ゲーム理論BASIC 第25回補足 -シンプルポーカーでの完全ベイジアン均衡-
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This document describes a simple poker game between two players. It shows the card hands and bets of each player over multiple rounds of the game. Strategies like calling and folding are used. Mathematical equations are also presented, analyzing the game theoretically in terms of probabilities and expected values for the players in each situation.
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ゲーム理論BASIC 第25回補足 -シンプルポーカーでの完全ベイジアン均衡-
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- 29. ̍ʘ $BMM ӕͩ 'PME ߱ , , ,, ̍ ʘ $BMM ӕͩ 'PME ߱ , ̍ , ʘ $BMM ӕͩ 'PME ߱ , ֬ͰΧʔυΛΓ·͢ ϕΠδΞϯήʔϜ
- 30. ̍ʘ $BMM ӕͩ 'PME ߱ , , ,, ̍ ʘ $BMM ӕͩ 'PME ߱ , ̍ , ʘ $BMM ӕͩ 'PME ߱ , ֬ͰΧʔυΛΓ·͢ ϕΠδΞϯήʔϜ ࢧ͞ΕΔ ࢧ͞ΕΔ
- 31. ̍ʘ $BMM ӕͩ 'PME ߱ , 1/3 2/3 2/3 1/3 ϓϨΠϠʔͷϚοΫεϛχઓུ F1 (p, q) = pq × 0 + (1 − p)q × 1/2 + p(1 − q) × 1 + (1 − p)(1 − q) × 0 = q 2 (1 − 3p) + p ϓϨΠϠʔͷϛχϚοΫεઓུ F1 (p, q) = p 2 (2 − 3q) + q 2 p 1 3 1 3 0 1 q 2 3 1 3 0 1 Fi Fi ϚοΫεϛχ ϛχϚοΫε ϓϨΠϠʔΛಘ͍ͯΕඞͣʮΛએݴʯ͠ ,Λಘ͍ͯΕ ͰʮͰ͋ΔͱӕΛͭ͘ʯͱ͍͏͜ͱʹͳΔ ϓϨΠϠʔͰ$BMM ର߅ ͠ ͰήʔϜ͔Β߱ΓΔ ֤ϓϨΠϠʔ͕͚͓ʹߧۉΔઓུΛͱΔͱ͢Ε ˎϓϨΠϠʔͷظརಘ ˎϓϨΠϠʔͷظརಘ ͱͳΓ ͜ͷήʔϜʹ͓͍ͯϓϨΠϠʔͷํ͕༗ར
- 32. ήʔϜ 1MBZFS , 1MBZFS , $BMM 'PME (−1, + 1) / 1MBZFS 1 2 1 2 $BMM 'PME (+2, − 2) (+1, − 1) (−2, + 2) (+1, − 1) (−1, + 1) ,
- 33. ੳ 1MBZFS , 1MBZFS , (−1, + 1) / 1MBZFS 1 2 1 2 $BMM 'PME (+2, − 2) (+1, − 1) (−2, + 2) (+1, − 1) (−1, + 1) , $BMM 'PME ϓϨΠϠʔͷใू߹ ্ͷ৴೦Λ ͱ͢Δ ϓϨΠϠʔͷߦಈઓུΛ ͱ͢Δ ϓϨΠϠʔͷใू߹ ʹ͓͚ΔظརಘΛٻΊΔ ϓϨΠϠʔ͜ΕΛ࠷େʹ͢Δ͜ͱΛߟ͑Δ u2 1 (r, 1 − r) (b2 , 1 − b2 ) u2 1 H2 (u2 1, (r,1 − r), (b1,u2 1, b2,u2 1)) = r{−2b2 + (−1)(1 − b2 )} + (1 − r){2b2 + (−1)(1 − b2 )} = (3 − 4r)b2 − 1 r 1 − r b2 1 − b2 b2 1 − b2 u2 1 u1 1 u1 2 b2 r 1 1 0 3 4 ͷʹΑΓ࠷దͳߦಈ ͕มԽ r b2
