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Parrondo’s coin-tossing games comprise two games, A A and B B . The result of game A A is determined by the toss of a fair coin. The result of game B B is determined by the toss of a p 0 p_0 -coin if capital is a multiple of r r , and by the toss of a p 1 p_1 -coin otherwise. In either game, the player wins one unit with heads and loses one unit with tails. Game B B is fair if ( 1 − p 0 ) ( 1 − p 1 ) r − 1 = p 0 p 1 r − 1 (1-p_0)(1-p_1)^{r-1}=p_0\,p_1^{r-1} . In a previous paper we showed that, if the parameters of game B B , namely r r , p 0 p_0 , and p 1 p_1 , are allowed to be arbitrary, subject to the fairness constraint, and if the two (fair) games A A and B B are played in an arbitrary periodic sequence, then the rate of profit can not only be positive (the so-called Parrondo effect), but also be arbitrarily close to 1 (i.e., 100%). Here we prove the same conclusion for a random sequence of the two games instead of a periodic one, that is, at each turn game A A is played with probability γ \gamma and game B B is played otherwise, where γ ∈ ( 0 , 1 ) \gamma \in (0,1) is arbitrary.

The game of baccarat has evolved from a parlor game played by French aristocrats in the first half of the 19th century to a casino game that generated over US$41 billion in revenue for the casinos of Macau in 2013. The parlor game was originally a three-person zero-sum game. Later in the 19th century it was simplified to a two-person zero-sum game. Early in the 20th century the parlor game became a casino game, no longer zero-sum. In the mid 20th century, the strategic casino game became a nonstrategic game, with players competing against the house instead of against each other. We argue that this evolution was motivated by both economic and game-theoretic considerations.

A Brownian ratchet is a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. A flashing Brownian ratchet is a process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, producing directed motion. These processes have been of interest to physicists and biologists for nearly 25 years. The flashing Brownian ratchet is the process that motivated Parrondo’s paradox, in which two fair games of chance, when alternated, produce a winning game. Parrondo’s games are relatively simple, being discrete in time and space. The flashing Brownian ratchet is rather more complicated. We show how one can study the latter process numerically using a random walk approximation.

The two-parameter Poisson--Dirichlet diffusion, introduced in 2009 by Petrov, extends the infinitely-many-neutral-alleles diffusion model, related to Kingman's one-parameter Poisson--Dirichlet distribution and to certain Fleming--Viot processes. The additional parameter has been shown to regulate the clustering structure of the population, but is yet to be fully understood in the way it governs the reproductive process. Here we shed some light on these dynamics by formulating a $K$-allele Wright--Fisher model for a population of size $N$, involving a uniform mutation pattern and a specific state-dependent migration mechanism. Suitably scaled, this process converges in distribution to a $K$-dimensional diffusion process as $N\to\infty$. Moreover, the descending order statistics of the $K$-dimensional diffusion converge in distribution to the two-parameter Poisson--Dirichlet diffusion as $K\to\infty$. The choice of the migration mechanism depends on a delicate balance between reinforcement and redistributive effects. The proof of convergence to the infinite-dimensional diffusion is nontrivial because the generators do not converge on a core. Our strategy for overcoming this complication is to prove \textit{a priori} that in the limit there is no "loss of mass", i.e., that, for each limit point of the sequence of finite-dimensional diffusions (after a reordering of components by size), allele frequencies sum to one., Comment: To appear in The Annals of Applied Probability

Assuming that cards are dealt with replacement from a single deck and that each of Player and Banker sees the total of his own two-card hand but not its composition, baccara is a 2 x 2 88 matrix game, which was solved by Kemeny and Snell in 1957. Assuming that cards are dealt without replacement from a d-deck shoe and that Banker sees the composition of his own two-card hand while Player sees only his own total, baccara is a 2 x 2 484 matrix game, which was solved by Downton and Lockwood in 1975 for d = 1, 2, . . . , 8. Assuming that cards are dealt without replacement from a d-deck shoe and that each of Player and Banker sees the composition of his own two-card hand, baccara is a 2 5 x 2 484 matrix game, which is solved herein for every positive integer d.

