Evolutionary cycles of cooperation and defection.

The main obstacle for the evolution of cooperation is that natural selection favors defection in most settings. In the repeated prisoner's dilemma, two individuals interact several times, and, in each round, they have a choice between cooperation and defection. We analyze the evolutionary dynamics of three simple strategies for the repeated prisoner's dilemma: always defect (ALLD), always cooperate (ALLC), and tit-for-tat (TFT). We study mutation-selection dynamics in finite populations. Despite ALLD being the only strict Nash equilibrium, we observe evolutionary oscillations among all three strategies. The population cycles from ALLD to TFT to ALLC and back to ALLD. Most surprisingly, the time average of these oscillations can be entirely concentrated on TFT. In contrast to the classical expectation, which is informed by deterministic evolutionary game theory of infinitely large populations, stochastic evolution of finite populations need not choose the strict Nash equilibrium and can therefore favor cooperation over defection.

[1]  J. Nash Equilibrium Points in N-Person Games. , 1950, Proceedings of the National Academy of Sciences of the United States of America.

[2]  P. Moran,et al.  The statistical processes of evolutionary theory. , 1963 .

[3]  A. W. F. Edwards,et al.  The statistical processes of evolutionary theory , 1963 .

[4]  R. Trivers The Evolution of Reciprocal Altruism , 1971, The Quarterly Review of Biology.

[5]  J. M. Smith,et al.  The Logic of Animal Conflict , 1973, Nature.

[6]  J M Smith,et al.  Evolution and the theory of games , 1976 .

[7]  D. E. Matthews Evolution and the Theory of Games , 1977 .

[8]  P. Taylor,et al.  Evolutionarily Stable Strategies and Game Dynamics , 1978 .

[9]  J. Riley,et al.  Evolutionary equilibrium strategies. , 1979, Journal of theoretical biology.

[10]  S. Smale The Prisoner's Dilemma and Dynamical Systems Associated to Non-Cooperative Games , 1980 .

[11]  Robert J. Aumann,et al.  Essays in game theory and mathematical economics in honor of Oskar Morgenstern , 1981 .

[12]  W. Hamilton,et al.  The Evolution of Cooperation , 1984 .

[13]  P. Molander The Optimal Level of Generosity in a Selfish, Uncertain Environment , 1985 .

[14]  Drew Fudenberg,et al.  The Folk Theorem in Repeated Games with Discounting or with Incomplete Information , 1986 .

[15]  M. Schaffer,et al.  Evolutionarily stable strategies for a finite population and a variable contest size. , 1988, Journal of theoretical biology.

[16]  J. Sanford,et al.  Applying the PDR principle to AIDS. , 1988, Journal of theoretical biology.

[17]  D. Fudenberg,et al.  Evolution and Cooperation in Noisy Repeated Games , 1990 .

[18]  L. Samuelson,et al.  Evolutionary stability in repeated games played by finite automata , 1992 .

[19]  M. Nowak,et al.  Tit for tat in heterogeneous populations , 1992, Nature.

[20]  Alvin E. Roth,et al.  The Early History of Experimental Economics , 1993, Journal of the History of Economic Thought.

[21]  M. Nowak,et al.  A strategy of win-stay, lose-shift that outperforms tit-for-tat in the Prisoner's Dilemma game , 1993, Nature.

[22]  Ross Cressman,et al.  Evolutionary Stability in the Finitely Repeated Prisoner 's Dilemma Game , 1996 .

[23]  Josef Hofbauer,et al.  Evolutionary Games and Population Dynamics , 1998 .

[24]  David B. Fogel,et al.  On the instability of evolutionary stable strategies in small populations , 1998 .

[25]  D P Kraines,et al.  Natural selection of memory-one strategies for the iterated prisoner's dilemma. , 2000, Journal of theoretical biology.

[26]  Sebastian J. Schreiber,et al.  Urn Models, Replicator Processes, and Random Genetic Drift , 2001, SIAM J. Appl. Math..

[27]  Drew Fudenberg,et al.  Evolutionary game dynamics in finite populations , 2004, Bulletin of mathematical biology.

[28]  D. Fudenberg,et al.  Emergence of cooperation and evolutionary stability in finite populations , 2004, Nature.

[29]  Karl Sigmund,et al.  Indirect reciprocity, image scoring, and moral hazard. , 2005, Proceedings of the National Academy of Sciences of the United States of America.