Conservative and Dissipative Polymatrix Replicators

TL;DR

This paper extends the theory of dissipative Lotka-Volterra systems to polymatrix replicators, providing insights into their attractors and embedding properties within Hamiltonian systems, applicable to diverse evolutionary game dynamics.

Contribution

It generalizes Redheffer's dissipative Lotka-Volterra theory to polymatrix replicators, including a reduction algorithm and attractor analysis.

Findings

Extension of dissipative theory to polymatrix replicators

Reduction algorithm for attractor localization

Embedding of attractor dynamics into Hamiltonian systems

Abstract

In this paper we address a class of replicator dynamics, referred as polymatrix replicators, that contains well known classes of evolutionary game dynamics, such as the symmetric and asymmetric (or bimatrix) replicator equations, and some replicator equations for $n$-person games. Polymatrix replicators form a simple class of algebraic o.d.e.'s on prisms (products of simplexes), which describe the evolution of strategical behaviours within a po\-pu\-lation stratified in $n\geq 1$ social groups. In the 80's Raymond Redheffer et al. developed a theory on the class of stably dissipative Lotka-Volterra systems. This theory is built around a reduction algorithm that "infers" the localization of the system' s attractor in some affine subspace. It was later proven that the dynamics on the attractor of such systems is always embeddable in a Hamiltonian Lotka-Volterra system. In this paper we extend these results to polymatrix replicators.