Fractionally subadditive

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.[1] The term fractionally-subadditive was given by Uriel Feige.[2]

Definition

There is a finite base set of items, ${\displaystyle M:=\{1,\ldots ,m\}}$.

There is a function ${\displaystyle v}$ which assigns a number to each subset of ${\displaystyle M}$.

The function ${\displaystyle v}$ is called fractionally-subadditive (or XOS) if there exists a collection of set functions, ${\displaystyle \{a_{1},\ldots ,a_{l}\}}$, such that:[3]

• Each ${\displaystyle a_{j}}$ is additive, i.e., it assigns to each subset ${\displaystyle X\subseteq M}$, the sum of the values of the items in ${\displaystyle X}$.
• The function ${\displaystyle v}$ is the pointwise maximum of the functions ${\displaystyle a_{j}}$. I.e, for every subset ${\displaystyle X\subseteq M}$:

${\displaystyle v(X)=\max _{j=1}^{l}a_{j}(X)}$

Equivalent Definition

The name fractionally subadditive comes from the following equivalent definition: a set function ${\displaystyle v}$ is fractionally subadditive if, for any ${\displaystyle S\subseteq M}$ and any collection ${\displaystyle \{\alpha _{i},T_{i}\}_{i=1}^{k}}$ with ${\displaystyle \alpha _{i}>0}$ and ${\displaystyle T_{i}\subseteq M}$ such that ${\displaystyle \sum _{T_{i}\ni j}\alpha _{i}\geq 1}$ for all ${\displaystyle j\in S}$, we have ${\displaystyle v(S)\leq \sum _{i=1}^{k}\alpha _{i}v(T_{i})}$. This can be viewed as a fractional relaxation of the definition of subadditivity.

Relation to other utility functions

Every submodular set function is XOS, and every XOS function is a subadditive set function.[1]

See also: Utility functions on indivisible goods.

References

1. Nisan, Noam (2000). "Bidding and allocation in combinatorial auctions". Proceedings of the 2nd ACM conference on Electronic commerce - EC '00. p. 1. doi:10.1145/352871.352872. ISBN 1581132727.
2. Feige, Uriel (2009). "On Maximizing Welfare when Utility Functions Are Subadditive". SIAM Journal on Computing. 39: 122–142. CiteSeerX 10.1.1.86.9904. doi:10.1137/070680977.
3. Christodoulou, George; Kovács, Annamária; Schapira, Michael (2016). "Bayesian Combinatorial Auctions". Journal of the ACM. 63 (2): 1. CiteSeerX 10.1.1.721.5346. doi:10.1145/2835172.