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 non-negative 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 when restricted to non-negative additive functions: 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 ${\displaystyle v(S)\leq \sum _{i=1}^{k}\alpha _{i}v(T_{i})}$.

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.

Etymology

The term XOS stands for XOR-of-ORs of Singleton valuations.^[4]

A Singleton valuation is a valuation function ${\displaystyle v(\cdot )}$ such that there exists a value ${\displaystyle w}$ and item ${\displaystyle i}$ such that ${\displaystyle v(S):=w}$ if and only if ${\displaystyle i\in S}$, and ${\displaystyle v(S):=0}$ otherwise. That is, a Singleton valuation has value ${\displaystyle w}$ for receiving item ${\displaystyle i}$ and has no value for any other items.

An OR of valuations ${\displaystyle \{v_{1}(\cdot ),v_{2}(\cdot ),\ldots ,v_{k}(\cdot )\}}$ interprets each ${\displaystyle v_{j}(\cdot )}$ as representing a distinct player. The OR of ${\displaystyle \{v_{1}(\cdot ),v_{2}(\cdot ),\ldots ,v_{k}(\cdot )\}}$ is a valuation function ${\displaystyle v(\cdot )}$ such that ${\displaystyle v(S):=\max _{S_{1},\ldots ,S_{k}{\text{ s.th. }}S_{j}\cap S_{\ell }=\emptyset \ \forall j,\ell {\text{ and }}\cup _{j}S_{j}=S}\{\sum _{j=1}^{k}v_{j}(S_{j})\}}$. That is, the OR of valuations ${\displaystyle \{v_{1}(\cdot ),v_{2}(\cdot ),\ldots ,v_{k}(\cdot )\}}$ is the optimal welfare that can be achieved by partitioning ${\displaystyle S}$ among players with valuations ${\displaystyle \{v_{1}(\cdot ),v_{2}(\cdot ),\ldots ,v_{k}(\cdot )\}}$. The term "OR" refers to the fact that any of the players ${\displaystyle \{v_{1}(\cdot ),v_{2}(\cdot ),\ldots ,v_{k}(\cdot )\}}$ can receive items. Observe that an OR of Singleton valuations is an additive function.

An XOR of valuations ${\displaystyle \{v_{1}(\cdot ),\ldots ,v_{k}(\cdot )\}}$ is a valuation function ${\displaystyle v(\cdot )}$ such that ${\displaystyle v(S):=\max _{j}\{v_{j}(S)\}}$. The term "XOR" refers to the fact that exactly one (an "exclusive or") of the players can receive items. Observe that an XOR of additive functions is XOS.