We consider a collective version of Parrondo's games with probabilities parameterized by ρ ∈ (0, 1) in which a fraction φ ∈ (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 [L. Dinis and J.M.R. Parrondo, Optimal strategies in collective Parrondo games, Europhys. Lett. 63 (2003), pp. 319–325] and Van den Broeck and Cleuren [C. Van den Broeck and B. Cleuren, Parrondo games with strategy, in Noise in Complex Systems and Stochastic Dynamics II, Z. Gingl, J.M. Sancho, L. Schimansky-Geier, and J. Kertesz, eds., SPIE, Bellingham, WA, 2004, pp. 109–118.] studied the asymptotic behaviour of this greedy strategy, which corresponds to a piecewise-linear discrete dynamical system in a subset of the plane, for ρ = 1/3 and three choices of φ. We study its asymptotic behaviour for all (ρ, φ) ∈ (0, 1) × (0, 1], finding that there is a globally asymptotically stable equilibrium if φ ≤ 2/3 and, typ...

The flashing Brownian ratchet is a stochastic process that alternates between two regimes, a one-dimensional Brownian motion and a Brownian ratchet, the latter being a one-dimensional diffusion process that drifts towards a minimum of a periodic asymmetric sawtooth potential. The result is directed motion. In the presence of a static homogeneous force that acts in the direction opposite to that of the directed motion, there is a reduction (or even a reversal) of the directed motion effect. Such a process may be called a tilted flashing Brownian ratchet. We show how one can study this process numerically, using a random walk approximation or, equivalently, using numerical solution of the Fokker–Planck equation. Stochastic simulation is another viable method.

I t was widely reported in the media that, on 23 May 2009, at the Borgata Hotel Casino & Spa in Atlantic City, Patricia DeMauro (spelled Demauro in some accounts), playing craps for only the second time, rolled the dice for four hours and 18 minutes, finally sevening out at the 154th roll. Initial estimates of the probability of a run at least this long (assuming fair dice and independent rolls) ranged from one chance in 3.5 billion [5] to one chance in 1.56 trillion [10]. Subsequent computations agreed on one chance in 5.6 (or 5.59) billion [2, 6, 9]. This established a new world record. The old record was held by the late Stanley Fujitake (118 rolls, 28 May 1989, California Hotel and Casino, Las Vegas) [1]. One might ask how reliable these numbers (118 and 154) are. In Mr. Fujitake’s case, casino personnel replayed the surveillance videotape to confirm the number of rolls and the duration of time (three hours and six minutes). We imagine that the same happened in Ms. DeMauro’s case. There is also a report that Mr. Fujitake’s record was broken earlier by a gentleman known only as The Captain (148 rolls, July 2005, Atlantic City) [8, Part 4]. However, this incident is not well documented (specifically, the exact date and casino name were not revealed) and was unknown to Borgata officials. In fact, a statistical argument has been offered [4, p. 480] suggesting that the story is apocryphal. Our aim in this article is not simply to derive a more accurate probability, but to show that this apparently prosaic problem involves some interesting mathematics, including Markov chains, matrix theory, generating functions, and Galois theory.

It is well known that the Kelly system of proportional betting, which maximizes the long-term geometric rate of growth of the gambler's fortune, minimizes the expected time required to reach a specified goal. Less well known is the fact that it maximizes the median of the gambler's fortune. This was pointed out by the author in a 1988 paper, but only under asymptotic assumptions that might cause one to question its applicability. Here we show that the result is true more generally, and argue that this is a desirable property of the Kelly system.

