ゲーム理論 BASIC 演習83 -アナウンスは効果あるか-
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1. The document presents a probability model involving random variables p, q, and r. 2. It defines functions s(r) and E(r) and analyzes their properties for different ranges of r. 3. The optimal values of r that maximize E(r) are determined to be 1/2 and between 1/2 and 11/18.
ゲーム理論 BASIC 演習83 -アナウンスは効果あるか-
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- 3. ं݊֬ೝήʔϜʹ͓͚Δࠞ߹ઓུφογϡߧۉ Ͱ͋Δ ͜͜Ͱ όεଆ͕֬ೝ͢Δ֬ΛࣄલʹΞφϯε͢Δͱ͍͏͜ͱΛߟ͑Δ ͜ͷࣄલΞφϯεͨ֬͠ೝ͢Δ֬Λ ͱͯ͠ ͜ΕΛ্ͬͨͰ٬ෆਖ਼͢Δ͠ͳ͍Λબ͢Δ ͜ͷͱ͖ ޙΖ͖ؼೲ๏ʹΑͬͯ ෦ήʔϜશߧۉΛٻΊΑ ώϯτɿόεଆͷ Λॴ༩ͱͯ͠ ٬ͷ࠷దߦಈΛߟ͑Δ ώϯτɿ٬ͷ࠷దߦಈΛߟྀͯ͠ όεଆ Λಈ͔ͯ͠ظརಘ͕࠷େͱͳΔ ΛٻΊΔ ((p,1 − p), (q,1 − q)) = (( 3 4 , 1 4 )( 1 2 , 1 2)) r(0 ≤ r ≤ 1) r(0 ≤ r ≤ 1) r(0 ≤ r ≤ 1) r േۚύλʔϯ ٬ʗόε ֬ೝ͠ͳ͍ ֬ೝ͢Δ ෆਖ਼͠ͳ͍ ෆਖ਼͢Δ ਖ਼نͷྉۚʴേۚ֬ೝίετ p 1 − p q 1 − q
- 4. ղ ώϯτɿόεଆͷ Λॴ༩ͱͯ͠ ٬ͷ࠷దߦಈΛߟ͑Δ ٬ͷߦಈઓུΛ ͱͯ͠ ෆਖ਼͠ͳ͍ ͕࠷దͱͳΔͷ ͷͱ͖ ෆਖ਼͢Δ ͕࠷దͱͳΔͷ ͷͱ͖ ෆਖ਼ෆਖ਼͠ͳ͍࠷ద ਖ਼֬ʹͱ͏ݴ ҙͷ Ͱ࠷ద ͱͳΔͷ ͷͱ͖ r(0 ≤ r ≤ 1) (s(r),1 − s(r)) (s(r) = 1) (−5)r + (−5)(1 − r) 0r + (−10)(1 − r) ⇔ r 1 2 (s(r) = 0) (−5)r + (−5)(1 − r) 0r + (−10)(1 − r) ⇔ r 1 2 0 ≤ s(r) ≤ 1 (−5)r + (−5)(1 − r) = 0r + (−10)(1 − r) ⇔ r = 1 2 േۚύλʔϯ ٬ʗόε ֬ೝ͠ͳ͍ ֬ೝ͢Δ ෆਖ਼͠ͳ͍ ෆਖ਼͢Δ r 1 − r s(r) 1 − s(r) όε r ٬ ෆਖ਼͠ͳ͍ɿs(r) ෆਖ਼͢Δɿ1 − s(r)
- 5. ղ ώϯτɿ٬ͷ࠷దߦಈΛߟྀͯ͠ όεଆ Λಈ͔ͯ͠ظརಘ͕࠷େͱͳΔ ΛٻΊΔ ٬ͷߦಈઓུΛ ͱͯ͠ ෆਖ਼͠ͳ͍ ͕࠷దͱͳΔͷ ͷͱ͖ ෆਖ਼͢Δ ͕࠷దͱͳΔͷ ͷͱ͖ ҙͷ Ͱ࠷ద ͱͳΔͷ ͷͱ͖ όεଆͷظརಘ ͷͱ͖ ٬ͷ࠷దߦಈ ͱͯ͠ ͷͱ͖ ٬ͷ࠷దߦಈ ͱͯ͠ ͷͱ͖ ٬ͷ࠷దߦಈ ͱͯ͠ r(0 ≤ r ≤ 1) r (s(r),1 − s(r)) (s(r) = 1) 0 ≤ r 1 2 (s(r) = 0) 1 2 r ≤ 1 0 ≤ s(r) ≤ 1 r = 1 2 