Three steps ahead

Heller, Yuval (2012): Three steps ahead.

Abstract

Experimental evidence suggest that people only use 1-3 iterations of strategic reasoning, and that some people systematically use less iterations than others. In this paper, we present a novel evolutionary foundation for these stylized facts. In our model, agents interact in finitely repeated Prisoner's Dilemma, and each agent is characterized by the number of steps he thinks ahead. When two agents interact, each of them has an independent probability to observe the opponent's type. We show that if this probability is not too close to 0 or 1, then the evolutionary process admits a unique stable outcome, in which the population includes a mixture of “naive” agents who think 1 step ahead, and “sophisticated” agents who think 2-3 steps ahead.

• Item Type: MPRA Paper
• Language: English
• Keywords: Indirect evolution, cognitive hierarchy, bounded forward-looking, Prisoner's Dilemma, Cooperation
• Subjects: D - Microeconomics > D0 - General > D03 - Behavioral Microeconomics: Underlying Principles; C - Mathematical and Quantitative Methods > C7 - Game Theory and Bargaining Theory > C73 - Stochastic and Dynamic Games ; Evolutionary Games ; Repeated Games

References

Andreoni, J., and J. Miller (1993): Rational cooperation in the finitely repeated prisoner's dilemma: Experimental evidence, The Economic Journal, 103(418), 570-585.

Beard, T., and R. Beil Jr (1994): Do people rely on the self-interested maximization of others: An experimental test, Management Science, pp. 252-262.

Bosch-Domenech, A., J. Montalvo, R. Nagel, and A. Satorra (2002): One,two,(three), innity,...: Newspaper and lab beauty-contest experiments, The American Economic Review, 92(5), 1687-1701.

Bruttel, L., W. Güth, and U. Kamecke (2012): Finitely repeated prisoners dilemma experiments without a commonly known end, International Journal of Game Theory, pp. 125.

Camerer, C. (2003): Behavioral game theory: Experiments in strategic interaction. Princeton University Press.

Camerer, C., T. Ho, and J. Chong (2004): A cognitive hierarchy model of games, The Quarterly Journal of Economics, 119(3), 861-898.

Chong, J., C. Camerer, and T. Ho (2005): Cognitive hierarchy: A limited thinking theory in games, Experimental Business Research, pp. 203-228.

Cooper, R., D. DeJong, R. Forsythe, and T. Ross (1996): Cooperation without reputation: experimental evidence from prisoner's dilemma games, Games and Economic Behavior, 12, 187-218.

Crawford, V. (2003): Lying for strategic advantage: Rational and boundedly rational misrepresentation of intentions, The American Economic Review, 93(1), 133-149.

Crawford, V., and N. Iriberri (2007): Level-k Auctions: Can a Nonequilibrium Model of Strategic Thinking Explain the Winner's Curse and Overbidding in Private-Value Auctions, Econometrica, 75(6), 1721-1770.

Dekel, E., J. C. Ely, and O. Yilankaya (2007): Evolution of Preferences, Review of Economic Studies, 74(3), 685-704.

Frenkel, S., Y. Heller, and R. Teper (2012): Endowment as a Blessing.

Gill, D., and V. Prowse (2012): Cognitive ability and learning to play equilibrium: A level-k analysis.

Güth, W., and M. Yaari (1992): Explaining Reciprocal Behavior in Simple Strategic Games: An Evolutionary Approach, in Explaining Process and Change: Approaches to Evolutionary Economics, ed. by U. Witt, pp. 2334. The University of Michigan Press, Ann Arbor.

Haviland, W., H. Prins, and D. Walrath (2007): Cultural anthropology: the human challenge. Wadsworth Pub Co.

Heifetz, A., and A. Pauzner (2005): Backward induction with players who doubt others' faultlessness, Mathematical Social Sciences, 50(3), 252-267.

Ho, T., C. Camerer, and K. Weigelt (1998): Iterated dominance and iterated best response in experimental" p-beauty contests", The American Economic Review, 88(4), 947-969.

Hyndman, K., A. Terracol, and J. Vaksmann (2012): Beliefs and (In) Stability in Normal-Form Games.

Jehiel, P. (2005): Analogy-based expectation equilibrium, Journal of Economic theory, 123(2), 81-104.

Johnson, E., C. Camerer, S. Sen, and T. Rymon (2002): Detecting failures of backward induction: Monitoring information search in sequential bargaining, Journal of Eco- nomic Theory, 104(1), 16-47.

Maynard Smith, J. (1982): Evolution and the theory of games.

McKelvey, R., and T. Palfrey (1995): Quantal response equilibria for normal form games, Games and Economic Behavior, 10(1), 638.

Mohlin, E. (2012): Evolution of theories of mind, Games and Economic Behavior, 75(1), 299-318.

Nagel, R. (1995): Unraveling in guessing games: An experimental study, The American Economic Review, 85(5), 1313-1326.

Nagel, R., and F. Tang (1998): Experimental results on the centipede game in normal form: an investigation on learning, Journal of Mathematical Psychology, 42(2), 356-384.

Rapoport, A., and W. Amaldoss (2004): Mixed strategies and iterative elimination of strongly dominated strategies: An experimental investigation of states of knowledge, Journal of Economic Behavior & Organization, 42(4), 483-521.

Robson, A. (2003): The evolution of rationality and the Red Queen, Journal of Economic Theory, 111(1), 122.

Rosenthal, R. (1981): Games of Perfect Information, Predatory Pricing and the Chain-Store Paradox., Journal of Economic Theory, 25(1), 92-100.

Selten, R., and R. Stoecker (1986): End behavior in sequences of nite Prisoner's Dilemma supergames A learning theory approach, Journal of Economic Behavior & Or- ganization, 7(1), 47-70.

Stahl, D., and P. Wilson (1994): Experimental evidence on players' models of other players, Journal of economic behavior & organization, 25(3), 309-327.

Stahl Dale, O. (1993): Evolution of Smartn Players, Games and Economic Behavior, 5(4), 604-617.

Stennek, J. (2000): The survival value of assuming others to be rational, International Journal of Game Theory, 29(2), 147-163.