Motivation: Microarray technology emerges as a powerful tool in life science. One major application of microarray technology is to identify differentially expressed genes under various conditions. Currently, the statistical methods to analyze microarray data are generally unsatisfactory, mainly due to the lack of understanding of the distribution and error structure of microarray data. Results: We develop a generalized likelihood ratio (GLR) test based on the two-component model proposed by Rocke and Durbin to identify differentially expressed genes from microarray data. Simulation studies show that the GLR test is more powerful than commonly used methods, like the fold-change method and the two-sample t-test. When applied to microarray data, the GLR test identifies more differentially expressed genes than the t-test, has a lower false discovery rate and shows more consistency over independently repeated experiments. Availability: The approach is implemented in software called GLR, which is freely available for downloading at http://www.cc.utah.edu/~jw27c60

Consider a wager that is more complicated than simply winning or losing the amount of the bet. For example, a pass line bet with double odds is such a wager, as is a bet on video poker using a specified drawing strategy. We are concerned with the probability that, in an independent sequence of identical wagers of this type, the gambler loses L or more betting units (i.e., the gambler is “ruined”) before he wins W or more betting units. Using an idea of Markov, Feller established upper and lower bounds on the probability of ruin, bounds that are often very close to each other. However, his formulation depends on finding a positive nontrivial root of the equation φ (ρ )=1, where φ is the probability generating function for the wager in question. Here we give simpler bounds, which rely on the first few moments of the specified wager, thereby making such gambler's ruin probabilities more easily computable.

The definition of a (discrete-time) martingale is generalized, allowing the process to be nonadapted, and it is shown that the optional stopping theorem still holds. This result was motivated by and is applied to the study of gambling systems, especially at games such as craps in which bets are not immediately resolved.

Petrov constructed a diffusion process in the Kingman simplex whose unique stationary distribution is the two-parameter Poisson-Dirichlet distribution of Pitman and Yor. We show that the subset of the simplex comprising vectors whose coordinates sum to 1 is the natural state space for the process. In fact, the complementary set acts like an entrance boundary.

Parrondo games with one-dimensional (1D) spatial dependence were introduced by Toral and extended to the two-dimensional (2D) setting by Mihailović and Rajković. [Formula: see text] players are arranged in an [Formula: see text] array. There are three games, the fair, spatially independent game [Formula: see text], the spatially dependent game [Formula: see text], and game [Formula: see text], which is a random mixture or non-random pattern of games [Formula: see text] and [Formula: see text]. Of interest is [Formula: see text] (or [Formula: see text]), the mean profit per turn at equilibrium to the set of [Formula: see text] players playing game [Formula: see text] (or game [Formula: see text]). Game [Formula: see text] is fair, so if [Formula: see text] and [Formula: see text], then we say the Parrondo effect is present. We obtain a strong law of large numbers (SLLN) and a central limit theorem (CLT) for the sequence of profits of the set of [Formula: see text] players playing game [Formula: see text] (or game [Formula: see text]). The mean and variance parameters are computable for small arrays and can be simulated otherwise. The SLLN justifies the use of simulation to estimate the mean. The CLT permits evaluation of the standard error of a simulated estimate. We investigate the presence of the Parrondo effect for both small arrays and large ones. One of the findings of Mihailović and Rajković was that “capital evolution depends to a large degree on the lattice size.” We provide evidence that this conclusion is partly incorrect. A paradoxical feature of the 2D game [Formula: see text] that does not appear in the 1D setting is that, for fixed [Formula: see text] and [Formula: see text], the mean function [Formula: see text] is not necessarily a monotone function of its parameters.

Fleming-Viot processes and Dawson-Watanabe processes are two classes of “superprocesses” that have received a great deal of attention in recent years. These processes have many properties in common. In this paper, we prove a result that helps to explain why this is so. It allows one to prove certain theorems for one class when they are true for the other. More specifically, we show that product moments of a Fleming-Viot process can be bounded above by the corresponding moments of the Dawson-Watanabe process with the same “underlying particle motion”, and vice versa except for a multiplicative constant. As an application, we establish existence and continuity properties of local time for certain Fleming-Viot processes.

