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ゲーム理論BASIC 第32回 -市場ゲームとコア-
- 2. ༏Ճ๏ੑΛຬͨ͢ήʔϜʹ͓͚ΔίΞ ෮श ަࢢͱࡁܦήʔϜ ڝ૪ߧۉ ڝ૪ͱߧۉ ڝ૪ͱߧۉίΞ ิ ίΞͷଘࡏίΞͷݶۃఆཧ
- 3. ༏Ճ๏ੑΛຬͨ͢ήʔϜʹ͓͚ΔίΞ ఆཧ ಛੑؔܗήʔϜ  ʹ͓͍ͯ  ͕༏Ճ๏ੑΛຬͨ͢ͱ͖ ίΞ  ͍͑ݴΕ ίΞఏ߹ܞཧੑΛຬͨ͢ͱ͍͏͜ͱ (N, v) v C(v) = {x ∈ 𝒜 | ∑ i∈S xi ≥ v(S), ∀S ⊂ N, S ≠ ∅, N} 1 2 3 x1 ≤ 20 x2 ≤ 30 x3 ≤ 40 ίΞ ఏ߹ܞཧੑ
- 4. ަࢢͱࡁܦήʔϜ ަࡁܦ ࡁܦओମ ϓϨΠϠʔ ͷू߹Λ  ͱ͢Δ ࡒͷछྨΛ  छྨͱ͠ ͜ͷࡒΛަ͢Δ ϓϨΠϠʔ  ͷࡒͷॳظอ༗ϕΫτϧΛ   ൪ͷࡒ՟ฎͰ͋Γ ҙʹׂՄೳͳࡒͱ͢Δ ࡒϕΫτϧ  ʹର͢ΔϓϨΠϠʔ  ͷޮ༻ؔҎԼͰ༩͑ΒΕΔͱ͢Δɿ  ͜Ε՟ฎʹؔͯ͠ઢ͋Ͱؔ༻ޮͳܗΓ ৡՄೳͳޮ༻Λͭ ৡՄೳͳޮ༻ΛͭަࡁܦΛ  Ͱද͢ N = {1,2,⋯, n} m + 1 i ∈ N wi = (w1 i , ⋯, wm i , wm+1 i ) ∈ ℜm+1 + m + 1 x = (x1 , ⋯xm , xm+1 ) ∈ ℜm+1 + i Ui(x1 , ⋯, xm , xm+1 ) = ui(x1 , ⋯, xm ) + xm+1 E = (N, {wi}i∈N, {ui}i∈N)
- 5. ަࢢͱࡁܦήʔϜ ࢢήʔϜ ৡՄೳͳޮ༻Λͭަࡁܦ  ͱ͢Δ ఏܞ  ͕ܗ͞Ε  ͷϝϯόʔ͚ͩͰަߦ͕ΘΕΔͱ͢Δ ࣮ݱՄೳͳࡒϕΫτϧ  ɹɹɹ  ఏܞ  ͷ૯ޮ༻ͷ࠷େΛ  ͱ͢Δͱ   E = (N, {wi}i∈N, {ui}i∈N) S ⊂ N S xi = (x1 i , ⋯xm i , xm+1 i ) ∑ i∈S xj i ≤ ∑ i∈S wj i , ∀j ∈ 1,2,⋯, m + 1 S v(S) v(S) = max (xi)i∈S {∑ i∈S ui(x1 i , ⋯, xm i ) + ∑ i∈S xm+1 i } s . t . ∑ i∈S xj i ≤ ∑ i∈S wj i , ∀j ∈ 1,2,⋯, m + 1 ఏަͰܞ
- 6. ަࢢͱࡁܦήʔϜ ࢢήʔϜ ৡՄೳͳޮ༻Λͭަࡁܦ  ͱ͢Δ ఏܞ  ͕ܗ͞Ε  ͷϝϯόʔ͚ͩͰަߦ͕ΘΕΔͱ͢Δ ࣮ݱՄೳͳࡒϕΫτϧ  ɹɹɹ  ఏܞ  ͷ૯ޮ༻ͷ࠷େΛ  ͱ͢Δͱ   E = (N, {wi}i∈N, {ui}i∈N) S ⊂ N S xi = (x1 i , ⋯xm i , xm+1 i ) ∑ i∈S xj i ≤ ∑ i∈S wj i , ∀j ∈ 1,2,⋯, m + 1 S v(S) v(S) = max (xi)i∈S {∑ i∈S ui(x1 i , ⋯, xm i ) + ∑ i∈S xm+1 i } s . t . ∑ i∈S xj i ≤ ∑ i∈S wj i , ∀j ∈ 1,2,⋯, m + 1 ఏަͰܞ   v(S) = max (xi)i∈S ∑ i∈S ui(x1 i , ⋯, xm i ) + ∑ i∈S wm+1 i s . t . ∑ i∈S xj i ≤ ∑ i∈S wj i , ∀j ∈ 1,2,⋯, m
