Abstract
<p style='text-indent:20px;'>We consider a model of cultural evolution for a strategy selection in a population of individuals who interact in a game theoretic framework. The evolution combines individual learning of the environment (population strategy profile), reproduction, proportional to the success of the acquired knowledge, and social transmission of the knowledge to the next generation. A mean-field type equation is derived that describes the dynamics of the distribution of cultural traits, in terms of the rate of learning, the reproduction rate and population size. We establish global well-posedness of the initial-boundary value problem for this equation and give several examples that illustrate the process of the cultural evolution.
Highlights
Evolutionary game theory, pioneered by Maynard Smith and Price [16] is a powerful tool that explains dominance of some behavioral traits as being uninvadable by other traits in the competition for Darwinian fitness points, when fitness is frequency dependent
The replicator equation governs the dynamics of reinforcement learning in repeated play of a game, see Borgers and Sarin [1], Fudenberg and Levine [3], Krishnedu et al [10], Perepelitsa [11]
Learning in games is an integral part of game theory that goes back to works of Robinson [13] and Shapley [14]
Summary
Evolutionary game theory, pioneered by Maynard Smith and Price [16] is a powerful tool that explains dominance of some behavioral traits as being uninvadable by other traits in the competition for Darwinian fitness points, when fitness is frequency dependent. Of Rock-Paper-Scissors game, we show how exponentially growing heterogeneous population can lock the cultural evolution in a suboptimal pure strategy, in contrast to both, the dynamics of the replicator and best-response equations. Determining asymptotic state for this type of evolution for an arbitrary game is problematic due to complicated dynamics and absence of entropy functionals It can be done in some cases, at least partially, as in the model with zero reproduction rate. An interaction is a round of the game between to random agents, say i and j, who play according to their priors Pit and Ptj, that is, using their best response strategies Based on that, they earn fitness points and update the learning priors.
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