  Recently there has been much interest in discrete forms of Brownian ratchets,
using a game-theoretic formalism. Using the approach pioneered by Parrondo, we
develop a new method for obtaining the stationary probabilities and probability
current for the case of discrete-time and discrete-space ratchets. We then use
this result to calculate the Parrondian ratchet efficiency in two cases:
firstly, for Parrondo's original system and, secondly, for a set of
probabilities derived from a discretized ratchet potential.
