Stochastic dynamics and Parrondo’s paradox

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 A 1 , … , A r . For every ρ = 1 , … , r there is prescribed a strict contraction Γ ρ : A ρ → M . A random walk ( X m ) m ∈ N 0 on M is then defined as follows. The starting position is X 0 = x 0 , where x 0 ∈ M is fixed, and if the walk at the m ’th step is at position X m ∈ M , then one chooses a ρ among the ρ with X m ∈ A ρ (with equal probability, say) and defines X m + 1 as Γ ρ ( X m ) . Associated with the walk is a gain φ ( X m ) in every round, where φ : M → R is a continuous function. The aim of the present investigations is the study of the expectation G m of φ ( X m ) as a function of m . Our main result states that the sequence ( G m ) is “eventually approximately periodic” provided that all A ρ are not only closed but also open in M : for every ε there is an l 0 ∈ N such that ( G m ) is l 0 -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 A 1 , A 2 , the contractions Γ 1 , Γ 2 are rather simple, but the expectations of the gains are not even Cesáro convergent.