Pure submodule

In mathematics, a function

$f\colon R^{k}\to R$

is supermodular if

$f(x\uparrow y)+f(x\downarrow y)\geq f(x)+f(y)$

for all x, y $\in$ R^k, where x $\uparrow$ y denotes the componentwise maximum and x $\downarrow$ y the componentwise minimum of x and y.

If −f is supermodular then f is called submodular, and if the inequality is changed to an equality the function is modular.

If f is twice continuously differentiable, then supermodularity is equivalent to the condition

${\frac {\partial ^{2}f}{\partial z_{i}\,\partial z_{j}}}\geq 0{\mbox{ for all }}i\neq j.$

Supermodularity in economics and game theory

The concept of supermodularity is used in the social sciences to analyze how one agent's decision affects the incentives of others.

Consider a symmetric game with a smooth payoff function $\,f\,$ defined over actions $\,z_{i}\,$ of two or more players $i\in {1,2,\dots ,N}$. Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval: $z_{i}\in [a,b]$. In this context, supermodularity of $\,f\,$ implies that an increase in player $\,i\,$'s choice $\,z_{i}\,$ increases the marginal payoff $df/dz_{j}$ of action $\,z_{j}\,$ for all other players $\,j\,$. That is, if any player $\,i\,$ chooses a higher $\,z_{i}\,$, all other players $\,j\,$ have an incentive to raise their choices $\,z_{j}\,$ too. Following the terminology of Bulow, Geanakoplos, and Klemperer (1985), economists call this situation strategic complementarity, because players' strategies are complements to each other. This is the basic property underlying examples of multiple equilibria in coordination games.

The opposite case of submodularity of $\,f\,$ corresponds to the situation of strategic substitutability. An increase in $\,z_{i}\,$ lowers the marginal payoff to all other player's choices $\,z_{j}\,$, so strategies are substitutes. That is, if $\,i\,$ chooses a higher $\,z_{i}\,$, other players have an incentive to pick a lower $\,z_{j}\,$.

For example, Bulow et al. consider the interactions of many imperfectly competitive firms. When an increase in output by one firm raises the marginal revenues of the other firms, production decisions are strategic complements. When an increase in output by one firm lowers the marginal revenues of the other firms, production decisions are strategic substitutes.

A standard reference on the subject is by Topkis.

Supermodular functions of subsets

Supermodularity and submodularity are also defined for functions defined over subsets of a larger set. Intuitively, a submodular function over the subsets demonstrates "diminishing returns". There are specialized techniques for optimizing submodular functions.

Let S be a finite set. A function $f\colon 2^{S}\to R$ is submodular if for any $A\subset B\subset S$ and $x\in S\setminus B$, $f(A\cup \{x\})-f(A)\geq f(B\cup \{x\})-f(B)$. For supermodularity, the inequality is reversed.

A simple illustrative example motivates this definition of submodular. Let S be a set of different foods, $M\subset S$ a meal, and $f(M)$ the "goodness" of that meal. Then A above is one meal, and B is A but with even more options. Let x be ice cream. Adding ice cream to a meal is always good, but it is best if there is not already a dessert. If A and B either both have a dessert or both do not, then adding ice cream to them is comparably good. But if A does not have dessert and B does, then the effect of adding ice cream is more pronounced in A.

The definition of submodularity can equivalently be formulated as

$f(A)+f(B)\geq f(A\cap B)+f(A\cup B)$

for all subsets A and B of S.

See also

• Pseudo-Boolean function
• Topkis's theorem
• Submodular set function