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6 Auctions with Interdependent Values
Problem 6.1 (A¢ liation) Suppose there are two bidders who receive private signals
X1and X2which are jointly distributed over the set
S=n(x1; x2)2[0;1]2:px1x2(x1)2o
with a uniform density. The bidders attach a common value V=1
2(X1+X2)to the
object.
a. Find symmetric equilibrium bidding strategies in both a first-price and a second-
price auction?
b. Calculate the expected revenues from both auctions and show that the revenue
in a second-price auction is greater than that in a first-price auction.
Solution. Part a. First, we apply Proposition 6.3 to find the equilibrium strategy
for the first-price auction.
The area of the set Sis
Z1
0px1x2
1dx1=1
3
so the density function of X1and X2is f(x1; x2) = 3. Let G( j x)be the distribution
of X2conditional on X1=xand let g( j x)be the associated conditional density
function, then we have
G(yjx) = Ry
x2f(s; x)ds
Rpx
pxx2
g(yjx) = f(y; x)
Rpx
pxx2
Therefore,
L(yjx) = exp Zx
y
g(tjt)
G(tjt)dt
1
25
Proposition 6.3 implies that the symmetric equilibrium strategies for the first-price
auction are given by
I(x) = Zx
0
v(y; y)dL(yjx)
y
Second, Proposition 6.1 implies the symmetric equilibrium strategies in the second-
price auction are given by
Part b. Let us first consider the first-price auction. The expected payment from
bidder iwith value xis the product of his bid and his probability of winning
mI(x) = I(x)G(xjx)
xln(1 x)xx2
So the expected revenue for the seller is
ERI= 2EmI(Xi)
= 2 Z1
0
mI(x)fi(x)dx
3
where the third equality comes from the fact that the density of the marginal distri-
bution of Xiis
x2
x2
In the second-price auction, the expected payment from the bidder with value x
is
26
So the corresponding expected revenue for the seller is
E[RII]=2EmII (Xi)
2
Thus, ERI< E RII.
Problem 6.2 (Lack of a¢ liation) Suppose there are two bidders whose private values
X1and X2which are jointly distributed over the set
S=n(x1; x2)2[0;1]2:x1+x21o
with a uniform density. Show that a first-price auction with this information structure
does not have a monotonic pure strategy equilibrium.
Solution. This proof is taken from Section 6 in Reny and Zamir (2004). Suppose
to the contrary that such an equilibrium exists, and denote the equilibrium bidding
strategies as 1()and 2().
Let us first show that x22(x2)for all x22[0;1). Suppose not and there
exists some ^x22[0;1) such that 2(^x2)>^x2. Since 2is nondecreasing, 2(x2)> x2
satisfies
therefore bidder 1 has negative payo¤. Third, if bidder 2 wins the object, he also has
negative payo¤ because 2(x2)> x2. So there is a positive probability that at least
For any "2(0;1), suppose bidder ihas value 1". He knows that the other
bidder’s value is below ", so are the other bidder’s bids. Thus i’s bid should be
Problem 6.3 (Bidding gap) Consider a common value second-price auction with
two bidders. The bidders receive signals X1and X2, respectively, and these are in-
dependently and uniformly distributed on [0;1]. Each bidder’s value for the object is
V=1
2(X1+X2).
a. If the seller sets a reserve price r > 0;show that there is no bid in the neigh-
borhood of the reserve price.
b. Find the optimal reserve price.
27
Solution. Part a. Let the symmetric equilibrium bidding function be (). If
bidder 1with value xbids br, his expected payo¤ is
(b; x) = Z1(b)
0
v(x; y)g(yjx)dy rG 1(r)jx
0
2dy r1(r)Z1(b)
1(r)
Maximizing it with respect to byields the first-order condition
x+1(b)
At symmetric equilibrium, b=(x)so the first-order condition becomes
x+x
So the equilibrium bidding strategy is
(x) = x
The corresponding payo¤ is
((x); x) = Zx
0
x+y
2dy r2Zx
r
ydy
=1
which means that ((x); x)0only when xp2r. Therefore only bidders with
values in p2r; 1will participate in the auction, and there is a gap between reserve
Part b. Suppose the object has no value to the seller. The expected revenue for
the seller from setting a reserve price r > 0is
p2r
So the optimal reserve price is 0.
Problem 6.4 (Common value auction with Pareto distribution) Suppose that two
bidders are competing in a first-price auction for an object with common value V;
which is distributed according to a Pareto distribution FV(v)=1v2over [1;1):
Prior to bidding, each bidder receives a signal Xiwhose distribution, conditional on
the realized common value V=v, is uniform over [0; v] ; that is, FXijV(xijv) =
28
(xi=v):Conditional on V=v; the signals X1and X2are independently distributed.
