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Global Dynamics in Infinitely Repeated Games with Additively Separable Continuous Payoffs

Author

Listed:
  • Takashi Kamihigashi

    (Research Institute for Economics & Business Administration (RIEB), Kobe University, Japan)

  • Taiji Furusawa

    (Graduate School of Economics, Hitotsubashi University, Japan)

Abstract

This paper studies a class of infinitely repeated games with two players in which the action space of each player is an interval, and the one-shot payoff of each player is additively separable in their actions. We define an immediately reactive equilibrium (IRE) as a pure-strategy subgame perfect equilibrium such that each player's action in each period is a stationary function of the other player's last action. We completely characterize IREs and their dynamics in terms of certain indifference curves. In a special case we establish a folk-type theorem using only IREs that are continuous and punish deviations in a minimal way. Our results are used to show that in a prisoners' dilemma game with observable mixed strategies, gradual cooperation occurs when the players are sufficiently patient, and that in a certain duopoly game, kinked demand curves emerge naturally.

Suggested Citation

  • Takashi Kamihigashi & Taiji Furusawa, 2007. "Global Dynamics in Infinitely Repeated Games with Additively Separable Continuous Payoffs," Discussion Paper Series 210, Research Institute for Economics & Business Administration, Kobe University.
    • Handle: RePEc:kob:dpaper:210
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    File URL: https://www.rieb.kobe-u.ac.jp/academic/ra/dp/English/dp210.pdf
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    References listed on IDEAS

