Journal of Mathematical Biology

, Volume 52, Issue 5, pp 667–681 | Cite as

Evolutionary game dynamics in a Wright-Fisher process

  • Lorens A. Imhof
  • Martin A. Nowak


Evolutionary game dynamics in finite populations can be described by a frequency dependent, stochastic Wright-Fisher process. We consider a symmetric game between two strategies, A and B. There are discrete generations. In each generation, individuals produce offspring proportional to their payoff. The next generation is sampled randomly from this pool of offspring. The total population size is constant. The resulting Markov process has two absorbing states corresponding to homogeneous populations of all A or all B. We quantify frequency dependent selection by comparing the absorption probabilities to the corresponding probabilities under random drift. We derive conditions for selection to favor one strategy or the other by using the concept of total positivity. In the limit of weak selection, we obtain the 1/3 law: if A and B are strict Nash equilibria then selection favors replacement of B by A, if the unstable equilibrium occurs at a frequency of A which is less than 1/3.


Frequency dependent process Stochastic game dynamics Finite populations 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  1. 1.Statistische AbteilungUniversität BonnBonnGermany
  2. 2.Program for Evolutionary Dynamics, Department of Mathematics, Department of Organismic and Evolutionary BiologyHarvard UniversityCambridgeUSA

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