ゲーム理論 BASIC 演習83 -アナウンスは効果あるか-

1. ήʔϜཧ࿦#4*$ԋश
Ξφ΢ϯε͸ޮՌ͋Δ͔ʁ ৐ं݊֬ೝ

2. ໰୊
ղ౴

3. ໰୊
৐ं݊֬ೝήʔϜʹ͓͚Δࠞ߹ઓུφογϡ‫ߧۉ‬͸ Ͱ͋Δ
͜͜Ͱ όεଆ͕֬ೝ͢Δ֬཰ΛࣄલʹΞφ΢ϯε͢Δͱ͍͏͜ͱΛߟ͑Δ
͜ͷࣄલΞφ΢ϯεͨ֬͠ೝ͢Δ֬཰Λ ͱͯ͠
͜ΕΛ஌্ͬͨͰ٬͸ෆਖ਼͢Δ͠ͳ͍Λબ୒͢Δ
͜ͷͱ͖ ‫ޙ‬Ζ޲͖‫ؼ‬ೲ๏ʹΑͬͯ
෦෼ήʔϜ‫׬‬શ‫ߧۉ‬Λ‫ٻ‬ΊΑ
ώϯτɿόεଆͷ Λॴ༩ͱͯ͠ ٬ͷ࠷దߦಈΛߟ͑Δ
ώϯτɿ٬ͷ࠷దߦಈΛߟྀͯ͠ όεଆ͸ Λಈ͔ͯ͠‫ظ‬଴རಘ͕࠷େ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
ෆਖ਼΋ෆਖ਼͠ͳ͍΋࠷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͸ Λಈ͔ͯ͠‫ظ‬଴རಘ͕࠷େͱͳΔ Λ‫ٻ‬ΊΔ
٬ͷߦಈઓུΛ ͱͯ͠
ෆਖ਼͠ͳ͍ ͕࠷దͱͳΔͷ͸ ͷͱ͖
ෆਖ਼͢Δ ͕࠷దͱͳΔͷ͸ ͷͱ͖
೚ҙͷ Ͱ࠷ద ͱͳΔͷ͸ ͷͱ͖
όεଆͷ‫ظ‬଴རಘ͸
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͸ Λಈ͔ͯ͠‫ظ‬଴རಘ͕࠷େ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͸ Λಈ͔ͯ͠‫ظ‬଴རಘ͕࠷େͱͳΔ Λ‫ٻ‬ΊΔ
ӈਤΑΓ όεଆͷ‫ظ‬଴རಘ࠷େͱͳΔͷ͸
͜ͷͱ͖ ٬ଆͷ࠷దߦಈ͸ෆਖ਼Λߦ͏ ͱͳΔ
ΛબͿͷͰ͋Ε͹ ͳΔ ΛબͼརಘΛߴ͘Ͱ͖Δ
ͭ·Γ Ξφ΢ϯεͯ͠͠·͏͜ͱͰ ෆਖ਼ߦҝ͕ԣߦ͢Δ͜ͱʹͳΔ
͜ͷརಘߏ଄ͩͱ Ξφ΢ϯε͠ͳ͍ํ͕ྑ͍
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
ਖ਼֬ʹ͸ӡૹίετΛߟ͑Δඞཁ͕͋ͬͨ
േۚ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. Ξφ΢ϯε͕༗ޮͳ৔߹Λ‫ݕ‬౼
േۚ͸ߟ͑Δ͕ όεଆʹͦͷേۚΛࢧ෷Θͣ ୈࡾऀʹ౉͢Α͏ͳརಘߏ଄ ͭ·Γόεଆͷേۚ෼Λརಘʹߟྀ͠ͳ͍
ͱ͢Ε͹Ͳ͏͔ʁφογϡ‫͓ߧۉ‬ΑͼΞφ΢ϯεͷޮՌΛ࣍ճԋशͱ͢Δ
േۚύλʔϯվ
٬ʗόε ֬ೝ͠ͳ͍ ֬ೝ͢Δ
ෆਖ਼͠ͳ͍
ෆਖ਼͢Δ
ਖ਼‫ن‬ͷྉۚʴേۚ֬ೝίετӡૹίετ

10. ήʔϜཧ࿦#4*$ԋश
Ξφ΢ϯε͸ޮՌ͋Δ͔ʁ ৐ं݊֬ೝ