Conservative and dissipative polymatrix replicators

Hassan Najafi  Alishah; Pedro  Duarte; Telmo  Peixe

doi:10.3934/jdg.2015.2.157

Article Contents

2015, Volume 2, Issue 2: 157-185. Doi: 10.3934/jdg.2015.2.157

This issue Previous Article Smale strategies for network prisoner's dilemma games Next Article On the hierarchical optimal control of a chain of distributed systems

Conservative and dissipative polymatrix replicators

1.
Departamento de Matemática, Instituto de Ciências Exatas, Universidade Federal de Minas Gerais, Belo Horizonte, 31270-901
2.
Departamento de Matemática and CMAF-CIO, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edificio C6, Piso 2, 1749-016 Lisboa
3.
Departamento de Matemática, Instituto Superior de Economia e Gestão and CMAF-CIO, Faculdade de Ciências, Universidade de Lisboa, Campo Grande, Edificio C6, Piso 2, 1749-016 Lisboa

Received: July 2015

Revised: October 2015

Published online: December 2015

Abstract / Introduction Related Papers Cited by

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 population stratified in $p\geq 1$ 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.

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Mathematics Subject Classification: Primary: 91A22, 37B25; Secondary: 70G45.

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