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ゲーム理論BASIC 第42回 -仁に関する証明1-
- 2. ఏܞͷෆຬͱϓϨΠϠʔͷ༏Ґੑ ෮श Χʔωϧ ෮श ఏܞͷෆຬ ෮श ͕ड༰తͰ͋Δ͜ͱͷఆٛ ෮श ਔͷఆٛ ෮श ਔΧʔωϧʹ·ؚΕΔ
- 3. ఏܞͷෆຬͱϓϨΠϠʔͷ༏Ґੑ ఆٛɿఏܞ  ͷෆຬ ಛੑؔܗήʔϜ  ʹ͓͍ͯ  ͱఏܞ  ΛͱΔͱ͖  Λ  ʹର͢Δఏܞ  ͷෆຬͱ͍͏ ਓͷϓϨΠϠʔ   ΛͱΓ  ͱ͓͘ ఆٛɿ࠷େෆຬ ਓͷϓϨΠϠʔ   ͱ  ʹ͍ͭͯ  Λ  ʹ͓͚ΔϓϨΠϠʔ  ͷ  ʹର͢Δ࠷େෆຬͱ͍͏ S (N, v) x ∈ 𝒜 (v) S ⊆ N e(S, x) = v(S) − ∑ i∈S xi x S i j ∈ N Tij = {S ⊆ N|i ∈ S, j ∉ S} i j ∈ N x ∈ 𝒜 (v) sij(x) = max S∈Tij e(S, x) x i j
- 4. ఏܞͷෆຬͱϓϨΠϠʔͷ༏Ґੑ ఆٛɿ༏Ґ ਓͷϓϨΠϠʔ   ͱ  ʹ͍ͭͯ   ͱͳΔͱ͖  ʹ͓͍ͯϓϨΠϠʔ   ΑΓ༏ҐͰ͋Δͱ͍͍  ͱॻ͘  Ͱ͋Ε ϓϨΠϠʔ   ʹରͯ͠རಘΛ͘Εͱ͍͑ͳ͍ ͳͥͳΒ  ఏܞΛΉΠϯηϯςΟϒ͕ͳ͘ͳΔͨΊ i j ∈ N x ∈ 𝒜 (v) sij(x) sji(x) xj v({j}) x i j i ≻x j xj ≤ v({j}) i j j ਓͷϓϨΠϠʔͷ ࠷େෆຬΛൺֱ
- 5. Χʔωϧͷఆٛ ఆٛɿߧۉঢ়ଶ ਓͷϓϨΠϠʔ   ͱ  ʹ͍ͭͯ  Ͱͳ͘  Ͱͳ͍ͱ͖  ʹ͓͍ͯϓϨΠϠʔ  ͱ  ߧۉঢ়ଶͰ͋Δͱ͍͍  ͱॻ͘ ఆٛɿΧʔωϧ ҙͷਓͷϓϨΠϠʔ͕ߧۉঢ়ଶʹ͋ΔΑ͏ͳͷશମ  ΛΧʔωϧͱ͍͏ i j ∈ N x ∈ 𝒜 (v) i ≻x j j ≻x i x i j i ∼x j 𝒦 (v) = {x ∈ 𝒜 (v)|i ∼x j ∀i, j ∈ N, i ≠ j}
- 6. ఏܞͷෆຬ ఆٛɿఏܞ  ͷෆຬ ಛੑؔܗήʔϜ  ʹ͓͍ͯ  ͱఏܞ  ΛͱΔͱ͖  Λ  ʹର͢Δఏܞ  ͷෆຬͱ͍͏  ʹର֤ͯ͠ఏܞ  ͷෆຬ  Λ େ͖ͳͷ͔ΒॱʹฒͨϕΫτϧΛ  ͱද͢ ͨͩ͠ ͯ͢ͷʹରͯ͠  ͱ  ʹؔͯ͠ෆຬͳͷͰআ֎ͯ͠ߟ͑Δ Ώ͑ʹ   ݸͷෆຬΛฒͨϕΫτϧʹͳΔ S (N, v) x ∈ 𝒜 (v) S ⊆ N e(S, x) = v(S) − ∑ i∈S xi x S x ∈ 𝒜 (v) S ⊆ N e(S, x) θ(x) N ∅ θ(x) 2n − 2
