ゲーム理論NEXT 戦略形協力ゲーム第9回 -結託耐性ナッシュ均衡関連定理-
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ゲーム理論NEXT 戦略形協力ゲーム第9回 -結託耐性ナッシュ均衡関連定理-
- 2. 1. ݁ୗੑφογϡ֬ͱߧۉఆతઓུͷؔΛཧղ 2 2. ಛघͳέʔεͰͷ݁ୗੑφογϡߧۉͷଘࡏ ຊಈըͷత
- 3. 1.ؔ࿈ఆཧ (a)ڧφογϡͳߧۉΒ݁ୗੑΛͭ ূ໌ (b)෦ήʔϜʹ͓͚Δ݁ୗੑφογϡߧۉͷଘࡏ ূ໌ 2. ಛघͳέʔεͰͷ݁ୗੑφογϡߧۉͷଘࡏ (a)෦ήʔϜ͕།ҰͷφογϡߧۉΛͭͱ͖ ূ໌ ࣍
- 4. ؔ࿈ఆཧ
- 5. ݁ୗੑφογϡߧۉͷఆٛ ݁ୗੑͱ੍ࣗڧݾత Λҙʹ༩͑ΒΕͨઓུͷͱ͢Δ ֤ ʹ͍ͭͯ ͕෦ήʔϜ ʹ͓͍ͯ݁ୗੑΛͭͱ Ͱ͋Δ͜ͱΛ͍͏ ʮ֤ఏܞ ͨͩ͠ ʹର͠ ͕෦ήʔϜ ʹ͓͍ͯ݁ୗੑΛͭʯ ɹɹͱ͍͏͜ͱ͕ఆٛ͞Εͨͱ͢Δ͜ͷͱ͖֤ఏܞ ͨͩ͠ ʹ͍ͭͯ B ͕෦ήʔϜ ʹ੍͓͍ͯࣗڧݾతͰ͋Δͱ ͷͯ͢ͷਅ෦ू߹ ʹର͠ ʹ͓͍ͯ݁ୗੑΛͭ͜ͱΛ͍͏ C ͕෦ήʔϜ ʹ͓͍ͯ݁ୗੑͰ͋Δͱ ʹ੍͓͍ͯࣗڧݾతͰ͋ͬͯ ͔͠ Λຬͨ͢Α͏ͳ ʹ͓͚Δ੍ࣗڧݾతͳ ͕ଘࡏ͠ͳ͍͜ͱΛ͍͏ x ∈ X i ∈ N xi G|xN{i} ∀yi ∈ Xi, ui(x) ≥ ui(yi, xN{i}) S ⊆ N 1 ≤ |S| m xS G|xNS S ⊆ N |S| = m xS G|xNS S T ⊊ S xT G|xNT xS G|xNS xS G|xNS ∀i ∈ S, ui(yS, xNS) ui(x) G|xNS yS ∈ XS ͕݁ୗੑΛͭͱ ͕ ʹ͓͍ͯ ݁ୗੑΛͭ͜ͱΛ͍͏ x ∈ X x G
- 6. ֬ఆతઓུͷఆٛ ֬ఆతઓུ ΛۭͰͳ͍ҙͷఏ ܞ Λҙʹ༩͑ΒΕͨઓུͷͱ͢Δ ͜ͷͱ͖ ͕ʹ͓͍ͯ֬ఆతઓུ Λͭͱ ͕ઓུ ͷͲͷਅ෦ू߹ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍ ͱ͖Λ͍͏ S ⊆ N x ∈ X S x yS ∈ XS yS S T ⊊ S (yS, xNS) N S T S T yS yT º ਓͰͷઓུ֬ఆతઓུ
- 7. ݁ୗੑφογϡ֬ͱߧۉఆతઓུͷؔఆཧূ໌ ݁ୗੑφογϡ֬ͱߧۉఆతઓུΛͨͳ͍ ߧۉ ઓུͷ ͕༩͑ΒΕ͍ͯΔ͜ͷͱ͖ҙͷ ʹ͍ͭͯ ͕෦ήʔϜ ʹ͓͍ͯ ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ ʹ͓͍ͯͰ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ͱ͘ʹ ͕ ʹ͓͍ͯ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ Ͱ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ x ∈ X S ⊆ N xS ∈ XS G|xNS T ⊆ S G|xNS x x G T ⊆ N x
