We consider the optimal stopping problem of maximizing the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length n, where m and n are predetermined integers such that 1 ≤ m < n. The optimal stopping rule of this problem has a nice interpretation, that is, it stops on the first success for which the sum of the m-fold multiplicative odds of success for the future trials is less than or equal to 1. This result can be viewed as a generalization of Bruss' (2000) odds theorem. Application will be made to the secretary problem. For more generality, we extend the problem in several directions in the same manner that Ferguson (2008) used to extend the odds theorem. We apply this extended result to the full-information analogue of the secretary problem, and derive the optimal stopping rule and the probability of win explicitly. The asymptotic results, as n tends to ∞, are also obtained via the planar Poisson process approach.
A finite number of candidates appear one-by-one in random order with all permutations equally likely. We are able, at any time, to rank the candidates that have so far appeared according to some order of preference. Each candidate may be classified into one of two types independent of the other candidates: available or unavailable. An unavailable candidate does not accept an offer of employment. The goal is to find a strategy that maximizes the probability of employing the best among the available candidates based on both the relative ranks and the availabilities observed so far. According to when the availability of a candidate can be ascertained, two models are considered. The availability is ascertained only by giving an offer of employment (MODEL 1), while the availability is ascertained just after the arrival of the candidate (MODEL 2).
We consider the secalled secretary problem, in which an offer may be declined by each applicant with a fixed k"own probability g (= 1 -p, O s{ g < 1) and the number of offering chances are at most m (2 1). The optimal strategy of this problem is derived and some asymptotic results are presented.Furthermore we briefly consider the case in which t・he acceptance probability depends on the number m of effering chances.
We consider the optimal stopping problem of maximizing the probability of stopping on any of the last m successes of a sequence of independent Bernoulli trials of length n, where m and n are predetermined integers such that 1 ≤ m < n. The optimal stopping rule of this problem has a nice interpretation, that is, it stops on the first success for which the sum of the m-fold multiplicative odds of success for the future trials is less than or equal to 1. This result can be viewed as a generalization of Bruss' (2000) odds theorem. Application will be made to the secretary problem. For more generality, we extend the problem in several directions in the same manner that Ferguson (2008) used to extend the odds theorem. We apply this extended result to the full-information analogue of the secretary problem, and derive the optimal stopping rule and the probability of win explicitly. The asymptotic results, as n tends to ∞, are also obtained via the planar Poisson process approach.
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