ゲーム理論BASIC 第14回 -展開形ゲーム定義-
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ゲーム理論BASIC 第14回 -展開形ゲーム定義-
- 3. 1. ల։ܗήʔϜ (1)ల։ܗήʔϜͷఆٛɿ (A) : ༗ (B) : ϓϨΠϠʔׂ (C) : ۮવख൪ͷ֬ (D) : ใׂ (E) : རಘؔ Γ = (K, P, p, U, h) K P p U h ࣍
- 5. ల։ܗήʔϜͷఆٛ લճ·Ͱ ઓུܗήʔϜͰಉ࣌ʹҙࢥܾఆ ࠓճ͔Β ߦಈܾఆʹॱং͕͋Δ߹ͷੳΛߦ͏ ϙΠϯτ ϓϨΠϠʔͷઓུʹߦಈܭը ७ઓུɾࠞ߹ઓུɾߦಈઓུ ֤ϓϨΠϠʔ͕ͭใΛදݱՄೳ ใू߹ AʘB ൿ ࣗന ൿ ࣗന नਓͷδϨϯϚ 01 02 03 (8, 8) (10, 0) (0, 10) (6, 6) A B B a1 1 a1 2 a2 1 a2 2 a2 3 a2 4 ల։ܗήʔϜ
- 6. ల։ܗήʔϜͷఆٛɿ Γ = (K, P, p, U, h) ల։ܗήʔϜ ϓϨΠϠʔͷू߹Λ ͱ͢Δ Λల։ܗήʔϜͱͿݺ ༗ ϓϨΠϠʔׂ ۮવख൪ͷ֬ ใׂ རಘؔ N = {1,2,3,⋯, n} Γ = (K, P, p, U, h) K = (V, E) P = [P0 , P1 , ⋯, Pn ] p U = [U0 , U1 , ⋯, Un ] h 01 02 03 06 04 05 07 w1 ࣗવ P1 P1 P2 P2 P2 P2 w2 w3 w4 w5 w6 w7 w8 a0 1 a0 2 a1 1 a1 2 a1 3 a1 4 a2 1 a2 2 a2 7 a2 8
- 7. ༗ K ༗ K = (V, E) 01 02 03 06 04 05 07 w1 ࣗવ P1 P1 P2 P2 P2 P2 w2 w3 w4 w5 w6 w7 w8 ͷू߹ ࢬͷू߹Ͱ͋Γ ͷͱ͖Λͷલͷ Λͷޙͷͱ ͼݺ B લͷΛͨͳ͍ͨͩҰͭଘࡏ͠ ࢝ͱͿݺ C ޙͷΛͨͳ͍ ऴ·ͨͱ Ϳݺ D ͷू߹Λ ͱ͢Δͱ Λख൪ͱ Ϳݺ E ҙͷख൪ ʹରͯ͠ ͦͷख൪ͱޙͷͱ݁ͿࢬΛબࢶ·ͨߦಈͱͼݺ બࢶશମΛ Ͱද͢ ҙͷऴ ʹର͔ͯ࢝͠ΒͷύεΛϓϨΠͱͿݺ ॳ͔ظΒऴ·Ͱͷύεͨͩͭ V E E = {e = xy|x, y ∈ V} xy ∈ E x y y x W X = VW x ∈ X A(x) w ∈ W ࢝ ख൪ ऴ બࢶ a0 1 a0 2 a1 1 a1 2 ·ͰͷϓϨΠ w6 ·ͰͷϓϨΠ w6 ·ͰͷϓϨΠ w6
- 8. ϓϨΠϠʔׂ P ϓϨΠϠʔׂ P = [P0 , P1 , ⋯, Pn ] 01 02 03 06 04 05 07 w1 ࣗવ P1 P1 P2 P2 P2 P2 w2 w3 w4 w5 w6 w7 w8 Ҏ֎ͷఴ͑ࣈϓϨΠϠʔΛද͠ ҙͷͭͷ ʹରͯ͠ ۮવख൪ͷू߹ͱ͢Δ ۮવख൪ͱ ϓϨΠϠʔͷҙࢥʹແؔʹܾఆ͢Δ X = P0 ∪ P1 ∪ ⋯ ∪ Pn Pi , Pj (i ≠ j) Pi ∩ Pj = ∅ Pi ≠ ∅ i = 1,2,⋯n P0 P0 = {01} P1 = {02,03} P2 = {04,05,06,07} a0 1 a0 2 a1 1 a1 2 P0
- 9. ۮવख൪ͷ֬ p ۮવख൪ͷ֬ p 01 02 03 06 04 05 07 w1 ࣗવ P1 P1 P2 P2 P2 P2 w2 w3 w4 w5 w6 w7 w8 ͯ͢ͷۮવख൪ ʹରͯ͠ ্ͷબࢶͷू߹ ͷ֬ ͕ఆΊΒΕ͍ͯΔ બࢶ ʹରͯͦ͠Εʹ༩͑Δ֬Λ ͱ͢Δͱ ͕Γཱͭ ۮવख൪ͷ֬ͷΛ ͱ͢Δ x ∈ P0 x A(x) px e ∈ A(x) px(e) ∑ e∈A(x) px(e) = 1, px(e) ≥ 0 p p = {px |x ∈ P0 } p01 (a0 1) = 1 2 a1 1 a1 2 P0 p01 (a0 2) = 1 2 a0 1 a0 2