- 7. ަࢢͱࡁܦήʔϜ ࢢήʔϜ ৡՄೳͳޮ༻Λͭަࡁܦ  ͱ͢Δ ఏܞ  ͕ܗ͞Ε  ͷϝϯόʔ͚ͩͰަߦ͕ΘΕΔͱ͢Δ ࣮ݱՄೳͳࡒϕΫτϧ  ɹɹɹ  ఏܞ  ͷ૯ޮ༻ͷ࠷େΛ  ͱ͢Δͱ   E = (N, {wi}i∈N, {ui}i∈N) S ⊂ N S xi = (x1 i , ⋯xm i , xm+1 i ) ∑ i∈S xj i ≤ ∑ i∈S wj i , ∀j ∈ 1,2,⋯, m + 1 S v(S) v(S) = max (xi)i∈S {∑ i∈S ui(x1 i , ⋯, xm i ) + ∑ i∈S xm+1 i } s . t . ∑ i∈S xj i ≤ ∑ i∈S wj i , ∀j ∈ 1,2,⋯, m + 1 ఏަͰܞ   v(S) = max (xi)i∈S ∑ i∈S ui(x1 i , ⋯, xm i ) + ∑ i∈S wm+1 i s . t . ∑ i∈S xj i ≤ ∑ i∈S wj i , ∀j ∈ 1,2,⋯, m ͜͜Λແࢹͯ͠࠷దղಉ͡ ࠷ద͜ͷ͚ࠩͩʹͳΔͷͰ ઓུతಉͰ͋Δಛੑ͕ؔ࡞ΕΔ
- 8. ަࢢͱࡁܦήʔϜ ࢢήʔϜ ৡՄೳͳޮ༻Λͭަࡁܦ  ͱ͢Δ ఏܞ  ͕ܗ͞Ε  ͷϝϯόʔ͚ͩͰަߦ͕ΘΕΔͱ͢Δ   ಛੑ্͕ؔͰهද͞ΕΔಛੑؔܗήʔϜ  ΛৡՄೳͳޮ༻ΛͭࢢήʔϜͱ͍͏ Ҏ߱ ࡒϕΫτϧ  Ͱͳ͘  ͱ՟ฎΛলུ͢Δ E = (N, {wi}i∈N, {ui}i∈N) S ⊂ N S v(S) = max (xi)i∈S ∑ i∈S ui(x1 i , ⋯, xm i ) s . t . ∑ i∈S xj i ≤ ∑ i∈S wj i , ∀j ∈ 1,2,⋯, m (N, v) x = (x1 , ⋯xm , xm+1 ) ∈ ℜm+1 + x = (x1 , ⋯xm ) ∈ ℜm +
- 9. ڝ૪ߧۉ ڝ૪ߧۉ ৡՄೳͳޮ༻Λͭަࡁܦ  ͱ͢Δ Ձ֨ϕΫτϧ  ͱࡒϕΫτϧ  ͷ  ͕ ڝ૪͋ͰߧۉΔͱ ͯ͢ͷϓϨΠϠʔ  ͱͯ͢ͷࡒϕΫτϧ  ʹର͠   ͕Γཱͭ͜ͱͰ͋Δ E = (N, {wi}i∈N, {ui}i∈N) p* = (p1* , ⋯, pm* ) ∈ ℜm + x* i = (x1* i , ⋯xm* i ) ∈ ℜm + (p*, {x* i }i∈N) i xi ∈ ℜm + ui(x* i ) − p* ⋅ (x* i − wi) ≥ ui(xi) − p* ⋅ (xi − wi) ∑ i∈N x* i = ∑ i∈N wi ࢢͰ ݁धڅҰக  ͷͱͰޮ༻࠷େԽ  ച٫͠  Λಘͯ  ߪೖ͠  Λࢧ͏ p* wi p* ⋅ wi xi p* ⋅ xi   p* ⋅ wi = m ∑ j=1 pj* wj i p* ⋅ xi = m ∑ j=1 pj* xj i
- 10. ڝ૪ͱߧۉ ఆཧ ৡՄೳͳޮ༻Λͭަࡁܦ  ͱ͠ ڝ૪ߧۉΛ  ͱ͢Δ ͜ͷͱ͖  Ͱఆٛ͞ΕΔརಘϕΫτϧ  ަࡁܦ  ͔Βߏ͞ΕΔࢢήʔϜ  ʹ͓͚ΔͰ͋Δ ূ໌  ڝ૪͔ͩߧۉΒ ҙͷ  ʹରͯ͠  Λຬͨ͢͜ͷӈลͷ  ʹରͯ͠  ͱ͢Ε  ͱͳΔͷͰݸਓ߹ཧੑΛຬͨ͢ E = (N, {wi}i∈N, {ui}i∈N) (p*, {x* i }i∈N) zi = ui(x* i ) − p* ⋅ (x* i − wi), i = 1,2,⋯, n z = (z1, ⋯ . zn) E (N, v) (p*, {x* i }i∈N) xi ∈ ℜm + ui(x* i ) − p* ⋅ (x* i − wi) ≥ ui(xi) − p* ⋅ (xi − wi) xi xi = wi zi = ui(x* i ) − p* ⋅ (x* i − wi) ≥ ui(wi) = v({i})   v(S) = max (xi)i∈S ∑ i∈S ui(x1 i , ⋯, xm i ) s . t . ∑ i∈S xj i ≤ ∑ i∈S wj i , ∀j ∈ 1,2,⋯, m ڝ૪ߧۉͱ͍͏
