Recognizing 1-Euclidean Preferences: An Alternative Approach

  • Edith Elkind
  • Piotr Faliszewski
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8768)

Abstract

We consider the problem of detecting whether a given election is 1-Euclidean, i.e., whether voters and candidates can be mapped to points on the real line so that voters’ preferences over the candidates are determined by the Euclidean distance. A recent paper by Knoblauch [14] shows that this problem admits a polynomial-time algorithm. Knoblauch’s approach relies on the fact that a 1-Euclidean election is necessarily single-peaked, and makes use of the properties of the respective candidate order to find a mapping of voters and candidates to the real line. We propose an alternative polynomial-time algorithm for this problem, which is based on the observation that a 1-Euclidean election is necessarily singe-crossing, and we use the properties of the respective voter order to find the desired mapping.

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References

  1. 1.
    Barberà, S., Moreno, B.: Top monotonicity: A common root for single peakedness, single crossing and the median voter result. Games and Economic Behavior 73(2), 345–359 (2011)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Bartholdi III, J., Trick, M.: Stable matching with preferences derived from a psychological model. Operations Research Letters 5(4), 165–169 (1986)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Betzler, N., Slinko, A., Uhlmann, J.: On the computation of fully proportional representation. Journal of Artificial Intelligence Research 47, 475–519 (2013)MATHMathSciNetGoogle Scholar
  4. 4.
    Black, D.: On the rationale of group decision-making. Journal of Political Economy 56(1), 23–34 (1948)CrossRefGoogle Scholar
  5. 5.
    Brandt, F., Brill, M., Hemaspaandra, E., Hemaspaandra, L.: Bypassing combinatorial protections: Polynomial-time algorithms for single-peaked electorates. In: AAAI 2010, pp. 715–722 (2010)Google Scholar
  6. 6.
    Bredereck, R., Chen, J., Woeginger, G.: A characterization of the single-crossing domain. Social Choice and Welfare 41(4), 989–998 (2013)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Chen, J., Pruhs, K., Woeginger, G.: Characterizations of the one-dimensional Euclidean domain. Manuscript (2014)Google Scholar
  8. 8.
    Elkind, E., Faliszewski, P., Skowron, P.: A characterization of the single-peaked single-crossing domain. In: AAAI 2014 (2014)Google Scholar
  9. 9.
    Elkind, E., Faliszewski, P., Slinko, A.: Clone structures in voters’ preferences. In: ACM EC 2012, pp. 496–513 (2012)Google Scholar
  10. 10.
    Escoffier, B., Lang, J., Öztürk, M.: Single-peaked consistency and its complexity. In: ECAI 2008, pp. 366–370 (2008)Google Scholar
  11. 11.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: The shield that never was: Societies with single-peaked preferences are more open to manipulation and control. Information and Computation 209(2), 89–107 (2011)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Grandmont, J.M.: Intermediate preferences and the majority rule. Econometrica 46(2), 317–330 (1978)CrossRefMATHMathSciNetGoogle Scholar
  13. 13.
    Hotelling, H.: Stability in competition. Economic Journal 39, 41–57 (1929)CrossRefGoogle Scholar
  14. 14.
    Knoblauch, V.: Recognizing one-dimensional euclidean preference profiles. Journal of Mathematical Economics 46, 1–5 (2010)CrossRefMATHMathSciNetGoogle Scholar
  15. 15.
    Mirrlees, J.: An exploration in the theory of optimal income taxation. Review of Economic Studies 38, 175–208 (1971)CrossRefMATHGoogle Scholar
  16. 16.
    Moulin, H.: On strategy-proofness and single peakedness. Public Choice 35(4), 437–455 (1980)CrossRefGoogle Scholar
  17. 17.
    Saporiti, A.: Strategy-proofness and single-crossing. Theoretical Economics 4(2), 127–163 (2009)Google Scholar
  18. 18.
    Skowron, P., Yu, L., Faliszewski, P., Elkind, E.: The complexity of fully proportional representation for single-crossing electorates. In: Vöcking, B. (ed.) SAGT 2013. LNCS, vol. 8146, pp. 1–12. Springer, Heidelberg (2013)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Edith Elkind
    • 1
  • Piotr Faliszewski
    • 2
  1. 1.University of OxfordOxfordUK
  2. 2.AGH UniversityKrakowPoland

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