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Baccara banque is a three-person zero-sum game parameterized by \(\theta\in(0,1)\). 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 . However, this solution exists only for certain \(\theta\). 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 \(\theta\).

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.

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.

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.

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 $N\to\infty$.

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.

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.

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., Published in at http://dx.doi.org/10.1214/09-AAP653 the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org)