 Supermodular set function

A set function is supermodular whenEquivalently, $-f$ is a submodular set function. On a finite ground set, this means increasing marginal increments: if $A\subseteq B$ and $x\notin B$, then ${\Delta }_{x}f(A)\le {\Delta }_{x}f(B)$.

For example, $f(S)=|S{|}^2$ has increment $2|S|+1$, which increases with the cardinality of $S$. In a convex cooperative game, supermodularity expresses increasing gains from joining larger coalitions; the corresponding marginal allocations lie in the core of a cooperative game.