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Monotone Comparative Statics for Games with Strategic Substitutes

  • Roy, Sunanda
  • Sabarwal, Tarun

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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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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  1. Milgrom, Paul & Roberts, John, 1994. "Comparing Equilibria," American Economic Review, American Economic Association, vol. 84(3), pages 441-59, June.
  2. Villas-Boas, J. Miguel, 1997. "Comparative Statics of Fixed Points," Journal of Economic Theory, Elsevier, vol. 73(1), pages 183-198, March.
  3. Rabah Amir & Val E. Lambson, 1998. "On the Effects of Entry in Cournot Markets," CIE Discussion Papers 1998-06, University of Copenhagen. Department of Economics. Centre for Industrial Economics.
  4. Aaron S. Edlin and Chris Shannon., 1995. "Strict Monotonicity in Comparative Statics," Economics Working Papers 95-238, University of California at Berkeley.
  5. Amir, Rabah, 1996. "Cournot Oligopoly and the Theory of Supermodular Games," Games and Economic Behavior, Elsevier, vol. 15(2), pages 132-148, August.
  6. Milgrom, Paul & Roberts, John, 1990. "Rationalizability, Learning, and Equilibrium in Games with Strategic Complementarities," Econometrica, Econometric Society, vol. 58(6), pages 1255-77, November.
  7. Federico Echenique., 2000. "Comparative Statics by Adaptive Dynamics and The Correspondence Principle," Economics Working Papers E00-273, University of California at Berkeley.
  8. 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.
  9. Milgrom, Paul & Shannon, Chris, 1994. "Monotone Comparative Statics," Econometrica, Econometric Society, vol. 62(1), pages 157-80, January.
  10. Kukushkin, Nikolai S., 1994. "A fixed-point theorem for decreasing mappings," Economics Letters, Elsevier, vol. 46(1), pages 23-26, September.
  11. John K.-H Quah, 2007. "The Comparative Statics of Constrained Optimization Problems," Econometrica, Econometric Society, vol. 75(2), pages 401-431, 03.
  12. Martin Jensen, 2010. "Aggregative games and best-reply potentials," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 43(1), pages 45-66, April.
  13. Shannon, Chris, 1995. "Weak and Strong Monotone Comparative Statics," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 5(2), pages 209-27, March.
  14. Sunanda Roy & Tarun Sabarwal, 2008. "On the (non-)lattice structure of the equilibrium set in games with strategic substitutes," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 37(1), pages 161-169, October.
  15. Echenique, Federico & Sabarwal, Tarun, 2003. "Strong comparative statics of equilibria," Games and Economic Behavior, Elsevier, vol. 42(2), pages 307-314, February.
  16. 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.
  17. 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.
  18. Roy, Sunanda & Sabarwal, T, 2008. "On the (Non)-Lattice Structure of Equilibrium Sets in Games with Strategic Substitutes," Staff General Research Papers Archive 13103, Iowa State University, Department of Economics.
  19. 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).
  20. Bulow, Jeremy I & Geanakoplos, John D & Klemperer, Paul D, 1985. "Multimarket Oligopoly: Strategic Substitutes and Complements," Journal of Political Economy, University of Chicago Press, vol. 93(3), pages 488-511, June.
  21. Bamon, Rodrigo & Fraysse, Jean, 1985. "Existence of Cournot Equilibrium in Large Markets," Econometrica, Econometric Society, vol. 53(3), pages 587-97, May.
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