Maximin share of a set function

Dependencies:

1. Set function
2. /analysis/sup-inf

$\newcommand{\Fcal}{\mathcal{F}}$ $\newcommand{\MMS}{\operatorname{MMS}}$ Let $\Omega$ be a set and $\Fcal \subseteq 2^{\Omega}$. Let $f: \Fcal \to \mathbb{R}$ be a function.

For any set $S \subseteq \Omega$, let $\Pi_n(S)$ denote the set of all $n$-partitions of $S$, i.e., all (unordered) tuples of the form $X = (X_1, X_2, \ldots, X_n)$ such that $X_i \in \Fcal$ for all $i$, $X_i \cap X_j = \emptyset$ for all $i \neq j$, and $\bigcup_{i=1}^n X_i = S$.

Then for any $S \subseteq \Omega$, $f$'s $n$-part maximin share of $S$ is defined as \[ \MMS_f^n(S) := \sup_{X \in \Pi_n(S)} \min_{j=1}^n f(X_j). \]

When $S$ is finite, we can replace the $\sup$ by $\max$.

When $\Omega$ is finite, we assume $\Fcal = 2^{\Omega}$, unless specified otherwise.

Dependency for:

1. Maximin share allocations

Info:

• Depth: 3
• Number of transitive dependencies: 4

Transitive dependencies:

1. /sets-and-relations/countable-set
2. /analysis/sup-inf
3. σ-algebra
4. Set function