15 Sequential Sales
Problem 15.1 (Power distribution) Consider a situation in which two identical ob-
jects are to be sold to three interested bidders in two auctions conducted sequentially.
Each bidder has use for at most one item— there is single-unit demand. Bidders’pri-
vate values are identically and independently distributed according to the distribution
F(x) = x2on [0;1] :
a. Find a symmetric equilibrium bidding strategy if a sequential first-price format
is used.
b. Find a symmetric equilibrium bidding strategy if a sequential second-price for-
mat is used.
c. Compare the distribution of prices in the two auctions under the first- and
second-price formats.
Solution. Part a. Using (15.4) with F(x) = x2, we find
I
2(x) = 1
F(x)Zx
0
ydF (y)
Using (15.6), we find
I
1(x) = 1
F2(x)Zx
0
2(y)dF 2(y)
=1
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Part b. By Proposition 15.3,
II
1(x) = 2
3x
II
Part c. Renumber the bidders so that X1> X2> X3. The distribution of prices
from the first-price sequential auction is given by
LI(p1; p2) = Pr maxfI
1(X1); I
1(X2); I
1(X3)g p1;maxfI
2(X2); I
2(X3)g p2
= Pr 8
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And the price distribution for the second-price sequential auction is
LII(p1; p2) = F3
:
It is clear that LILII so prices in the second-price sequential auction stochastically
dominate those from the first-price.
Problem 15.2 (Multiunit demand) Consider a situation in which two identical ob-
jects are to be sold to two interested bidders in two second-price auctions conducted
sequentially. Bidders have multiunit demand with values determined as follows. Each
bidder draws two values Z1and Z2from the uniform distribution F(z) = zon [0;1] :
The bidder’s value for the first unit is X1= max fZ1; Z2gand his marginal value for
the second unit is X2= min fZ1; Z2g:(This is just an instance the multiuse model
discussed in Chapter 13.)
a. Show that the following strategy constitutes a symmetric equilibrium of the
sequential second-price format:
i. in the first auction, bid 1xi
1; xi
2=1
2xi
1; and
ii. in the second auction, bid truthfully— that is, bidder ibids xi
2if he won
the first auction; otherwise he bids xi
1.
b. Show that the sequence of equilibrium prices (P1; P2)is a submartingale—that
is, E[P2jP1=p1]p1and with positive probability, the inequality is strict.
Solution. Part a. We will show that 1(x1; x2) = 1
2x1is the unique strictly
increasing strategy that only depends on the first-period value. Because the first-
period strategy in the proposed equilibrium depends only on the first-period value,
expected payoff is
+(t;x) = Zt
0
(x1(y1))d[F2(y1)] + Zx2
0
(x2y1)d[F2(y1)]
0
The first line shows his expected payoff if the first round is won; and the second,
his expected payoff from losing the first round. As long as bidder 2 has not won
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payoff is
(t;x) = Zt
0
(x1(y1))d[F2(y1)] + Zx2
0
(x2y1)d[F2(y1)]
tZy1
0
On the first line is his expected payoff from winning the first round; on the second,
from losing to an opposing bid y1> x1; and on the third, from losing to y1x1. In
other tgives weakly lower payoff. The first-order conditions are
@+(t; x)
0and @(t; x)
0:
Written out, these are
2f(x1)F(x1)(x1(x1)) 2f(x1)Zx1
0
(x1y2)f(y2)dy20and
0
Putting these together and re-arranging gives a formula for :
Plugging in F(x) = xand f(x) = 1 gives (x1) = 1
2x1, as desired. This demonstrates
that satisfies the necessary first-order condition. However, we need to show that
bidding according to is optimal when the other player is playing . So we must
show that (x1)gives weakly higher payoff than bidding as type tx2. The payoff
from tx2is
(t;x) = Zt
0
(x1(y1))d[F2(y1)] + Zt
0
(x2y1)d[F2(y1)]
tZy1
0
where each line has the same interpretation as for , the only difference being that
now, upon winning in the first round, he knows y1t. Taking the derivative and
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Y1Y2the other bidder’s values. There are three equilikely orderings of these values,
so we can express the desired expectation in terms of them:
E[P2jP1=p1] = 1
3E[P2jX1> X2> Y1> Y2; P1=p1]
+1
In the first case P1=(Y1) = 1
2Y1and P2=Y1. Because is strictly increasing,
knowing P1=p1means knowing that the price tomorrow will be ;1(p1) = 2p1.
32p1. Plugging
these into the equation above, we have
E[P2jP1=p1] = 1
Problem 15.3 (Multiunit demand) Consider the same environment as in the previ-
ous problem.
a. Show that the following strategy also constitutes a symmetric equilibrium of
the second-price format:
i. in the first auction, bid 1xi
1; xi
2=xi
2; and
ii. in the second auction, bid truthfully.
b. What can you say about the resulting sequence of equilibrium prices (P1; P2)?
Solution. Part a. The following argument is similar to that found in Problem 15.2.
We will show that 1(x1; x2) = x2is the unique strictly increasing strategy that only
payoff from bidding as type t2(x2; x1)in the first round is
+(t;x) = 2 Zt
0Z1
y2
(x1(y2))dF (y1)dF (y2)
+ 2 Zx2
(x2y1)dF (y1)dF (y2)
tZ1
y2
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The first two lines are the payoff from winning the first round; and the last is the
payoff from losing it. The payoff from bidding t < x2is
(t;x) = 2 Zt
0Z1
y2
(x1(y2))dF (y1)dF (y2)
+ 2 Zt
(x2y1)dF (y1)dF (y2)
tZ1
y2
The first two lines are the payoff from winning the first round; and the last is the
payoff from losing it. The first-order conditions lead to a characterization of :
Z1
x2
x2
This is clearly satisfied by (x2) = x2. We still need to show that the payoff from
Part b. Consider the three possible orderings identified in Problem 15.2.a, above:
So the price sequence in this equilibrium is increasing.
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