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Noise in Complex Systems and Stochastic Dynamics II, edited by Zoltán Gingl,\udJosé M. Sancho, Lutz Schimansky-Geier, Janos Kertesz.We extend a recently developed relation between the master equation describing the Parrondo’s games and the\udformalism of the Fokker–Planck equation to the case in which the games are modied with the introduction of\ud“self–transition probabilities”. This accounts for the possibility that the capital can neither increase nor decrease\udduring a game. Using this exact relation, we obtain expressions for the stationary probability and current (games\udgain) in terms of an eective potential. We also demonstrate that the expressions obtained are nothing but a\uddiscretised version of the equivalent expressions in terms of the solution of the Fokker–Planck equation with\udmultiplicative noise.Peer reviewe

Exact ratchet description of Parrondo's games with self-transitions