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Analysis of the “Hiring Above the Median” Selection Strategy for the Hiring Problem

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Abstract

This paper gives a precise mathematical analysis of the behaviour of “hiring above the median” strategies for a problem in the context of “on-line selection under uncertainty” that is known (at least in computer science related literature) as the “hiring problem”. Here a sequence of candidates is interviewed sequentially and based on the “score” of the current candidate an immediate decision whether to hire him or not has to be made. For “hiring above the median” selection rules, a new candidate will be hired if he has a score better than the median score of the already recruited candidates. Under the natural probabilistic model assuming that the ranks of the first n candidates are forming a random permutation, we show exact and asymptotic results for various quantities of interest to describe the dynamics of the hiring process and the quality of the hired staff. In particular we can characterize the limiting distribution of the number of hired candidates of a sequence of n candidates, which reveals the somewhat surprising effect, that slight changes in the selection rule, i.e., assuming the “lower” or the “upper” median as the threshold, have a strong influence on the asymptotic behaviour. Thus we could extend considerably previous analyses (Krieger et al., Ann. Appl. Probab., 17:360–385, 2007; Broder et al., Proceedings of the 19th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pp. 1184–1193, ACM/SIAM, New York/Philadelphia, 2008 and Archibald and Martinez, Proceedings of the 21st International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2009), Discrete Mathematics and Theoretical Computer Science, pp. 63–76, 2009) of such selection rules. Furthermore, we discuss connections between the hiring process and the Chinese restaurant process introduced by Pitman (Combinatorial Stochastic Processes, Springer, Berlin, 2006).

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Notes

  1. As pointed out in [5], Lake Wobegon is a fictional town, where “the women are strong, the men are good looking, and all the children are above average”. The considered strategies match this term in the sense that each recruited candidate, at least at the time when he is hired, is above “average”.

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Acknowledgements

The authors thank the anonymous referees and the editors for many valuable comments.

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Authors and Affiliations

  1. Departament de Llenguatges i Sistemes Informàtics, Universitat Politècnica de Catalunya, Jordi Girona, 1-3, 08034, Barcelona, Spain

    Ahmed Helmi

  2. Institut für Diskrete Mathematik und Geometrie, Technische Universität Wien, Wiedner Hauptstr. 8-10/104, 1040, Wien, Austria

    Alois Panholzer

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  1. Ahmed Helmi
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  2. Alois Panholzer
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Correspondence to Alois Panholzer.

Additional information

An extended abstract of this work appeared as [14] in the “Proceedings of ANALCO 2012”.

This work started when the first author was visiting the second author in a short stay supported by an FPI grant from the Spanish Ministry of Science and Innovation. The research of the first author was supported by a project TIN2010-17254 (FRADA) from the Spanish Ministry of Science and Innovation. The second author was supported by the Austrian Science Foundation FWF, grants S9608-N23 and P25337-N23.

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Helmi, A., Panholzer, A. Analysis of the “Hiring Above the Median” Selection Strategy for the Hiring Problem. Algorithmica 66, 762–803 (2013). https://doi.org/10.1007/s00453-012-9727-2

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