- 10. ใׂ U ใׂ U = [U0 , U1 , ⋯, Un ] 01 02 03 06 04 05 07 w1 ࣗવ P1 P1 P2 P2 P2 P2 w2 w3 w4 w5 w6 w7 w8 ϓϨΠϠʔ͕ࣗͷख൪ʹ͓͍ͯ ͦΕҎલͷήʔϜʹରͯ͠ͲΜͳใΛ͔ͭɺΛදݱ Λใू߹ͱ Ϳݺ ͷҙͷͭͷใू߹ ʹରͯ͠ Pi = ⋃ u∈Ui u u Ui u, u′ u ∩ u′ = ∅ U0 = {u0 1} = {{01}} P0 = ⋃ u∈U0 u = {01} U1 = {u1 1} = {{02,03}} P1 = ⋃ u∈U1 u = {02,03} U2 = {u2 1, u2 2} = {{04,05}, {06,07}} P2 = ⋃ u∈U2 u = {04,05,06,07} a0 1 a0 2 a1 1 a1 2 u1 1 u2 1 u2 2 u0 1
- 11. ใׂ U = [U0 , U1 , ⋯, Un ] 01 02 03 06 04 05 07 w1 ࣗવ P1 P1 P2 P2 P2 P2 w2 w3 w4 w5 w6 w7 w8 ϓϨΠϠʔ͕ࣗͷख൪ʹ͓͍ͯ ͦΕҎલͷήʔϜʹରͯ͠ͲΜͳใΛ͔ͭɺΛදݱ Λใू߹ͱ Ϳݺ ͷҙͷͭͷใू߹ ʹରͯ͠ Pi = ⋃ u∈Ui u u Ui u, u′ u ∩ u′ = ∅ U0 = {u0 1} = {{01}} P0 = ⋃ u∈U0 u = {01} U1 = {u1 1} = {{02,03}} P1 = ⋃ u∈U1 u = {02,03} U2 = {u2 1, u2 2} = {{04,05}, {06,07}} P2 = ⋃ u∈U2 u = {04,05,06,07} a0 1 a0 2 a1 1 a1 2 u1 1 u2 1 u2 2 u0 1 ϓϨΠϠʔ ใू߹ ʹ౸ୡͨ͜͠ͱΘ͔Δ͕ ͱ ͲͪΒͷʹ͍Δ͔Θ͔Βͳ͍ ࣗવ͕ͲͪΒͷબΛબΜͩͷ͔Θ͔Βͳ͍ u1 1 02 03 ใׂ U
- 12. ใׂ U = [U0 , U1 , ⋯, Un ] 01 02 03 06 04 05 07 w1 ࣗવ P1 P1 P2 P2 P2 P2 w2 w3 w4 w5 w6 w7 w8 ϓϨΠϠʔ͕ࣗͷख൪ʹ͓͍ͯ ͦΕҎલͷήʔϜʹରͯ͠ͲΜͳใΛ͔ͭɺΛදݱ Λใू߹ͱ Ϳݺ ͷҙͷͭͷใू߹ ʹରͯ͠ Pi = ⋃ u∈Ui u u Ui u, u′ u ∩ u′ = ∅ U0 = {u0 1} = {{01}} P0 = ⋃ u∈U0 u = {01} U1 = {u1 1} = {{02,03}} P1 = ⋃ u∈U1 u = {02,03} U2 = {u2 1, u2 2} = {{04,05}, {06,07}} P2 = ⋃ u∈U2 u = {04,05,06,07} a0 1 a0 2 a1 1 a1 2 u1 1 u2 1 u2 2 u0 1 ϓϨΠϠʔ ใू߹ ʹ౸ୡͨ͜͠ͱΘ͔Δ͕ ͱ ͲͪΒͷʹ͍Δ͔Θ͔Βͳ͍ u2 1 04 05 ใׂ U
- 13. ใׂ U = [U0 , U1 , ⋯, Un ] ల։ܗήʔϜͷఆٛ 01 02 03 06 04 05 07 w1 ࣗવ P1 P1 P2 P2 P2 P2 w2 w3 w4 w5 w6 w7 w8 ϓϨΠϠʔ͕ࣗͷख൪ʹ͓͍ͯ ͦΕҎલͷήʔϜʹରͯ͠ͲΜͳใΛ͔ͭɺΛදݱ Λใू߹ͱ Ϳݺ ͷҙͷͭͷใू߹ ʹରͯ͠ ಉ͡ใू߹ʹ·ؚΕΔख൪ಉ͡બࢶΛͭ ಉ͡ҙࢥܾఆʹ໘͢Δ Pi = ⋃ u∈Ui u u Ui u, u′ u ∩ u′ = ∅ A(u) = {e ∈ A(x)|x ∈ u} a0 1 a0 2 a1 1 a1 2 u1 1 u2 1 u2 2 u0 1 a1 1 a1 2 ใׂ U
