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

Title
A discrete dynamical system for the greedy strategy at collective Parrondo games
Author(s)
이지연S. N. Ethier[S. N. Ethier]
Keywords
PARADOX
Issue Date
201112
Publisher
TAYLOR & FRANCIS LTD
Citation
DYNAMICAL SYSTEMS-AN INTERNATIONAL JOURNAL, v.26, no.4, pp.401 - 424
Abstract
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.
URI
http://hdl.handle.net/YU.REPOSITORY/24146http://dx.doi.org/10.1080/14689367.2011.596523
ISSN
1468-9367
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이과대학 > 통계학과 > Articles
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