- 7. ͕ड༰తͰ͋Δ͜ͱͷఆٛ ఆٛɿड༰త ̎ͭͷ  ʹରͯ͠ ͋Δ  ͕ଘࡏͯ͠    ͱͳΔͱ͖   ΑΓड༰తͰ͋Δͱ͍͍  ͱද͢  ͷҙຯɿ   ͩͱ  ͱͳΔͨΊ   Ұఆ x, y ∈ 𝒜 (v) k, 1 ≤ k ≤ 2n − 4 θl(x) = θl(y) ∀l = 1,⋯, k, θk+1(x) θk+1(y) x y x ≫ y 2n − 4 θl(x) = θl(y) ∀l = 1,⋯,2n − 3 θ2n−2(x) = θ2n−2(y) n−2 ∑ l=1 θl(x) = ∑ S⊂N,S≠N,∅ ( v(S) − ∑ i∈S xi) = ∑ S⊂N,S≠N,∅ v(S) − ∑ S⊂N,S≠N,∅ ∑ i∈S xi = ∑ S⊂N,∅ v(S) − n−1 ∑ |S|=1 n−1C|S|−1 ∑ i∈N xi = ∑ S⊂N,∅ v(S) − n−1 ∑ |S|=1 n−1C|S|−1v(N)
- 9. ਔΧʔωϧʹ·ؚΕΔ ఆཧ ήʔϜ  ʹ͓͍ͯ ਔΛ  ΧʔωϧΛ  ͱ͢Δͱ  Ͱ͋Δ ূ໌ ҙͷ  ΛͱΔ͜Ε͕  ͱ͢Δͱ ͋Δ  Ͱ༏Ґͷؔ  ͱͳΔ ͢ͳΘͪ   ͜ͷͱ͖ेখ͍͞  ʹରͯ͠    ͱͳΔ  ͕ͱΕΔ ݸਓ߹ཧੑ શମ߹ཧੑΛຬͨ͢ (N, v) 𝒩 (v) 𝒦 (v) 𝒩 (v) ⊆ 𝒦 (v) x ∈ 𝒩 (v) x ∉ 𝒦 (v) i, j ∈ N i ≻x j sij(x) sji(x) xj v({j}) ϵ 0 yk = xk + ϵ (k = i) xk − ϵ (k = j) xk (k ≠ i, j) sij(y) = sij(x) − ϵ sji(x) + ϵ = sji(y) xj yj v({j}) y ∈ 𝒜 (v) ఆٛɿ༏Ґ
- 10. ήʔϜ  ʹ͓͍ͯ ਔΛ  ΧʔωϧΛ  ͱ͢Δͱ  Ͱ͋Δ ূ໌ ҙͷ  ΛͱΔ͜Ε͕  ͱ͢Δͱ ͋Δ  Ͱ༏Ґͷؔ  ͱͳΔ ͢ͳΘͪ   ͜ͷͱ͖ेখ͍͞  ʹରͯ͠    ͱͳΔ  ͕ͱΕΔ ݸਓ߹ཧੑ શମ߹ཧੑΛຬͨ͢ (N, v) 𝒩 (v) 𝒦 (v) 𝒩 (v) ⊆ 𝒦 (v) x ∈ 𝒩 (v) x ∉ 𝒦 (v) i, j ∈ N i ≻x j sij(x) sji(x) xj v({j}) ϵ 0 yk = xk + ϵ (k = i) xk − ϵ (k = j) xk (k ≠ i, j) sij(y) = sij(x) − ϵ sji(x) + ϵ = sji(y) xj yj v({j}) y ∈ 𝒜 (v) ఆٛɿ࠷େෆຬ
- 11.   Ͱ  ͷཁૉͱ͢Δ ༏Ґͷؔ  ͱ  ͷఆ͔ٛΒ  ิ ෆຬΛେ͖͍ॱ͔ΒฒΔ    T = {S ∈ 2N (Tij ∪ Tji)|e(S, x) ≥ sij(x)} t = |T| T sij(x) = max S∈Tij e(S, x) sji(x) = max S∈Tji e(S, x) T θt+1(x) = sij(x) θ1(x), ⋯, θt(x) e(S,x)≥sij(x), S∈2N (Tij∪Tji) θt+1(x) sij(x) θt+2(x), ⋯ ਔΧʔωϧʹ·ؚΕΔ ఆٛɿ࠷େෆຬ
