Abstract
Given a ranking of elements of a set and given a disjoint partition of the same set, the ranking does not generally imply a total order of the partition. In this paper, we introduce the Kendall-\(\tau \)partition ranking, a linear order of the subsets of the partition which follows from the given ranking. We prove that, under certain assumptions, the Kendall-\(\tau \) partition ranking is robust, in the sense that it remains the same when removing subsets of the partition. Then, we give several results (properties) concerning the adequacy of the ranking for ordering the partition and we prove that the integrality gap of the 0–1 problem associated to the Kendall-\(\tau \) partition ranking tends to 8/7. Additionally, we provide a comparison of the new ranking with respect to the mean and median based scores from a theoretical and empirical point of view. Finally, a real application with data from the Programme for International Student Assessment is presented, where countries are ordered based on their school rankings.
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Notes
A preference matrix \(M = (m_{ij})\) defines a partial order (tournament) in V , i.e., candidate i is preferred to candidate j if \(m_{ij} > m_{ji}\). A tournament provides a linear order of candidates in V if it is transitive, i.e., if \(m_{ij} > m_{ji}\) and \(m_{jk} > m_{kj}\) it implies \(m_{ik} > m_{ki}\), the opposite is known as the “the Condorcet paradox”.
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Acknowledgements
The authors thank the financial support from the Spanish Ministry for Economy and Competitiveness (Ministerio de Economía, Industria y Competitividad), the State Research Agency (Agencia Estatal de Investigación) and the European Regional Development Fund (Fondo Europeo de Desarrollo Regional) under Grants MTM2016-79765-P (AEI/FEDER, UE) and PGC2018-099428-B-100. We would also like to thank an anonymous referee and the corresponding associate editor for providing constructive comments and help.
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Aparicio, J., Landete, M. & Monge, J.F. A linear ordering problem of sets. Ann Oper Res 288, 45–64 (2020). https://doi.org/10.1007/s10479-019-03473-y
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DOI: https://doi.org/10.1007/s10479-019-03473-y