Supermodular function
In mathematics, a function is supermodular if for all formula_3, formula_4, where formula_5 denotes the componentwise maximum and formula_6 the componentwise minimum of formula_3 and formula_8. 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  formula_10 defined over actions formula_11 of two or more players formula_12. Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval: formula_13. In this context, supermodularity of formula_10 implies that an increase in player formula_15's choice formula_11 increases the marginal payoff formula_17 of action formula_18 for all other players formula_19. That is, if any player formula_15 chooses a higher formula_11, all other players formula_19 have an incentive to raise their choices formula_18 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 formula_10 corresponds to the situation of strategic substitutability. An increase in formula_11 lowers the marginal payoff to all other player's choices formula_18, so strategies are substitutes. That is, if formula_15 chooses a higher formula_11, other players have an incentive to pick a "lower" formula_18. 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. Submodular 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 formula_30 is submodular if for any formula_31 and formula_32, formula_33. For supermodularity, the inequality is reversed. The definition  of submodularity can equivalently be formulated as for all subsets "A" and "B" of "S". Theory and enumeration algorithms for finding local and global maxima (minima) of submodular (supermodular) functions can be found in B. Goldengorin. European Journal of Operational Research 198(1):102-112, DOI: 10.1016/j.ejor.2008.08.022