Auction Theory, 2nd Edition

9.2 Two-Bidder Auctions 131

p

v

i

v

j

X

j

Values

v

i

5 p

v

j

5 p

X

i

X

j

FIGURE 9.1 Single crossing condition.

the same and maximal. The single crossing condition is depicted in Figure 9.1.

In the left-hand panel, the slope of v

j

with respect to x

j

is greater than that of v

i

whenever they have the same value. The right-hand panel depicts the iso-value

or “indifference” curves of the two bidders, both at the same level, p. Whenever

they intersect, bidder j’s indifference curve is steeper, so they can cross at most

once.

It will be convenient to denote the partial derivative of bidder i’s valuation

with respect to bidder j’s signal by v



ij

—that is,

v



ij

(

x

)

≡

∂v

i

∂x

j

(

x

)

(9.2)