Auction Theory, 2nd Edition

264 Multiple Objects and Interdependent Values

SEPARABILITY AND EFFICIENCY

Suppose that set K of distinct objects, labeled a, b, c, ..., and so on, are for sale.

Each buyer receives a multidimensional signal

x

i

=



x

i

a



a∈K

with the interpretation that x

i

a

is the one-dimensional information received by

buyer i that pertains to object a. Buyers’ valuations for the different objects are

informationally separable—that is, the value of an object a ∈K to buyer i is of

the form

v

i

a



x

1

a

,x

2

a

,..., x

N

a



In other words, the value to i of obtaining object a depends on the signals of all

buyers that pertain only to object a, and not on signals x

j

b

, say.

We also need that for each a ∈K, the valuations v

1

a

,v

2

a

,..., v

N

a

satisfy the

single crossing condition.

Suppose further ...