For the gambler’s ruin problem with two players starting with the same amount of money, we show the playing time is stochastically maximized when the games are fair.

This paper is concerned with developing low variance simulation estimators of probabilities related to the sum of Bernoulli random variables. It shows how to utilize an identity used in the Chen-Stein approach to bounding Poisson approximations to obtain low variance estimators. Applications and numerical examples in such areas as pattern occurrences, generalized coupon collecting, system reliability, and multivariate normals are presented. We also consider the problem of estimating the probability that a positive linear combination of Bernoulli random variables is greater than some specified value, and present a simulation estimator that is always less than the Markov inequality bound on that probability.

We consider a competition involving $r$ teams, where each individual game involves two teams, and where each game between teams $i$ and $j$ is won by $i$ with probability $P_{i,j} = 1 - P_{j,i}$. We suppose that $i$ and $j$ are scheduled to play $n(i,j)$ games and say that the team that wins the most games is the winner of the competition. We show that the conditional probability that $i$ is the winner, given that $i$ wins $k$ games, is increasing in $k$. We bound the tail probability of the number of wins of the winning team. We consider the special case where $P_{i,j} = {v_i}/{(v_i + v_j)}$, and obtain structural results on the probability that team $i$ is the winner. We give efficient simulation approaches for computing the probability that team $i$ is the winner, and the conditional probability given the number of wins of $i$.

There is a set of n bandits and at every stage, two of the bandits are chosen to play a game, with the result of a game being learned. In the “weak regret problem,” we suppose there is a “best” bandit that wins each game it plays with probability at least p > 1/2, with the value of p being unknown. The objective is to choose bandits to maximize the number of times that one of the competitors is the best bandit. In the “strong regret problem”, we suppose that bandit i has unknown value vi, i = 1, …, n, and that i beats j with probability vi/(vi + vj). One version of strong regret is interested in maximizing the number of times that the contest is between the players with the two largest values. Another version supposes that at any stage, rather than choosing two arms to play a game, the decision maker can declare that a particular arm is the best, with the objective of maximizing the number of stages in which the arm with the largest value is declared to be the best. In the weak regret problem, we propose a policy and obtain an analytic bound on the expected number of stages over an infinite time frame that the best arm is not one of the competitors when this policy is employed. In the strong regret problem, we propose a Thompson sampling type algorithm and empirically compare its performance with others in the literature.

We consider a friendship model in which each member of a community has a latent value such that the probability that any two individuals are friends is a function of their latent values. We consider such questions as does information that i and j are both friends with k make it more likely that i and j are themselves friends. Among other things, we show that for fixed sets S and T, the more friends that i has in S, then the stochastically more friends i has in T. We consider how a variation of the friendship paradox applies to our model. We also study the distribution of the number of friendless individuals in the community and derive a bound on the total variation distance between it and a Poisson with the same mean.

Suppose there are n players in an ongoing competition, with player i having value vi, and suppose that a game between i and j is won by i with probability vi/(vi + vj). Consider the winner plays competition where in each stage two players play a game, and the winner keeps playing in the next game. We consider two models for choosing its opponent, analyze both models as Markov chains, and determine their stationary probabilities as well as other quantities of interest.

Suppose there are n players, with player i having value vi > 0, and suppose that a game between i and j is won by i with probability vi/(vi + vj). In the winner plays random knockout tournament, we suppose that the players are lined up in a random order; the first two play, and in each subsequent game the winner of the last game plays the next in line. Whoever wins the game involving the last player in line, is the tournament winner. We give bounds on players’ tournament win probabilities and make some conjectures. We also discuss how simulation can be efficiently employed to estimate the win probabilities.