Social Choice and Welfare

, Volume 41, Issue 4, pp 989–998 | Cite as

A characterization of the single-crossing domain

  • Robert Bredereck
  • Jiehua ChenEmail author
  • Gerhard J. Woeginger
Original Paper


We characterize single-crossing preference profiles in terms of two forbidden substructures, one of which contains three voters and six (not necessarily distinct) alternatives, and one of which contains four voters and four (not necessarily distinct) alternatives. We also provide an efficient way to decide whether a preference profile is single-crossing.


Social Choice Choice Rule Social Choice Function Condorcet Winner Strategic Vote 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.



This research was started and partially conducted during the Schloss Dagstuhl seminar 12101 on “Computation and Incentives in Social Choice”. We are grateful to the organizers of this seminar (Edith Elkind, Christian Klamler, Jeffrey Rosenschein, M. Remzi Sanver) and to the Dagstuhl staff for providing a stimulating atmosphere. Robert Bredereck is supported by the DFG, research project PAWS, NI 369/10. Jiehua Chen is supported by the Studienstiftung des Deutschen Volkes. Gerhard Woeginger acknowledges support by the Netherlands Organization for Scientific Research (NWO), Grant 639.033.403, and by DIAMANT (an NWO mathematics cluster).


  1. Abello J (1991) The weak Bruhat order of \(\text{ S}_\Sigma, \) consistent sets, and Catalan numbers. SIAM J Discrete Math 4(1):1–16CrossRefGoogle Scholar
  2. Ballester MA, Haeringer G (2011) A characterization of the single-peaked domain. Soc Choice Welf 36(2):305–322Google Scholar
  3. Barberà S, Jackson MO (2004) Choosing how to choose: self-stable majority rules and constitutions. Quart J Econ 119(3):1011–1048CrossRefGoogle Scholar
  4. Barberà S, Moreno B (2011) Top monotonicity: a common root for single peakedness, single crossing and the median voter result. Games Econ Behav 73(2):345–359CrossRefGoogle Scholar
  5. Black D (1948) On the rationale of group decision-making. J Politic Econ 56(1):23–34CrossRefGoogle Scholar
  6. Bóna M (2004) Combinatorics of permutations. Chapman and Hall/CRC, Boca RatonCrossRefGoogle Scholar
  7. Booth KS, Lueker GS (1976) Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms. J Comput Syst Sci 13(3):335–379CrossRefGoogle Scholar
  8. Demange G (1994) Intermediate preferences and stable coalition structures. J Math Econ 23(1):45–58CrossRefGoogle Scholar
  9. Elkind E, Faliszewski P, Slinko A (2012) Clone structures in voters’ preferences. In: Proceedings of the 13th ACM conference on electronic commerce, pp 496–513Google Scholar
  10. Epple D, Platt GJ (1998) Equilibrium and local redistribution in an urban economy when households differ in both preferences and incomes. J Urban Econ 43(1):23–51CrossRefGoogle Scholar
  11. Galambos Á, Reiner V (2008) Acyclic sets of linear orders via the Bruhat orders. Soc Choice Welf 30(2):245–264CrossRefGoogle Scholar
  12. Gans JS, Smart M (1996) Majority voting with single-crossing preferences. J Public Econ 59(2):219–237CrossRefGoogle Scholar
  13. Grandmont JM (1978) Intermediate preferences and the majority rule. Econometrica 46(2):317–330CrossRefGoogle Scholar
  14. Hoffman AJ, Kolen AWJ, Sakarovitch M (1985) Totally-balanced and greedy matrices. SIAM J Alg Discret Methods 6(4):721–730Google Scholar
  15. Inada K (1969) The simple majority decision rule. Econometrica 37(3):490–506CrossRefGoogle Scholar
  16. Kung FC (2006) An algorithm for stable and equitable coalition structures with public goods. J Public Econ Theory 8(3):345–355CrossRefGoogle Scholar
  17. Kuratowski K (1930) Sur le problème des courbes gauches en topologie. Fundamenta Mathematicae 15:271–283Google Scholar
  18. Lekkerkerker CG, Boland JC (1962) Representation of a finite graph by a set of intervals on the real line. Fundamenta Mathematicae 51:45–64Google Scholar
  19. Meltzer AH, Richard SF (1981) A rational theory of the size of government. J Politic Econ 89(5):914–927CrossRefGoogle Scholar
  20. Moulin H (1980) On strategy-proofness and single peakedness. Public Choice 35(4):437–455CrossRefGoogle Scholar
  21. Roberts KWS (1977) Voting over income tax schedules. J Public Econ 8(3):329–340CrossRefGoogle Scholar
  22. Rothstein P (1990) Order restricted preferences and majority rule. Soc Choice Welf 7(4):331–342CrossRefGoogle Scholar
  23. Saporiti A (2009) Strategy-proofness and single-crossing. Theor Econ 4(2):127–163Google Scholar
  24. Saporiti A, Tohmé F (2006) Single-crossing, strategic voting and the median choice rule. Soc Choice Welf 26(2):363–383CrossRefGoogle Scholar
  25. Westhoff F (1977) Existence of equilibria in economies with a local public good. J Econ Theory 14(1):84–112CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Robert Bredereck
    • 1
  • Jiehua Chen
    • 1
    Email author
  • Gerhard J. Woeginger
    • 2
  1. 1.Institut fuer Softwaretechnik und Theoretische Informatik, TU BerlinBerlinGermany
  2. 2.Department of Mathematics and Computer ScienceTU EindhovenEindhovenThe Netherlands

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