Solution Manual For Auction Threory Third Edition by Alexey KUshnur and Jun Xiao Pass With Good Grade A+

Solution Manual For Auction Threory by Alexey KUshnur and
Jun Xiao Pass With Good Grade A+
SolutionsManual for AUCTI fd fd fd




ON THEORY* fd




Alexey Kushnir and Jun Xiao fd fd fd fd




August 2009 fd




Contents
2 Private Value Auctions: A First Look ....................................................................... 2
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3 The Revenue Equivalence Principle ............................................................................. 8
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4 Qualifications and Extensions ..................................................................................... 11
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5 Mechanism Design....................................................................................................... 17
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6 Auctions with Interdependent Values........................................................................ 25
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8 Asymmetries and Other Complications .................................................................... 34
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9 Efficiency and the English Auction........................................................................... 40
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10 Mechanism Design with Interdependent Values ....................................................... 43
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11 Bidding Rings...............................................................................................................48
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13 Equilibrium and Efficiency with Private Values....................................................... 52
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15 Sequential Sales ........................................................................................................... 55
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16 Nonidential Objects .................................................................................................... 60
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17 Packages and Positions ............................................................................................... 62
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f d V. Krishna, Auction fheory (2nd. Ed.), Elsevier, 2009.
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1

,2 Private Value Auctions: A First Look fd fd fd fd fd




Problem 2.1 (Pomer distribution) Suppose there are tmo bidders mith private values t
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hat are distributed independently according to the distribution F (x) = xa over [0, 1] m
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here a > 0. Find symmetric equilibrium bidding strategies in a first−price auction.
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Solution. Since N = 2, G(x) = F (x) = xa. Thus, using the formula on page 16 of t
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he text, fd



∫ ∫ x a
β I (x) = x — x G (y)dy = x — ya dy = a x
f d f d
f d f d
fd f df d f d


O G(x) O x 1 +a
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fd
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Problem 2.2 (Pareto distribution) Suppose there are tmo bidders mith private values t
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hat are distributed independently according to a Pareto distribution F (x) = 1 —
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(x + 1)—
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2 over [0, ∞). Find symmetric equilibrium bidding strategies in a first−price auction.
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Shom by direct computation that the expected revenues in a first− and second− price auc
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tion are the same. fd fd fd




Solution. Again, since N = 2, G (x) = F (x) = 1 — (x + 1)—2. Thus,
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∫ xf d Gfd(y)
I
β (x) = x— dy
Of d f d Gfd(x)
f d f d f d f d f d fd




∫ xf d 1 —2
— (y + 1)
x—
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= f d f d fd
dy
x
O 1 — (x + 1)—2
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=
x+2 fd fd




In the first-price auction, the expected revenue of the seller is
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E RI 2E mI (x)
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f d = f d f d f d fd


f d fd


= 2E∫ G
∞
f (x) × βI (x)
d fd fd fd fd

x
f d
fd fd



= 2 1 — (x + 1)—2
f d f d fd fd fd fd fd 2 (x + 1)—3 dx
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O x+2 fd fd



= f d f d 1/3

Let Y2 be the second highest value, and its density is ƒ2 (y) = 2 (1 — F (y)) g (y)
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(see Appendix C). fd fd



In a second-price auction, the expected revenue of the seller is
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f d f d f d


E RII = E [Y2] f d f d fd


∫ ∞ fd
fd


= y2 (y + 1)—2 2 (y + 1)—3 dy
f d fd fd fd fd fd fd fd fd

O
= f d f d 1/3

Therefore, the expected revenues in the two auctions are the same.
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2

,Problem 2.3 (Stochastic dominance) Gonsider an N −bidder first−price auction mith i
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ndependent private values. Get β be the symmetric equilibrium bidding strategy mhen
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mhich each bidder’s value is distributed according to F on [0, c] . Similarly, let β∗ be th
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e equilibrium strategy mhen each bidder’s value distribution is F∗ on [0, c∗] .
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a· Shom that if F∗ dominates F in termsof the reverse hazard rate (see Appendix
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B for a definition) then for all x ∈ [0, c] , β∗(x) ≥ β (x) .
b· By considering F (x) = 3x — x2 on [0, 1(3 — √ 5)] and F∗ (x) = 3x — 2x2 on
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∗ 2
0, 21 , shom that the condition that F first−order stochastically dominates F is not
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sufficient to guarantee that β∗(x) ≥ β(x) . fd fd fd fd fd fd fd fd fd




Solution. Part a. Because G (x) = F (x)N—1 and g (x) = (N — 1) F (x)N—
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2 ƒ (x) , the symmetric equilibrium in Proposition 2.2 could be rewritten as follows
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∫ x f d

1
β (x) = yg (y) dy
f d f d f d f d fd




G (x)
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O
∫ x
fd
f d


1
= y (N — 1) F (x)N—2 ƒ (x) dy fd fd fd fd fd fd fd


[F (x)]N—
∫
f d


1 f d f d O
x ƒ (y)
= (N — 1) y dy
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F (y)
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O
∫
fd


x
= (N — 1) fdfd fd fd yσ (y) dy fd fd


O

where σ (x) is the reverse hazard rate. Similarly, we have
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∫ x
β∗ (x) = (N — 1) fd fd fd fd fd yσ∗ (y) dy fd


O

So it is easy to see that if F∗ dominates F in terms of reverse hazard rate, then
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σ∗ (y) ≥ σ (y) for all y ∈ [0, c] . Therefore β∗ (x) ≥ β (x) for all x ∈ [0, c].
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Part b. Obviously, F ∗ (x) F (x),
≤ so F ∗ stochastically dominates F . The distrib
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utions F and F ∗ are illustrated in Figure S2.1, where the solid line represents F and t
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he dashed line represents F∗.
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3

, 1.0

0.9

0.8

0.7

0.6

0.5

0.4

0.3

0.2

0.1

0.0
0.0 0.1 0.2 0.3 0.4 0.5
x
Figure S2.1 fd




Suppose there are two bidders, thenfd fd fd fd fd




∫ xf d Gfd(y)
β (x) = x— dy
Of d f d Gfd(x)
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∫ xf d 3y y2
—
= f d f d x—
dy fd


O 3x — x2 fd fd
fd




1 2x — 9
x
fd fd fd

=
6 x—3
fd

fd fd
f d f d


for x ∈ 0, 1 3 —
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√ f d f d
2
fd5 . Similarly,
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xfd3y
β∗ (x) fd f d f d = f d f d x ∫ fd
— 2y2 f d

dy
— O 3x — 2x2 fd fd
fd



x
1
= (8x — 9)
6 2x — 3
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f d √ f d f d


for x ∈ 0,21 . It is easy to see that β∗ (x) < β (x) for x ∈ (0, 1 2 3 — 5 ]. The bidding
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strategies β and β∗ are plotted in Figure S2.2, where β is the solid line and β∗ is the d
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ashed line. fd




4