- 14. ใׂ U = [U0 , U1 , ⋯, Un ] 01 02 03 P1 ϓϨΠϠʔ͕ࣗͷख൪ʹ͓͍ͯ ͦΕҎલͷήʔϜʹରͯ͠ͲΜͳใΛ͔ͭɺΛදݱ Λใू߹ͱ Ϳݺ ͷҙͷͭͷใू߹ ʹରͯ͠ ಉ͡ใू߹ʹ·ؚΕΔख൪ಉ͡બࢶΛͭ ಉ͡ҙࢥܾఆʹ໘͢Δ ใू߹ಉ͡ϓϨΠʹճҎ্ަΘΒͳ͍ ҙࢥܾఆͨ͠ࣄ࣮ΛΕͳ͍ Pi = ⋃ u∈Ui u u Ui u, u′ u ∩ u′ = ∅ A(u) = {e ∈ A(x)|x ∈ u} u1 1 0x 0y w1 02 ใׂ U
- 15. ใׂ U = [U0 , U1 , ⋯, Un ] 01 02 03 06 04 05 07 w1 ࣗવ P1 P1 P2 P2 P2 P2 w2 w3 w4 w5 w6 w7 w8 ϓϨΠϠʔ͕ࣗͷख൪ʹ͓͍ͯ ͦΕҎલͷήʔϜʹରͯ͠ͲΜͳใΛ͔ͭɺΛදݱ ʹ·ؚΕΔใू߹ ͨͩͭͷۮવख൪ΛؚΉ U0 u a0 1 a0 2 a1 1 a1 2 u1 1 u2 1 u2 2 u0 1 a1 1 a1 2 ใׂ U
- 16. རಘؔ h རಘؔ h 01 02 03 06 04 05 07 w1 ࣗવ P1 P1 P2 P2 P2 P2 w2 w3 w4 w5 w6 w7 w8 ήʔϜͷ ͷҙͷऴ ʹରͯ͠ རಘϕΫτϧ ΛରԠͤ͞Δ ୈΛϓϨΠϠʔͷརಘͱ͢Δ ɾɾɾ K w ∈ W h(w) = (h1 (w), h2 (w), ⋯, hn (w)) i i h(w1) = (h1 (w1), h2 (w1)) = (8, 8) h(w2) = (h1 (w2), h2 (w2)) = (10, 0) h(w7) = (h1 (w7), h2 (w7)) = (2, 0) h(w8) = (h1 (w8), h2 (w8)) = (4, 4) a0 1 a0 2 a1 1 a1 2 u1 1 u2 1 u2 2 u0 1 a1 1 a1 2 (8, 8) (10, 0) (0, 10) (6, 6) (−6, − 6) (0, 2) (2, 0) (4, 4)
- 17. ల։ܗήʔϜͷఆٛ ల։ܗήʔϜ ϓϨΠϠʔͷू߹Λ ͱ͢Δ Λల։ܗήʔϜͱͿݺ ༗ ϓϨΠϠʔׂ ۮવख൪ͷ֬ ใׂ རಘؔ N = {1,2,3,⋯, n} Γ = (K, P, p, U, h) K = (V, E) P = [P0 , P1 , ⋯, Pn ] p U = [U0 , U1 , ⋯, Un ] h u1 1 u2 1 u2 2 u0 1 01 02 03 06 04 05 07 w1 ࣗવ P1 P1 P2 P2 P2 P2 w2 w3 w4 w5 w6 w7 w8 a0 1 a0 2 a1 1 a1 2 a2 1 a2 2 (8, 8) (10, 0) (0, 10) (6, 6) (−6, − 6) (0, 2) (2, 0) (4, 4) a1 1 a1 2 a2 3 a2 4
- 18. ల։ܗήʔϜͷఆٛ ల։ܗήʔϜ ϓϨΠϠʔͷू߹Λ ͱ͢Δ Λల։ܗήʔϜͱͿݺ ༗ ϓϨΠϠʔׂ ۮવख൪ͷ֬ ใׂ རಘؔ N = {1,2,3,⋯, n} Γ = (K, P, p, U, h) K = (V, E) P = [P0 , P1 , ⋯, Pn ] p U = [U0 , U1 , ⋯, Un ] h u1 1 u2 1 u2 2 u0 1 N P1 P1 P2 P2 P2 P2 a0 1 a0 2 a1 1 a1 2 a2 1 a2 2 a2 3 a2 4 (8, 8) (10, 0) (0, 10) (6, 6) (−6, − 6) (0, 2) (2, 0) (4, 4) a1 1 a1 2