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Consider a gambler who on each bet either wins 1 with probability p or loses 1 with probability q=1-p, with the results of successive bets being independent. The gambler will stop betting when they are either up k or down k. Letting N be the number of bets made, we show that N is a new better than used random variable. Moreover, we show that if k is even then N/2 has an increasing failure rate, and if k is odd then (N+1)/2 has an increasing failure rate. 

Consider a set of n players. We suppose that each game involves two players, that there is some unknown player who wins each game it plays with a probability greater than 1/2, and that our objective is to determine this best player. Under the requirement that the policy employed guarantees a correct choice with a probability of at least some specified value, we look for a policy that has a relatively small expected number of games played before decision. We consider this problem both under the assumption that the best player wins each game with a probability of at least some specified value >1/2, and under a Bayesian assumption that the probability that player i wins a game against player j is its value divided by the sum of the values, where the values are the unknown values of n independent and identically distributed exponential random variables. In the former case, we propose a policy where chosen pairs play a match that ends when one of them has had a specified number of wins more than the other; in the latter case, we propose a Thompson sampling type rule. 

We consider the problem of finding, through adaptive sampling, which of n populations (arms) has the largest mean. Our objective is to determine a rule which identifies the best arm with a fixed minimum confidence using as few observations as possible.We study such problems when the population distributions are either Bernoulli or normal. We take a Bayesian approach that assumes that the unknown means are the values of independent random variables having a common specified distribution. We propose to use the classical vector at a time rule, which samples each remaining arm once in each round, eliminating arms whose cumulative sum falls k below that of another arm. We show how this rule can be implemented and analyzed in our Bayesian setting and how it can be improved by early elimination. We also propose and analyze a variant of the classical play the winner algorithm. Numerical results show that these rules perform quite well, even when considering cases where the set of means do not look like they come from the specified prior.