ゲーム理論 BASIC 演習オークション関連補足 - 収入同値定理 -

1. ήʔϜཧ࿦#4*$ԋशิ଍
ऩೖಉ஋ఆཧ

2. ‫ڞ‬௨ઃఆ
֤ΦʔΫγϣϯͷϧʔϧ֬ೝ
‫ظ‬଴ࢧ෷ؔ਺͔Β‫ظ‬଴ऩೖ‫ࢉܭ‬

3. ೖࡳऀ͸ ਓͱ͢Δ֤ೖࡳऀ͸ͭͷग़඼෺ʹର͢ΔධՁ஋Λ΋ͭͱ͢Δ
ͨͩ͠ ࣗ෼ͷධՁ஋͸Θ͔Δ͕ଞͷೖࡳऀͷධՁ஋͸Θ͔Βͳ͍ͱ͢Δ
ೖࡳऀ ͷධՁ஋ ͸ ಠཱͳ֬཰ม਺Ͱ ͷൣғͰҰ༷෼෍͍ͯ͠Δͱ͢Δ
֤ϓϨΠϠʔ͸‫͍ޓ‬ͷ֬཰෼෍Λ஌͍ͬͯΔ΋ͷͱ͢Δ
͕‫ͳີݫ‬୯ௐ૿Ճؔ਺ʹै͏΋ͷͱ͠ ධՁ஋͕ߴ͍ํ͕མࡳͰ͖Δ
·ͨ ೖࡳऀͷ‫ظ‬଴ࢧ෷͍ؔ਺ͱͯ͠ ͱ͠ Ͱ͋Δͱ͢Δ
ධՁ஋ Λ΋ͭೖࡳऀ͕ධՁ஋ ͷϑϦΛͨ͠৔߹ͷ‫ظ‬଴རಘؔ਺͸
ରশϕΠδΞϯφογϡ‫ͳͱߧۉ‬Δͱ͖ Ͱ‫ظ‬଴རಘ࠷େͱͳΔͨΊ
ΑΓ
2
i ∈ N = {1,2} vi
[0,1]
bi
(vi
) = b(vi
) b(0) = 0
Q(x) Q(0) = 0
vi
x
Ei
(x|vi
) = P(x vj
)vi
− Q(x) = xvi
− Q(x)
x = vi
dEi
(x|vi
)
dx
|x=vi = vi
−
dQ(x)
dx
|x=vi = vi
− Q′

(vi
) = 0 ⇒ Q′

(vi
) = vi
Q(vi
) =
∫
Q′

(vi
)dvi
=
∫
vi
dvi
=
vi2
2
+ C Q(0) = 0 Q(vi
) =
vi2
2
‫ڞ‬௨ઃఆ
࠷΋ߴ͍ೖࡳֹΛͨ͠ਓ ධՁ஋ͷߴ͍ਓ ͕
མࡳ͢ΔϧʔϧͰ͋Ε͹ ࢧ෷͍ͷϧʔϧʹؔΘΒͣ
ରশϕΠδΞϯφογϡ‫͚͓ʹߧۉ‬Δ‫ظ‬଴ࢧ෷ֹ͸
ͱͳΔ
Q(vi
) =
vi2
2

4. ϑΝʔετϓϥΠεΦʔΫγϣϯ 'JSTU1SJDFVDUJPO ͷϧʔϧ͸ҎԼͷ௨ΓʹͳΔɿ
ɾϧʔϧɿ࠷΋ߴ͍Ձ֨Λೖࡳͨ͠ਓ͕མࡳ ߪೖ ͢ΔΦʔΫγϣϯͰ͋Δ·ͨམࡳऀ͕ͦͷೖࡳֹΛࢧ෷͏
ɹɾམࡳͨ͠৔߹ɿ མࡳͰ͖ͳ͔ͬͨ৔߹ɿ
ɾೖࡳऀ ͷ‫ظ‬଴ࢧ෷ؔ਺ɿ
ɾ ΑΓ
ηΧϯυϓϥΠεΦʔΫγϣϯ 4FDPOE1SJDFVDUJPO ͷϧʔϧ͸ҎԼͷ௨ΓʹͳΔɿ
ɾϧʔϧɿ࠷΋ߴ͍Ձ֨Λೖࡳͨ͠ਓ͕མࡳ ߪೖ ͢ΔΦʔΫγϣϯͰ͋Δ·ͨམࡳऀ͸൪໨ʹେ͖͍ೖࡳֹΛࢧ෷͏
ɹɾམࡳͨ͠৔߹ɿ མࡳͰ͖ͳ͔ͬͨ৔߹ɿ
ɾೖࡳऀ ͷ‫ظ‬଴ࢧ෷ؔ਺ɿ
ɾ ͕ऑࢧ഑ઓུ
ui
(b|vi
) = vi
− b ui
(b|vi
) = 0
i Q(x) = P(x vj
)b(x) = xb(x)
Q(vi
) =
vi2
2
vi2
2
= vi
b(vi
) ⇔ b(vi
) =
vi
2
ui
(b|vi
) = vi
− bj
ui
(b|vi
) = 0
i Q(x) =
∫
x
0
vj
dvj
=
x2
2
b(vi
) = vi
֤ΦʔΫγϣϯͷϧʔϧ֬ೝ

