Sequential search and selection problem under uncertainty

This paper formulates and discusses a series of sequential decision problems of the following common structure: A decision alternative of multiple attributes – that is, a job, an employee, or an investment alternative – is to be selected within a certain fixed length of time. An unknown number of alternatives are presented sequentially, either deterministically or in a random manner. The decision maker can rank all the alternatives from best to worst without ties, and the decision to accept or reject an alternative is based solely on the relative ranks of those alternatives evaluated so far. The nonparametric sequential decision problem is first studied for a model involving a discrete time period and then generalized in terms of continuous time. Also considered is a variant of this problem involving a Bayesian estimation of (1) the uncertain probability of having an alternative at a given stage in the discrete-time model and (2) the arrival rate of alternatives in the continuous-time model. The optimal selection strategy that maximizes the probability of selecting the absolute best alternative is illustrated with the job search problem and the single-machine job assignment problem.