The finitely repeated Prisoners’ 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 ”Convincer” strategies insist with two rounds of initial cooperation, trying to establish more cooperative play in the game, while the ”Follower” 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.
References
[1]
Handbook of Computational Economics, 1st; Tesfatsion, L.S., Judd, K.L., Eds.; Elsevier: Amsterdam, The Netherlands, 2006; Volume Vol. 2.
[2]
Arthur, W.B.; Holland, J.H.; Lebaron, B.; Palmer, R.; Tayler, P. Asset pricing under endogenous expectations in an artificial stock market model. In The Economy as an Evolving Complex System II; Arthur, W.B., Durlauf, S.N., Lane, D.A., Eds.; Addison-Wesley: Boston, MA, USA, 1997; pp. 15–44.
[3]
Samanidou, E.; Zschischang, E.; Stauffer, D.; Lux, T. Agent-based models of financial markets. Rep. Prog. Phys. 2007, 70, 409–450, doi:10.1088/0034-4885/70/3/R03.
[4]
Bosetti, V.; Carraro, C.; Galeotti, M.; Massetti, E.; Tavoni, M. A world induced technical change hybrid model. The Energy Journal 2006, 0, 13–38.
[5]
Parker, D.C.; Manson, S.M.; Janssen, M.A.; Hoffmann, M.J.; Deadman, P. Multi-agent systems for the simulation of land-use and land-cover change: A review. Ann. Assoc. Am. Geogr. 2003, 93, 314–337, doi:10.1111/1467-8306.9302004.
[6]
Simon, H.A. A Behavioral Model of Rational Choice. Models of Man, Social and Rational: Mathematical Essays on Rational Human Behavior in a Social Setting; Wiley: New York, NY, USA, 1957; pp. 99–118.
[7]
Camerer, C. Behavioral Game Theory: Experiments in Strategic Interaction; The Roundtable series in behavioral economics; Russell Sage Foundation: New York, NY, USA, 2003.
[8]
Aumann, R.J. Rule-Rationality versus Act-Rationality. In Discussion paper series; Center for Rationality and Interactive Decision Theory, Hebrew University: Jerusalem, Israel, 2008.
[9]
Binmore, K. Modeling rational players: Part I. Economics and Philosophy 1987, 3, 179–214, doi:10.1017/S0266267100002893.
[10]
Binmore, K. Modeling rational players: Part II. Economics and Philosophy 1988, 4, 9–55, doi:10.1017/S0266267100000328.
[11]
Nash, J.F. Equilibrium points in n-person games. In Proceedings of the National Academy of Sciences of the United States of America; 1950; 36, pp. 48–49.
[12]
Pettit, P.; Sugden, R. The backward induction paradox. J. Phil. 1989, 86, 169–182.
Basu, K. The traveler’s dilemma: Paradoxes of rationality in game theory. Am. Econ. Rev. 1994, 84, 391–395.
[15]
Selten, R.; Stoecker, R. End behavior in sequences of finite Prisoner’s Dilemma supergames A learning theory approach. J. Econ. Behav. Organ. 1986, 7, 47–70, doi:10.1016/0167-2681(86)90021-1.
[16]
Schuster, P.; Sigmund, K. Replicator dynamics. J. Theor. Biol. 1983, 100, 533–538, doi:10.1016/0022-5193(83)90445-9.
[17]
Hofbauer, J. The selection mutation equation. J. Math. Biol. 1985, 23, 41–53, doi:10.1007/BF00276557.
[18]
Nachbar, J.H. Evolution in the finitely repeated prisoner’s dilemma. J. Econ. Behav. Organ. 1992, 19, 307–326, doi:10.1016/0167-2681(92)90040-I.
[19]
Cressman, R. Evolutionary stability in the finitely repeated Prisoner’s Dilemma game. J. Econ. Theory 1996, 68, 234–248, doi:10.1006/jeth.1996.0012.
[20]
Cressman, R. Evolutionary Dynamics and Extensive Form Games; MIT Press Series on Economic Learning and Social Evolution; MIT Press: Cambridge, MA, USA, 2003.
[21]
Noldeke, G.; Samuelson, L. An evolutionary analysis of backward and forward induction. Games Econ. Behav. 1993, 5, 425–454, doi:10.1006/game.1993.1024.
[22]
Cressman, R.; Schlag, K. The dynamic (in)stability of backwards induction. J. Econ. Theory 1998, 83, 260–285, doi:10.1006/jeth.1996.2465.
[23]
Binmore, K.; Samuelson, L. Evolutionary drift and equilibrium selection. Rev. Econ. Stud. 1999, 66, 363–393.
[24]
Hart, S. Evolutionary dynamics and backward induction. Games Econ. Behav. 2002, 41, 227–264.
[25]
Hofbauer, J.; Sandholm, W.H. Survival of dominated strategies under evolutionary dynamics. Theor. Econ. 2011, 6, 341–377, doi:10.3982/TE771.
[26]
Gintis, H.; Cressman, R.; Ruijgrok. Subgame perfection in evolutionary dynamics with recurrent perturbations. In Handbook of Research on Complexity; Barkley Rosser, J., Ed.; Edward Elgar Publishing: Northampton, MA, USA, 2009; pp. 353–368.
[27]
Ponti, G. Cycles of learning in the centipede game. Games Econ. Behav. 2000, 30, 115–141, doi:10.1006/game.1998.0707.
[28]
Rosenthal, R. Games of perfect information, predatory pricing and the chain-store paradox. J. Econ. Theory 1981, 25, 92–100, doi:10.1016/0022-0531(81)90018-1.
[29]
Aumann, R.J. Backward induction and common knowledge of rationality. Games Econ. Behav. 1995, 8, 6–19, doi:10.1016/S0899-8256(05)80015-6.
[30]
Aumann, R.J. Reply to Binmore. Games Econ. Behav. 1996, 17, 138–146, doi:10.1006/game.1996.0099.
[31]
Binmore, K. Rationality and backward induction. J. Econ. Methodol. 1997, 4, 23–41, doi:10.1080/13501789700000002.
[32]
Gintis, H. Towards a renaissance of economic theory. J. Econ. Behav. Organ. 2010, 73, 34–40, doi:10.1016/j.jebo.2008.09.012.
