Evolutionary game dynamics in a Wright-Fisher process

Abstract

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.

Active linking in evolutionary games

“…For detailed reviews of the replicator equation and other approaches to evolutionary game dynamics, see Fudenberg and Tirole (1991), Weibull (1995), Samuelson (1997), Cressman (2003), Sigmund (1998, 2003), Gintis (2000) and Nowak and Sigmund (2004). in finite populations Imhof and Nowak, 2006;Taylor et al, 2004;Fudenberg et al, 2006;Traulsen et al, 2006a,b), in spatially extended systems (Nowak and May, 1992;Nakamaru et al, 1998;Killingback and Doebeli, 1996;van Baalen and Rand, 1998;Irwin and Taylor, 2001;Hauert and Doebeli, 2004;Ifti et al, 2004;Nakamaru and Iwasa, 2005;Jansen and van Baalen, 2006) or on graphs (Lieberman et al, 2005;Santos et al, , 2006a. Taylor and Nowak (2006) analyze a scenario where the interaction rate does depend on the strategies.…”

Continuous Probabilistic Analysis to Evolutionary Game Dynamics in Finite Populations

“…Game theory, in which only selfish behaviors will be rewarded, shares the common dilemma as the evolution of cooperation in nature. Therefore, during the last few decades, game theory has been a central tool for understanding the origin of cooperation (Maynard Smith, 1982;Weibull, 1995;Hofbauer and Sigmund, 1998;Nowak and Sigmund, 2004;Nowak, 2006). Actually, there are a variety of game-theoretic models used to analyze cooperative dilemma, such as Prisoner's dilemma (PD) game, snow-drift (SD) game (or Hawk-dove game), and public goods games.…”

“…Over the past few decades, several mechanisms have been proposed to successfully overcome the dilemma. Nowak (2006) reviewed the related studies and categorized there mechanisms as five rules: kin selection, direct reciprocity, indirect reciprocity, network reciprocity, and group selection. Generally, any game-theoretic E-mail address: gaomeng03@hotmail.com. model of cooperative evolution will be related to the five rules.…”

“…For structured population, the seminal work by Nowak and May (1992) spawned a large number of investigations of "games on grids." Although spatial structure does not always favor cooperation, it is considered to be of relevance to "kin selection" (Doebeli and Hauert, 2004, and references therein;Nowak, 2006). However, for finite populations, one inevitable problem in analysis of evolutionary game dynamics is the analytical intractability.…”

“…However, for finite populations, one inevitable problem in analysis of evolutionary game dynamics is the analytical intractability. Thus, a stochastic approach is required in studying evolutionary game dynamics in finite populations Taylor et al, 2004;Imhof and Nowak, 2006;Traulsen et al 2006aTraulsen et al , 2006b.…”