Abstract
We analyze a class of imitation dynamics with mutations for games with any finite number of actions, and give conditions for the selection of a unique equilibrium as the mutation rate becomes small and the population becomes large. Our results cover the multiple-action extensions of the aspiration-and-imitation process of Binmore and Samuelson [Muddling through: noisy equilibrium selection, J. Econ. Theory 74 (1997) 235–265] and the related processes proposed by Benaı¨m and Weibull [Deterministic approximation of stochastic evolution in games, Econometrica 71 (2003) 873–903] and Traulsen et al. [Coevolutionary dynamics: from finite to infinite populations, Phys. Rev. Lett. 95 (2005) 238701], as well as the frequency-dependent Moran process studied by Fudenberg et al. [Evolutionary game dynamics in finite populations with strong selection and weak mutation, Theoretical Population Biol. 70 (2006) 352–363]. We illustrate our results by considering the effect of the number of periods of repetition on the selected equilibrium in repeated play of the prisoner's dilemma when players are restricted to a small set of simple strategies.
Highlights
We study a class of imitation dynamics in large populations playing an n n game
We assume that if only two strategies are present in the population, the probabilities of the possible transitions depend only on the current payo¤s to these strategies, and that the expected motion is in the direction of the better response; this is the sense in which the dynamics are "monotone. 1" we assume that a small mutation term makes the system ergodic
This class of dynamics encompasses various models that have been studied in the literature, mainly for 2 2 games; e.g. the aspirationand-imitation process of Binmore and Samuelson [3], the related imitation processes proposed by Benaïm and Weibull [2], Björnerstedt and Weibull [5] and Traulsen et al [24], and the frequency-dependent Moran process introduced by Nowak et al [20]
Summary
We study a class of imitation dynamics in large populations playing an n n game. Our main assumptions are that at every time step, at most one agent changes his strategy, and that this agent may only imitate a strategy that is currently in use. For any ...nite number m of repetitions, the system converges to the state where all agents play "Always Defect," but with an in...nite number of repetitions the "Always Defect" is weakly dominated, so its share of the population goes to 0, and the long-run distribution is some combination of the strategies "Tit for Tat" and "Perfect Tit-for-tat." We show that the conclusion from large-population limit of the case m = 1 is robust in the sense that it applies to the case of a ...nite but large number of repetitions and a range of "intermediate" population sizes This proves a conjecture of Imhof et al [16]. We state and prove a more general version of this result, relating the large-population limit in one game to the invariant distributions of ...nite-population versions of "nearby" games
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