- 8. ݁ୗੑφογϡ֬ͱߧۉఆతઓུͷؔఆཧ ݁ୗੑφογϡ֬ͱߧۉఆతઓུΛͨͳ͍ ߧۉ ઓུͷ ͕༩͑ΒΕ͍ͯΔ͜ͷͱ͖ҙͷ ʹ͍ͭͯ ͕෦ήʔϜ ʹ͓͍ͯ ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ ʹ͓͍ͯͰ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ͱ͘ʹ ͕ ʹ͓͍ͯ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ Ͱ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ؔ࿈ఆཧ ڧφογϡߧۉ݁ୗੑΛͭ ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ x ∈ X S ⊆ N xS ∈ XS G|xNS T ⊆ S G|xNS x x G T ⊆ N x x ∈ X S ⊆ N G|xNS
- 9. ڧφογϡͳߧۉΒ݁ୗੑΛͭূ໌ ݁ୗੑφογϡ֬ͱߧۉఆతઓུΛͨͳ͍ ߧۉ ઓུͷ ͕༩͑ΒΕ͍ͯΔ͜ͷͱ͖ҙͷ ʹ͍ͭͯ ͕෦ήʔϜ ʹ͓͍ͯ ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ ʹ͓͍ͯͰ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ͱ͘ʹ ͕ ʹ͓͍ͯ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ Ͱ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ؔ࿈ఆཧ ڧφογϡߧۉ݁ୗੑΛͭ ূ໌ ઓུͷ Λڧφογϡ͢ͱߧۉΔ͜ͷͱ͖ڧφογϡߧۉͷఆٛΑΓ ͲΜͳఏܞ Λߟ͑ͯʹ͓͍ͯઓུΛͨͳ͍ ઓུΛͨͳ͍ͷͳΒ ֬ఆతઓུͨͳ͍ Ώ͑ʹͲΜͳఏܞ Λߟ͑ͯ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍͜ͱΑΓ ఆཧ͔Β݁ୗੑφογϡ͋ͰߧۉΔ ূ໌ऴྃ x ∈ X S ⊆ N xS ∈ XS G|xNS T ⊆ S G|xNS x x G T ⊆ N x x ∈ X S ⊆ N x S ⊆ N x ڧφογϡߧۉͷఆٛ ઓུͷ ͕ڧφογϡ͋ͰߧۉΔͱɺ͍͔ͳΔఏܞ ʹ͓͍ͯઓུΛͨͳ͍͜ͱΛ͍͏ x ∈ X S ⊆ N x ֬ఆతઓུ ΛۭͰͳ͍ҙͷఏ ܞ Λҙʹ༩͑ΒΕͨઓུͷͱ͢Δ ͜ͷͱ͖ ͕ʹ͓͍ͯ֬ఆతઓུ Λͭͱ ͕ઓུ ͷͲͷਅ෦ू߹ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍ S ⊆ N x ∈ X S x yS ∈ XS yS S T ⊊ S (yS, xNS)
- 10. ݁ୗੑφογϡ֬ͱߧۉఆతઓུΛͨͳ͍ ߧۉ ઓུͷ ͕༩͑ΒΕ͍ͯΔ͜ͷͱ͖ҙͷ ʹ͍ͭͯ ͕෦ήʔϜ ʹ͓͍ͯ ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ ʹ͓͍ͯͰ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ͱ͘ʹ ͕ ʹ͓͍ͯ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ Ͱ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ ূ໌ ςΩετ͔ΒͷҾ༻ Λ ʹ͓͍ͯଞͷ͍͔ͳΔ֬ఆతઓུʹࢧ͞Εͳ͍֬ఆతઓུͱ͢Δ ͢Δͱ ͲͷΑ͏ͳ෦ू߹ ෦ήʔϜ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍ Ώ͑ʹ্ͷఆཧ͔Β ෦ήʔϜ ʹ͓͍ͯ݁ୗੑΛͭ ɾɾɾʁͳ͔ͥཧղͰ͖ͳ͍ͳͥʁ x ∈ X S ⊆ N xS ∈ XS G|xNS T ⊆ S G|xNS x x G T ⊆ N x x ∈ X S ⊆ N G|xNS yS ∈ XS S T ⊊ S G|xNS yS G|xNS ෦ήʔϜʹ͓͚Δ݁ୗੑφογϡߧۉͷଘࡏূ໌
