Analysis of "Hiring Above the Median": A "Lake Wobegon" Strategy for The Hiring Problem

The hiring problem is a recent research problem, which has been introduced and studied first by Broder et al. [2] in 2008. It belongs to the category of on-line decision making under uncertainty. In such kind of research, the input is a sequence of instances and a decision must be taken for each instance depending on the instances seen so far while no information on the future is available. The hiring problem can be considered as a natural extension of the well-known secretary problem [4], where the employer is now looking for many candidates rather than only one (as it is the case for the secretary problem). Here the goal is to design some hiring strategy to meet the demands of the employer, which essentially are to obtain a good quality staff at a reasonable hiring rate, which is a main difference to the secretary problem, where an optimization policy, namely the demand of hiring the best candidate, occurs. Broder et al. introduced two so-called "Lake Wobegon strategies", namely "hiring above the current mean" and "hiring above the current median", applied for a continuous probabilistic model for the sequence of scores of the candidates. Archibald and Martinez [1] have reformulated the problem for a discrete model that considers the relative ranks amongst candidates as it is the case in the secretary problem. For this model in [1] the authors studied two general strategies, namely "hiring above the m-th best candidate", and "hiring above the median" (and other quantiles). In this work we give a detailed study of the "hiring above the median" strategy under this discrete model for the input sequence of scores of the candidates. This strategy processes the sequence of candidates as follows: hire the first interviewed candidate, and then any coming candidate is hired if and only if his rank is better than the rank of the median of the already hired staff, and discarded otherwise. Compared to the previous work of [1] we use a somewhat different recursive approach for a study of the "hiring above the median" strategy leading to rather explicit results. The key ingredients are to take into account the score of the median (the so-called threshold candidate) of the hired staff during the hiring process as well as to distinguish between two cases according to the parity of the size of the hiring set. Considering the transition probabilities during the hiring process yields, for fundamental hiring quantities, a system of linear recurrences that can be translated into a system of partial differential equations for the corresponding generating functions. In order to solve the PDEs appearing it turned out to be crucial to find suitable normalization factors of the studied recursive sequences, such that one of the corresponding generating functions itself reduces to a first order linear PDE. The exact solutions obtained for the differential equations yield to a rather detailed description of the exact probability distributions together with limiting distribution results for various hiring quantities, which might lead to a fairly good understanding of the "hiring above the median" hiring process. In particular we obtained results for the number of hired candidates, the score of the last hired candidate, the index of the last hired candidate, the distance between the last two hirings, the score of the best discarded candidate, and the number of hired candidates conditioned on the score of the first candidate. We also give the probability that a given score is getting hired in a sequence of n candidates.