State-Space Visualisation and Fractal Properties of Parrondo's Games
  In Parrondo's games, the apparently paradoxical situation occurs where individually losing games combine to win. The basic formulation and definitions of Parrondo's games are described in Harmer et al.. These games have recently gained considerable attention as they are physically motivated and have been related to physical systems such as the Brownian ratchet, lattice gas automata and spin systems. Various authors have pointed outinterest in these games for areas as diverse as biogenesis, political models, small-world networks, economics and population genetics. In this chapter, we will first introduce the relevant properties of Markov transition operators and then introduce some terminology and visualisation techniques from the theory of dynamical systems. We will then use these tools, later in the chapter, to define and investigate some interesting properties of Parrondo's games. 
A. Allison, C.E.M. Pearce, D. Abbott