A discrete dynamical system for the greedy strategy at collective Parrondo games

TL;DR

This paper analyzes a collective greedy strategy in Parrondo's games using a discrete dynamical system, revealing conditions for stable equilibrium or limit cycles across all parameter ranges.

Contribution

It extends previous studies by characterizing the asymptotic behavior of the system for all parameter values, including stability conditions and the existence of limit cycles.

Findings

Stable equilibrium for phi <= 2/3

Limit cycles typically for phi > 2/3

Partial conjectures on stability for phi > 2/3

Abstract

We consider a collective version of Parrondo's games with probabilities parametrized by rho in (0,1) in which a fraction phi in (0,1] of an infinite number of players collectively choose and individually play at each turn the game that yields the maximum average profit at that turn. Dinis and Parrondo (2003) and Van den Broeck and Cleuren (2004) studied the asymptotic behavior of this greedy strategy, which corresponds to a piecewise-linear discrete dynamical system in a subset of the plane, for rho=1/3 and three choices of phi. We study its asymptotic behavior for all (rho,phi) in (0,1)x(0,1], finding that there is a globally asymptotically stable equilibrium if phi<=2/3 and, typically, a unique (asymptotically stable) limit cycle if phi>2/3 ("typically" because there are rare cases with two limit cycles). Asymptotic stability results for phi>2/3 are partly conjectural.