Auction Theory, 2nd Edition

16.4 Bundling 231

so the substitute property implies that x

i

(

·

)

is a subadditive function over the set

of packages. When we say that the objects in K are substitutes, we mean that all

buyers consider them as such.

In analogous fashion, we will say that the objects are complements if the

marginal value of obtaining a particular object a is larger if the set of objects

already in hand is “larger.” Formally, buyer i considers the objects in K to be

complements if for all a ∈K and packages S and T not containing a , such that

S ⊂T ,

x

i

(

S ∪

{

a

}

)

−x

i

(

S

)

≤x

i

(

T ∪

{

a

}

)

−x

i

(

T

)

(16.8)

This is equivalent to requiring that for all packages S and T,

x

i

(

S

)

+x

i

(

T

)

≤x

i

(

S ∪T

)

+