Fractionally subadditive valuation

A set function is called fractionally subadditive, or XOS (not to be confused with OXS), 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, $$M := \{1,\ldots,m\}$$.

There is a function $$v$$ which assigns a number to each subset of $$M$$.

The function $$v$$ is called fractionally subadditive (or XOS) if there exists a collection of set functions, $$\{a_1,\ldots,a_l\}$$, such that:
 * Each $$a_j$$ is additive, i.e., it assigns to each subset $$X\subseteq M$$, the sum of the values of the items in $$X$$.
 * The function $$v$$ is the pointwise maximum of the functions $$a_j$$. I.e, for every subset $$X\subseteq M$$:
 * $$v(X) = \max_{j=1}^l a_j(X)$$

Equivalent Definition
The name fractionally subadditive comes from the following equivalent definition: a set function $$ v $$ is fractionally subadditive if, for any $$ S\subseteq M$$ and any collection $$ \{\alpha_i, T_i\}_{i=1}^k$$ with $$\alpha_i > 0$$ and $$T_i\subseteq M$$ such that $$ \sum_{T_i \ni j} \alpha_i \ge 1$$ for all $$j\in S$$, we have $$ v(S) \le \sum_{i=1}^k \alpha_i v(T_i) $$.

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.