Bayes' Model of the Best-Choice Problem with Disorder

We consider the best-choice problem with disorder and imperfect observation. The decision-maker observes sequentially a known number of i.i.d random variables from a known distribution with the object of choosing the largest. At the random time the distribution law of observations is changed. The random variables cannot be perfectly observed. Each time a random variable is sampled the decision-maker is informed only whether it is greater than or less than some level specified by him. The decision-maker can choose at most one of the observation. The optimal rule is derived in the class of Bayes' strategies. 1. Introduction In the papers we consider the following best-choice problem with disorder and imperfect observations. A decision-maker observes sequentially iid random variables . The observations are from a continuous distribution law (state ). At the random time , the distribution law of observations is changed to continuous distribution function (i.e., the disorder happen—state ). The moment of the disorder has a geometric distribution with parameter . The observer knows parameters , , and , but the exact moment is unknown. At each time in which a random variable is sampled, the observer has to make a decision to accept (and stop the observation process) or reject the observation (and continue the observation process). If the decision-maker decided to accept at step ( ), she receives as the payoff the value of the random variable discounted by the factor , where . The random variables cannot be perfectly observed. The decision-maker is only informed whether the observation is greater than or less than some level specified by her. The aim of the decision-maker is to maximize the expected value of the accepted discounted observation. We find the solution in the class of the following strategies. At each moment ( ), the observer estimates the a posterior probability of the current state and specifies the threshold . The decision-maker accepts the observation if and only if it is greater than the corresponding threshold . This problem is the generalization of the best-choice problem [1, 2] and the quickest determination of the change-point (disorder) problem [3–5]. The best-choice problems with imperfect information were treated in [6–8]. Only few papers related to the combined best-choice and disorder problem are published [9–11]. Yoshida [9] considered the full-information case and found the optimal stopping rule which maximizes the probability that accepted value is the largest of all random variables for a given integer . Closely related work to