13 Equilibrium and E¢ ciency with Private Values
Problem 13.1 (Uniform price auction) Consider a three-unit uniform-price auction
with two bidders. Each bidder’s value vector Xi=Xi
1; Xi
2; Xi
3is independently and
identically distributed on the set X=fx2[0;1]3:x1x2x3gaccording to a
density function fsuch that the marginal distributions are:
F1(x1)=(x1)2
F2(x2) = (2 x2)x2
F3is left unspecified. Show that the bidding strategy (x1; x2; x3)=(x1;(x2)2;0)
constitutes a symmetric equilibrium of the uniform-price auction.
Solution. Suppose one bidder with values (x1; x2; x3)bids (b1; b2; b3)where 1b1
b2b30:Let (y1; y2; y3)be the values for the other bidder. The expected payo¤
of the bidder is
= Zfy1<b3g
[x1+x2+x33y1]dF (y1; y2)
which is equivalent to
= Zfy1<b3g
[x1+x2+x33y1]dF (y1; y2)
and then to
= x1b2+Zfy1<b3g
[x33y1+ 2b3]dF1(y1)
52
Finally,
= x1b2+Zb3
0
(x33y1+ 2b3) 2y1dy1
+Zpb2
0x22 (y2)2+b2(2 2y2)dy2
+Zpb3
where the second equality comes from b1>0and y1y2
2. So the marginal payo¤s
are @
@b1
= 0
@
@b2
=x2b1
2
211b
1
2
2
@
1
2
It is easy to check that every marginal payo¤ is zero when (b1; b2; b3) = x1;(x2)2;0:
Moreover, @
Problem 13.2 (Uncertain supply) Consider a multiunit uniform-price auction with
Nbidders each of whom has use for one unit only. At the time of bidding, the actual
number of units that will be available for sale is uncertain and could range anywhere
between 1and Kwhere K < N: Show that it is a weakly dominant strategy for each
bidder to bid his or her value.
Solution. The expected payo¤ for one bidder with value vbids bis
K
X
k=1
where Pr (k)is the probability of kunit supply and (b; v ; k)is his expected payo¤
conditional on kunits of supply. In section 13.4, we already know it is weakly
dominant strategy for him to bid his value without supply uncertainty, so
53
so K
X
k=1
K
X
k=1
which means it is also weakly dominant strategy to bid his value with uncertain
supply.
Problem 13.3 (Multiple equilibria) Consider a two-unit uniform-price auction with
two bidders. Each bidder’s value vector Xiis identically and independently distributed
so that the marginal distributions of the values of both goods is uniform, that is,
F1(x1) = x1and F2(x2) = x2:Show that for any increasing function (z)such that
0(z)z; the bidding strategy (x1; x2)=(x1; (x1)) constitutes a symmetric
equilibrium.
Solution. Suppose bidder 1 has value (x1; x2)and the other bidder’s strategy is
(x1; x2) = (x1; (x1)). We already know that he bids x1for the first unit in Section
13.4, so suppose he deviates to (x1; b);then his expected payo¤ is:
0
b
1(b)
The marginal expected payo¤ is
@
@b = 2(x1b) + 1
0(1(b)) 1(x1b)1(b)b
1
It is easy to check that when b=(x1);@
@b = 0. Moreover, if b>(x1),@
@b <0and
if b<(x1),@
54