State-Space Visualisation and Fractal Properties of Parrondo’s Games

Parrondo’s games are essentially Markov games. They belong to the same class as Snakes and Ladders. The important distinguishing feature of Parrondo’s games is that the transition probabilities may vary in time. It is as though “snakes,” “ladders” and “dice” were being added and removed while the game was still in progress. Parrondo’s games are not homogeneous in time and do not necessarily settle down to an equilibrium. They model non-equilibrium processes in physics. We formulate Parrondo’s games as an inhomogeneous sequence of Markov transition operators, with rewards. Parrondo’s “paradox” is shown to be equivalent to saying that the expected value of the reward, from the whole process, is not a linear function of the Markov operators. When we say that a game is “winning” or “losing” then we must be careful to include the whole process in our definition of the word “‘game.” Specifically, we must include the time varying probability vector in our calculations. We give practical rules for calculating the expected value of the return from sequences of Parrondo’s games. We include a worked example and a comparison between the theory and a simulation. We apply visualisation techniques, from physics and engineering, to an inhomogeneous Markov process and show that the limiting set or “attractor” of this process has fractal geometry. This is in contrast to the relevant theory for homogeneous Markov processes where the stable, equilibrium limiting set is a single point in the state space. We show histograms of simulations and describe methods for calculating the capacity dimension and the moments of the fractal attractors. We indicate how to construct optimal forms of Parrondo’s games and describe a symmetrical family of games which includes the optimal form, as a limiting case. We investigate the fractal geometry of the attractors for this symmetrical family of games. The resulting geometry is very interesting, even beautiful.