- 11. ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ ূ໌ Λ ʹ͓͍ͯଞͷ͍͔ͳΔ֬ఆతઓུʹࢧ͞Εͳ͍֬ఆతઓུͱ͢Δ ఏܞΛߟ͑ͨ߹ͷࢧͷఆٛ ͕ Λࢧ͢Δͱ ҙͷ ʹରͯ͠ ͱͳΔͱ͖Λ͍͏ ͜ΕΛߟྀ͢Δͱ ҙͷ ʹରͯ͠ ͱͳΔΑ͏ͳ ֬ఆతઓུ ͕ଘࡏ͠ͳ͍͜ͱʹͳΔ x ∈ X S ⊆ N G|xNS yS ∈ XS S y′ S ∈ XS yS ∈ XS xNS ∈ XNS ∀i ∈ S, ui(y′ S, xNS) ui(yS, xNS) yS ∈ XS xNS ∈ XNS ∀i ∈ S, ui(y′ S, xNS) ui(yS, xNS) y′ S ∈ XS ෦ήʔϜʹ͓͚Δ݁ୗੑφογϡߧۉͷଘࡏূ໌
- 12. ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ ূ໌ Λ ʹ͓͍ͯଞͷ͍͔ͳΔ֬ఆతઓུʹࢧ͞Εͳ͍֬ఆతઓུͱ͢Δ ఏܞΛߟ͑ͨ߹ͷࢧͷఆٛ ͕ Λࢧ͢Δͱ ҙͷ ʹରͯ͠ ͱͳΔͱ͖Λ͍͏ ͜ΕΛߟྀ͢Δͱ ҙͷ ʹରͯ͠ ͱͳΔΑ͏ͳ ֬ఆతઓུ ͕ଘࡏ͠ͳ͍͜ͱʹͳΔ x ∈ X S ⊆ N G|xNS yS ∈ XS S y′ S ∈ XS yS ∈ XS xNS ∈ XNS ∀i ∈ S, ui(y′ S, xNS) ui(yS, xNS) yS ∈ XS xNS ∈ XNS ∀i ∈ S, ui(y′ S, xNS) ui(yS, xNS) y′ S ∈ XS ෦ήʔϜʹ͓͚Δ݁ୗੑφογϡߧۉͷଘࡏূ໌ N S yS xNS S T yT º N S y′ S xNS S T yT º S S yS y′ S Λࢧ͢ΔΑ͏ͳ ֬ఆతઓུ ͕ଘࡏ yS ∈ XS y′ S ∈ XS ֬ఆతઓུ ΛۭͰͳ͍ҙͷఏ ܞ Λҙʹ༩͑ΒΕͨઓུͷͱ͢Δ ͜ͷͱ͖ ͕ʹ͓͍ͯ֬ఆతઓུ Λͭͱ ͕ઓུ ͷͲͷਅ෦ू߹ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍ S ⊆ N x ∈ X S x yS ∈ XS yS S T ⊊ S (yS, xNS)
- 13. ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ ূ໌ Λ ʹ͓͍ͯଞͷ͍͔ͳΔ֬ఆతઓུʹࢧ͞Εͳ͍֬ఆతઓུͱ͢Δ ఏܞΛߟ͑ͨ߹ͷࢧͷఆٛ ͕ Λࢧ͢Δͱ ҙͷ ʹରͯ͠ ͱͳΔͱ͖Λ͍͏ ͜ΕΛߟྀ͢Δͱ ҙͷ ʹରͯ͠ ͱͳΔΑ͏ͳ ֬ఆతઓུ ͕ଘࡏ͠ͳ͍͜ͱʹͳΔ x ∈ X S ⊆ N G|xNS yS ∈ XS S y′ S ∈ XS yS ∈ XS xNS ∈ XNS ∀i ∈ S, ui(y′ S, xNS) ui(yS, xNS) yS ∈ XS xNS ∈ XNS ∀i ∈ S, ui(y′ S, xNS) ui(yS, xNS) y′ S ∈ XS ෦ήʔϜʹ͓͚Δ݁ୗੑφογϡߧۉͷଘࡏূ໌ N S yS xNS S T yT º N S y′ S xNS S T yT º S S yS y′ S Λࢧ͢ΔΑ͏ͳ ֬ఆతઓུ ͕ଘࡏ yS ∈ XS y′ S ∈ XS ֬ఆతઓུ ΛۭͰͳ͍ҙͷఏ ܞ Λҙʹ༩͑ΒΕͨઓུͷͱ͢Δ ͜ͷͱ͖ ͕ʹ͓͍ͯ֬ఆతઓུ Λͭͱ ͕ઓུ ͷͲͷਅ෦ू߹ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍ S ⊆ N x ∈ X S x yS ∈ XS yS S T ⊊ S (yS, xNS) Λࢧ͢ΔΑ͏ͳ ֬ఆతઓུ ͕ଘࡏ͠ͳ͍ͷͰ͋Ε ͜Ε෦ήʔϜ ʹ͓͍ͯ ͕֬ఆతઓུΛͨͳ͍͜ͱΛҙຯ͢Δ yS ∈ XS y′ S ∈ XS G|xNS S º
