Pairwise comparison and selection temperature in evolutionary game dynamics.

Recently, the frequency-dependent Moran process has been introduced in order to describe evolutionary game dynamics in finite populations. Here, an alternative to this process is investigated that is based on pairwise comparison between two individuals. We follow a long tradition in the physics community and introduce a temperature (of selection) to account for stochastic effects. We calculate the fixation probabilities and fixation times for any symmetric 2 x 2 game, for any intensity of selection and any initial number of mutants. The temperature can be used to gauge continuously from neutral drift to the extreme selection intensity known as imitation dynamics. For some payoff matrices the distribution of fixation times can become so broad that the average value is no longer very meaningful.

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

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

[3]  C. Hauert,et al.  Game theory and physics , 2005 .

[4]  L. Imhof,et al.  Stochasticity and evolutionary stability. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[5]  Jörgen W. Weibull,et al.  Evolutionary Game Theory , 1996 .

[6]  T. Antal,et al.  Fixation of Strategies for an Evolutionary Game in Finite Populations , 2005, Bulletin of mathematical biology.

[7]  Michael Doebeli,et al.  Spatial structure often inhibits the evolution of cooperation in the snowdrift game , 2004, Nature.

[8]  V. Eguíluz,et al.  Cooperation and the Emergence of Role Differentiation in the Dynamics of Social Networks1 , 2005, American Journal of Sociology.

[9]  F. C. Santos,et al.  Scale-free networks provide a unifying framework for the emergence of cooperation. , 2005, Physical review letters.

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

[11]  E. Hill.  Journal of Theoretical Biology , 1961, Nature.

[12]  M. Kimura.  Evolutionary Rate at the Molecular Level , 1968, Nature.

[13]  M. Nowak,et al.  Evolutionary game dynamics in a Wright-Fisher process , 2006, Journal of mathematical biology.

[14]  B A Huberman,et al.  Evolutionary games and computer simulations. , 1993, Proceedings of the National Academy of Sciences of the United States of America.

[15]  M. Nowak,et al.  Evolutionary games and spatial chaos , 1992, Nature.

[16]  G. Szabó,et al.  Evolutionary prisoner's dilemma game on a square lattice , 1997, cond-mat/9710096.

[17]  Jordan B. Pollack,et al.  Effects of Finite Populations on Evolutionary Stable Strategies , 2000, GECCO.

[18]  M. Nowak,et al.  Evolutionary Dynamics of Biological Games , 2004, Science.

[19]  M A Nowak,et al.  Spatial games and the maintenance of cooperation. , 1994, Proceedings of the National Academy of Sciences of the United States of America.

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

[21]  Arne Traulsen,et al.  Stochastic payoff evaluation increases the temperature of selection. , 2007, Journal of theoretical biology.

[22]  L. Blume.  The Statistical Mechanics of Strategic Interaction , 1993 .

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

[24]  P. Taylor,et al.  Fitness and evolutionary stability in game theoretic models of finite populations , 2004, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[25]  W. Godwin.  Article in Press , 2000 .

[26]  D. Fudenberg,et al.  Evolutionary cycles of cooperation and defection. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[27]  R. Axelrod,et al.  Evolutionary Dynamics , 2004 .

[28]  Drew Fudenberg,et al.  Imitation Processes with Small Mutations , 2004, J. Econ. Theory.

[29]  Erwin Frey,et al.  Coexistence versus extinction in the stochastic cyclic Lotka-Volterra model. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[30]  Samuel Karlin,et al.  A First Course on Stochastic Processes , 1968 .

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

[32]  F. C. Santos,et al.  Graph topology plays a determinant role in the evolution of cooperation , 2006, Proceedings of the Royal Society B: Biological Sciences.

[33]  M. Nowak,et al.  Stochastic dynamics of invasion and fixation. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

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

[35]  M. Kimura,et al.  An introduction to population genetics theory , 1971 .

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

[37]  Arne Traulsen,et al.  Coevolutionary dynamics in large, but finite populations. , 2006, Physical review. E, Statistical, nonlinear, and soft matter physics.

[38]  M. Nowak,et al.  The evolution of stochastic strategies in the Prisoner's Dilemma , 1990 .

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