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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 twice continuously differentiable, then supermodularity is equivalent to the condition<ref>The equivalence between the definition of supermodularity and its calculus formulation is sometimes called [[Topkis's theorem|Topkis' characterization theorem]]. See {{cite journal |first=Paul |last=Milgrom |first2=John |last2=Roberts |year=1990 |title=Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities |journal=[[Econometrica]] |volume=58 |issue=6 |pages=1255–1277 [p. 1261] |jstor=2938316 |doi=10.2307/2938316 }}</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 concept of supermodularity is used in the social sciences to analyze how one [[Agent (economics)|agent's]] decision affects the incentives of others.
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 |first=Jeremy I. |last=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 }}</ref> This is the basic property underlying examples of [[General equilibrium#Uniqueness|multiple equilibria]] in [[coordination game]]s.<ref>{{cite journal |first=Russell |last=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 }}</ref>
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>.
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A supermodular [[utility function]] is often related to [[complementary goods]]. However, this view is disputed.<ref>{{Cite journal|doi=10.1016/j.jet.2008.06.004 |title=Supermodularity and preferences |journal=[[Journal of Economic Theory]] |volume=144 |issue=3 |pages=1004 |year=2009 |last1=Chambers |first1=Christopher P. |last2=Echenique |first2=Federico |citeseerx=10.1.1.122.6861 }}</ref>
==Submodular functions of subsets==
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