Auction Theory, 2nd Edition

2.4 Revenue Comparison 17

0

Uniform Distribution Exponential Distribution (

5

2)

X

1

1

0

X

3

1

2

1

2





FIGURE 2.2 Equilibria of two-bidder symmetric first-price auctions.

In this case, the equilibrium strategy calls upon a bidder to bid a constant fraction

of his value. For the case of two bidders, the equilibrium bidding strategy is

depicted in the left-hand panel of Figure 2.2. 

Example 2.2. Values are exponentially distributed on [0,∞), and there are only

two bidders.

If F(x) =1 −exp(−λx), for some λ>0, and N =2, then

β

I

(x) = x −



x

0

F(y)

F(x)

dy

=

1

λ

−

x exp(−λx)

1−exp(−λx)

As a particular instance, consider the case where λ =2 so that E[X]=

1

2

. The

equilibrium bidding ...