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ゲーム理論NEXT 戦略形協力ゲーム第7回 -確定的離反戦略-
1. The document discusses game theory concepts related to coalitional games including subsets, characteristic functions, and improvements. 2. It also covers the core concept which examines whether any group has an incentive to break off on its own given the current allocation. 3. Examples are provided to illustrate coalitional games and the core including settings with 3 players and various subsets.
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ゲーム理論NEXT 戦略形協力ゲーム第7回 -確定的離反戦略-
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- 5. ʢ෮शʣ෦ήʔϜ ෦ήʔϜ ઓུͷ ͱ ఏܞ ʹରͯ͠ ʹΑͬͯ Λݻఆͯ͠ಘΒΕΔ෦ήʔϜ Λද͢ ͨͩ͠ ͷ߹ݩͷήʔϜΛද͢ͱ͢Δ ྫ Λݻఆ x ∈ X S ⊆ N G|xNS xNS G′ = (S, {Xi}i∈S, {ui}i∈S) S = N N = {A, B, C} {a2, b2, c1} ∈ X S = {A, C} xN{A,C} = xB = b2 AʘB AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 $͕ c2 a1
- 6. ʢ෮शʣ݁ୗੑͷఆٛ ݁ୗੑͱ੍ࣗڧݾత Λҙʹ༩͑ΒΕͨઓུͷͱ͢Δ ֤ ʹ͍͕ͭͯ෦ήʔϜ 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
- 7. ֬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
- 8. ֬ఆతઓུͷఆٛ ֬ఆతઓུ ΛۭͰͳ͍ҙͷఏ ܞ Λҙʹ༩͑ΒΕͨઓུͷͱ͢Δ ͜ͷͱ͖ ͕ʹ͓͍ͯ֬ఆతઓུ Λͭͱ ͕ઓུ ͷͲͷਅ෦ू߹ ʹ͓͍ͯ֬ఆతઓུΛͨͳ͍ ͱ͖Λ͍͏ ͷͱ͖͜ͷਅ෦ू߹ۭू߹Ͱ͋Γɺ֬ఆతઓུΛͨͳ͍ Ώ͑ʹʹ͓͚Δઓུ Λͭͱ͢Ε֬ఆతͰ͋Δ ͷͱ͖͜ͷਅ෦ू߹ Ͱ͋Δ Ώ͑ʹʹ͓͚Δઓུ Λͪ ͕֬ఆతઓུΛͨͳ͍ͷͰ͋Ε ֬ఆతͰ͋Δ ͜ͷΑ͏ʹ࠶ؼతʹܾ·Δ S ⊆ N x ∈ X S x yS ∈ XS yS S T ⊊ S (yS, xNS) S = {i} x y{i} S = {i, j} {i} {j} x y{i,j} {i} {j} |S| 1 ਓͰͷઓུ֬ఆతઓུ Ͱఆٛ͞Ε͍ͯΔ S = {i}
- 9. ݁ୗੑφογϡͱߧۉͷؔ ݁ୗੑ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
- 10. ݁ୗੑφογϡͱߧۉͷؔ ݁ୗੑφογϡ֬ͱߧۉఆతઓུΛͨͳ͍ ߧۉ ઓུͷ ͕༩͑ΒΕ͍ͯΔ͜ͷͱ͖ҙͷ ʹ͍ͭͯ ͕෦ήʔϜ ʹ͓͍ͯ ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ ʹ͓͍ͯͰ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ͱ͘ʹ ͕ ʹ͓͍ͯ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ Ͱ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ݁ୗੑφογϡߧۉ ʹ͍ͭͯߟ͑Δ ෦ఏܞ ʹ͍ͭͯߟ͑Δ ʹ͍ͭͯઓུΛͨͳ͍͔Β ֬ఆతઓུΛͨͳ͍ ֬ఆతઓུΛͨͳ͍ x ∈ X S ⊆ N xS ∈ XS G|xNS T ⊆ S G|xNS x x G T ⊆ N x x = (a2, b2, c1) {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C} {A} {B}{C} AʘB AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 a1 $͕ c2