- 11. ڝ૪ͱߧۉ  ূ໌ଓ શମ߹ཧੑΛ֬ೝ͢Δ ࡒϕΫτϧ  ͕  Λ࣮͢ݱΔͱ͢Δ ͢ͳΘͪ   ͜͜Ͱ  ʹରͯ͠ΛͱΔͱ   zi = ui(x* i ) − p* ⋅ (x* i − wi), i = 1,2,⋯, n y = (y1, ⋯, yn) v(N) v(N) = ∑ i∈N ui(y1 i , ⋯, ym i ) ∑ i∈N yi ≤ ∑ i∈N wi zi, i = 1,2,⋯, n ∑ i∈N zi = ∑ i∈N { ui(x1* i , ⋯, xm* i ) − p* ⋅ (x* i − wi) } ≥ ∑ i∈N { ui(y1 i , ⋯, ym i ) − p* ⋅ (yi − wi) } = ∑ i∈N ui(y1 i , ⋯, ym i ) − p*⋅ (∑ i∈N yi − ∑ i∈N wi) ≥ ∑ i∈N ui(y1 i , ⋯, ym i ) = v(N)   v(S) = max (xi)i∈S ∑ i∈S ui(x1 i , ⋯, xm i ) s . t . ∑ i∈S xj i ≤ ∑ i∈S wj i , ∀j ∈ 1,2,⋯, m  ڝ૪ߧۉ (p*, {x* i }i∈N)
- 12. ڝ૪ͱߧۉ  ূ໌ଓ શମ߹ཧੑΛ֬ೝ͢Δ धڅҰக݅ΑΓ   Ώ͑ʹ  zi = ui(x* i ) − p* ⋅ (x* i − wi), i = 1,2,⋯, n ∑ i∈N zi = ∑ i∈N { ui(x1* i , ⋯, xm* i ) − p* ⋅ (x* i − wi) } = ∑ i∈N ui(x1* i , ⋯, xm* i ) − p*⋅ (∑ i∈N x* i − ∑ i∈N wi) = ∑ i∈N ui(x1* i , ⋯, xm* i ) ≤ v(N) ∑ i∈N zi = v(N)  ∑ i∈N x* i = ∑ i∈N wi   v(N) = max (xi)i∈N ∑ i∈N ui(x1 i , ⋯, xm i ) = ∑ i∈N ui(y1 i , ⋯, ym i ) s . t . ∑ i∈N xj i ≤ ∑ i∈N wj i , ∀j ∈ 1,2,⋯, m
- 13. ڝ૪ͱߧۉίΞ ఆཧ ڝ૪ߧۉίΞʹଐ͢Δ ূ໌  Λڝ૪ڝ ͠ͱߧۉ૪ߧۉΛ  ͱ͢Δ ҙͷఏܞ  ʹରͯ͠ ࡒϕΫτϧ  ͕  Λ࣮͢ݱΔͱ͢Δ ͢ͳΘͪ   ͜ͷͱ͖  ͱͳΓ ఏ߹ܞཧੑΛຬͨ͢ (p*, {x* i }i∈N) zi = ui(x* i ) − p* ⋅ (x* i − wi), i = 1,2,⋯, n S ⊂ N y = (yi)i∈S v(S) v(S) = ∑ i∈S ui(y1 i , ⋯, ym i ) ∑ i∈S yi ≤ ∑ i∈S wi ∑ i∈S zi = ∑ i∈S ui(x* i ) − p* ⋅ ∑ i∈S (x* i − wi) ≥ ∑ i∈S ui(yi) − p* ⋅ ∑ i∈S (yi − wi) ≥ ∑ i∈S ui(yi) = v(S)  ڝ૪ߧۉ (p*, {x* i }i∈N) ڝ૪Ͱߧۉͷ ϓϨΠϠʔؒͷࣗൃతͳަবʹΑͬͯࡒΛަ͢Δͱ͖ ίΞͱ͍͏҆ఆঢ়ଶʹΑΓୡՄೳ