Cooperation, but no reciprocity: Individual strategies in the repeated Prisoner's Dilemma

Breitmoser, Yves (2013): Cooperation, but no reciprocity: Individual strategies in the repeated Prisoner's Dilemma.

Abstract

In the repeated Prisoner's Dilemma, predictions are notoriously difficult. Recently, however, Blonski, Ockenfels, and Spagnolo (2011, BOS) showed that experimental subjects predictably cooperate when the discount factor exceeds a particular threshold. I show that this threshold implies existence of an equilibrium robust to two standard refinement assumptions (utility perturbations and imperfect monitoring). The equilibrium is "Semi-Grim": Cooperate after mutual cooperation, defect after mutual defection, randomize otherwise. Testing six resulting predictions on existing data, comprising 37.000 observations, I then find that subjects indeed play Semi-Grim strategies, and switch to cooperation in round 1, when the former turn into equilibria -- at the BOS-threshold.

• Item Type: MPRA Paper
• Language: English
• Keywords: Repeated Prisoner's Dilemma, experiment, equilibrium selection, cooperative behavior, reciprocity, belief-free equilibria, robustness
• Subjects: C - Mathematical and Quantitative Methods > C9 - Design of Experiments > C92 - Laboratory, Group Behavior; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C73 - Stochastic and Dynamic Games ; Evolutionary Games ; Repeated Games; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C72 - Noncooperative Games

References

Abreu, D., Pearce, D., and Stacchetti, E. (1990). Toward a theory of discounted repeated games with imperfect monitoring. Econometrica, 58(5):1041–1063.

Axelrod, R. (1980a). Effective choice in the prisoner’s dilemma. Journal of Conflict Resolution, 24(1):3–25.

Axelrod, R. (1980b). More effective choice in the prisoner’s dilemma. Journal of Conflict Resolution, 24(3):379–403.

Bhaskar, V., Mailath, G., and Morris, S. (2008). Purification in the infinitely-repeated prisoners’ dilemma. Review of Economic Dynamics, 11(3):515–528.

Biernacki, C., Celeux, G., and Govaert, G. (2000). Assessing a mixture model for clustering with the integrated completed likelihood. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(7):719–725.

Blonski, M., Ockenfels, P., and Spagnolo, G. (2011). Equilibrium selection in the repeated prisoner’s dilemma: Axiomatic approach and experimental evidence. American Economic Journal: Microeconomics, 3(3):164–192.

Breitmoser, Y. (2012). Strategic reasoning in p-beauty contests. Games and Economic Behavior, 75(2):555–569.

Breitmoser, Y., Tan, J., and Zizzo, D. (2010). Understanding perpetual r&d races. Economic Theory, 44(3):445–467.

Bruttel, L. and Kamecke, U. (2012). Infinity in the lab. how do people play repeated games? Theory and Decision, 72(2):205–219.

Capra, C., Goeree, J., Gomez, R., and Holt, C. (1999). Anomalous behavior in a traveler’s dilemma? American Economic Review, 89(3):678–690.

Choi, S., Gale, D., and Kariv, S. (2008). Sequential equilibrium in monotone games: A theory-based analysis of experimental data. Journal of Economic Theory, 143(1):302–330.

Dal Bo, P. (2005). Cooperation under the shadow of the future: Experimental evidence from infinitely repeated games. American Economic Review, 95(5):1591–1604.

Dal Bo, P. and Fréchette, G. (2011). The evolution of cooperation in infinitely repeated games: Experimental evidence. American Economic Review, 101(1):411–429.

Doraszelski, U. and Escobar, J. F. (2010). A theory of regular markov perfect equilibria in dynamic stochastic games: Genericity, stability, and purification. Theoretical Economics, 5:369–402.

Duffy, J. and Ochs, J. (2009). Cooperative behavior and the frequency of social interaction. Games and Economic Behavior, 66(2):785–812.

Ely, J., Hörner, J., and Olszewski, W. (2005). Belief-free equilibria in repeated games. Econometrica, 73(2):377–415.

Ely, J. and Välimäki, J. (2002). A robust folk theorem for the prisoner’s dilemma. Journal of Economic Theory, 102(1):84–105.