Verify that the following strategy constitutes a symmetric equilibrium of the first-price
auction
(x) = 2
3max fx; 1g(1 + max fx; 1g2)
Solution. Part a. Let the symmetric equilibrium bidding function be (). If
bidder 1with value xbids br, his expected payo¤ is
(b; x) = Z1(b)
0
v(x; y)g(yjx)dy rG 1(r)jx
Maximizing with respect to byields the first-order condition
x+1(b)
At a symmetric equilibrium, b=(x)so the first-order condition becomes
x+x
The corresponding payo¤ is
((x); x) = Zx
0
x+y
2dy r2Zx
r
ydy
=1
which means that ((x); x)0only when xp2r. Therefore only bidders with
p2r
So the optimal reserve price is 0.
29
The density function of joint distribution of X1; X2; V is
so the the density of joint distribution of X1and X2is
h(x1; x2) = Z+1
maxfx1;x2;1g
h(x1; x2; v)dv
=Z+1
2v5dv
and the density function of Vconditional on X1=x1and X2=x2is
h(vjx1; x2) = h(x1; x2; v)
The distribution of X2given X1=x1is
G(x2jx1) = Zx2
0
h(x1; t)dt =Zx2
0
1
2max fx1; t; 1g4dt
and its density function is
and we also have
v(x; y)E[VjX1=x; X2=y]
=Z+1
vh (vjx; y)dv
Note that v(y; y)is no longer strictly increasing in yand (x)is not strictly
increasing neither, so the proof of Proposition 6.3 does not apply to this question.
30
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0
1
2
3
4
y
Figure S6.1
0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0
0
1
2
3
4
x
Figure S6.2
31
First, if a bidder with value x > 1bids (z)where z > x; his expected payo¤ is
(z; x) = Zz
0
[v(x; y)(z)] g(yjx)dy
=Zz
04
3max fx; yg 2z
31 + z2g(yjx)dy
=Zx
It is easy to see that
@ (z; x)
So we have
Similarly, if bidder 1 with value x > 1bids (z)where 1< z < x, his expected
payo¤ is
(z; x) = Zz
04
3max fx; yg 2z
31 + z2g(yjx)dy
=Zz
Then we have
If bidder 1 with value x > 1bids (z)where 0z < 1, he can only win the
object in a tie and his expected payo¤ is
(z; x) = 1
2Z1
04
3x4
31
2x4dy
1
1
32
so
(z; x)< (x; x)for 0z < 1< x (10)
Therefore (8), (9) and (10) imply that the bidder with value x > 1bids (x).
Second, let us consider the bidder with value 0x1. If he bids (z)where
z2[0;1], his expected payo¤ is
(z; x) = 1
so
(z; x) = (x; x)for 0z1and 0x1(11)
If the bidder bids (z)where z > 1, his expected payo¤ is
(z; x) = Zz
0v(x; y)2z
31 + z2g(yjx)dy
=Z1
therefore
(z; x)< (1; x) = 0 for z > 1and 0x1(12)
Hence (11) and (12) imply that the bidder with value 0x1bids (x).
Problem 6.5 (Discrete values and signals) Consider a common value first-price auc-
tion with two bidders. The common value Vcan take on only two values, 0and 1:
Prior to the auction, each bidder ireceives a signal Xiwhich can also take on only
two values, 0and 1:The joint distribution of the three random variables V,X1and
X2is:
Pr [V=v; X1=x1; X2=x2] = 8
<
:
2=9if x1=x2=v
1=18 if x1=x26=v
1=9if x16=x2
Show that the following strategy constitutes a symmetric equilibrium.
a. A bidder with signal Xi= 0 bids E[VjX1= 0; X2= 0] = 1
5:
b. A bidder with signal Xi= 1 bids randomly over the interval 1
5;8
15 according
to the distribution
G(b) = 4
55b1
45b
(Since the signals and values are discrete, the equilibrium is in mixed strategies.)
33
Solution. The distribution in the question could be illustrated in the table below
V X1X2Probability
1 1 1 2/9
1 0 0 1/18
1 1 0 1/9
First of all, given bidder 2’s strategy, bidder 1 can not win by bidding below
bidder 2’s lowest bid, 1
If bidder 1 with value X1= 0 bids b21
5;8
15 , his expected payo¤ is
(b; 0) = 1
18 (1 b) + 1
9(1 b)G(b) + 2
9(0 b) + 1
9(0 b)G(b)
=1
Because b21
5;8
15 ;the expected payo¤ is negative unless b=1
5, hence bidder 1
with value 0 will not deviate from the strategy and bid 1
5:
If bidder 1 has value X1= 1, we need to show that any bid in 1
5;8
15 gives him
the same expected payo¤. Suppose the bidder bid any b21
5;8
15 , his expected payo¤
is
(b; 1) = 2
9(1 b)G(b) + 1
9(1 b) + 1
18 (0 b)G(b) + 1
9(0 b)
=2
So bidder with value 1 also does not have any incentive to deviate. So the strategy
constitutes a symmetric equilibrium.
8 Asymmetries and Other Complications
Problem 8.1 (Reserve price) Consider a first-price auction with Nbidders who
have private values X1; : : : ; XN:The vector (X1; : : : ; XN)is distributed over [0;1]N
according to a density function fwhich is a¢ liated and symmetric. Suppose that the
seller sets a small reserve price r > 0:Show that the following constitutes a symmetric
34
