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S. N. Ethier

Publications and source records attributed to S. N. Ethier.

At least 19 recordsLinked to original sources
arXiv2026

Long-term behavior of casino games

We study the asymptotic behavior of the ratio of total return (or total profit) to total amount bet in a casino game. While the limit is well understood when the sequence of wagers is independent and identically distributed, here we consider the case in which bet sizes vary over time and may depend on past outcomes. We propose a general framework that yields such results under mild conditions on the conditional expectations of bets, returns, and profits. The set-up applies to many casino games (including compound games and those in which wagers are not immediately resolved), expressing the long-term behavior in terms of intrinsic parameters, namely return to player (RTP) and house advantage (HA). As an application, we examine the roulette win documented in Leigh's (1976) Thirteen against the Bank and attempt to quantify the likelihood that the story is true.

math.PR↗
arXiv2020

How strong can the Parrondo effect be? II

Parrondo's coin-tossing games comprise two games, $A$ and $B$. The result of game $A$ is determined by the toss of a fair coin. The result of game $B$ is determined by the toss of a $p_0$-coin if capital is a multiple of $r$, and by the toss of a $p_1$-coin otherwise. In either game, the player wins one unit with heads and loses one unit with tails. Game $B$ is fair if $(1-p_0)(1-p_1)^{r-1}=p_0\,p_1^{r-1}$. In a previous paper we showed that, if the parameters of game $B$, namely $r$, $p_0$, and $p_1$, are allowed to be arbitrary, subject to the fairness constraint, and if the two (fair) games $A$ and $B$ 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 $A$ is played with probability $γ$ and game $B$ is played otherwise, where $γ\in(0,1)$ is arbitrary.

math.PR↗
arXiv2019

The tilted flashing Brownian ratchet

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 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.

math.PR↗
arXiv2019

How strong can the Parrondo effect be?

If the parameters of the original Parrondo games $A$ and $B$ are allowed to be arbitrary, subject to a fairness constraint, and if the two (fair) games $A$ and $B$ are played in an arbitrary periodic sequence, then the rate of profit can not only be positive, it can be arbitrarily close to 1 (i.e., 100%).

math.PR↗
arXiv2017

The flashing Brownian ratchet and Parrondo's paradox

A Brownian ratchet is a one-dimensional diffusion process that drifts toward 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.

math.PR↗
arXiv2016

Optimal conditional expectation at the video poker game Jacks or Better

There are 134,459 distinct initial hands at the video poker game Jacks or Better, taking suit exchangeability into account. A computer program can determine the optimal strategy (i.e., which cards to hold) for each such hand, but a complete list of these strategies would require a book-length manuscript. Instead, a hand-rank table, which fits on a single page and reproduces the optimal strategy perfectly, was found for Jacks or Better as early as the mid 1990s. Is there a systematic way to derive such a hand-rank table? We show that there is indeed, and it involves finding the exact optimal conditional expected return, given the initial hand. In the case of Jacks or Better (paying 800, 50, 25, 9, 6, 4, 3, 2, 1, 0), this is a random variable with 1,153 distinct values, of which 766 correspond to garbage hands for which it is optimal to draw five new cards. We describe the hands corresponding to each of the remaining 387 values of the optimal conditional expected return (sorted from largest to smallest) and show how this leads readily to an optimal strategy hand-rank table for Jacks or Better. Of course, the method applies to other video poker games as well.

math.OC↗
arXiv2015

Parrondo games with two-dimensional spatial dependence

Parrondo games with one-dimensional spatial dependence were introduced by Toral and extended to the two-dimensional setting by Mihailović and Rajković. $MN$ players are arranged in an $M\times N$ array. There are three games, the fair, spatially independent game $A$, the spatially dependent game $B$, and game $C$, which is a random mixture or nonrandom pattern of games $A$ and $B$. Of interest is $μ_B$ (or $μ_C$), the mean profit per turn at equilibrium to the set of $MN$ players playing game $B$ (or game $C$). Game $A$ is fair, so if $μ_B\le0$ and $μ_C>0$, then we say the Parrondo effect is present. We obtain a strong law of large numbers and a central limit theorem for the sequence of profits of the set of $MN$ players playing game $B$ (or game $C$). 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 incorrect. Part of the evidence is that, under certain conditions, the means $μ_B$ and $μ_C$ converge as $M,N\to\infty$. Proof requires that a related spin system on ${\bf Z}^2$ be ergodic. However, our sufficient conditions for ergodicity are rather restrictive.

math.PR↗
arXiv2015

The evolution of the game of baccarat

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.