E(r) = 3rs(r) + 0(1 − r)s(r) + (−2)r(1 − s(r)) + 7(1 − r)(1 − s(r)) = 12rs(r) − 9r − 7s(r) + 7 0 ≤ r 1 2 s(r) = 1 E(r) = 12r − 9r − 7 + 7 = 3r 1 2 r ≤ 1 s(r) = 0 E(r) = 12r ⋅ 0 − 9r − 7 ⋅ 0 + 7 = 7 − 9r r = 1 2 0 ≤ s(r) ≤ 1 E ( 1 2 ) = 12 1 2 s(1/2) − 9 1 2 − 7s(1/2) + 7 = − s(1/2) + 5 2 േۚύλʔϯ ٬ʗόε ֬ೝ͠ͳ͍ ֬ೝ͢Δ ෆਖ਼͠ͳ͍ ෆਖ਼͢Δ r 1 − r s(r) 1 − s(r)
- 6. ղ ώϯτɿ٬ͷ࠷దߦಈΛߟྀͯ͠ όεଆ Λಈ͔ͯ͠ظརಘ͕࠷େͱͳΔ ΛٻΊΔ ͷͱ͖ ٬ͷ࠷దߦಈ ͱͯ͠ ͷͱ͖ ٬ͷ࠷దߦಈ ͱͯ͠ ͷͱ͖ ٬ͷ࠷దߦಈ ͱͯ͠ r(0 ≤ r ≤ 1) r 0 ≤ r 1 2 s(r) = 1 E(r) = 12r − 9r − 7 + 7 = 3r 1 2 r ≤ 1 s(r) = 0 E(r) = 12r ⋅ 0 − 9r − 7 ⋅ 0 + 7 = 7 − 9r r = 1 2 0 ≤ s(r) ≤ 1 E ( 1 2) = 12 1 2 s(1/2) − 9 1 2 − 7s(1/2) + 7 = − s(1/2) + 5 2 r E 1 2 3 2 −2 1 5 2 E(r) = 3r E(r) = 7 − 9r s(r) = 0 11 18 7 9 s(r) = 1
- 7. ղ ώϯτɿ٬ͷ࠷దߦಈΛߟྀͯ͠ όεଆ Λಈ͔ͯ͠ظརಘ͕࠷େͱͳΔ ΛٻΊΔ ӈਤΑΓ όεଆͷظརಘ࠷େͱͳΔͷ ͜ͷͱ͖ ٬ଆͷ࠷దߦಈෆਖ਼Λߦ͏ ͱͳΔ ΛબͿͷͰ͋Ε ͳΔ ΛબͼརಘΛߴ͘Ͱ͖Δ ͭ·Γ Ξφϯεͯ͠͠·͏͜ͱͰ ෆਖ਼ߦҝ͕ԣߦ͢Δ͜ͱʹͳΔ ͜ͷརಘߏͩͱ Ξφϯε͠ͳ͍ํ͕ྑ͍ r(0 ≤ r ≤ 1) r r = 1 2 s(r) = 0 s(r) = 1 1 2 r′  11 18 r′  r E 1 2 3 2 −2 1 5 2 E(r) = 3r E(r) = 7 − 9r s(r) = 0 11 18 7 9 s(r) = 1 േۚύλʔϯ ٬ʗόε ֬ೝ͠ͳ͍ ֬ೝ͢Δ ෆਖ਼͠ͳ͍ ෆਖ਼͢Δ
- 8. గਖ਼ ਖ਼֬ʹӡૹίετΛߟ͑Δඞཁ͕͋ͬͨ േۚύλʔϯ ٬ʗόε ֬ೝ͠ͳ͍ ֬ೝ͢Δ ෆਖ਼͠ͳ͍ ෆਖ਼͢Δ ਖ਼نͷྉۚʴേۚ֬ೝίετ േۚύλʔϯ ٬ʗόε ֬ೝ͠ͳ͍ ֬ೝ͢Δ ෆਖ਼͠ͳ͍ ෆਖ਼͢Δ ྉۚӡૹίετ ֬ೝͷίετ ӡૹίετ ਖ਼نͷྉۚʴേۚ֬ೝίετӡૹίετ ((p,1 − p), (q,1 − q)) = (( 7 10 , 3 10)( 1 2 , 1 2 )) ((p,1 − p), (q,1 − q)) = (( 3 4 , 1 4 )( 1 2 , 1 2)) p 1 − p q 1 − q
- 9. Ξφϯε͕༗ޮͳ߹Λݕ౼ േۚߟ͑Δ͕ όεଆʹͦͷേۚΛࢧΘͣ ୈࡾऀʹ͢Α͏ͳརಘߏ ͭ·ΓόεଆͷേۚΛརಘʹߟྀ͠ͳ͍ ͱ͢ΕͲ͏͔ʁφογϡ͓ߧۉΑͼΞφϯεͷޮՌΛ࣍ճԋशͱ͢Δ േۚύλʔϯվ ٬ʗόε ֬ೝ͠ͳ͍ ֬ೝ͢Δ ෆਖ਼͠ͳ͍ ෆਖ਼͢Δ ਖ਼نͷྉۚʴേۚ֬ೝίετӡૹίετ