In Toral's games, at each turn one member of an ensemble of $N\ge2$ players is selected at random to play. He plays either game $A'$, which involves transferring one unit of capital to a second randomly chosen player, or game $B$, which is an asymmetric game of chance whose rules depend on the player's current capital, and which is fair or losing. Game $A'$ is fair (with respect to the ensemble's total profit), so the \textit{Parrondo effect} is said to be present if the random mixture $\gamma A'+(1-\gamma)B$ (i.e., play game $A'$ with probability $\gamma$ and play game $B$ otherwise) is winning. Toral demonstrated the Parrondo effect for $\gamma=1/2$ using computer simulation. We prove it, establishing a strong law of large numbers and a central limit theorem for the sequence of profits of the ensemble of players for each $\gamma\in(0,1)$. We do the same for the nonrandom pattern of games $(A')^r B^s$ for all integers $r,s\ge1$. An unexpected relationship between the random-mixture case and the nonrandom-pattern case occurs in the limit as $N\to\infty$.

Stochastic models for gene frequencies can be viewed as probability-measure-valued processes. Fleming and Viot introduced a class of processes that arise as limits of genetic models as the population size and the number of possible genetic types tend to infinity. In general, the topology on the process values in which these limits exist is the topology of weak convergence; however, convergence in the weak topology is not strong enough for many genetic applications. A new topology on the space of finite measures is introduced in which convergence implies convergence of the sizes and locations of atoms, and conditions are given under which genetic models converge in this topology. As an application, Kingman's Poisson-Dirichlet limit is extended to models with selection.

Fleming and Viot [Indiana Univ. Math. J., 28 (1979), pp. 817–843] introduced a class of probability-measure-valued diffusion processes that has attracted the interest of both pure and applied probabilists. This paper surveys the subject of Fleming–Viot processes as it relates to population genetics. Topics include:1. Introduction.2. Some measure-valued Markov chains.2.1. A diploid model. 2.2. The Wright–Fisher model.2.3. A Moran model.3. The Fleming–Viot process: characterization.4. Convergence.5. Ergodicity.6. An infinite particle system.7. Bounded mutation operators.8. Reversibility.9. Examples.9.1. Continuous-state stepwise-mutation model.9.2. Infinitely-many-neutral-alleles model.9.3. Infinitely-many-neutral-alleles model with ages.9.4. Two-locus model with recombination.9.5. n-locus model with gene conversion.9.6. Infinitely-many-sites model without recombination.9.7. Infinitely-many-neutral-alleles model with allelic genealogies.

This chapter is concerned with finding the probability that, in an independent sequence of identical wagers, the gambler loses L or more units (that is, he is ruined) before he wins W or more units. In Section 7.1, we treat the case of even-money payoffs by deriving an explicit formula. In Section 7.2, we consider more-general integer-valued payoffs. Formulas are available in some special cases, but in general a computational algorithm is needed. Finally, Section 7.3 allows arbitrary payoffs, and here we resort to establishing upper and lower bounds on the probability of interest.

Since 1990 a number of new casino games have been introduced, and here we consider two of the most successful such games, Let It Ride and Three Card Poker (Sections 16.1 and 16.2, respectively). Each is house banked and poker based, and each offers a greater advantage to the house than do the traditional games of craps, faro, baccarat, trente et quarante, and twenty-one. The analysis of each game is straightforward in principle but exhausting in practice.

The reader is assumed to be familiar with basic probability, and here we provide the definitions and theorems, without proofs, for easy reference. We restrict our attention to discrete random variables but not necessarily to discrete sample spaces. A number of examples are worked out in detail, and problems are provided for those who need additional review.

Trente et quarante (“thirty and forty” in French) is an elegant card game that dates at least as far back as the 17th century and is still played in Monte Carlo. It has been studied by Poisson, De Morgan, Bertrand, and Thorp, among others. In Section 20.1 we describe the four betting opportunities and evaluate the associated probabilities and house advantages. In Section 20.2 we analyze the card-counting potential of trente et quarante.

This chapter is concerned with a betting system for superfair games, known as optimal proportional play or the Kelly system, that maximizes the long-term geometric rate of growth of the gambler’s fortune. Section 10.1 considers the case in which a single superfair betting opportunity is available at each coup. Section 10.2 assumes that multiple betting opportunities, at least one of which is superfair, are available at each coup. Finally, Section 10.3 establishes several optimality properties of the Kelly system.