Stochastic dynamics and Parrondo’s paradox |
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Authors: | Ehrhard Behrends |
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Institution: | Fachbereich Mathematik und Informatik, Freie Universität Berlin, Arnimallee 2-6, D-14 195 Berlin, Germany |
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Abstract: | 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), and M is written as the union of certain closed subsets A1,…,Ar. For every ρ=1,…,r there is prescribed a strict contraction Γρ:Aρ→M. A random walk (Xm)m∈N0 on M is then defined as follows. The starting position is X0=x0, where x0∈M is fixed, and if the walk at the m’th step is at position Xm∈M, then one chooses a ρ among the ρ with Xm∈Aρ (with equal probability, say) and defines Xm+1 as Γρ(Xm). Associated with the walk is a gainφ(Xm) in every round, where φ:M→R is a continuous function.The aim of the present investigations is the study of the expectation Gm of φ(Xm) as a function of m. Our main result states that the sequence (Gm) is “eventually approximately periodic” provided that all Aρ are not only closed but also open in M: for every ε there is an l0∈N such that (Gm) is l0-periodic up to an error of at most ε for sufficiently large m. 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 M is the unit interval. M is written as the union of two closed subsets A1,A2, the contractions Γ1,Γ2 are rather simple, but the expectations of the gains are not even Cesáro convergent. |
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Keywords: | Stochastic dynamical system Stochastic game Fractal Parrondo&rsquo s paradox |
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