Auction Theory, 2nd Edition

134 Efficiency and the English Auction

β

2

(

x

2

)

>β

3

(

x

3

)

, so bidder 3 drops out first. Now if bidder 1’s signal is x

1

−ε,

bidder 3 has the highest value and it is inefficient for him to drop out. This is a

contradiction, so there cannot be an efficient equilibrium.



The single crossing condition is a bilateral condition—it is separately

applied to pairs of bidders—and, as we have seen, is not sufficient to guarantee

that the English auction has an efficient equilibrium once there are three or more

bidders. We now introduce a multilateral extension of the single crossing con-

dition, called the average crossing condition, that links the valuations of all the

bidders ...