Abstract
In several decision-making problems, alternatives should be ranked on the basis of paired comparisons between them. We present an axiomatic approach for the universal ranking problem with arbitrary preference intensities, incomplete and multiple comparisons. In particular, two basic properties—independence of irrelevant matches and self-consistency—are considered. It is revealed that there exists no ranking method satisfying both requirements at the same time. The impossibility result holds under various restrictions on the set of ranking problems, however, it does not emerge in the case of round-robin tournaments. An interesting and more general possibility result is obtained by restricting the domain of independence of irrelevant matches through the concept of macrovertex.
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Notes
IIM is the abbreviation of independence of irrelevant matches, an axiom to be discussed in Sect. 3.1.
While \(m_{ij} \in \{ 0; 1 \}\) for all \(X_i,X_j \in N\) allows for \(m=0\), it leads to a meaningless ranking problem without any comparison.
Some of their differences are highlighted by González-Díaz et al. (2014).
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Acknowledgements
We thank Sándor Bozóki for useful advice. Anonymous reviewers provided valuable comments and suggestions on earlier drafts. The research was supported by OTKA grant K 111797 and by the MTA Premium Post Doctorate Research Program.
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Csató, L. An impossibility theorem for paired comparisons. Cent Eur J Oper Res 27, 497–514 (2019). https://doi.org/10.1007/s10100-018-0572-5
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DOI: https://doi.org/10.1007/s10100-018-0572-5