A set function is called fractionally subadditive (or XOS) if it is the maximum of several additive set functions. This valuation class was defined, and termed XOS, by Noam Nisan, in the context of combinatorial auctions. The term fractionally subadditive was given by Uriel Feige.
Definition
There is a finite base set of items, .
There is a function which assigns a number to each subset of .
The function is called fractionally subadditive (or XOS) if there exists a collection of set functions, , such that: Each is additive, i.e., it assigns to each subset , the sum of the values of the items in . The function is the pointwise maximum of the functions . I.e, for every subset :
Equivalent Definition
The name fractionally subadditive comes from the following equivalent definition: a set function is fractionally subadditive if, for any and any collection with and such that for all , we have .
Relation to other utility functions
Every submodular set function is XOS, and every XOS function is a subadditive set function.
See also: Utility functions on indivisible goods.