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A Foundation for Markov Equilibria in Infinite Horizon Perfect Information Games

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  • V. Bhaskar

    ()
    (Department of Economics, University College, London)

  • George J. Mailath

    ()
    (Department of Economics, University of Pennsylvania)

  • Stephen Morris

    ()
    (Department of Economics, Princeton University)

Abstract

We study perfect information games with an infinite horizon played by an arbitrary number of players. This class of games includes infinitely repeated perfect information games, repeated games with asynchronous moves, games with long and short run players, games with overlapping generations of players, and canonical non-cooperative models of bargaining. We consider two restrictions on equilibria. An equilibrium is purifiable if close by behavior is consistent with equilibrium when agents' payoffs at each node are perturbed additively and independently. An equilibrium has bounded recall if there exists K such that at most one player's strategy depends on what happened more than K periods earlier. We show that only Markov equilibria have bounded memory and are purifiable. Thus if a game has at most one long-run player, all purifiable equilibria are Markov.

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Bibliographic Info

Paper provided by Penn Institute for Economic Research, Department of Economics, University of Pennsylvania in its series PIER Working Paper Archive with number 12-043.

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Length: 29 pages
Date of creation: 29 Oct 2012
Date of revision:
Handle: RePEc:pen:papers:12-043

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Keywords: Markov; bounded recall; purification;

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References

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  1. V. Bhaskar & George J. Mailath & Stephen Morris, 2008. "Purification in the Infinitely-Repeated Prisoners' Dilemma," Review of Economic Dynamics, Elsevier for the Society for Economic Dynamics, Elsevier for the Society for Economic Dynamics, vol. 11(3), pages 515-528, July.
  2. George Mailath & Wojciech Olszewski, 2008. "Folk theorems with Bounded Recall under(Almost) Perfect Monitoring," Discussion Papers, Northwestern University, Center for Mathematical Studies in Economics and Management Science 1462, Northwestern University, Center for Mathematical Studies in Economics and Management Science.
  3. Mailath, George J & Samuelson, Larry, 2001. "Who Wants a Good Reputation?," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 68(2), pages 415-41, April.
  4. George J Mailath & Stephen Morris, 1999. "Repeated Games with Almost Public Monitoring," Levine's Working Paper Archive 2107, David K. Levine.
  5. Mas-Colell, Andreu & Whinston, Michael D. & Green, Jerry R., 1995. "Microeconomic Theory," OUP Catalogue, Oxford University Press, Oxford University Press, number 9780195102680, October.
  6. Rubinstein, Ariel, 1982. "Perfect Equilibrium in a Bargaining Model," Econometrica, Econometric Society, Econometric Society, vol. 50(1), pages 97-109, January.
  7. Roger Lagunoff & Akihiko Matsui, 1997. "Asynchronous Choice in Repeated Coordination Games," Econometrica, Econometric Society, Econometric Society, vol. 65(6), pages 1467-1478, November.
  8. Doraszelski, Ulrich & Escobar, Juan, 2010. "A theory of regular Markov perfect equilibria in dynamic stochastic games: genericity, stability, and purification," Theoretical Economics, Econometric Society, Econometric Society, vol. 5(3), September.
  9. Maskin, Eric & Tirole, Jean, 2001. "Markov Perfect Equilibrium: I. Observable Actions," Journal of Economic Theory, Elsevier, Elsevier, vol. 100(2), pages 191-219, October.
  10. Mailath, George J. & Samuelson, Larry, 2006. "Repeated Games and Reputations: Long-Run Relationships," OUP Catalogue, Oxford University Press, Oxford University Press, number 9780195300796, October.
  11. Morris, Stephen & Ui, Takashi, 2004. "Best response equivalence," Games and Economic Behavior, Elsevier, Elsevier, vol. 49(2), pages 260-287, November.
  12. Bhaskar, V., 1994. "Informational Constraints and the Overlapping Generations Model: Folk and Anti-Folk Theorems," Papers, Tilburg - Center for Economic Research 9485, Tilburg - Center for Economic Research.
  13. Matsui Akihiko & Matsuyama Kiminori, 1995. "An Approach to Equilibrium Selection," Journal of Economic Theory, Elsevier, Elsevier, vol. 65(2), pages 415-434, April.
  14. Maskin, Eric & Tirole, Jean, 1988. "A Theory of Dynamic Oligopoly, II: Price Competition, Kinked Demand Curves, and Edgeworth Cycles," Econometrica, Econometric Society, Econometric Society, vol. 56(3), pages 571-99, May.
  15. Roger Lagunoff & Akihiko Matsui, . ""An 'Anti-Folk Theorem' for a Class of Asynchronously Repeated Games''," CARESS Working Papres, University of Pennsylvania Center for Analytic Research and Economics in the Social Sciences 95-15, University of Pennsylvania Center for Analytic Research and Economics in the Social Sciences.
  16. Livshits, Igor, 2002. "On non-existence of pure strategy Markov perfect equilibrium," Economics Letters, Elsevier, Elsevier, vol. 76(3), pages 393-396, August.
  17. Jeheil Phillippe, 1995. "Limited Horizon Forecast in Repeated Alternate Games," Journal of Economic Theory, Elsevier, Elsevier, vol. 67(2), pages 497-519, December.
  18. Bhaskar, V. & Vega-Redondo, Fernando, 2002. "Asynchronous Choice and Markov Equilibria," Journal of Economic Theory, Elsevier, Elsevier, vol. 103(2), pages 334-350, April.
  19. D. Fudenberg & David K. Levine, 1989. "Reputation and Equilibrium Selection in Games with a Patient Player," Levine's Working Paper Archive 508, David K. Levine.
  20. Chatterjee, Kalyan & Bhaskar Dutta & Debraj Ray & Kunal Sengupta, 1993. "A Noncooperative Theory of Coalitional Bargaining," Review of Economic Studies, Wiley Blackwell, Wiley Blackwell, vol. 60(2), pages 463-77, April.
  21. Abhinay Muthoo & Kenneth Shepsle, 2010. "Information, institutions and constitutional arrangements," Public Choice, Springer, Springer, vol. 144(1), pages 1-36, July.
  22. Maskin, Eric & Tirole, Jean, 1987. "A theory of dynamic oligopoly, III : Cournot competition," European Economic Review, Elsevier, Elsevier, vol. 31(4), pages 947-968, June.
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Cited by:
  1. Marina Azzimonti, 2011. "Barriers to Investment in Polarized Societies," American Economic Review, American Economic Association, American Economic Association, vol. 101(5), pages 2182-2204, August.
  2. Herings P.J.J. & Meshalkin A. & Predtetchinski A., 2012. "A Folk Theorem for Bargaining Games," Research Memorandum, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR) 056, Maastricht University, Maastricht Research School of Economics of Technology and Organization (METEOR).
  3. Hannu Salonen & Hannu Vartiainen, 2011. "On the Existence of Markov Perfect Equilibria in Perfect Information Games," Discussion Papers, Aboa Centre for Economics 68, Aboa Centre for Economics.

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