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We study the behavior of a stochastic variant of replicator dynamics intwo-agent zero-sum games. We characterize the statistics of such systems bytheir invariant measures which can be shown to be entirely supported on theboundary of the space of mixed strategies. Depending on the noise strength wecan furthermore characterize these invariant measures by finding accumulationof mass at specific parts of the boundary. In particular, regardless of themagnitude of noise, we show that any invariant probability measure is a convexcombination of Dirac measures on pure strategy profiles, which correspond tovertices/corners of the agents' simplices. Thus, in the presence of stochasticperturbations, even in the most classic zero-sum settings, such as MatchingPennies, we observe a stark disagreement between the axiomatic prediction ofNash equilibrium and the evolutionary emergent behavior derived by anassumption of stochastically adaptive, learning agents.

A stochastic variant of replicator dynamics in zero-sum games and its invariant measures