5. શһࢧ෷͍ΦʔΫγϣϯͷϧʔϧ͸ҎԼͷ௨ΓʹͳΔɿ
ɾ࠷΋ߴ͍Ձ֨Λೖࡳͨ͠ਓ͕མࡳ ߪೖ ͢ΔΦʔΫγϣϯͰ͋Δͨͩ͠ ࢀՃऀશһ͕ೖࡳֹ௨Γͷࢧ෷͍Λߦ͏
ɹɾམࡳͨ͠৔߹ɿ མࡳͰ͖ͳ͔ͬͨ৔߹ɿ
ɾೖࡳऀ ͷ‫ظ‬଴ࢧ෷ؔ਺ɿ
ɾ ΑΓ
αϯλΫϩʔεΦʔΫγϣϯ 4BOUB$MBVTVDUJPO ͷϧʔϧ͸ҎԼͷ௨ΓʹͳΔɿ
ɾ࠷΋ߴ͍Ձ֨Λೖࡳͨ͠ਓ͕མࡳ ߪೖ ͢ΔΦʔΫγϣϯͰ͋Δ·ͨམࡳऀ͕ͦͷೖࡳֹΛࢧ෷͏
ɾམࡳ͢Δ͠ͳ͍ʹؔΘΒͣ Λೖࡳͨ͠ೖࡳऀʹ ͕ࢧ෷ΘΕΔ
ɹɾམࡳͨ͠৔߹ɿ མࡳͰ͖ͳ͔ͬͨ৔߹ɿ
ɾೖࡳऀ ͷ‫ظ‬଴ࢧ෷ؔ਺ɿ
ɾ ΑΓ
ui
(b|vi
) = vi
− b ui
(b|vi
) = − b
i Q(x) = 1 ⋅ b(x) = b(x)
Q(vi
) =
vi2
2
vi2
2
= b(vi
) ⇔ b(vi
) =
vi2
2
b S(b) =
∫
b
0
xdx =
b2
2
ui
(b|vi
) = vi
− b +
b2
2
ui
(b|vi
) =
b2
2
i Q(x) = P(x vj
)b(x) −
b(x)2
2
= xb(x) −
b(x)2
2
Q(vi
) =
vi2
2
vi2
2
= vi
b(vi
) −
b(vi
)
2
2
⇔ b(vi
)
2
− 2vi
b(vi
) + vi2
= 0 ⇔ (b(vi
) − vi
)2
= 0 ⇔ b(vi
) = vi
֤ΦʔΫγϣϯͷϧʔϧ֬ೝ

6. ഊऀࢧ෷͍ΦʔΫγϣϯ 4BE-PTFSVDUJPO ͷϧʔϧ͸ҎԼͷ௨ΓʹͳΔɿ
ɾ࠷΋ߴ͍Ձ֨Λೖࡳͨ͠ਓ͕མࡳ ߪೖ ͢ΔΦʔΫγϣϯͰ͋ΔམࡳͰ͖ͳ͔ͬͨਓ͕ࣗ਎ͷೖࡳֹΛࢧ෷͏
ɹɾམࡳͨ͠৔߹ɿ མࡳͰ͖ͳ͔ͬͨ৔߹ɿ
ɾೖࡳऀ ͷ‫ظ‬଴ࢧ෷ؔ਺ɿ
ɾ ΑΓ
ui
(b|vi
) = vi
ui
(b|vi
) = − b
i Q(x) = P(x vj
)b(x) = (1 − x)b(x)
Q(vi
) =
vi2
2
vi2
2
= (1 − vi
)b(vi
) ⇔ b(vi
) =
vi2
2(1 − vi)
֤ΦʔΫγϣϯͷϧʔϧ֬ೝ

7. ϑΝʔετϓϥΠεΦʔΫγϣϯ 'JSTU1SJDFVDUJPO
ೖࡳऀ ͷ‫ظ‬଴ࢧ෷ؔ਺ɿ ΑΓ
‫ظ‬଴ऩೖ͸
ଞͷΦʔΫγϣϯ΋ Ͱ͋Δ͜ͱ͔Β ౳͍݁͠ՌΛಘΔ
i Q(x) = P(x vj
)b(x) = xb(x) = x
x
2
2
∫
1
0
Q(vi
)dvi
= 2
∫
1
0
vi
b(vi
)dvi
= 2
∫
1
0
vi vi
2
dvi
= 2
[
vi3
6 ]
1
0 =
1
3
Q(vi
) =
vi2
2
‫ظ‬଴ࢧ෷ؔ਺͔Β‫ظ‬଴ऩೖ‫ࢉܭ‬

8. ήʔϜཧ࿦#4*$ԋशิ଍
ऩೖಉ஋ఆཧ