11 Bidding Rings
Problem 11.1 (Maximal loss from collusion) Consider a second-price auction with
N2bidders. Each bidder’s private value Xiis independently and uniformly dis-
tributed according on [0;1] :
a. First, suppose bidders bid individually— that is, there is no bidding ring. As
a benchmark, find the expected revenue of the seller if he sets an optimal reserve
price r>0(as in Chapter 2).
b. Now suppose that the Nbidders form a perfectly functioning bidding ring. Find
the expected revenue of the seller if he sets an optimal reserve price r >0in
the face of such collusion.
c. Show that for all n; the optimal revenue with collusion is at least one-half of
the optimal revenue without collusion; that is, >1
2.
Solution. Part a. Each player’s expected payment under a reserve price rcan be
found using (2.9) and G(y) = Fn1(y):
mII(x; r) = rN+Zx
r
(N1)yN1dy
=1
The seller’s expected revenue is the sum of the bidders’expected payments:
E[R] = NZ1
r
mII(y; r)dy
The seller maximizes this to find the optimal reserve price r=1
2. We could also
find this by noting that the seller values the object at x0= 0 and the hazard rate
(x) = 1=(1 x)is increasing in x, so (2.12) is a su¢ cient condition for an optimal
reserve price r. Plugging this into the expression above, we find the seller’s expected
revenue under the optimal reserve price:
Part b. Because the ring consists of all bidders, it does not need to protect itself
from outside bids. If no bidder values the object at least at r, the ring will not bid.
If any ring member has a value at or above r, the ring will submit a bid of r. Hence,
the seller’s expected revenue from setting a reserve price of ris
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1)N+1=N .
Part c. Because r is optimal, the revenue from setting r=1
2instead is less
than or equal to (with equality at N= 1). The first line below uses this fact.
1
21
211
2N1
2
1
2
N+N1
N+ 1
>0
Problem 11.2 (Collusion in first-price auctions) Consider a first-price auction with
three bidders. Bidder 1’s value X1=3
4with probability 3
4and X1=1
2with probability
1
4:Bidders 2and 3have fixed and commonly known values. Specifically, x2= 1 and
x3=1
4:
a. Find an equilibrium of the first-price auction when the three bidders act inde-
pendently. (Note: Since values are discrete, this will be in mixed strategies).
b. Now suppose that bidders 1and 2form a cartel. While the cartel cannot con-
trol the bids submitted by its members, it can arrange transfers and recommend bids
.Further, suppose that the values of its members become commonly known among the
cartel once it is formed.
i. Find an equilibrium with the cartel, assuming that bidder 3acts indepen-
dently.
ii. Is it possible for the cartel to ensure that only one member submits a bid?
Solution. Part a. Suppose bidder 2 always bids b2=3
4and wins, for a payo¤ of
0.25. Bidder 2 is already winning, so bidding more is not better. The strategies of
bidders 1 and 3 must ensure that bidding less is also a bad idea. Let bidder 3 always
bid zero. Suppose that bidding b < 3
4gives 2 an expected payo¤ of b=3<0:25. Then
bidder 1’s mixed strategy F1(b)can be solved for b2[0;3=4] by setting
and so for b2[0;3=4]
1b
It is straightforward to verify that F1is indeed a distribution function and that
bidders 1 and 3 cannot do better than follow the strategies given above. There are
many other equilibria leading to the same outcome (allocation of the object and
payments). For example, bidder 1’s strategy may depend on his value.
Part b.i. Suppose the cartel makes no transfers and always lets bidder 2 have
the object. The following argument follows that of part a. Bidder 2 bids b2=x1for
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a payo¤ of 1x1; and bidder 1 uses the mixed strategies F1(bjx1)so that 2 gets
payo¤ b(1 x1)=x1<1x1from bidding b < x1:
F1bjx1=1
2=b
1bfor b2[0;1=2]
F1bjx1=3
Again, let bidder 3 drop out: b3= 0. It remains to verify that bidders 1 and 3 are
behaving optimally and that F1( j x1)is a distribution function for each x1.
Part b.ii. No, the cartel cannot enforce a one-bidder policy. Suppose bidder 1
has value x1and does not submit a bid. If the support of bidder 3’s strategy extends
to y > 1
4, then bidder 1 will bid b1< y, trading o¤ a lower payment and a lower
Problem 11.3 (PAKT) A single object is to be sold via a second-price auction to
two bidders whose private values Xiare drawn independently from the uniform distri-
bution on [0;1]. Suppose that the bidders form a cartel. Find the equilibrium bidding
strategies in the preauction knockout (PAKT).
Solution. First, consider the second-price PAKT. As argued in the book, this is
an incentive-compatible and individually rational direct mechanism. Hence truth-
ful reporting is an equilibrium. Next, suppose the bidders use a first-price PAKT.
Equilibrium bidding strategies are given in Proposition 11.3:
4x
Problem 11.4 (Collusion-proof mechanism) A single object is to be sold to two bid-
ders with private values drawn independently from the uniform distribution on [0;1].
The following mechanism is used to sell the object. Each bidder isubmits a bid bi:
Suppose that bi> bj:Then the loser, bidder j, is asked to pay a fixed amount 1
3to the
seller. The winner, bidder i, is awarded the object and asked to pay bito the losing
bidder j. (If there is a tie, either bidder is assigned the role of a winner.)
a. Find a symmetric equilibrium of this mechanism assuming that the bidders act
noncooperatively.
b. Can the two bidders gain by forming a cartel and colluding against the seller?
Solution. Part a. Suppose is a symmetric equilibrium. If bidder 1 bids (z)
when his value is x, his expected payo¤ in this equilibrium is
Zz
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The first-order condition evaluated at z=xis
This can be solved using the method of integrating factors. First, rearrange so that
the ’s are on one side:
Next, multiply by x2= exp R2
xdxg.
x20(x)+2x(x) = x2+x=3
d
Integration gives the solution:
Part b. No. Roughly speaking, the bidders cannot take the seller’s piece of
the pie; nor can they make the pie larger. Therefore they have nothing to gain
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