We extend a recently developed relation between the master equation describing the Parrondo's games and the formalism of the Fokker-Planck equation to the case in which the games are modified with the introduction of "self-transition probabilities." This accounts for the possibility that the capital can neither increase nor decrease during a game. Using this exact relation, we obtain expressions for the stationary probability and current (games gain) in terms of an effective potential. We also demonstrate that the expressions obtained are nothing but a discretised version of the equivalent expressions in terms of the solution of the Fokker-Planck equation with multiplicative noise.
We study the relation between the discrete-time version of the
flashing ratchet known as Parrondo's games and a compression technique used very recently with thermal ratchets for evaluating the transfer of information -- negentropy -- between the Brownian
particle and the source of fluctuations. We present some results concerning different versions of Parrondo's games showing, for each case, a good qualitative agreement between the gain and the inverse of the entropy.
We discuss in detail two recently proposed relations between the Parrondo's games and the Fokker-Planck equation describing the flashing ratchet as the overdamped motion of a particle in a potential landscape. In both cases it is possible to relate exactly the probabilities of the games to the potential in which the overdamped particle moves. We will discuss under which conditions current-less potentials correspond to fair games and vie versa.