Evolutionary Exploration of the Finitely Repeated Prisonersâ€™ Dilemmaâ€”The Effect of Out-of-Equilibrium Play
The finitely repeated Prisonersâ€™ Dilemma is a good illustration of the discrepancy between the strategic behaviour suggested by a game-theoretic analysis and the behaviour often observed among human players, where cooperation is maintained through most of the game. A game-theoretic reasoning based on backward induction eliminates strategies step by step until defection from the first round is the only remaining choice, reflecting the Nash equilibrium of the game. We investigate the Nash equilibrium solution for two different sets of strategies in an evolutionary context, using replicator-mutation dynamics. The first set consists of conditional cooperators, up to a certain round, while the second set in addition to these contains two strategy types that react differently on the first round action: The â€ Convincerâ€ strategies insist with two rounds of initial cooperation, trying to establish more cooperative play in the game, while the â€ Followerâ€ strategies, although being first round defectors, have the capability to respond to an invite in the first round. For both of these strategy sets, iterated elimination of strategies shows that the only Nash equilibria are given by defection from the first round. We show that the evolutionary dynamics of the first set is always characterised by a stable fixed point, corresponding to the Nash equilibrium, if the mutation rate is sufficiently small (but still positive). The second strategy set is numerically investigated, and we find that there are regions of parameter space where fixed points become unstable and the dynamics exhibits cycles of different strategy compositions. The results indicate that, even in the limit of very small mutation rate, the replicator-mutation dynamics does not necessarily bring the system with Convincers and Followers to the fixed point corresponding to the Nash equilibrium of the game. We also perform a detailed analysis of how the evolutionary behaviour depends on payoffs, game length, and mutation rate.
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Please report citation or reference errors to , or , if you are the registered author of the cited work, log in to your RePEc Author Service profile, click on "citations" and make appropriate adjustments.:
- Basu, Kaushik, 1994. "The Traveler's Dilemma: Paradoxes of Rationality in Game Theory," American Economic Review, American Economic Association, vol. 84(2), pages 391-95, May.
- Aumann, Robert J., 1996. "Reply to Binmore," Games and Economic Behavior, Elsevier, vol. 17(1), pages 138-146, November.
- Ponti, Giovanni, 2000.
"Cycles of Learning in the Centipede Game,"
Games and Economic Behavior,
Elsevier, vol. 30(1), pages 115-141, January.
- Giovanni Ponti, 1996. "Cycles of Learning in the Centipede Game," Discussion Papers 96-22 ISSN 1350-6722, University College London, Department of Economics.
- Giovanni Ponti, . "Cycles Of Learning In The Centipede Game," ELSE working papers 024, ESRC Centre on Economics Learning and Social Evolution.
- Nachbar, John H., 1992. "Evolution in the finitely repeated prisoner's dilemma," Journal of Economic Behavior & Organization, Elsevier, vol. 19(3), pages 307-326, December.
- Rosemarie Nagel & Antoni Bosch-Domènech & Albert Satorra & José García Montalvo, 1999.
"One, two, (three), infinity: Newspaper and lab beauty-contest experiments,"
Economics Working Papers
438, Department of Economics and Business, Universitat Pompeu Fabra.
- Antoni Bosch-Domènech & José G. Montalvo & Rosemarie Nagel & Albert Satorra, 2002. "One, Two, (Three), Infinity, ...: Newspaper and Lab Beauty-Contest Experiments," American Economic Review, American Economic Association, vol. 92(5), pages 1687-1701, December.
- Albert Satorra & Antoni Bosch-Domenech & Jose Garcia-Montalvo & Rosemarie Nagel, 2002. "One, two, (three), infinity: Newspaper and lab beauty-contest experiments," Artefactual Field Experiments 00011, The Field Experiments Website.
- Selten, Reinhard & Stoecker, Rolf, 1986. "End behavior in sequences of finite Prisoner's Dilemma supergames A learning theory approach," Journal of Economic Behavior & Organization, Elsevier, vol. 7(1), pages 47-70, March.
- Rosenthal, Robert W., 1981. "Games of perfect information, predatory pricing and the chain-store paradox," Journal of Economic Theory, Elsevier, vol. 25(1), pages 92-100, August.
- Binmore, Ken, 1987. "Modeling Rational Players: Part I," Economics and Philosophy, Cambridge University Press, vol. 3(02), pages 179-214, October.
- Noldeke Georg & Samuelson Larry, 1993.
"An Evolutionary Analysis of Backward and Forward Induction,"
Games and Economic Behavior,
Elsevier, vol. 5(3), pages 425-454, July.
- Noeldecke,Georg & Samuelson,Larry, . "An evolutionary analysis of backward and forward induction," Discussion Paper Serie B 228, University of Bonn, Germany.
- G. Noldeke & L. Samuelson, 2010. "An Evolutionary Analysis of Backward and Forward Induction," Levine's Working Paper Archive 538, David K. Levine.
- Ken Binmore & Larry Samuelson, 1999. "Evolutionary Drift and Equilibrium Selection," Review of Economic Studies, Oxford University Press, vol. 66(2), pages 363-393.
- Aumann, Robert J., 1995. "Backward induction and common knowledge of rationality," Games and Economic Behavior, Elsevier, vol. 8(1), pages 6-19.
- E. Samanidou & E. Zschischang & D. Stauffer & T. Lux, 2007. "Agent-based Models of Financial Markets," Papers physics/0701140, arXiv.org.
- Ken Binmore & Larry Samuelson, 2010. "Evolutionary Drift and Equilibrium Selection," Levine's Working Paper Archive 390, David K. Levine.
- R. Cressman, K.H. Schlag, 1995.
"The Dynamic (In)Stability of Backwards Induction,"
Discussion Paper Serie B
347, University of Bonn, Germany.
- Hart, Sergiu, 2002.
"Evolutionary dynamics and backward induction,"
Games and Economic Behavior,
Elsevier, vol. 41(2), pages 227-264, November.
- Binmore, Ken, 1988. "Modeling Rational Players: Part II," Economics and Philosophy, Cambridge University Press, vol. 4(01), pages 9-55, April.
- Valentina Bosetti, Carlo Carraro, Marzio Galeotti, Emanuele Massetti, Massimo Tavoni, 2006. "A World induced Technical Change Hybrid Model," The Energy Journal, International Association for Energy Economics, vol. 0(Special I), pages 13-38.
- Ken Binmore, 1997. "Rationality and backward induction," Journal of Economic Methodology, Taylor & Francis Journals, vol. 4(1), pages 23-41.
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