A Problem of Optimal Choice and Assignment

Within the conventional framework of the so-called secretary problem, we introduce two decision makers, one of whom is dominant. The objectives are to select the two best objects and immediately to assign them to the decision makers so that the dominant one obtains the best object. Optimal strategies for selection and assignment, and the maximum probability of achieving the objectives, are derived. We also investigate an important suboptimal strategy: the dominant decision maker acts to maximize the probability of obtaining the very best object while, independently, the other decision maker acts to maximize the probability of selecting the second best. Although it has intuitive appeal, this suboptimal strategy is rather poor; indeed, the two decision makers may trade with one another the objects they select without affecting their chances of obtaining the desired objects. An interesting by-product of this study is that a simple solution in closed form is obtained for the problem of selecting the second best object.