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Individuals belonging to two large populations are repeatedly randomly matched to play a cyclic \(2\times 2\) game such as Matching Pennies. Between matching rounds, individuals sometimes change their strategy after observing a finite sample of other outcomes within their population. Individuals from the same population follow the same behavioral rule. In the resulting discrete time dynamics the unique Nash equilibrium is unstable. However, for sample sizes greater than one, we present an imitation rule where long run play cycles closely around the equilibrium.

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Hofbauer, J., Schlag, K. Sophisticated imitation in cyclic games. J Evol Econ 10, 523–543 (2000).

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