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

作者：Ethier S N; Lee Jiyeon^*

来源：Dynamical Systems, 2011, 26(4): 401-424.

DOI：10.1080/14689367.2011.596523

• Parrondo's paradox
• collective games
• greedy strategy
• discrete dynamical system
• globally asymptotically stable equilibrium
• unstable equilibrium
• asymptotically stable limit cycle

We consider a collective version of Parrondo's games with probabilities parameterized by rho is an element of (0, 1) in which a fraction phi is an element of (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 [L. Dinis 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 rho = 1/3 and three choices of phi. We study its asymptotic behaviour for all (rho, phi) is an element of (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 > 42/3 ('typically' because there are rare cases with two limit cycles). Asymptotic stability results for phi > 42/3 are partly conjectural.

• 出版日期2011