- 11. ݁ୗੑφογϡͱߧۉͷؔ ݁ୗੑφογϡ֬ͱߧۉఆతઓུΛͨͳ͍ ߧۉ ઓུͷ ͕༩͑ΒΕ͍ͯΔ͜ͷͱ͖ҙͷ ʹ͍ͭͯ ͕෦ήʔϜ ʹ͓͍ͯ ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ ʹ͓͍ͯͰ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ͱ͘ʹ ͕ ʹ͓͍ͯ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ Ͱ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ݁ୗੑφογϡߧۉ ʹ͍ͭͯߟ͑Δ ʹ͍ͭͯઓུΛͨͳ͍͔Β ֬ఆతઓུΛͨͳ͍ ֬ఆతઓུΛͨͳ͍ ʹ͍ͭͯઓུΛͨͳ͍͔Β ֬ఆతઓུΛͨͳ͍ x ∈ X S ⊆ N xS ∈ XS G|xNS T ⊆ S G|xNS x x G T ⊆ N x x = (a2, b2, c1) {A, B} {A, C}{B, C} {A, B, C} AʘB AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 a1 $͕ c2
- 12. ݁ୗੑφογϡͱߧۉͷؔ ݁ୗੑφογϡ֬ͱߧۉఆతઓུΛͨͳ͍ ߧۉ ઓུͷ ͕༩͑ΒΕ͍ͯΔ͜ͷͱ͖ҙͷ ʹ͍ͭͯ ͕෦ήʔϜ ʹ͓͍ͯ ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ ʹ͓͍ͯͰ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ͱ͘ʹ ͕ ʹ͓͍ͯ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ Ͱ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ݁ୗੑφογϡͰߧۉͳ͍ ʹ͍ͭͯߟ͑Δ ෦ఏܞ ʹ͍ͭͯߟ͑Δ ʹ͍ͭͯઓུΛͨͳ͍͔Β ֬ఆతઓུΛͨͳ͍ ֬ఆతઓུΛͨͳ͍ x ∈ X S ⊆ N xS ∈ XS G|xNS T ⊆ S G|xNS x x G T ⊆ N x x = (a1, b1, c1) {A}, {B}, {C}, {A, B}, {A, C}, {B, C}, {A, B, C} {A} {B}{C} AʘB AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 a1 $͕ c2
- 13. ݁ୗੑφογϡͱߧۉͷؔ ݁ୗੑφογϡ֬ͱߧۉఆతઓུΛͨͳ͍ ߧۉ ઓུͷ ͕༩͑ΒΕ͍ͯΔ͜ͷͱ͖ҙͷ ʹ͍ͭͯ ͕෦ήʔϜ ʹ͓͍ͯ ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ ʹ͓͍ͯͰ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ͱ͘ʹ ͕ ʹ͓͍ͯ݁ୗੑφογϡ͋ͰߧۉΔͱ ҙͷ෦ఏܞ ͕ Ͱ֬ఆతઓུΛͨͳ͍͜ͱͰ͋Δ ݁ୗੑφογϡͰߧۉͳ͍ ʹ͍ͭͯߟ͑Δ ʹ͍ͭͯઓུ Λͭ͞Βʹ ʹର͠ ͱʹઓུΛͨͳ͍ ͭ·Γ֬ఆతઓུΛͨͳ͍ Ώ͑ʹઓུ ֬ఆతͰ͋Δ x ∈ X S ⊆ N xS ∈ XS G|xNS T ⊆ S G|xNS x x G T ⊆ N x x = (a1, b1, c1) {A, B} y{A,B} = (a2, b2) x′ = (a2, b2, c1) {A} {B} y{A,B} = (a2, b2) AʘB AʘB $͕ c1 a1 a2 a2 b1 b2 b1 b2 a1 $͕ c2 ֬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) y{A,B} = (a2, b2) º º
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- 15. See you nexttimeʂ ήʔϜཧ /95 ࣍ճɿ֬ఆతઓུͱ݁ୗੑφογϡߧۉͷؔূ໌