In mathematics, a function
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is supermodular if
for all x, 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
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
The opposite case of submodularity of
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 supermodular utility function is often related to complementary goods. However, this view is disputed.
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
The definition of supermodularity can equivalently be formulated as
for all subsets A and B of S.