[Submitted on 3 Jun 2009 (v1), last revised 15 Oct 2010 (this version, v2)]
Download PDFDan from Communication Group 34 takes you through the rules and mathematics of Casino slot Machines!Hub Video: https://www.youtube.com/watch?v=bOl-ycNvSE.
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.
Markov Process
Submission history
Use Of Markov Chain
From: S. N. Ethier [view email] [via VTEX proxy][v1] Wed, 3 Jun 2009 21:46:45 UTC (165 KB)
[v2]Fri, 15 Oct 2010 12:07:17 UTC (124 KB)
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