Supermodular

In mathematics, a function:$fcolon\; R^k\; o\; R$is supermodular if:$f(x\; lor\; y)\; +\; f(x\; land\; y)\; geq\; f(x)\; +\; f(y)$for all "x", "y" $isin$ "R"^"k", where "x" $vee$ "y" denotes the componentwise maximum and "x" $wedge$ "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 smooth, then supermodularity is equivalent to the condition [The equivalence between the definition of supermodularity and its calculus formulation is sometimes called "Topkis' Characterization Theorem". See Paul Milgrom and John Roberts (1990), 'Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities', "Econometrica" 58 (6), page 1261.]

:$frac\{partial\; ^2\; f\}\{partial\; z\_i\; partial\; z\_j\}\; geq\; 0\; mbox\{\; for\; all\; \}\; i\; eq\; j.$

upermodularity 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 $i\; in\; \{1,2,...,N\}$. 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 $frac\{df\}\{dz\_j\}$ 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. [Jeremy I. Bulow, John D. Geanakoplos, and Paul D. Klemperer (1985), 'Multimarket oligopoly: strategic substitutes and strategic complements'. "Journal of Political Economy" 93, pp. 488-511.] This is the basic property underlying examples of multiple equilibria in coordination games. [Russell Cooper and Andrew John (1988), 'Coordinating coordination failures in Keynesian models.' "Quarterly Journal of Economics" 103 (3), pp. 441-63.]

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 [Donald M. Topkis (1998), Supermodularity and Complementarity, Princeton University Press.] .

ee also

* Topkis's theorem

Notes and references