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Abstract: The antique Mills Futurity slot machine has two unusual features. First, if aplayer loses 10 times in a row, the 10 lost coins are returned. Second, thepayout distribution varies from coup to coup in a manner that is nonrandom andperiodic with period 10. It follows that the machine is driven by a 100-stateirreducible period-10 Markov chain. Here, we evaluate the stationarydistribution of the Markov chain, and this leads to a strong law of largenumbers and a central limit theorem for the sequence of payouts. Following asuggestion of Pyke In Mathematical Statistics and Applications: Festschriftfor Constance van Eeden 2003 185-216 Institute of Mathematical Statistics,we address the question of whether there exists a two-armed version of this``one-armed bandit- that obeys Parrondo-s paradox. More precisely, is theresuch a machine with the property that the casino can honestly advertise thatboth arms are fair, yet when players alternate arms in certain random ornonrandom ways, the casino makes money in the long run? The answer is aqualified yes. Although this ``history-dependent- game is conceptually simplerthan the original such games of Parrondo, Harmer and Abbott Phys. Rev. Lett.2000 85 5226-5229, it is nearly as complicated analytically, and openproblems remain.

Autor: S. N. Ethier, Jiyeon Lee

Fuente: https://arxiv.org/

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