- 14. ݁ୗੑφογϡ֬ͱߧۉఆతઓུΛͨͳ͍ ߧۉ ઓུͷ ͕༩͑ΒΕ͍ͯΔ͜ͷͱ͖ҙͷ ʹ͍ͭͯ ͕෦ήʔϜ ʹ͓͍ͯ ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ ʹ͓͍ͯͰ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ͱ͘ʹ ͕ ʹ͓͍ͯ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ Ͱ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ ূ໌ Λ ʹ͓͍ͯଞͷ͍͔ͳΔ֬ఆతઓུʹࢧ͞Εͳ͍֬ఆతઓུͱ͢Δ ͜Ε෦ήʔϜ ʹ͓͍ͯ ͕ઓུΛͨͳ͍͜ͱΛҙຯ͢Δ ·ͨ ͲͷΑ͏ͳ෦ू߹ ෦ήʔϜ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍ ͜Ε֬ఆతઓུͷఆٛΑΓ͑ݴΔ x ∈ X S ⊆ N xS ∈ XS G|xNS T ⊆ S G|xNS x x G T ⊆ N x x ∈ X S ⊆ N G|xNS yS ∈ XS S G|xNS S T ⊊ S G|xNS ෦ήʔϜʹ͓͚Δ݁ୗੑφογϡߧۉͷଘࡏূ໌ ֬ఆతઓུ ΛۭͰͳ͍ҙͷఏ ܞ Λҙʹ༩͑ΒΕͨઓུͷͱ͢Δ ͜ͷͱ͖ ͕ʹ͓͍ͯ֬ఆతઓུ Λͭͱ ͕ઓུ ͷͲͷਅ෦ू߹ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍ S ⊆ N x ∈ X S x yS ∈ XS yS S T ⊊ S (yS, xNS)
- 15. ݁ୗੑφογϡ֬ͱߧۉఆతઓུΛͨͳ͍ ߧۉ ઓུͷ ͕༩͑ΒΕ͍ͯΔ͜ͷͱ͖ҙͷ ʹ͍ͭͯ ͕෦ήʔϜ ʹ͓͍ͯ ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ ʹ͓͍ͯͰ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ͱ͘ʹ ͕ ʹ͓͍ͯ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ Ͱ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ ূ໌ Λ ʹ͓͍ͯଞͷ͍͔ͳΔ֬ఆతઓུʹࢧ͞Εͳ͍֬ఆతઓུͱ͢Δ ͜Ε෦ήʔϜ ʹ͓͍ͯ ͕ઓུΛͨͳ͍͜ͱΛҙຯ͢Δ ·ͨ ͲͷΑ͏ͳ෦ू߹ ෦ήʔϜ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍ ͜Ε֬ఆతઓུͷఆٛΑΓ͑ݴΔ ͜ΕΒΑΓ ͲͷΑ͏ͳ෦ू߹ ෦ήʔϜ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍ͨΊ ্هఆཧ͔Β ෦ήʔϜ ʹ͓͍ͯ ݁ୗੑΛͭ ূ໌ऴྃ x ∈ X S ⊆ N xS ∈ XS G|xNS T ⊆ S G|xNS x x G T ⊆ N x x ∈ X S ⊆ N G|xNS yS ∈ XS S G|xNS S T ⊊ S G|xNS T ⊆ S G|xNS yS ∈ XS G|xNS ෦ήʔϜʹ͓͚Δ݁ୗੑφογϡߧۉͷଘࡏূ໌
