Hamiltonian evolutionary games

Abstract

Abstract. We introduce a class of o.d.e.'s that generalizes to polymatrix games the replicator equations on symmetric and asymmetric games. We also introduce a new class of Poisson structures on the phase space of these systems, and characterize the corresponding subclass of Hamiltonian polymatrix replicator systems. This extends known results for symmetric and asymmetric replicator systems.

Conservative replicator and Lotka-Volterra equations in the context of Dirac\big-isotropic structures

“…The constant of motion forX B is the primitive of the 1-form (1 + n−1 i=1 e ui ) D t η q . This result relaxes the skew-symmetric condition on B which is required at [2].…”

“…Our results: In this paper we consider replicator equations and study their constants of motion and Hamiltonian character using Dirac and big-isotropic structures. Our work here is a continuation of what is done in [2] where for a more general class of equations, i.e. polymatrix replicator equations, a class of poisson structures on their phase space was introduced and the corresponding subclass of Hamiltonian polymatrix replicators were characterised.…”

“…where r i = a in − a nn and a ij = a ij − a nj , see Theorem 5.1. Equations (2) are known as Lotka-Volterra equations. Lotka-Volterra equations were introduced independently by Alfred Lotka [13] and Vito Volterra [18] to model the evolution of biological and chemical ecosystems.…”

“…It is shown in [2] that restriction of a given replicator equation X A with payoff matrix A and a formal equilibrium point q = (q 1 , . .…”

“…Considering other geometries such as poisson , presymplectic, Dirac and big-isotropic helps to extend the scope of the systems that could be described in a Hamiltonian way. For examples of Hamiltonian systems in the context of poisson geometry see references mentioned in [2,3], for Hamiltonian systems in the context of presymplectic geometry see references mentioned in [1] and for Hamiltonian systems described using Dirac structures see [6,19].…”