2 Private Value Auctions: A First Look
Problem 2.1 (Power distribution) Suppose there are two bidders with private values
that are distributed independently according to the distribution F(x) = xaover [0;1]
where a > 0:Find symmetric equilibrium bidding strategies in a first-price auction.
Solution. Since N= 2,G(x) = F(x) = xa:Thus, using the formula on page 16 of
the text,
G(y)
ya
Problem 2.2 (Pareto distribution) Suppose there are two bidders with private values
that are distributed independently according to a Pareto distribution F(x) = 1
(x+ 1)2over [0;1). Find symmetric equilibrium bidding strategies in a first-price
auction. Show by direct computation that the expected revenues in a first- and second-
price auction are the same.
Solution. Again, since N= 2,G(x) = F(x) = 1 (x+ 1)2. Thus,
G(y)
In the first-price auction, the expected revenue of the seller is
Let Y2be the second highest value, and its density is f2(y) = 2 (1 F(y)) g(y)
(see Appendix C).
In a second-price auction, the expected revenue of the seller is
ERII=E[Y2]
Therefore, the expected revenues in the two auctions are the same.
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Problem 2.3 (Stochastic dominance) Consider an N-bidder first-price auction with
independent private values. Let be the symmetric equilibrium bidding strategy when
which each bidder’s value is distributed according to Fon [0; !]:Similarly, let be
the equilibrium strategy when each bidder’s value distribution is Fon [0; !]:
a. Show that if Fdominates Fin termsof the reverse hazard rate (see Appendix
B for a definition) then for all x2[0; !]; (x)(x):
b. By considering F(x)=3xx2on [0;1
2(3 p5)] and F(x)=3x2x2on
0;1
2, show that the condition that Ffirst-order stochastically dominates Fis not
su¢ cient to guarantee that (x)(x):
Solution. Part a. Because G(x) = F(x)N1and g(x) = (N1) F(x)N2f(x);the
symmetric equilibrium in Proposition 2.2 could be rewritten as follows
(x) = 1
G(x)Zx
0
yg (y)dy
0
where (x)is the reverse hazard rate. Similarly, we have
0
So it is easy to see that if Fdominates Fin terms of reverse hazard rate, then
Part b. Obviously, F(x)F(x), so Fstochastically dominates F. The
distributions Fand Fare illustrated in Figure S2.1, where the solid line represents
Fand the dashed line represents F.
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0.0 0.1 0.2 0.3 0.4 0.5
0.0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0
x
Figure S2.1
Suppose there are two bidders, then
G(y)
for x20;1
23p5:Similarly,
3y2y2
for x20;1
2:It is easy to see that (x)< (x)for x2(0;1
23p5]:The bidding
strategies and are plotted in Figure S2.2, where is the solid line and is the
dashed line.
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0.0 0.1 0.2 0.3 0.4 0.5
0.00
0.05
0.10
0.15
0.20
x
Figure S2.2
Problem 2.4 (Mixed auction) Consider an N-bidder auction which is a “mixture”
of a first- and second-price auction in the sense that the highest bidder wins and pays
a convex combination of his own bid and the second-highest bid. Precisely, there is a
fixed 2(0;1) such that upon winning, bidder ipays bi+ (1 ) (maxj6=ibj):Find
a symmetric equilibrium bidding strategy in such an auction when all bidders’values
are distributed according to F:
Solution. Suppose all bidders other than 1follow the strategy : The expected
payoff of bidder ifrom bidding bwhen his value is xis
(b; x)) = G1(b)[xb (1 )E[(Y1)j(Y1)b1]]
0(y)g(y)dy
0
Maximizing this with respect to byields the first-order condition:
0 = g1(b)
At a symmetric equilibrium, b=(x), so the first-order condition becomes
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Using G(x)(1=)1as the integrating factor, the solution to the above differential
equation is easily seen to be
(x) = 1
G(x)Zx
0
yg(y)dy
where GG1= and g=G0
.
Problem 2.5 (Resale) Consider a two-bidder first-price auction in which bidders’
values are distributed according to F: Let be the symmetric equilibrium (as derived
in Proposition 2.2). Now suppose that after the auction is over, both the losing and
winning bids are publicly announced. In addition, there is the possibility of post-
auction resale: The winner of the auction may, if he so wishes, offer the object to the
other bidder at a fixed “take-it-or-leave-it” price of p: If the other bidder agrees, then
the object changes hands and the losing bidder pays the winning bidder p. Otherwise,
the object stays with the winning bidder and no money changes hands. The possibility
of post-auction resale in this manner is commonly known to both bidders prior to
participating in the auction. Show that remains an equilibrium even if resale is
allowed. In particular, show that a bidder with value xcannot gain by bidding an
amount b>(x)even when he has the option of reselling the object to the other
bidder.
Solution. First, let us consider the resale stage. Suppose bidder 1 wins the auction
and the announced bids are b1and b2. Hence bidder 1 can recover bidder 2’s private
value by x2=1
Second, now we move to the auction stage. Let (x) = 1
F(x)Rx
0yf(y)dy be the
payoff is
1(z; x) = (x(z)) F(z)if xz
If x2< z xthere is no resale. If z > x x2;bidder 1 does not offer to bidder
2 and his payoff remains the same. If zx2x, bidder 1 sells to bidder 2 and the
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payoff after resale is x2(z):Note that
(x(z)) F(x) + Zz
x
(y(z)) f(y)dy
so we have
1(z; x) = (x(z)) F(z)if xz
which is not more than 1(x; x). So no deviation strictly increases a bidder’s payoff
and (x)is still an equilibrium in the presence of resale.
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