ゲーム理論 BASIC 演習98 -リスク回避的な入札者 -

1. ήʔϜཧ࿦#4*$ԋश
ϦεΫճආతͳೖࡳऀ

2. ໰୊
ղ౴

3. ϑΝʔετϓϥΠεΦʔΫγϣϯ 'JSTU1SJDFVDUJPO ͷ΋ͱͰΦʔΫγϣϯ͕࣮ࢪ͞ΕΔ
ೖࡳऀ͸ ਓͱ͢Δ֤ೖࡳऀ͸ͭͷग़඼෺ʹର͢ΔධՁ஋Λ΋ͭͱ͢Δ
ͨͩ͠ ࣗ෼ͷධՁ஋͸Θ͔Δ͕ଞͷೖࡳऀͷධՁ஋͸Θ͔Βͳ͍ͱ͢Δ
ೖࡳऀ ͷධՁ஋ ͸ ಠཱͳ֬཰ม਺Ͱ ͷൣғͰҰ༷෼෍͍ͯ͠Δͱ͢Δ
֤ೖࡳऀ͸‫͍ޓ‬ͷ֬཰෼෍Λ஌͍ͬͯΔ΋ͷͱ͢Δ
ϑΝʔετϓϥΠεΦʔΫγϣϯ 'JSTU1SJDFVDUJPO ͷϧʔϧ͸ҎԼͷ௨ΓʹͳΔɿ
ɾ࠷΋ߴ͍Ձ֨Λೖࡳͨ͠ਓ͕མࡳ ߪೖ ͢ΔΦʔΫγϣϯͰ͋Δ·ͨ མࡳͨ͠ਓ͕ࣗ਎ͷೖࡳֹΛࢧ෷͏
ࠓճ ֤ೖࡳऀ͸ϦεΫճආతͰ͋Δͱ͠ ಉҰͷޮ༻ؔ਺Λ΋ͭͱ͢Δ ͢ͳΘͪ ʹ͍ͭͯ
Ώ͑ʹ
ɹɾམࡳͨ͠৔߹ɿ
ɹɾམࡳͰ͖ͳ͔ͬͨ৔߹ɿ
֤ೖࡳऀ͕ࣗ਎ͷλΠϓ͕ ͷͱ͖
Λೖࡳ͢Δ͜ͱ͕ରশϕΠδΞϯφογϡ‫ͳͱߧۉ‬Δͱ͖ͷ ͷ஋Λ‫ٻ‬ΊΑ
ϑΝʔετϓϥΠεΦʔΫγϣϯͷ΋ͱͰͷओ࠵ऀଆͷ‫ظ‬଴ऩೖΛ‫ٻ‬ΊΑͨͩ͠ ओ࠵ऀଆ͸ϦεΫதཱతͰ͋Δͱ͢Δ
2
i ∈ N = {1,2} vi
[0,1]
i ∈ N = {1,2} ui
(x) = u(x) = x
u(b|vi
) = vi
− b
u(b|vi
) = 0
vi
b(vi
) = kvi
, k 0 k
໰୊

4. ֤ೖࡳऀ͕ࣗ਎ͷλΠϓ͕ ͷͱ͖
Λೖࡳ͢Δ͜ͱ͕ରশϕΠδΞϯφογϡ‫ͳͱߧۉ‬Δͱ͖ͷ ͷ஋Λ‫ٻ‬ΊΑ
೚ҙͷೖࡳऀ ʹ͍ͭͯߟ͑Δ͜ͷೖࡳऀͷධՁ஋͸ ͱ͢Δ
΋͏ਓͷೖࡳऀ ͕ ΛͱΔͱ͢Δ
Λೖࡳͨ͠৔߹ʹ
མࡳͰ͖Δ֬཰͸
Ώ͑ʹ‫ظ‬଴རಘ͸
Ұ֊৚݅ΑΓ
Ώ͑ʹ Ͱ͋Δ‫͜ʹٯ‬ͷઓུͷͱ͖ ্‫ͱه‬ಉٞ͡࿦Ͱ ͱͳΔ͔Β Ͱ͋Δ
vi
b(vi
) = kvi
, k 0 k
i ∈ {1,2} vi
∈ [0,1]
j ≠ i b(vj
) = kvj
b(vi
) = x
P(x kvj
) = P
(
x
k
vj
)
=
x
k
E(x) =
x
k
vi
− x
E′