- 12.     Ͱ͋Ε   ΑΓ   Ͱ͋Γ  Ͱ͋Δ͔Β  yk = xk + ϵ (k = i) xk − ϵ (k = j) xk (k ≠ i, j) sij(y) = sij(x) − ϵ sji(x) + ϵ = sji(y) xj yj v({j}) S ∈ 2N (Tij ∪ Tji) e(S, y) = e(S, x) sij(y) = sij(x) − ϵ T′  = {S ∈ 2N (Tij ∪ Tji)|e(S, y) ≥ sij(y)} = {S ∈ 2N (Tij ∪ Tji)|e(S, x) ≥ sij(x) − ϵ} T = {S ∈ 2N (Tij ∪ Tji)|e(S, x) ≥ sij(x)} t = |T| ≤ |T′  | ਔΧʔωϧʹ·ؚΕΔ ఆٛɿ࠷େෆຬ       θ1(x), ⋯, θt(x) θt+1(x) sij(x) θt+2(x), ⋯ θ1(y), ⋯, θt(y) θt+1(y) θt+2(y), ⋯   Ҏ߱ʹଘࡏ sij(y) θt+1(y)  Ͱ͋Ε  Ώ͑ʹ   ͱಉ͡ S ∈ 2N (Tij ∪ Tji) e(S, y) = e(S, x) θ1(y), ⋯, θt(y) θ1(x), ⋯, θt(x)
- 13.    ·ͨ  ൪ͷෆຬʹ͍ͭͯ  Ͱ͋Ε   Ͱ͋ΔͨΊʣ  Ͱ͋Ε   Ͱ͋Ε  ͕ͨͬͯ͠ ͯ͢ͷ  ʹؔͯ͠   ͱͳΔ Ώ͑ʹ  ͱͳΓ  ͕ਔͰ͋Δ͜ͱʹໃ६ yk = xk + ϵ (k = i) xk − ϵ (k = j) xk (k ≠ i, j) sij(y) = sij(x) − ϵ sji(x) + ϵ = sji(y) xj yj v({j}) t + 1 S ∈ 2N (Tij ∪ Tji) e(S, y) = e(S, x) sij(x) T = {S ∈ 2N (Tij ∪ Tji)|e(S, x) ≥ sij(x)} S ∈ Tij e(S, y) = e(S, x) − ϵ ≤ sij(x) − ϵ = sij(y) sij(x) S ∈ Tji e(S, y) = e(S, x) + ϵ ≤ sji(x) + ϵ sij(x) − ϵ = sij(y) sij(x) l = 1,⋯, t θl(y) = θl(x) θt+1(y) sij(x) = θt+1(x) y ≫ x x ਔΧʔωϧʹ·ؚΕΔ ఆٛɿ࠷େෆຬ       θ1(x), ⋯, θt(x) θt+1(x) sij(x) θt+2(x), ⋯ θ1(y), ⋯, θt(y) θt+1(y) θt+2(y), ⋯ e(S, y)