In mathematics, a function
is supermodular if
for all , , where denotes the componentwise maximum and the componentwise minimum of and .
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 defined over actions of two or more players . Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval: . In this context, supermodularity of implies that an increase in player 's choice increases the marginal payoff of action for all other players . That is, if any player chooses a higher , all other players have an incentive to raise their choices 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 corresponds to the situation of strategic substitutability. An increase in lowers the marginal payoff to all other player's choices , so strategies are substitutes. That is, if chooses a higher , other players have an incentive to pick a lower .
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.
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 is submodular if for any and , . For supermodularity, the inequality is reversed.
The definition of submodularity can equivalently be formulated as
for all subsets A and B of S.
- Pseudo-Boolean function
- Topkis's theorem
- Submodular set function
- Utility functions on indivisible goods
Notes and references
- The equivalence between the definition of supermodularity and its calculus formulation is sometimes called Topkis' characterization theorem. See Milgrom, Paul; Roberts, John (1990). "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities". Econometrica. 58 (6): 1255–1277 [p. 1261]. doi:10.2307/2938316. JSTOR 2938316.
- Bulow, Jeremy I.; Geanakoplos, John D.; Klemperer, Paul D. (1985). "Multimarket Oligopoly: Strategic Substitutes and Complements". Journal of Political Economy. 93 (3): 488–511. CiteSeerX 10.1.1.541.2368. doi:10.1086/261312.
- Cooper, Russell; John, Andrew (1988). "Coordinating coordination failures in Keynesian models". Quarterly Journal of Economics. 103 (3): 441–463. doi:10.2307/1885539. JSTOR 1885539.
- Chambers, Christopher P.; Echenique, Federico (2009). "Supermodularity and preferences". Journal of Economic Theory. 144 (3): 1004. CiteSeerX 10.1.1.122.6861. doi:10.1016/j.jet.2008.06.004.