(x) =
1
k
vi
− x +
x
k
(−1)
2 vi − x
= 0 ⇔ vi
− x =
x
2 vi − x
⇔ 2(vi
− x) = x ⇔ x =
2
3
vi
b(vi
) =
2
3
vi
b(vj
) =
2
3
vj
k =
2
3
ղ౴
x
k
1
vj
1
0
x
k

5. ϑΝʔετϓϥΠεΦʔΫγϣϯͷ΋ͱͰͷओ࠵ऀଆͷ‫ظ‬଴ऩೖΛ‫ٻ‬ΊΑ
ͨͩ͠ ओ࠵ऀଆ͸ϦεΫதཱతͰ͋Δͱ͢Δ
೚ҙͷೖࡳऀ ʹର͠ Λೖࡳ͢Δ͜ͱ͸
ରশϕΠδΞϯφογϡ‫Ͱߧۉ‬ ͜ͷ΋ͱͰͷ‫ظ‬଴ऩೖΛߟ͑Δ
ධՁ஋ ͷೖࡳऀͷ‫ظ‬଴ࢧ෷ֹ͸ ͳͷ͔ͩΒ ओ࠵ऀଆ͔Β͢Δͱ
͜Ε͸‫ظ‬଴ऩೖͱͳΔ
‫ظ‬଴ऩೖ͸
ิ଍ ͜ͷ‫ظ‬଴ऩೖ͸ϦεΫதཱతͳೖࡳऀͷ৔߹ΑΓ΋ߴ͍
·ͨ ηΧϯυϓϥΠεΦʔΫγϣϯʹ͓͚Δ‫ظ‬଴རಘΑΓ΋ߴ͘ͳΔ ࣍ճ֬ೝ
i ∈ {1,2} b(vi
) =
2
3
vi
vi
∈ [0,1] Q(vi
) =
b(vi
)
k
b(vi
) =
2
3
vi2
E =
∑
i∈{1,2}
∫
1
0
Q(vi
)dvi
=
∑
i∈{1,2}
∫
1
0
2
3
vi2
dvi
=
∑
i∈{1,2}
[
2
9
vi3
]
1
0 =
4
9
1
3
ղ౴

6. ϑΝʔετϓϥΠεΦʔΫγϣϯͷ΋ͱͰͷओ࠵ऀଆͷ‫ظ‬଴ऩೖΛ‫ٻ‬ΊΑ
ͨͩ͠ ओ࠵ऀଆ͸ϦεΫதཱతͰ͋Δͱ͢Δ
೚ҙͷೖࡳऀ ʹର͠ Λೖࡳ͢Δ͜ͱ͸
ରশϕΠδΞϯφογϡ‫Ͱߧۉ‬ ͜ͷ΋ͱͰͷ‫ظ‬଴ऩೖΛߟ͑Δ
͸ ͰҰ༷෼෍͍ͯ͠Δऩೖ͕ ҎԼ ͱͳΔ֬཰ ෼෍ؔ਺ ͸
ऩೖ͕ ҎԼ
Ώ͑ʹऩೖͷ֬཰ີ౓ؔ਺͸
‫ظ‬଴ऩೖ͸
i ∈ {1,2} b(vi
) =
2
3
vi
b(vi
) =
2
3
vi
[
0,
2
3]
x 0 ≤ x ≤
2
3
P( x ) = P(x ≥
2
3
vi
and x ≥
2
3
vj
) = P(x ≥
2
3
vi
)P(x ≥
2
3
vj
) =
3x
2
⋅
3x
2
=
9x2
4
9x
2
E =
∫
2
3
0
x
9x
2
dx =
[
9
6
x3
]
2
3
0
=
9
6(
2
3 )
3
=
4
9
ผղ
x 2
3
b(vi
)
3
2
0
3x
2

7. ήʔϜཧ࿦#4*$ԋश
ϦεΫճආతͳೖࡳऀ