Engle-Warnick, J. and Slonim, R. (2004). The evolution of strategies in a repeated trust game. Journal of Economic Behavior and Organization, 55(4):553–573.

Engle-Warnick, J. and Slonim, R. (2006). Inferring repeated-game strategies from actions: evidence from trust game experiments. Economic Theory, 28(3):603–632.

Erev, I. and Roth, A. (2001). Simple reinforcement learning models and reciprocation in the prisoner’s dilemma game. In Gigerenzer, G. and Selten, R., editors, Bounded rationality: The adaptive toolbox, pages 215–231. MIT Press.

Evans, R. and Maskin, E. (1989). Efficient renegotiation-proof equilibria in repeated games. Games and Economic Behavior, 1:361–369.

Fey, M., McKelvey, R., and Palfrey, T. (1996). An experimental study of constant-sum centipede games. International Journal of Game Theory, 25(3):269–287.

Fudenberg, D. and Maskin, E. (1986). The folk theorem in repeated games with discounting or with incomplete information. Econometrica, 54(3):533–554.

Fudenberg, D., Rand, D., and Dreber, A. (2012). Slow to anger and fast to forgive: Cooperation in an uncertain world. American Economic Review.

Goeree, J., Holt, C., and Laury, S. (2002a). Private costs and public benefits: Unraveling the effects of altruism and noisy behavior. Journal of Public Economics, 83(2):255–276.

Goeree, J., Holt, C., and Palfrey, T. (2002b). Quantal response equilibrium and overbidding in private-value auctions. Journal of Economic Theory, 104(1):247–272.

Imhof, L., Fudenberg, D., and Nowak, M. (2007). Tit-for-tat or win-stay, lose-shift? Journal of Theoretical Biology, 247(3):574–580.

Kandori, M. (2002). Introduction to repeated games with private monitoring. Journal of Economic Theory, 102(1):1–15.

Luce, R. (1959). Individual choice behavior. Wiley New York.

McKelvey, R. and Palfrey, T. (1995). Quantal response equilibria for normal form games. Games and Economic Behavior, 10(1):6–38.

McLachlan, G. and Peel, D. (2000). Finite Mixture Models. Wiley series in probability and statistics.

Murnighan, J. and Roth, A. (1983). Expecting continued play in prisoner’s dilemma games. Journal of Conflict Resolution, 27(2):279–300.

Nowak, M., Sigmund, K., et al. (1993). A strategy of win-stay, lose-shift that outperforms tit-for-tat in the prisoner’s dilemma game. Nature, 364(6432):56–58.

Powell, M. (2006). The newuoa software for unconstrained optimization without derivatives. Large-Scale Nonlinear Optimization, pages 255–297.

Press, W. H. and Dyson, F. J. (2012). Iterated prisoner's dilemma contains strategies that dominate any evolutionary opponent. Proceedings of the National Academy of Sciences, 109(26):10409–10413.

Rapoport, A. and Mowshowitz, A. (1966). Experimental studies of stochastic models for the prisoner’s dilemma. Behavioral Science, 11(6):444–458.

Roth, A. and Murnighan, J. (1978). Equilibrium behavior and repeated play of the prisoner’s dilemma. Journal of Mathematical Psychology, 17(2):189–198.

Stahl, D. et al. (1991). The graph of prisoners’ dilemma supergame payoffs as a function of the discount factor. Games and Economic Behavior, 3(3):368–384.

Stahl, D. and Wilson, P. (1994). Experimental evidence on players’ models of other players. Journal of Economic Behavior and Organization, 25(3):309–327.

Stahl, D. and Wilson, P. (1995). On players’ models of other players: Theory and experimental evidence. Games and Economic Behavior, 10(1):218–254.

Stewart, A. J. and Plotkin, J. B. (2012). Extortion and cooperation in the prisoner's dilemma. Proceedings of the National Academy of Sciences, 109(26):10134–10135.

Van Damme, E. (1989). Renegotiation-proof equilibria in repeated prisoners’ dilemma. Journal of Economic Theory, 47(1):206–217.