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On the (non-)lattice structure of the equilibrium set in games with strategic substitutes

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  • Sunanda Roy

    ()

  • Tarun Sabarwal

    ()

Abstract

This paper studies models where the optimal response functions under consideration are non-increasing in endogenous variables, and weakly increasing in exogenous parameters. Such models include games with strategic substitutes, and include cases where additionally, some variables may be strategic complements. The main result here is that the equilibrium set in such models is a non-empty, complete lattice, if, and only if, there is a unique equilibrium. Indeed, for a given parameter value, a pair of distinct equilibria are never comparable. Therefore, with multiple equilibria, some of the established techniques for exhibiting increasing equilibria or computing equilibria that use the largest or smallest equilibrium, or that use the lattice structure of the equilibrium set do not apply to such models. Moreover, there are no ranked equilibria in such models. Additionally, the analysis here implies a new proof and a slight generalization of some existing results. It is shown that when a parameter increases, no new equilibrium is smaller than any old equilibrium. (In particular, in n-player games of strategic substitutes with real-valued action spaces, symmetric equilibria increase with the parameter.)

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Bibliographic Info

Article provided by Springer in its journal Economic Theory.

Volume (Year): 37 (2008)
Issue (Month): 1 (October)
Pages: 161-169

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Handle: RePEc:spr:joecth:v:37:y:2008:i:1:p:161-169

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Keywords: Monotone comparative statics; Non-increasing functions; Never increasing correspondences; Strategic substitutes; Equilibrium set; C60; C61; C62; C72;

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  15. Zhou Lin, 1994. "The Set of Nash Equilibria of a Supermodular Game Is a Complete Lattice," Games and Economic Behavior, Elsevier, vol. 7(2), pages 295-300, September.
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Cited by:
  1. Roy, Sunanda & Sabarwal, Tarun, 2005. "Monotone Comparative Statics for Games with Strategic Substitutes," MPRA Paper 4709, University Library of Munich, Germany, revised 04 Sep 2007.
  2. Roy, Sunanda & Sabarwal, Tarun, 2010. "Characterizing Stability Properties in Games with Strategic Substitutes," Staff General Research Papers 32009, Iowa State University, Department of Economics.
  3. Andrew Monaco & Tarun Sabarwal, 2012. "Games with Strategic Heterogeneity," WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS 201240, University of Kansas, Department of Economics, revised Nov 2012.

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