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

Abstract

For 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.