Auction Theory, 2nd Edition

8.4 Reserve Prices and Entry Fees 127

Pinkse and Tan (2005) point to a curious phenomenon in first-price auctions

with affiliated private values; an increase in the number of bidders may cause

a bidder with a given value to actually decrease his bid. Specifically, suppose

that there is random variable Z and each bidder’s value X

i

is independently

drawn from a distribution F

X|Z

(x |z ). Thus bidders’ values are conditionally

independent. Under weak assumptions, the values X

1

,X

2

,..., X

N

are affiliated.

As the number of bidders increases, a bidder ascribes greater probabilities to

low values of Z and thus of the rival’s values as well. This may well overwhelm

the competitive effect of the increased number of bidders and cause him to bid

lower. Problem 8.2 explores ...