S {i} R n − s − 1 r − s − 1 0 n − s − 1 r s + 1 n ∫ 1 0 xr−1 dx = [ xr r ] 1 0 = 1 r
- 13. γϟʔϓϨΠͷಋग़ ͜͜Ͱ ͱ͓͘ͱ r′  = r − s − 1 n ∑ r=s+1 ∫ 1 0 ( n − s − 1 r − s − 1) (−1)r−s xr−1 dx = n−s−1 ∑ r′  =0 ∫ 1 0 ( n − s − 1 r′  ) (−1)r′  +1 xr′  +s dx = − n−s−1 ∑ r′  =0 ∫ 1 0 ( n − s − 1 r′  ) (−1)r′  xr′  +s dx = − ∫ 1 0 n−s−1 ∑ r′  =0 ( n − s − 1 r′  ) (−1)r′  xr′  +s dx = − ∫ 1 0 n−s−1 ∑ r′  =0 ( n − s − 1 r′  ) (−x)r′  xs dx = − ∫ 1 0 (1 − x)n−s−1 xs dx ೋ߲ఆཧ ΛҰͭ֎ͩ͠
- 14. γϟʔϓϨΠͷಋग़ ͱ͢Ε ෦ੵ͔Β n ∑ r=s+1 ∫ 1 0 ( n − s − 1 r − s − 1) (−1)r−s xr−1 dx = − ∫ 1 0 (1 − x)n−s−1 xs dx I(s, n − s − 1) = ∫ 1 0 (1 − x)n−s−1 xs dx I(s, n − s − 1) = ∫ 1 0 xs (1 − x)n−s−1 dx = [ xs+1 s + 1 (1 − x)n−s−1 ] 1 0 − ∫ 1 0 xs+1 s + 1 (n − s − 1)(1 − x)n−s−2 (−1)dx = n − s − 1 s + 1 ∫ 1 0 xs+1 (1 − x)n−s−2 dx = n − s − 1 s + 1 I(s + 1,n − s − 2) = n − s − 1 s + 1 n − s − 2 s + 2 I(s + 2,n − s − 3) = ⋯ = n − s − 1 s + 1 n − s − 2 s + 2 ⋯ 1 n − 1 I(n − 1,0) = n − s − 1 s + 1 n − s − 2 s + 2 ⋯ 1 n − 1 ∫ 1 0 xn−1 dx = n − s − 1 s + 1 n − s − 2 s + 2 ⋯ 1 n − 1 [ xn n ] 1 0 = n − s − 1 s + 1 n − s − 2 s + 2 ⋯ 1 n − 1 1 n = (n − s − 1)!s! n! ূ໌ੵͷඍ͔Β লུ ∫ b a f(x)g(x)dx = [ f(x)G(x) ] b a − ∫ b a f′  (x)G(x)dx G′  (x) = g(x)
- 15. γϟʔϓϨΠͷಋग़ ʹ͍ͭͯ i ∉ S αi(S) = ∑ R ⊇ S ∪ {i} R ⊆ N 1 r (−1)r−s = n ∑ r=s+1 1 r (−1)r−s ( n − s − 1 r − s − 1) = n ∑ r=s+1 (−1)r−s ( n − s − 1 r − s − 1) 1 r = n ∑ r=s+1 (−1)r−s ( n − s − 1 r − s − 1)∫ 1 0 xr−1 dx = n ∑ r=s+1 ∫ 1 0 ( n − s − 1 r − s − 1) (−1)r−s xr−1 dx = − (n − s − 1)!s! n! ψi(v) = ∑ S:S⊆N,i∉S s!(n − s − 1)! n! (v(S ∪ {i}) − v(S)) Ώ͑ʹ ψi(v) = ∑ S ⊆ N i ∉ S αi(S)(−v(S ∪ {i}) + v(S)) = ∑ S ⊆ N i ∉ S − (n − s − 1)!s! n! (−v(S ∪ {i}) + v(S)) = ∑ S ⊆ N i ∉ S (n − s − 1)!s! n! (v(S ∪ {i}) − v(S))