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    1. Bhaskar, V. & Vega-Redondo, Fernando, 2002. "Asynchronous Choice and Markov Equilibria," Journal of Economic Theory, Elsevier, vol. 103(2), pages 334-350, April.
    2. Maskin, Eric & Tirole, Jean, 1987. "A theory of dynamic oligopoly, III : Cournot competition," European Economic Review, Elsevier, vol. 31(4), pages 947-968, June.
    3. Samuelson, Larry, 1987. "Non-trivial subgame perfect duopoly equilibria can be supported by continuous reaction functions," Economics Letters, Elsevier, vol. 24(3), pages 207-211.
    4. Friedman, James W, 1973. "On Reaction Function Equilibria," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 14(3), pages 721-734, October.
    5. James W. Friedman, 1971. "A Non-cooperative Equilibrium for Supergames," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 38(1), pages 1-12.
    6. Green, Edward J & Porter, Robert H, 1984. "Noncooperative Collusion under Imperfect Price Information," Econometrica, Econometric Society, vol. 52(1), pages 87-100, January.
    7. Stanford, William G., 1986. "Subgame perfect reaction function equilibria in discounted duopoly supergames are trivial," Journal of Economic Theory, Elsevier, vol. 39(1), pages 226-232, June.
    8. Friedman James W. & Samuelson Larry, 1994. "Continuous Reaction Functions in Duopolies," Games and Economic Behavior, Elsevier, vol. 6(1), pages 55-82, January.
    9. Roger Lagunoff & Akihiko Matsui, 1997. "Asynchronous Choice in Repeated Coordination Games," Econometrica, Econometric Society, vol. 65(6), pages 1467-1478, November.
    10. Radner, Roy, 2003. "Viscous demand," Journal of Economic Theory, Elsevier, vol. 112(2), pages 189-231, October.
      • Roy Radner, 1999. "Viscous Demand," Working Papers 99-10, New York University, Leonard N. Stern School of Business, Department of Economics.
    11. Abreu, Dilip, 1988. "On the Theory of Infinitely Repeated Games with Discounting," Econometrica, Econometric Society, vol. 56(2), pages 383-396, March.
    12. Friedman James W. & Samuelson Larry, 1994. "An Extension of the Folk Theorem with Continuous Reaction Functions," Games and Economic Behavior, Elsevier, vol. 6(1), pages 83-96, January.
    13. Drew Fudenberg & Eric Maskin, 2008. "The Folk Theorem In Repeated Games With Discounting Or With Incomplete Information," World Scientific Book Chapters, in: Drew Fudenberg & David K Levine (ed.), A Long-Run Collaboration On Long-Run Games, chapter 11, pages 209-230, World Scientific Publishing Co. Pte. Ltd..
    14. Maskin, Eric & Tirole, Jean, 1988. "A Theory of Dynamic Oligopoly, II: Price Competition, Kinked Demand Curves, and Edgeworth Cycles," Econometrica, Econometric Society, vol. 56(3), pages 571-599, May.
    15. Langlois, Jean-Pierre P. & Sachs, Jonathan A., 1993. "Existence and local stability of Pareto superior reaction function equilibria in discounted supergames," Journal of Mathematical Economics, Elsevier, vol. 22(3), pages 199-221.
    16. Maskin, Eric & Tirole, Jean, 1988. "A Theory of Dynamic Oligopoly, I: Overview and Quantity Competition with Large Fixed Costs," Econometrica, Econometric Society, vol. 56(3), pages 549-569, May.
    17. Yoon, Kiho, 2001. "A Folk Theorem for Asynchronously Repeated Games," Econometrica, Econometric Society, vol. 69(1), pages 191-200, January.
    18. James W. Friedman, 1976. "Reaction Functions as Nash Equilibria," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 43(1), pages 83-90.
    19. Partha Dasgupta & Douglas Gale & Oliver Hart & Eric Maskin (ed.), 1992. "Economic Analysis of Markets and Games: Essays in Honor of Frank Hahn," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262541599, December.
    20. Maskin, Eric & Tirole, Jean, 2001. "Markov Perfect Equilibrium: I. Observable Actions," Journal of Economic Theory, Elsevier, vol. 100(2), pages 191-219, October.
    21. J. W. Friedman, 1968. "Reaction Functions and the Theory of Duopoly," The Review of Economic Studies, Review of Economic Studies Ltd, vol. 35(3), pages 257-272.
    22. Takashi Kamihigashi & Santanu Roy, 2006. "Dynamic optimization with a nonsmooth, nonconvex technology: the case of a linear objective function," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 29(2), pages 325-340, October.
    23. Friedman, James W. & Samuelson, Larry, 1990. "Subgame perfect equilibrium with continuous reaction functions," Games and Economic Behavior, Elsevier, vol. 2(4), pages 304-324, December.
    24. Robson, Arthur J, 1986. "The Existence of Nash Equilibria in Reaction Functions for Dynamic Models of Oligopoly," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 27(3), pages 539-544, October.
    25. Corchon, Luis C., 1994. "Comparative statics for aggregative games the strong concavity case," Mathematical Social Sciences, Elsevier, vol. 28(3), pages 151-165, December.
    26. Rand, David, 1978. "Exotic phenomena in games and duopoly models," Journal of Mathematical Economics, Elsevier, vol. 5(2), pages 173-184, September.
    27. Haller, Hans & Lagunoff, Roger, 2010. "Markov Perfect equilibria in repeated asynchronous choice games," Journal of Mathematical Economics, Elsevier, vol. 46(6), pages 1103-1114, November.
    28. Sen, Debapriya, 2004. "The kinked demand curve revisited," Economics Letters, Elsevier, vol. 84(1), pages 99-105, July.
    29. Abreu, Dilip, 1986. "Extremal equilibria of oligopolistic supergames," Journal of Economic Theory, Elsevier, vol. 39(1), pages 191-225, June.
    30. Stahl, Dale II, 1991. "The graph of Prisoners' Dilemma supergame payoffs as a function of the discount factor," Games and Economic Behavior, Elsevier, vol. 3(3), pages 368-384, August.
    31. Furusawa, Taiji & Kawakami, Toshikazu, 2008. "Gradual cooperation in the existence of outside options," Journal of Economic Behavior & Organization, Elsevier, vol. 68(2), pages 378-389, November.
    32. Kalai, Ehud & Samet, Dov & Stanford, William, 1988. "A Note on Reactive Equilibria in the Discounted Prisoner's Dilemma and Associated Games," International Journal of Game Theory, Springer;Game Theory Society, vol. 17(3), pages 177-186.
    33. Kranton, Rachel E, 1996. "The Formation of Cooperative Relationships," The Journal of Law, Economics, and Organization, Oxford University Press, vol. 12(1), pages 214-233, April.
    34. Watson, Joel, 1999. "Starting Small and Renegotiation," Journal of Economic Theory, Elsevier, vol. 85(1), pages 52-90, March.
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