Supermodular function: Difference between revisions

Revision as of 07:45, 18 April 2010

Revision as of 09:35, 30 June 2010

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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<ref>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.</ref>

:<math> \frac{\partial ^2 f}{\partial z_i\, \partial z_j} \geq 0 \mbox{ for all } i \neq j.</math>

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The opposite case of submodularity of <math>\,f\,</math> corresponds to the situation of [[strategic complements|strategic substitutability]]. An increase in <math>\,z_i\,</math> lowers the marginal payoff to all other player's choices <math>\,z_j\,</math>, so strategies are substitutes. That is, if <math>\,i\,</math> chooses a higher <math>\,z_i\,</math>, other players have an incentive to pick a ''lower'' <math>\,z_j\,</math>.

For example, Bulow et al. consider the interactions of many [[Imperfect_competitionImperfect competition|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<ref>Donald M. Topkis (1998), Supermodularity and Complementarity, Princeton University Press.</ref>.

==Supermodular functions of subsets==

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==Notes and references==

<references />

==External Linkslinks==

{{DEFAULTSORT:Supermodular Function}}

[[Category:Order theory]]