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A class of dissimilarity semimetrics for preference relations

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  • Hiroki Nishimura
  • Efe A. Ok
Abstract
We propose a class of semimetrics for preference relations any one of which is an alternative to the classical Kemeny-Snell-Bogart metric. (We take a fairly general viewpoint about what constitutes a preference relation, allowing for any acyclic order to act as one.) These semimetrics are based solely on the implications of preferences for choice behavior, and thus appear more suitable in economic contexts and choice experiments. In our main result, we obtain a fairly simple axiomatic characterization for the class we propose. The apparently most important member of this class (at least in the case of finite alternative spaces), which we dub the top-difference semimetric, is characterized separately. We also obtain alternative formulae for it, and relative to this metric, compute the diameter of the space of complete preferences, as well as the best transitive extension of a given acyclic preference relation. Finally, we prove that our preference metric spaces cannot be isometically embedded in a Euclidean space.

Suggested Citation

  • Hiroki Nishimura & Efe A. Ok, 2022. "A class of dissimilarity semimetrics for preference relations," Papers 2203.04418, arXiv.org.
    • Handle: RePEc:arx:papers:2203.04418
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    References listed on IDEAS

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    1. Can, Burak, 2014. "Weighted distances between preferences," Journal of Mathematical Economics, Elsevier, vol. 51(C), pages 109-115.
    2. Can, Burak & Storcken, Ton, 2018. "A re-characterization of the Kemeny distance," Journal of Mathematical Economics, Elsevier, vol. 79(C), pages 112-116.
    3. Kazuhiro Hara & Efe A. Ok & Gil Riella, 2019. "Coalitional Expected Multi‐Utility Theory," Econometrica, Econometric Society, vol. 87(3), pages 933-980, May.
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    5. Wade D. Cook & Lawrence M. Seiford, 1978. "Priority Ranking and Consensus Formation," Management Science, INFORMS, vol. 24(16), pages 1721-1732, December.
    6. Nishimura, Hiroki & Ok, Efe A., 2016. "Utility representation of an incomplete and nontransitive preference relation," Journal of Economic Theory, Elsevier, vol. 166(C), pages 164-185.
    7. Erick Moreno-Centeno & Adolfo R. Escobedo, 2016. "Axiomatic aggregation of incomplete rankings," IISE Transactions, Taylor & Francis Journals, vol. 48(6), pages 475-488, June.
    8. Mauricio Ribeiro & Gil Riella, 2017. "Regular preorders and behavioral indifference," Theory and Decision, Springer, vol. 82(1), pages 1-12, January.
    9. Eliaz, Kfir & Ok, Efe A., 2006. "Indifference or indecisiveness? Choice-theoretic foundations of incomplete preferences," Games and Economic Behavior, Elsevier, vol. 56(1), pages 61-86, July.
    10. Jamison, Dean T & Lau, Lawrence J, 1973. "Semiorders and the Theory of Choice," Econometrica, Econometric Society, vol. 41(5), pages 901-912, September.
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    Cited by:

    1. Alfio Giarlotta & Angelo Petralia & Stephen Watson, 2022. "Semantics meets attractiveness: Choice by salience," Papers 2204.08798, arXiv.org, revised Aug 2022.

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