The emergence of rational behavior in the presence of stochastic perturbations

We study repeated games where players use an exponential learning scheme in order to adapt to an ever-changing environment. If the game’s payoffs are subject to random perturbations, this scheme leads to a new stochastic version of the replicator dynamics that is quite different from the “aggregate shocks” approach of evolutionary game theory. Irrespective of the perturbations’ magnitude, we find that strategies which are dominated (even iteratively) eventually become extinct and that the game’s strict Nash equilibria are stochastically asymptotically stable. We complement our analysis by illustrating these results in the case of congestion games.

💡 Research Summary

This paper investigates how rational behavior can emerge in repeated games when payoffs are subject to stochastic perturbations. The authors adopt an exponential learning rule, whereby each player adjusts the weight of a strategy proportionally to its realized payoff, with a constant learning rate. Assuming that payoff fluctuations follow independent white‑noise processes, the continuous‑time limit yields a novel stochastic replicator dynamics:

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