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

TL;DR

This paper analyzes a stochastic version of replicator dynamics in zero-sum games, revealing that invariant measures are supported on pure strategies and highlighting differences from Nash equilibrium predictions.

Contribution

It characterizes the invariant measures of stochastic replicator dynamics in zero-sum games, showing they are supported on pure strategies and depend on noise strength.

Findings

Invariant measures are supported on boundary pure strategies.

Noise strength influences the distribution of invariant measures.

Stochastic dynamics differ from Nash equilibrium predictions.

Abstract

We study the behavior of a stochastic variant of replicator dynamics in two-agent zero-sum games. We characterize the statistics of such systems by their invariant measures which can be shown to be entirely supported on the boundary of the space of mixed strategies. Depending on the noise strength we can furthermore characterize these invariant measures by finding accumulation of mass at specific parts of the boundary. In particular, regardless of the magnitude of noise, we show that any invariant probability measure is a convex combination of Dirac measures on pure strategy profiles, which correspond to vertices/corners of the agents' simplices. Thus, in the presence of stochastic perturbations, even in the most classic zero-sum settings, such as Matching Pennies, we observe a stark disagreement between the axiomatic prediction of Nash equilibrium and the evolutionary emergent behavior derived by an assumption of stochastically adaptive, learning agents.