- 16. ݁ୗੑφογϡ֬ͱߧۉఆతઓུͷؔఆཧ ݁ୗੑφογϡ֬ͱߧۉఆతઓུΛͨͳ͍ ߧۉ ઓུͷ ͕༩͑ΒΕ͍ͯΔ͜ͷͱ͖ҙͷ ʹ͍ͭͯ ͕෦ήʔϜ ʹ͓͍ͯ ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ ʹ͓͍ͯͰ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ͱ͘ʹ ͕ ʹ͓͍ͯ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ Ͱ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ؔ࿈ఆཧ ڧφογϡߧۉ݁ୗੑΛͭ ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ ؔ࿈ఆཧ ʹґଘͨ͠Ͱܗͷ ෦ήʔϜ ʹ͓͚Δ݁ୗੑφογϡ͋ͰߧۉΔ x ∈ X S ⊆ N xS ∈ XS G|xNS T ⊆ S G|xNS x x G T ⊆ N x x ∈ X S ⊆ N G|xNS x ∈ X G|xNS
- 17. ݁ୗੑφογϡ֬ͱߧۉఆతઓུͷؔఆཧ ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ x ∈ X S ⊆ N G|xNS AʘB ̌ AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1 AʘB AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1 ͷઓུ S = {A, B} ͷ֬ఆతઓུ S = {A, B} ͷͱ͖෦ήʔϜ ʹ͓͍ͯ ݁ୗੑΛͭ S = {A, B} G|xC y{A,B} = (a2, b2)
- 18. ݁ୗੑφογϡ֬ͱߧۉఆతઓུͷؔఆཧ ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ x ∈ X S ⊆ N G|xNS AʘB AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1 AʘB AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1 ͷઓུ S = {C} ͷ֬ఆతઓུ S = {C} ͷͱ͖෦ήʔϜ ʹ͓͍ͯ ݁ୗੑΛͭ S = {C} G|x{A,B} y{C} = (c2)
- 19. ݁ୗੑφογϡ֬ͱߧۉఆతઓུͷؔఆཧ ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ x ∈ X S ⊆ N G|xNS AʘB ̌ AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1 શମͷήʔϜʹ͓͚Δ݁ୗੑφογϡߧۉͳ͍ AʘB ̌ AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1
- 20. ݁ୗੑφογϡ֬ͱߧۉఆతઓུͷؔఆཧ ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ x ∈ X S ⊆ N G|xNS AʘB ̌ AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1 શମͷήʔϜʹ͓͚Δ݁ୗੑφογϡߧۉͳ͍ AʘB ̌ AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1
