Parrondo Games with Spatial Dependence and a Related Spin System

Title
Parrondo Games with Spatial Dependence and a Related Spin System
Author(s)
이지연Stewart Ethier[Stewart Ethier]
Keywords
PARADOX
Issue Date
201306
Publisher
POLYMAT
Citation
MARKOV PROCESSES AND RELATED FIELDS, v.19, no.2, pp.163 - 194
Abstract
Toral introduced so-called cooperative Parrondo games, in which there are N >= 3 players arranged in a circle. At each turn one player is randomly chosen to play. He plays either game A or game B. Game A results in a win or loss of one unit based on the toss of a fair coin. Game B results in a win or loss of one unit based on the toss of a biased coin, with the amount of the bias depending on whether none, one, Or two of the player's two nearest neighbors have won their most recent games. Game A is fair, so the games are said to exhibit the Parrondo effect if game B is losing or fair and the random mixture C := (1/2)(A + B) is winning. With mu(N)(B) (resp., mu(N)(C)) denoting the mean profit per turn to the ensemble of N players always playing game B (resp., C), we give sufficient conditions for lim(N ->infinity) mu(N)(B) to exist and show that lim(N ->infinity) mu(N)(C) nearly always exists, with the limits expressible in terms of a parameterized spin system on the one-dimensional integer lattice. For a particular choice of the parameters, we show that the Parrondo effect (i.e., mu(N)(B) <= 0 and mu(N)(C) > 0) is present in the N-player model if and only if N is even. For the same choice of the parameters, we show that, with a suitable interpretation and for certain initial distributions, the Parrondo effect is present in the spin system if and only if N is even, N being the number of consecutive players whose collective profit is tracked.
URI
http://hdl.handle.net/YU.REPOSITORY/30010
ISSN
1024-2953
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이과대학 > 통계학과 > Articles
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