- 34. ੳ 1MBZFS , 1MBZFS , (−1, + 1) / 1MBZFS 1 2 1 2 $BMM 'PME (+2, − 2) (+1, − 1) (−2, + 2) (+1, − 1) (−1, + 1) , $BMM 'PME ϓϨΠϠʔͷߦಈઓུΛ ͱ͢Δ ϓϨΠϠʔͷใू߹ ʹ͓͚ΔظརಘΛٻΊΔ ϓϨΠϠʔ͜ΕΛ࠷େʹ͢Δ͜ͱΛߟ͑Δ͕ ϓϨΠϠʔͷ ʹରͯ͠ ͱ͢Δͷ͕࠷ద ((b1 1, 1 − b1 1), (b1 2, 1 − b1 2)) u1 1 H1 (u1 1, (r,1 − r), (b1,u1 1, b2,u1 1)) = (−1)b1 1 + 2(1 − b1 1)b2 + 1(1 − b1 1)(1 − b2 ) = − (b2 + 2)b1 1 + b2 + 1 b2 , 0 ≤ b2 ≤ 1 b1 1 = 0 r 1 − r b2 1 − b2 b2 1 − b2 u2 1 u1 1 u1 2 b1 1 1 − b1 1 b1 2 1 − b1 2
- 35. ੳ 1MBZFS , 1MBZFS , (−1, + 1) / 1MBZFS 1 2 1 2 $BMM 'PME (+2, − 2) (+1, − 1) (−2, + 2) (+1, − 1) (−1, + 1) , $BMM 'PME ϓϨΠϠʔͷߦಈઓུΛ ͱ͢Δ ϓϨΠϠʔͷใू߹ ʹ͓͚ΔظརಘΛٻΊΔ ϓϨΠϠʔ͜ΕΛ࠷େʹ͢Δ͜ͱΛߟ͑Δ͕ ϓϨΠϠʔͷ ʹରͯ͠ ͱ͢Δͷ͕࠷ద ((b1 1, 1 − b1 1), (b1 2, 1 − b1 2)) u1 1 H1 (u1 1, (r,1 − r), (b1,u1 1, b2,u1 1)) = (−1)b1 1 + 2(1 − b1 1)b2 + 1(1 − b1 1)(1 − b2 ) = − (b2 + 2)b1 1 + b2 + 1 b2 , 0 ≤ b2 ≤ 1 b1 1 = 0 r 1 − r b2 1 − b2 b2 1 − b2 u2 1 u1 1 u1 2 0 1 b1 2 1 − b1 2
- 36. ੳ 1MBZFS , 1MBZFS , (−1, + 1) / 1MBZFS 1 2 1 2 $BMM 'PME (+2, − 2) (+1, − 1) (−2, + 2) (+1, − 1) (−1, + 1) , $BMM 'PME ϓϨΠϠʔͷߦಈઓུΛ ͱ͢Δ ϓϨΠϠʔͷใू߹ ʹ͓͚ΔظརಘΛٻΊΔ ϓϨΠϠʔ͜ΕΛ࠷େʹ͢Δ͜ͱΛߟ͑Δ ((b1 1, 1 − b1 1), (b1 2, 1 − b1 2)) u1 2 H1 (u1 1, (r,1 − r), (b1,u1 1, b2,u1 1)) = (−1)b1 2 + (−2)(1 − b1 2)b2 + 1(1 − b1 2)(1 − b2 ) = (3b2 − 2)b1 2 + 1 − 3b2 r 1 − r b2 1 − b2 b2 1 − b2 u2 1 u1 1 u1 2 0 1 b1 2 1 − b1 2 b1 2 b2 1 1 0 2 3 ͷʹΑΓ࠷దͳߦಈ ͕มԽ b2 b1 2
- 37. ੳ 1MBZFS , 1MBZFS , (−1, + 1) / 1MBZFS 1 2 1 2 $BMM 'PME (+2, − 2) (+1, − 1) (−2, + 2) (+1, − 1) (−1, + 1) , $BMM 'PME r 1 − r b2 1 − b2 b2 1 − b2 u2 1 u1 1 u1 2 0 1 b1 2 1 − b1 2 b1 2 b2 1 1 0 2 3 ͷʹΑΓ࠷దͳߦಈ ͕มԽ b2 b1 2 b2 r 1 1 0 3 4 ͷʹΑΓ࠷దͳߦಈ ͕มԽ r b2
