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Monotone Comparative Statics for Games with Strategic Substitutes

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  • Roy, Sunanda
  • Sabarwal, Tarun

Abstract

This paper studies comparative statics of equilibria in models where the optimal responses under consideration are (weakly) decreasing in endogenous variables, and (weakly) increasing in exogenous parameters. Such models include parameterized games of strategic substitutes. The analysis provides a sufficient condition for existence of increasing equilibria at a higher parameter value. This condition is presented first for best-response functions; it can be translated easily to payoff functions with one-dimensional individual strategy spaces, and it has a natural analogue to best-response correspondences. The condition is tight in the sense that with a weakenened condition, the same result may not obtain. The results here apply to asymmetric equilibria, and are applied to two classes of examples -- Cournot duopoly and tournaments. Moreover, sufficient conditions are presented to exhibit strong comparative statics of equilibria (that is, every equilibrium at a higher parameter value is greater than a given equilibrium at a lower parameter value), and to show existence of increasing equilibrium selections.

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Bibliographic Info

Paper provided by University Library of Munich, Germany in its series MPRA Paper with number 4709.

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Date of creation: May 2005
Date of revision: 04 Sep 2007
Handle: RePEc:pra:mprapa:4709

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Keywords: Monotone comparative statics; Weakly decreasing functions; Strategic substitutes; Payoff functions;

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References

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  1. John K.-H Quah, 2007. "The Comparative Statics of Constrained Optimization Problems," Econometrica, Econometric Society, Econometric Society, vol. 75(2), pages 401-431, 03.
  2. AMIR, Rabah, 1994. "Cournot Oligopoly and the Theory of Supermodular Games," CORE Discussion Papers, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE) 1994013, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  3. Lippman, Steven A. & Mamer, John W. & McCardle, Kevin F., 1987. "Comparative statics in non-cooperative games via transfinitely iterated play," Journal of Economic Theory, Elsevier, vol. 41(2), pages 288-303, April.
  4. Federico Echenique, 1999. "Comparative Statics by Adaptative Dynamics and the Correspondence Principle," Documentos de Trabajo (working papers) 2099, Department of Economics - dECON.
  5. Roy, Sunanda & Sabarwal, Tarun, 2006. "On the (non-)lattice structure of the equilibrium set in games with strategic substitutes," MPRA Paper 4120, University Library of Munich, Germany, revised 23 May 2007.
  6. Rabah Amir, 2000. "On the Effects of Entry in Cournot Markets," Econometric Society World Congress 2000 Contributed Papers 1475, Econometric Society.
  7. Dubey, Pradeep & Haimanko, Ori & Zapechelnyuk, Andriy, 2006. "Strategic complements and substitutes, and potential games," Games and Economic Behavior, Elsevier, vol. 54(1), pages 77-94, January.
  8. Bulow, Jeremy I & Geanakoplos, John D & Klemperer, Paul D, 1985. "Multimarket Oligopoly: Strategic Substitutes and Complements," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 93(3), pages 488-511, June.
  9. Milgrom, Paul & Roberts, John, 1994. "Comparing Equilibria," American Economic Review, American Economic Association, vol. 84(3), pages 441-59, June.
  10. Martin Jensen, 2010. "Aggregative games and best-reply potentials," Economic Theory, Springer, vol. 43(1), pages 45-66, April.
  11. 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.
  12. Bamon, Rodrigo & Fraysse, Jean, 1985. "Existence of Cournot Equilibrium in Large Markets," Econometrica, Econometric Society, Econometric Society, vol. 53(3), pages 587-97, May.
  13. Shannon, Chris, 1995. "Weak and Strong Monotone Comparative Statics," Economic Theory, Springer, vol. 5(2), pages 209-27, March.
  14. Roy, Sunanda & Sabarwal, T, 2008. "On the (Non)-Lattice Structure of Equilibrium Sets in Games with Strategic Substitutes," Staff General Research Papers 13103, Iowa State University, Department of Economics.
  15. Villas-Boas, J. Miguel, 1997. "Comparative Statics of Fixed Points," Journal of Economic Theory, Elsevier, vol. 73(1), pages 183-198, March.
  16. Milgrom, Paul & Shannon, Chris, 1994. "Monotone Comparative Statics," Econometrica, Econometric Society, Econometric Society, vol. 62(1), pages 157-80, January.
  17. Milgrom, Paul & Roberts, John, 1990. "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities," Econometrica, Econometric Society, Econometric Society, vol. 58(6), pages 1255-77, November.
  18. Vives, X., 1988. "Nash Equilibrium With Strategic Complementarities," UFAE and IAE Working Papers 107-88, Unitat de Fonaments de l'Anàlisi Econòmica (UAB) and Institut d'Anàlisi Econòmica (CSIC).
  19. Edlin, Aaron S. & Shannon, Chris, 1998. "Strict Monotonicity in Comparative Statics," Journal of Economic Theory, Elsevier, vol. 81(1), pages 201-219, July.
  20. Echenique, Federico & Sabarwal, Tarun, 2003. "Strong comparative statics of equilibria," Games and Economic Behavior, Elsevier, vol. 42(2), pages 307-314, February.
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Cited by:
  1. Peter Arendorf Bache & Anders Laugesen, 2013. "Monotone Comparative Statics for the Industry Composition," Economics Working Papers, School of Economics and Management, University of Aarhus 2013-10, School of Economics and Management, University of Aarhus.
  2. Roy, Sunanda & Sabarwal, Tarun, 2012. "Characterizing stability properties in games with strategic substitutes," Games and Economic Behavior, Elsevier, vol. 75(1), pages 337-353.
  3. Acemoglu, Daron & Jensen, Martin Kaae, 2013. "Aggregate comparative statics," Games and Economic Behavior, Elsevier, vol. 81(C), pages 27-49.
  4. Shirai, Koji, 2010. "An existence theorem for Cournot-Walras equilibria in a monopolistically competitive economy," Journal of Mathematical Economics, Elsevier, vol. 46(6), pages 1093-1102, November.
  5. Eric Hoffmann, 2013. "Global Games Selection in Games with Strategic Substitutes or Complements," WORKING PAPERS SERIES IN THEORETICAL AND APPLIED ECONOMICS 201308, University of Kansas, Department of Economics.

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