math.OC↗
arXiv2015

Counting toroidal binary arrays, II

We derive formulas for $(i)$ the number of toroidal $n\times n$ binary arrays, allowing rotation of rows and/or columns as well as matrix transposition, and $(ii)$ the number of toroidal $n\times n$ binary arrays, allowing rotation and/or reflection of rows and/or columns as well as matrix transposition.

math.CO↗
arXiv2014

Parrondo games with spatial dependence, III

We study Toral's Parrondo games with $N$ players and one-dimensional spatial dependence as modified by Xie et al. Specifically, we use computer graphics to sketch the Parrondo and anti-Parrondo regions for $3\le N\le 9$. Our work was motivated by a recent paper of Li et al., who applied a state space reduction method to this model, reducing the number of states from $2^N$ to $N+1$. We show that their reduced Markov chains are inconsistent with the model of Xie et al.

math.PR↗
arXiv2014

On the three-person game baccara banque

Baccara banque is a three-person zero-sum game parameterized by $θ\in(0,1)$. A study of the game by Downton and Lockwood claimed that the Nash equilibrium is of only academic interest. Their preferred alternative is what we call the independent cooperative equilibrium. But this solution exists only for certain $θ$. A third solution, which we call the correlated cooperative equilibrium, always exists. Under a "with replacement" assumption as well as a simplifying assumption concerning the information available to one of the players, we derive each of the three solutions for all $θ$.

math.OC↗
arXiv2014

A property of Petrov's diffusion

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 do not sum to 1 acts like an entrance boundary for the diffusion.

math.PR↗
arXiv2013

A game-theoretic analysis of baccara chemin de fer

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.

cs.GT↗
arXiv2013

Counting toroidal binary arrays

A formula for the number of toroidal m x n binary arrays, allowing rotation of the rows and/or the columns but not reflection, is known. Here we find a formula for the number of toroidal m x n binary arrays, allowing rotation and/or reflection of the rows and/or the columns.

math.CO↗
arXiv2012

Parrondo games with spatial dependence and a related spin system, II

Let game B be Toral's cooperative Parrondo game with (one-dimensional) spatial dependence, parameterized by N (3 or more) and p_0,p_1,p_2,p_3 in [0,1], and let game A be the special case p_0=p_1=p_2=p_3=1/2. Let mu_B (resp., mu_(1/2,1/2)) denote the mean profit per turn to the ensemble of N players always playing game B (resp., always playing the randomly mixed game (1/2)(A+B)). In previous work we showed that, under certain conditions, both sequences converge and the limits can be expressed in terms of a parameterized spin system on the one-dimensional integer lattice. Of course one can get similar results for mu_(gamma,1-gamma) corresponding to gamma A+(1-gamma)B for 0<gamma<1. In this paper we replace the random mixture with the nonrandom periodic pattern A^r B^s, where r and s are positive integers. We show that, under certain conditions, mu_[r,s], the mean profit per turn to the ensemble of N players repeatedly playing the pattern A^r B^s, converges to the same limit that mu_(gamma,1-gamma) converges to, where gamma:=r/(r+s). For a particular choice of the probability parameters, namely p_0=1, p_1=p_2 in (1/2,1), and p_3=0, we show that the Parrondo effect (i.e., mu_B is nonpositive and mu_[r,s] is positive) is present if and only if N is even, at least when s=1.

math.PR↗
arXiv2012

Parrondo games with spatial dependence, II

Let game B be Toral's cooperative Parrondo game with (one-dimensional) spatial dependence, parameterized by N (3 or more) and p_0, p_1, p_2, p_3 in [0,1], and let game A be the special case p_0=p_1=p_2=p_3=1/2. In previous work we investigated mu_B and mu_(1/2,1/2), the mean profits per turn to the ensemble of N players always playing game B and always playing the randomly mixed game (1/2)(A+B). These means were computable for N=3,4,5,...,19, at least, and appeared to converge as N approaches infinity, suggesting that the Parrondo region (i.e., the region in which mu_B is nonpositive and mu_(1/2,1/2) is positive) has nonzero volume in the limit. The convergence was established under certain conditions, and the limits were expressed in terms of a parameterized spin system on the one-dimensional integer lattice. In this paper we replace the random mixture with the nonrandom periodic pattern A^r B^s, where r and s are positive integers. We show that mu_[r,s], the mean profit per turn to the ensemble of N players repeatedly playing the pattern A^r B^s, is computable for N=3,4,5,...,18 and r+s=2,3,4, at least, and appears to converge as N approaches infinity, albeit more slowly than in the random-mixture case. Again this suggests that the Parrondo region (mu_B is nonpositive and mu_[r,s] is positive) has nonzero volume in the limit. Moreover, we can prove this convergence under certain conditions and identify the limits.

math.PR↗