- 38. ੳ 1MBZFS , 1MBZFS , (−1, + 1) / 1MBZFS 1 2 1 2 $BMM 'PME (+2, − 2) (+1, − 1) (−2, + 2) (+1, − 1) (−1, + 1) , $BMM 'PME r 1 − r b2 1 − b2 b2 1 − b2 u2 1 u1 1 u1 2 0 1 b1 2 1 − b1 2 b1 2 b2 1 1 0 2 3 b2 r 1 1 0 3 4 ͜ͷͱ͖ ͕࠷ద r 3 4 ⇔ b1 2 2 3 b2 = 0 r = 1 2 1 2 + 1 − b1 2 2 1 2 2 3
- 39. ੳ 1MBZFS , 1MBZFS , (−1, + 1) / 1MBZFS 1 2 1 2 $BMM 'PME (+2, − 2) (+1, − 1) (−2, + 2) (+1, − 1) (−1, + 1) , $BMM 'PME r 1 − r b2 1 − b2 b2 1 − b2 u2 1 u1 1 u1 2 0 1 b1 2 1 − b1 2 b1 2 b2 1 1 0 2 3 b2 r 1 1 0 3 4 ͜ͷͱ͖ ͕࠷ద r 3 4 ⇔ b1 2 2 3 b2 = 0 1 2 2 3 r = 1 2 − b1 2
- 40. ੳ 1MBZFS , 1MBZFS , (−1, + 1) / 1MBZFS 1 2 1 2 $BMM 'PME (+2, − 2) (+1, − 1) (−2, + 2) (+1, − 1) (−1, + 1) , $BMM 'PME r 1 − r b2 1 − b2 b2 1 − b2 u2 1 u1 1 u1 2 0 1 b1 2 1 − b1 2 b1 2 b2 1 1 0 2 3 b2 r 1 1 0 3 4 ͜ͷͱ͖ҙͷ ͕࠷ద r = 3 4 ⇔ b1 2 = 2 3 b2 r = 1 2 − b1 2 1 2 2 3
- 41. ੳ 1MBZFS , 1MBZFS , (−1, + 1) / 1MBZFS 1 2 1 2 $BMM 'PME (+2, − 2) (+1, − 1) (−2, + 2) (+1, − 1) (−1, + 1) , $BMM 'PME r 1 − r b2 1 − b2 b2 1 − b2 u2 1 u1 1 u1 2 0 1 b1 2 1 − b1 2 b1 2 b2 1 1 0 2 3 b2 r 1 1 0 3 4 ͜ͷͱ͖ ͕࠷ద r 3 4 ⇔ b1 2 2 3 b2 = 1 1 2 2 3 r = 1 2 − b1 2
- 42. ੳ 1MBZFS , 1MBZFS , (−1, + 1) / 1MBZFS 1 2 1 2 $BMM 'PME (+2, − 2) (+1, − 1) (−2, + 2) (+1, − 1) (−1, + 1) , $BMM 'PME 3 4 u2 1 u1 1 u1 2 0 1 1 3 b1 2 b2 1 1 0 2 3 b2 r 1 1 0 3 4 1 2 2 3 ͕࠷దͳઓུͷ b1 2 = 2 3 , b2 = 2 3 2 3 1 4 r = 1 2 − 2/3 = 3 4 2 3 1 3
- 43. ੳ 1MBZFS , 1MBZFS , (−1, + 1) / 1MBZFS 1 2 1 2 $BMM 'PME (+2, − 2) (+1, − 1) (−2, + 2) (+1, − 1) (−1, + 1) , $BMM 'PME ͚͓ʹߧۉΔϓϨΠϠʔͷظརಘ E1 = 1 2 {2 × 1 × 2 3 + 1 × 1 × 1 3 } + 1 2 {(−2) × 1 3 × 2 3 + 1 × 1 3 × 1 3 + (−1) × 2 3 } = 1 3 3 4 u2 1 u1 1 u1 2 0 1 1 3 2 3 1 4 2 3 1 3 ̍ʘ $BMM ӕͩ 'PME ߱ , 1/3 2/3 2/3 1/3 ֤ϓϨΠϠʔ͕͚͓ʹߧۉΔઓུΛͱΔͱ͢Ε ˎϓϨΠϠʔͷظརಘ ࢀߟɿϕΠδΞϯφογϡߧۉ r = 1 2 − 2/3 = 3 4