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

Abstract

We consider a collective version of Parrondo's games with probabilities parameterized by ρ ∈ (0, 1) in which a fraction φ ∈ (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. Dinís and Parrondo [L. Dinís and J.M.R. Parrondo, Optimal strategies in collective Parrondo games, Europhys. Lett. 63 (2003), pp. 319–325] and Van den Broeck and Cleuren [C. Van den Broeck and B. Cleuren, Parrondo games with strategy, in Noise in Complex Systems and Stochastic Dynamics II, Z. Gingl, J.M. Sancho, L. Schimansky-Geier, and J. Kertesz, eds., SPIE, Bellingham, WA, 2004, pp. 109–118.] studied the asymptotic behaviour of this greedy strategy, which corresponds to a piecewise-linear discrete dynamical system in a subset of the plane, for ρ = 1/3 and three choices of φ. We study its asymptotic behaviour for all (ρ, φ) ∈ (0, 1) × (0, 1], finding that there is a globally asymptotically stable equilibrium if φ ≤ 2/3 and, typically, a unique (asymptotically stable) limit cycle if φ > 2/3 (‘typically’ because there are rare cases with two limit cycles). Asymptotic stability results for φ > 2/3 are partly conjectural.