Stochastic dynamics and Parrondo’s paradox

The Spanish physicist Juan Parrondo has provided two stochastic losing games such that for certain stochastic combinations one may obtain a winning game. If a large number of players are involved and if they try to play such that their gain in the next round is maximized one arrives at the problem of investigating a random walk on a certain space of measures.The appropriate abstract setting is as follows. There is given a compact metric space (M,d)(M,d), and MM is written as the union of certain closed subsets A1,…,ArA1,…,Ar. For every ρ=1,…,rρ=1,…,r there is prescribed a strict contraction Γρ:Aρ→MΓρ:Aρ→M. A random walk (Xm)m∈N0(Xm)m∈N0 on MM is then defined as follows. The starting position is X0=x0X0=x0, where x0∈Mx0∈M is fixed, and if the walk at the mm’th step is at position Xm∈MXm∈M, then one chooses a ρρ among the ρρ with Xm∈AρXm∈Aρ (with equal probability, say) and defines Xm+1Xm+1 as Γρ(Xm)Γρ(Xm). Associated with the walk is a gain  φ(Xm)φ(Xm) in every round, where φ:M→Rφ:M→R is a continuous function.The aim of the present investigations is the study of the expectation GmGm of φ(Xm)φ(Xm) as a function of mm. Our main result states that the sequence (Gm)(Gm) is “eventually approximately periodic” provided that all AρAρ are not only closed but also open in MM: for every εε there is an l0∈Nl0∈N such that (Gm)(Gm) is l0l0-periodic up to an error of at most εε for sufficiently large mm. In fact it turns out that the behaviour of our process can be described well with a finite Markov chain.In the general case, however, the process might behave rather chaotically. We give an example where MM is the unit interval. MM is written as the union of two closed subsets A1,A2A1,A2, the contractions Γ1,Γ2Γ1,Γ2 are rather simple, but the expectations of the gains are not even Cesáro convergent.