From Darwin to Poincar\'e and von Neumann: Recurrence and Cycles in Evolutionary and Algorithmic Game Theory

TL;DR

This paper characterizes the behavior of replicator dynamics in zero-sum evolutionary games, showing recurrence in the presence of interior Nash equilibria and convergence to boundaries otherwise, linking evolutionary and algorithmic game theory.

Contribution

It provides a complete characterization of replicator dynamics in zero-sum games and extends recurrence results to generalized game classes, unifying evolutionary and algorithmic game theory.

Findings

Replicator dynamics are recurrent if and only if an interior Nash equilibrium exists.

In absence of interior equilibria, strategies outside equilibrium support vanish.

Two degrees of freedom suffice to establish periodicity in these dynamics.

Abstract

Replicator dynamics, the continuous-time analogue of Multiplicative Weights Updates, is the main dynamic in evolutionary game theory. In simple evolutionary zero-sum games, such as Rock-Paper-Scissors, replicator dynamic is periodic \cite{zeeman1980population}, however, its behavior in higher dimensions is not well understood. We provide a complete characterization of its behavior in zero-sum evolutionary games. We prove that, if and only if, the system has an interior Nash equilibrium, the dynamics exhibit Poincar\'{e} recurrence, i.e., almost all orbits come arbitrary close to their initial conditions infinitely often. If no interior equilibria exist, then all interior initial conditions converge to the boundary. Specifically, the strategies that are not in the support of any equilibrium vanish in the limit of all orbits. All recurrence results furthermore extend to a class of games that generalize both graphical polymatrix games as well as evolutionary games, establishing a unifying link between evolutionary and algorithmic game theory. We show that two degrees of freedom, as in Rock-Paper-Scissors, is sufficient to prove periodicity.