Auction Theory, 2nd Edition

88 Auctions with Interdependent Values

the purposes of this chapter, the following three implications of affiliation are

sufficient.

First, define, as usual, the random variables Y

1

,Y

2

,..., Y

N−1

to be the

largest, second-largest, … , smallest from among X

2

,X

3

,..., X

N

. If the variables

X

1

,X

2

,..., X

N

are affiliated, then the variables X

1

,Y

1

,..., Y

N−1

are also affiliated.

Second, let G(·|x) denote the distribution of Y

1

conditional on X

1

=x. Then

the fact that X

1

and Y

1

are affiliated implies that if x



> x, then G(·|x



) dominates

G(·|x) in terms of the reverse hazard rate—that is, for all y,

g( y |x



)

G( y |x



)

≥

g( y |x)

G( y |x)

(6.1)

Third, if γ is any increasing function, ...