Hiring above the m-th Best Candidate: A Generalization of Records in Permutations

The hiring problem is a simple model of on-line decisionmaking under uncertainty. As in many other such models, the input is a sequence of instances and a decision must be taken for each instance depending on the subsequence examined so far, while nothing is known about the future. One famous example of on-line decision-making the secretary problem, formally introduced in the early sixties. Broder et al. (2008) introduced the hiring problem as an extension of the secretary problem. Instead of selecting only one candidate, we are looking for selecting (hiring) many candidates to grow up a small company. In this context, a hiring strategy should meet two demands: to hire candidates at some reasonable rate and to improve the average quality of the hired staff. Soon afterwards, Archibald and Mart́ınez (2009) introduced a discrete model of the hiring problem where candidates seen so far could be ranked from best to worst without the need to know their absolute quality scores. Hence the sequence of candidates could be modeled as a random permutation. Two general families of hiring strategies were introduced: hiring above the m-th best candidate and hiring in the top P% quantile (for instance, P = 50 is hiring above the median). In this paper we consider only hiring above the m-th best candidate. We introduce new hiring parameters that describe the dynamics of the hiring process, like the distance between the last two hirings, and the quality of the hired staff, like the score of the best discarded candidate. While Archibald and Mart́ınez made systematic use of analytic combinatorics techniques (Flajolet, Sedgewick, 2008) in their analysis, we use here a different approach to study the various hiring parameters related associated to the hiring process. We are able to obtain explicit formulas for the probability distribution or the probability generating function of the random variables of interest in a rather direct way. The explicit nature of our results also allows a very detailed study of their asymptotic behaviour. Adding our new results to those of Archibald and Mart́ınez leads to a very precise ? This work started when the first author was visiting the third author in a short stay supported by an FPI grant from the Spanish Ministry of Science. The first and the second authors were supported by project TIN2010-17254 (FRADA) from the Spanish Ministry of Science and Innovation. The third author was supported by the Austrian Science Foundation FWF, grant S9608-N23. quantitative characterization of the hiring above the m-th best candidate strategy. This might prove very useful in applications of the hiring process, e.g., in data stream algorithms.