Content deleted Content added
No edit summary |
m Bot: lowercase wikilinks to uppercase sections |
||
Line 16:
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>\frac{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, Geanakoplos, and Klemperer (1985), economists call this situation [[strategic complements|strategic complementarity]], because players' strategies are complements to each other.<ref>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.</ref> This is the basic property underlying examples of [[General equilibrium#
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>.
| |||