Breitmoser, Yves (2012): Cooperation, but no reciprocity: Individual strategies in the repeated Prisoner's Dilemma.
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A recent advance in our understanding of repeated PDs is the detection of a threshold d* at which laboratory subjects start to cooperate predictively. This threshold is substantially above the classic threshold "existence of Grim equilibrium" and has been characterized axiomatically by Blonski, Ockenfels, and Spagnolo (2011, BOS). In this paper, I derive its behavioral foundations. First, I show that the threshold is equivalent to existence of a "Semi-Grim" equilibrium s_cc>s_cd=s_dc>s_dd. It is cooperative (s_cc>0.5), non-reciprocal (s_cd=s_dc), and robust to imperfect monitoring ("belief-free"). Next, I show that the no-reciprocity condition s_cd=s_dc also follows from robustness to random-utility perturbations (logit equilibrium). Finally, I re-analyze strategies in four recent experiments and find that the majority of subjects indeed plays Semi-Grim when it is an equilibrium strategy, which explains d*'s predictive success.
|Item Type:||MPRA Paper|
|Original Title:||Cooperation, but no reciprocity: Individual strategies in the repeated Prisoner's Dilemma|
|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
|Depositing User:||Yves Breitmoser|
|Date Deposited:||05. Oct 2012 16:23|
|Last Modified:||01. Mar 2013 02:10|
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.
Barlo, M., Carmona, G., and Sabourian, H. (2009). Repeated games with one-memory. Journal of Economic Theory, 144(1):312–336.
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.
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., Rand, D., and Dreber, A. (2012). Slow to anger and fast to forgive: Cooperation in an uncertain world. American Economic Review.
Fudenberg, D. and Yamamoto, Y. (2010). Repeated games where the payoffs and monitoring structure are unknown. Econometrica, 78(5):1673–1710.
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.
Govindan, S. and Wilson, R. (2006). Sufficient conditions for stable equilibria. Theoretical Economics, 1(2):167–206.
Govindan, S. and Wilson, R. (2012). Axiomatic equilibrium selection for generic two-player games. Econometrica, 80(4):1639–1699.
Hörner, J. and Lovo, S. (2009). Belief-free equilibria in games with incomplete information. Econometrica, 77(2):453–487.
Hörner, J. and Olszewski, W. (2006). The folk theorem for games with private almost-perfect monitoring. Econometrica, 74(6):1499–1544.
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.
Kandori, M. (2011). Weakly belief-free equilibria in repeated games with private monitoring. Econometrica, 79(3):877–892.
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 out-performs 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.
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.
Sabourian, H. (1998). Repeated games with m-period bounded memory (pure strategies). Journal of Mathematical Economics, 30(1):1–35.
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.