S. N. Ethier — Page 2

arXiv2012

Parrondo games with spatial dependence and a related spin system

Toral introduced so-called cooperative Parrondo games, in which there are N players (3 or more) arranged in a circle. At each turn one player is randomly chosen to play. He plays either game A or game B, depending on the strategy. Game A results in a win or loss of one unit based on the toss of a fair coin. Game B results in a win or loss of one unit based on the toss of a biased coin, with the amount of the bias depending on whether none, one, or two of the player's two nearest neighbors have won their most recent games. Game A is fair, so the games are said to exhibit the Parrondo effect if game B is losing or fair and the random mixture C:=(1/2)(A+B) is winning. With mu_B (resp., mu_C) denoting the mean profit per turn to the ensemble of N players always playing game B (resp., C), we give sufficient conditions for lim mu_B (as N goes to infinity) to exist and show that lim mu_C nearly always exists, with the limits expressible in terms of a parameterized spin system on the one-dimensional integer lattice. For a particular choice of the parameters, we show that the Parrondo effect (i.e., mu_B is nonpositive and mu_C is positive) is present in the N-player model if and only if N is even. For the same choice of the parameters, we show that, with a suitable interpretation and for certain initial distributions, the Parrondo effect is present in the spin system if and only if N is even, N being the number of consecutive players whose collective profit is tracked.

math.PR↗

arXiv2012

Parrondo games with spatial dependence

Toral introduced so-called cooperative Parrondo games, in which there are N players (3 or more) arranged in a circle. At each turn one player is randomly chosen to play. He plays either game A or game B. Game A results in a win or loss of one unit based on the toss of a fair coin. Game B results in a win or loss of one unit based on the toss of a biased coin, with the amount of the bias depending on whether none, one, or two of the player's two nearest neighbors have won their most recent games. Game A is fair, so the games are said to exhibit the Parrondo effect if game B is losing or fair and the random mixture (1/2)(A+B) is winning. With the parameter space being the unit cube, we investigate the region in which the Parrondo effect appears. Explicit formulas can be found if N=3,4,5,6 and exact computations can be carried out if N=7,8,9,...,19, at least. We provide numerical evidence suggesting that the Parrondo region has nonzero volume in the limit as N approaches infinity.

math.PR↗

arXiv2011

A discrete dynamical system for the greedy strategy at collective Parrondo games

We consider a collective version of Parrondo's games with probabilities parametrized by rho in (0,1) in which a fraction phi in (0,1] of an infinite number of players collectively choose and individually play at each turn the game that yields the maximum average profit at that turn. Dinis and Parrondo (2003) and Van den Broeck and Cleuren (2004) studied the asymptotic behavior of this greedy strategy, which corresponds to a piecewise-linear discrete dynamical system in a subset of the plane, for rho=1/3 and three choices of phi. We study its asymptotic behavior for all (rho,phi) in (0,1)x(0,1], finding that there is a globally asymptotically stable equilibrium if phi<=2/3 and, typically, a unique (asymptotically stable) limit cycle if phi>2/3 ("typically" because there are rare cases with two limit cycles). Asymptotic stability results for phi>2/3 are partly conjectural.

math.PR↗

arXiv2011

Parrondo's paradox via redistribution of wealth

Toral (2002) considered an ensemble of N\geq2 players. In game B a player is randomly selected to play Parrondo's original capital-dependent game. In game A' two players are randomly selected without replacement, and the first transfers one unit of capital to the second. Game A' is fair (with respect to total capital), game B is losing (or fair), and the random mixture γA'+(1-γ)B is winning, as was demonstrated by Toral for γ=1/2 using computer simulation. We prove this, establishing a strong law of large numbers and a central limit theorem for the sequence of profits of the ensemble of players for each γ\in(0,1). We do the same for the nonrandom pattern of games (A')^r B^s for all integers r,s\geq1. An unexpected relationship between the random-mixture case and the nonrandom-pattern case occurs in the limit as N\rightarrow\infty.

math.PR↗

arXiv2010

A Markovian slot machine and Parrondo's paradox

The antique Mills Futurity slot machine has two unusual features. First, if a player loses 10 times in a row, the 10 lost coins are returned. Second, the payout distribution varies from coup to coup in a manner that is nonrandom and periodic with period 10. It follows that the machine is driven by a 100-state irreducible period-10 Markov chain. Here, we evaluate the stationary distribution of the Markov chain, and this leads to a strong law of large numbers and a central limit theorem for the sequence of payouts. Following a suggestion of Pyke [In Mathematical Statistics and Applications: Festschrift for 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 there such a machine with the property that the casino can honestly advertise that both arms are fair, yet when players alternate arms in certain random or nonrandom ways, the casino makes money in the long run? The answer is a qualified yes. Although this ``history-dependent'' game is conceptually simpler than the original such games of Parrondo, Harmer and Abbott [Phys. Rev. Lett. (2000) 85 5226--5229], it is nearly as complicated analytically, and open problems remain.

math.PR↗

arXiv2009

A world record in Atlantic City and the length of the shooter's hand at craps

It was widely reported in the media that, on 23 May 2009, at the Borgata Hotel Casino & Spa in Atlantic City, Patricia DeMauro, playing craps for only the second time, rolled the dice for four hours and 18 minutes, finally sevening out at the 154th roll, a world record. Initial estimates of the probability of this event were erroneous, but consensus was reached within days: one chance in 5.6 billion. More generally, what is P(L \ge n), where the random variable L denotes the length of the shooter's hand (154 in Ms. DeMauro's case) and n is a positive integer? It is well known that these probabilities can be derived recursively or by Markov chain methods. Our aim here is to give an explicit closed-form expression for them, showing that the distribution of L is a linear combination (not a convex combination) of four geometric distributions.

math.PR↗

arXiv2009

Limit theorems for Parrondo's paradox

That there exist two losing games that can be combined, either by random mixture or by nonrandom alternation, to form a winning game is known as Parrondo's paradox. We establish a strong law of large numbers and a central limit theorem for the Parrondo player's sequence of profits, both in a one-parameter family of capital-dependent games and in a two-parameter family of history-dependent games, with the potentially winning game being either a random mixture or a nonrandom pattern of the two losing games. We derive formulas for the mean and variance parameters of the central limit theorem in nearly all such scenarios; formulas for the mean permit an analysis of when the Parrondo effect is present.

math.PR↗