Random Walks in the Presence of Absorbing Barriers in Parrondo's Games

Abstract

Communicated by Stefania ResidoriFor a multi-agent spatial Parrondo's model that it is composed of games A and B, we use the discrete time Markov chains to derive the probability transition matrix. Then, we respectively deduce the stationary distribution for games A and B played individually and the randomized combination of game A + B. We notice that under a specific set of parameters, two absorbing states instead of a fixed stationary distribution exist in game B. However, the randomized game A + B can jump out of the absorbing states of game B and has a fixed stationary distribution because of the "agitating" role of game A. Moreover, starting at different initial states, we deduce the probabilities of absorption at the two absorbing barriers.

Parrondo Games with Two-Dimensional Spatial Dependence

“…Most works on these games (Refs. [1,3,[9][10][11][12][13][14][15][16][17][18]) have been concerned with the one-dimensional model, and until now only Mihailović and Rajković [2] have discussed the two-dimensional model. See Abbott [19] for the most recent review of Parrondo's paradox.…”