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Cooperation in Repeated Games when the Number of Stages is Not Commonly Known

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  • Abraham Neyman

Abstract

An exponentially small departure from the common knowledge assumption on the number T of repetitions of the prisoners' dilemma already enables cooperation. More generally, with such a departure, any feasible individually rational outcome of any one-shot game can be approximated by an equilibrium of a finitely repeated version of that game. The departure from common knowledge is small in the following sense:(1) the players know T with precision +/-K; (2) with probability 1 - epsilon, the players know T precisely; moreover, this knowledge is mutual of order epsilon T; and (3) the deviation of T from its finite expectation is exponentially small.

Suggested Citation

  • Abraham Neyman, 1999. "Cooperation in Repeated Games when the Number of Stages is Not Commonly Known," Econometrica, Econometric Society, vol. 67(1), pages 45-64, January.
    • Handle: RePEc:ecm:emetrp:v:67:y:1999:i:1:p:45-64
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    Citations

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    Cited by:

    1. Lisa Bruttel & Werner Güth & Ulrich Kamecke, 2012. "Finitely repeated prisoners’ dilemma experiments without a commonly known end," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(1), pages 23-47, February.
    2. Vi Cao, 2022. "An epistemic approach to explaining cooperation in the finitely repeated Prisoner’s Dilemma," International Journal of Game Theory, Springer;Game Theory Society, vol. 51(1), pages 53-85, March.
    3. Hitoshi Matsushima, 2012. "Finitely Repeated Prisoners' Dilemma With Small Fines: The Penance Contract," The Japanese Economic Review, Japanese Economic Association, vol. 63(3), pages 333-347, September.
    4. Ambrus, Attila & Pathak, Parag A., 2011. "Cooperation over finite horizons: A theory and experiments," Journal of Public Economics, Elsevier, vol. 95(7), pages 500-512.
    5. Francesca Barigozzi & Renaud Bourlès & Dominique Henriet & Giuseppe Pignataro, 2017. "Pool size and the sustainability of optimal risk-sharing agreements," Theory and Decision, Springer, vol. 82(2), pages 273-303, February.
    6. Andrea Gallice & Ignacio Monzón, 2019. "Co-operation in Social Dilemmas Through Position Uncertainty," The Economic Journal, Royal Economic Society, vol. 129(621), pages 2137-2154.
    7. Reichhuber, Anke & Camacho, Eva & Requate, Till, 2009. "A framed field experiment on collective enforcement mechanisms with Ethiopian farmers," Environment and Development Economics, Cambridge University Press, vol. 14(5), pages 641-663, October.
    8. Abraham Neyman, 2012. "The value of two-person zero-sum repeated games with incomplete information and uncertain duration," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(1), pages 195-207, February.
    9. Conlon, John R., 2003. "Hope springs eternal: learning and the stability of cooperation in short horizon repeated games," Journal of Economic Theory, Elsevier, vol. 112(1), pages 35-65, September.
    10. Gilad Bavly, 2017. "Uncertainty in the traveler’s dilemma," International Journal of Game Theory, Springer;Game Theory Society, vol. 46(1), pages 1-12, March.
    11. Binenbaum, Eran & Pardey, Philip G., 2005. "Collective Action in Plant Breeding," 2005 Annual meeting, July 24-27, Providence, RI 19530, American Agricultural Economics Association (New Name 2008: Agricultural and Applied Economics Association).
    12. Hans-Theo Normann & Brian Wallace, 2012. "The impact of the termination rule on cooperation in a prisoner’s dilemma experiment," International Journal of Game Theory, Springer;Game Theory Society, vol. 41(3), pages 707-718, August.
    13. Monderer, Dov & Tennenholtz, Moshe, 1999. "Distributed Games," Games and Economic Behavior, Elsevier, vol. 28(1), pages 55-72, July.
    14. Seale, Darryl A. & Arend, Richard J. & Phelan, Steven, 2006. "Modeling alliance activity: Opportunity cost effects and manipulations in an iterated prisoner's dilemma with exit option," Organizational Behavior and Human Decision Processes, Elsevier, vol. 100(1), pages 60-75, May.
    15. Renault, Regis, 2000. "Privately Observed Time Horizons in Repeated Games," Games and Economic Behavior, Elsevier, vol. 33(1), pages 117-125, October.
    16. Gilad Bavly, 2011. "Elasticity of Games," Discussion Paper Series dp592, The Federmann Center for the Study of Rationality, the Hebrew University, Jerusalem.
    17. Neyman, Abraham, 2017. "Continuous-time stochastic games," Games and Economic Behavior, Elsevier, vol. 104(C), pages 92-130.
    18. Francois Cochard & Anthony Ziegelmeyer & Kene Boun My, 2005. "The Regulation of Nonpoint Emissions in the Laboratory: A Stress Test of the Ambient Tax Mechanism," Papers on Strategic Interaction 2005-37, Max Planck Institute of Economics, Strategic Interaction Group.
    19. Eran Binenbaum, 2008. "Incentive Issues In R&D Consortia: Insights From Applied Game Theory," Contemporary Economic Policy, Western Economic Association International, vol. 26(4), pages 636-650, October.
    20. Gagen, Michael, 2013. "Isomorphic Strategy Spaces in Game Theory," MPRA Paper 46176, University Library of Munich, Germany.
    21. Marlats, Chantal, 2019. "Perturbed finitely repeated games," Mathematical Social Sciences, Elsevier, vol. 98(C), pages 39-46.
    22. Committee, Nobel Prize, 2005. "Robert Aumann's and Thomas Schelling's Contributions to Game Theory: Analyses of Conflict and Cooperation," Nobel Prize in Economics documents 2005-1, Nobel Prize Committee.
    23. Rose Lai & Ko Wang & Jing Yang, 2007. "Stickiness of Rental Rates and Developers’ Option Exercise Strategies," The Journal of Real Estate Finance and Economics, Springer, vol. 34(1), pages 159-188, January.
    24. Gaël GIRAUD & Sonia WEYERS, 2003. "Strategic Market Games with a Finite Horizon and Incomplete," Working Papers of BETA 2003-04, Bureau d'Economie Théorique et Appliquée, UDS, Strasbourg.

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