Auction Theory, 2nd Edition

13.4 Uniform-Price Auctions 195

Demand reduction (or bid shading) can be severe when the number of

bidders is small relative to the number of units for sale. An extreme case of this

phenomenon occurs in the following example.

Example 13.1. There are two units for sale and two bidders with value vectors

X that are identically and independently distributed according to the density

function f (x) =2 on

X ={x ∈

[

0,1

]

2

: x

1

≥x

2

}.

With this distribution of values, a symmetric equilibrium of the uniform-

price auction is β

1

(x

1

,x

2

) =x

1

and β

2

(x

1

,x

2

) =0. In every realization, each

bidder wins one unit and the price is zero!

To see this, suppose that bidder 2 is following ...