Parrondo games as disordered systems

Parrondo's paradox refers to the counter-intuitive situation where a winning strategy results from a suitable combination of losing ones. Simple stochastic games exhibiting this paradox have been introduced around the turn of the millennium. The common setting of these Parrondo games is that two rules, $A$ and $B$, are played at discrete time steps, following either a periodic pattern or an aperiodic one, be it deterministic or random. These games can be mapped onto 1D random walks. In capital-dependent games, the probabilities of moving right or left depend on the walker's position modulo some integer $K$. In history-dependent games, each step is correlated with the $Q$ previous ones. In both cases the gain identifies with the velocity of the walker's ballistic motion, which depends non-linearly on model parameters, allowing for the possibility of Parrondo's paradox. Calculating the gain involves products of non-commuting Markov matrices, which are somehow analogous to the transfer matrices used in the physics of 1D disordered systems. Elaborating upon this analogy, we study a paradigmatic Parrondo game of each class in the neutral situation where each rule, when played alone, is fair. The main emphasis of this systematic approach is on the dependence of the gain on the remaining parameters and, above all, on the game, i.e., the rule pattern, be it periodic or aperiodic, deterministic or random. One of the most original sides of this work is the identification of weak-contrast regimes for capital-dependent and history-dependent Parrondo games, and a detailed quantitative investigation of the gain in the latter scaling regimes.