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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
8
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
55
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
57
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
58
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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