- 21. ݁ୗੑφογϡ֬ͱߧۉఆతઓུͷؔఆཧ ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ x ∈ X S ⊆ N G|xNS AʘB ̌ AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1 શମͷήʔϜʹ͓͚Δ݁ୗੑφογϡߧۉͳ͍ AʘB ̌ AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1
- 22. ݁ୗੑφογϡ֬ͱߧۉఆతઓུͷؔఆཧ ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ x ∈ X S ⊆ N G|xNS AʘB ̌ AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1 શମͷήʔϜʹ͓͚Δ݁ୗੑφογϡߧۉͳ͍ AʘB ̌ AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1
- 23. ݁ୗੑφογϡ֬ͱߧۉఆతઓུͷؔఆཧ ؔ࿈ఆཧ ઓུͷ ʹ͓͍ͯ ֬ఆతઓུΛͭͱ͢Δ ͜ͷͱ͖ ෦ήʔϜ ʹ͓͍ͯ݁ୗੑφογϡ͕ߧۉଘࡏ͢Δ x ∈ X S ⊆ N G|xNS AʘB AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1 AʘB AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1 ͷઓུ S = N ͷ֬ఆతઓུ S = N ීวతͳ݁ୗੑφογϡߧۉͷଘࡏఆཧʁΒΕ͍ͯͳ͍
- 25. ෦ήʔϜ͕།ҰͷφογϡߧۉΛͭͱ͖ ఆཧ ઓུͷ ͕༩͑ΒΕͨͱ͖ ͷۭͰͳ͍ҙͷ෦ू߹ ʹ͍ͭͯ ෦ήʔϜ །ҰͷφογϡߧۉΛͭͱ͢Δͱ ͷ݁ୗੑφογϡ͋ͰߧۉΔ x ∈ X N S ⊆ N G|xNS x G
- 26. ෦ήʔϜ͕།ҰͷφογϡߧۉΛͭͱ͖ ఆཧ ઓུͷ ͕༩͑ΒΕͨͱ͖ ͷۭͰͳ͍ҙͷ෦ू߹ ʹ͍ͭͯ ෦ήʔϜ །ҰͷφογϡߧۉΛͭͱ͢Δͱ ͷ݁ୗੑφογϡ͋ͰߧۉΔ ূ໌ ͷͱ͖Λߟ͑Δ෦ήʔϜ Ͱ͋Δ͔Β ԾఆΑΓ ͷ།Ұͷφογϡ͋ͰߧۉΔ ·ͨ ͍͔ͳΔ෦ू߹ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍ ࣮ࡍ Λߟ͑Δ͜Ε͕ʹ͓͚Δઓུͱ͢Δ Ծఆ͔Β ෦ήʔϜ ʹ͓͚Δ།Ұͷφογϡ͋ͰߧۉΔͨΊ φογϡͰߧۉͳ͍ ͢Δͱ ͋ΔϓϨΠϠʔ ʹ͍ͭͯ ʹ͓͍ͯઓུ ਓఏͳܞͷͰ ͢ͳΘͪ֬ఆతઓུΛͭ ֬ఆతͰͳ͍ Ώ͑ʹ ͍͔ͳΔ෦ू߹ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍ͷ͔ͩΒ ݁ୗੑφογϡ͋ͰߧۉΔ ʢূ໌ऴྃ x ∈ X N S ⊆ N G|xNS x G S = N G|xNN = G x ∈ X G S ⊆ N x yS ≠ xS x xS G|xNS yS i ∈ S (yS, xNS) yS S ⊆ N x x N x N x S S yS i y′ {i}
- 28. See you next timeʂ ήʔϜཧ /95 ࣍ճɿ७ਮަήʔϜ࠶ͼ