Social Choice and Welfare

, Volume 48, Issue 2, pp 409–432 | Cite as

The one-dimensional Euclidean domain: finitely many obstructions are not enough

  • Jiehua Chen
  • Kirk R. Pruhs
  • Gerhard J. Woeginger
Original Paper


We show that one-dimensional Euclidean preference profiles can not be characterized in terms of finitely many forbidden substructures. This result is in strong contrast to the case of single-peaked and single-crossing preference profiles, for which such finite characterizations have been derived in the literature.



This research has been supported by COST Action IC1205 on Computational Social Choice. Jiehua Chen acknowledges support by the Studienstiftung des Deutschen Volkes. Kirk Pruhs is supported in part by NSF Grants CCF-1115575, CNS-1253218, CCF-1421508, and an IBM Faculty Award. Gerhard Woeginger acknowledges support by the Zwaartekracht NETWORKS grant of NWO, and by the Alexander von Humboldt Foundation, Bonn, Germany.


  1. Ballester MA, Haeringer G (2011) A characterization of the single-peaked domain. Soc Choice Welf 36:305–322CrossRefGoogle Scholar
  2. Barberà S, Jackson MO (2004) Choosing how to choose: self-stable majority rules and constitutions. Q J Econ 119:1011–1048CrossRefGoogle Scholar
  3. Bartholdi J III, Trick MA (1986) Stable matching with preferences derived from a psychological model. Oper Res Lett 5:165–169CrossRefGoogle Scholar
  4. Black D (1948) On the rationale of group decision-making. J Polit Econ 56:23–34CrossRefGoogle Scholar
  5. Bogomolnaia A, Laslier JF (2007) Euclidean preferences. J Math Econ 43:87–98CrossRefGoogle Scholar
  6. Brams SJ, Jones MA, Kilgour DM (2002) Single-peakedness and disconnected coalitions. J Theor Polit 14:359–383CrossRefGoogle Scholar
  7. Bredereck R, Chen J, Woeginger GJ (2013) A characterization of the single-crossing domain. Soc Choice Welf 41:989–998CrossRefGoogle Scholar
  8. Bredereck R, Chen J, Woeginger GJ (2016) Are there any nicely structured preference profiles nearby? Math Soc Sci 79:61–73CrossRefGoogle Scholar
  9. Coombs C (1964) A theory of data. Wiley, New YorkGoogle Scholar
  10. Demange G (1994) Intermediate preferences and stable coalition structures. J Math Econ 23:45–58CrossRefGoogle Scholar
  11. Diamond PA, Stiglitz JE (1974) Increases in risk and in risk aversion. J Econ Theory 8:337–360CrossRefGoogle Scholar
  12. Doignon J, Falmagne J (1994) A polynomial time algorithm for unidimensional unfolding representations. J Algorithms 16:218–233CrossRefGoogle Scholar
  13. Elkind E, Faliszewski P, Slinko AM (2012) Clone structures in voters’ preferences. In: Proceedings of the 13th ACM conference on electronic commerce (EC’12), pp 496–513Google Scholar
  14. Elkind E, Lackner M (2014) On detecting nearly structured preference profiles. In: Proceedings of the 28th AAAI conference on artificial intelligence (AAAI’2014), pp 661–667Google Scholar
  15. Epple D, Platt GJ (1998) Equilibrium and local redistribution in an urban economy when households differ in both preferences and incomes. J Urban Econ 43:23–51CrossRefGoogle Scholar
  16. Escoffier B, Lang J, Öztürk M (2008) Single-peaked consistency and its complexity. In: Proceedings of the 18th European conference on artificial intelligence (ECAI’08), pp 366–370Google Scholar
  17. Földes S, Hammer PL (1977) Split graphs. In: Proceedings of the eighth southeastern conference on combinatorics, graph theory and computing, pp 311–315Google Scholar
  18. Gans JS, Smart M (1996) Majority voting with single-crossing preferences. J Public Econ 59:219–237CrossRefGoogle Scholar
  19. Grandmont J-M (1978) Intermediate preferences and majority rule. Econometrica 46:317–330CrossRefGoogle Scholar
  20. Hoffman AJ, Kolen AWJ, Sakarovitch M (1985) Totally-balanced and greedy matrices. SIAM J Algebraic Discrete Methods 6:721–730CrossRefGoogle Scholar
  21. Hotelling H (1929) Stability in competition. Econ J 39(153):41–57CrossRefGoogle Scholar
  22. Inada K (1969) The simple majority rule. Econometrica 37:490–506CrossRefGoogle Scholar
  23. Karlin S (1968) Total positivity. Stanford University Press, LondonGoogle Scholar
  24. Knoblauch V (2010) Recognizing one-dimensional Euclidean preference profiles. J Math Econ 46:1–5CrossRefGoogle Scholar
  25. Kung F-C (2006) An algorithm for stable and equitable coalition structures with public goods. J Public Econ Theory 8:345–355CrossRefGoogle Scholar
  26. Kuratowski K (1930) Sur le problème des courbes gauches en topologie. Fundam Math 15:271–283Google Scholar
  27. Lekkerkerker C, Boland D (1962) Representation of finite graphs by a set of intervals on the real line. Fundam Math 51:45–64Google Scholar
  28. Meltzer AH, Richard SF (1981) A rational theory of the size of government. J Polit Econ 89:914–927CrossRefGoogle Scholar
  29. Mirrlees JA (1971) An exploration in the theory of optimal income taxation. Rev Econ Stud 38:175–208CrossRefGoogle Scholar
  30. Moulin H (1980) On strategy-proofness and single peakedness. Public Choice 35:437–455CrossRefGoogle Scholar
  31. Roberts KWS (1977) Voting over income tax schedules. J Public Econ 8:329–340CrossRefGoogle Scholar
  32. Westhoff F (1977) Existence of equilibria in economies with a local public good. J Econ Theory 14:84–112CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  • Jiehua Chen
    • 1
  • Kirk R. Pruhs
    • 2
  • Gerhard J. Woeginger
    • 3
  1. 1.Department of Software Engineering and Theoretical Computer ScienceTU BerlinBerlinGermany
  2. 2.Computer Science DepartmentUniversity of PittsburghPittsburghUSA
  3. 3.Department of Mathematics and Computer ScienceTU EindhovenEindhovenThe Netherlands

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