Supermodular function: Difference between revisions

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Consider a [[symmetric game]] with a smooth payoff function <math>\,f</math> defined over actions <math>\,z_i</math> of two or more players <math>i \in {1,2,\dots,N}</math>. Suppose the action space is continuous; for simplicity, suppose each action is chosen from an interval: <math>z_i \in [a,b]</math>. In this context, supermodularity of <math>\,f</math> implies that an increase in player <math>\,i</math>'s choice <math>\,z_i</math> increases the marginal payoff <math>df/dz_j</math> of action <math>\,z_j</math> for all other players <math>\,j</math>. That is, if any player <math>\,i</math> chooses a higher <math>\,z_i</math>, all other players <math>\,j</math> have an incentive to raise their choices <math>\,z_j</math> too. Following the terminology of Bulow, [[John Geanakoplos|Geanakoplos]], and [[Paul Klemperer|Klemperer]] (1985), economists call this situation [[strategic complements|strategic complementarity]], because players' strategies are complements to each other.<ref>{{cite journal |first1=Jeremy I. |last1=Bulow |first2=John D. |last2=Geanakoplos |first3=Paul D. |last3=Klemperer |year=1985 |title=Multimarket Oligopoly: Strategic Substitutes and Complements |journal=[[Journal of Political Economy]] |volume=93 |issue=3 |pages=488–511 |doi=10.1086/261312 |citeseerx=10.1.1.541.2368 |s2cid=154872708 }}</ref> This is the basic property underlying examples of [[General equilibrium#Uniqueness|multiple equilibria]] in [[coordination game]]s.<ref>{{cite journal |first1=Russell |last1=Cooper |first2=Andrew |last2=John |year=1988 |title=Coordinating coordination failures in Keynesian models |journal=[[Quarterly Journal of Economics]] |volume=103 |issue=3 |pages=441–463 |doi=10.2307/1885539 |jstor=1885539 |url=http://cowles.yale.edu/sites/default/files/files/pub/d07/d0745-r.pdf }}</ref>

The opposite case of submodularitysupermodularity of <math>\,f</math>, called submodularity, 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 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.