Auction Theory, 2nd Edition

9.4 Three or More Bidders 135

p

v

i

v

v

j

X

j

X

j

Values

v

i

5

p

v

j

5

p

v

5

p

X

i

FIGURE 9.2 Average crossing condition.

p. In that case, the average value is also p and the left-hand panel depicts the

inequalities in (9.5) directly. As in Figure 9.1, the right-hand panel of Figure 9.2

shows the three iso-value curves and their relative slopes in (x

j

,x

i

) space.

9.4 THREE OR MORE BIDDERS

The main result of this chapter is as follows:

Proposition 9.2. Suppose that the valuations v satisfy the average crossing

condition. Then there exists an ex post equilibrium of the English auction that

is efficient.

Before proceeding with the proof of Proposition 9.2, it is worthwhile to

understand ...