The Complexity of Fully Proportional Representation for Single-Crossing Electorates

  • Piotr Skowron
  • Lan Yu
  • Piotr Faliszewski
  • Edith Elkind
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8146)


We study the complexity of winner determination in single-crossing elections under two classic fully proportional representation rules—Chamberlin–Courant’s rule and Monroe’s rule. Winner determination for these rules is known to be NP-hard for unrestricted preferences. We show that for single-crossing preferences this problem admits a polynomial-time algorithm for Chamberlin–Courant’s rule, but remains NP-hard for Monroe’s rule. Our algorithm for Chamberlin–Courant’s rule can be modified to work for elections with bounded single-crossing width. To circumvent the hardness result for Monroe’s rule, we consider single-crossing elections that satisfy an additional constraint, namely, ones where each candidate is ranked first by at least one voter (such elections are called narcissistic). For single-crossing narcissistic elections, we provide an efficient algorithm for the egalitarian version of Monroe’s rule.


Social Choice Preference Order Vote Rule Proportional Representation Assignment Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  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)MathSciNetCrossRefzbMATHGoogle 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)MathSciNetCrossRefzbMATHGoogle 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)zbMATHGoogle Scholar
  4. 4.
    Bredereck, R., Chen, J., Woeginger, G.: A characterization of the single-crossing domain. Social Choice and Welfare (to appear, 2012)Google Scholar
  5. 5.
    Chamberlin, B., Courant, P.: Representative deliberations and representative decisions: Proportional representation and the borda rule. American Political Science Review 77(3), 718–733 (1983)CrossRefGoogle Scholar
  6. 6.
    Cornaz, D., Galand, L., Spanjaard, O.: Bounded single-peaked width and proportional representation. In: Proceedings of the 20th European Conference on Artificial Intelligence, pp. 270–275 (2012)Google Scholar
  7. 7.
    Cornaz, D., Galand, L., Spanjaard, O.: Kemeny elections with bounded single-peaked or single-crossing width. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence, pp. 76–82 (2013)Google Scholar
  8. 8.
    Elkind, E., Faliszewski, P., Slinko, A.: Clone structures in voters’ preferences. In: Proceedings of the 13th ACM Conference on Electronic Commerce, pp. 496–513 (2012)Google Scholar
  9. 9.
    Lu, T., Boutilier, C.: Budgeted social choice: From consensus to personalized decision making. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence, pp. 280–286 (2011)Google Scholar
  10. 10.
    Mirrlees, J.: An exploration in the theory of optimal income taxation. Review of Economic Studies 38, 175–208 (1971)CrossRefzbMATHGoogle Scholar
  11. 11.
    Monroe, B.: Fully proportional representation. American Political Science Review 89(4), 925–940 (1995)CrossRefGoogle Scholar
  12. 12.
    Potthoff, R., Brams, S.: Proportional representation: Broadening the options. Journal of Theoretical Politics 10(2), 147–178 (1998)CrossRefGoogle Scholar
  13. 13.
    Procaccia, A., Rosenschein, J., Zohar, A.: On the complexity of achieving proportional representation. Social Choice and Welfare 30(3), 353–362 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Saporiti, A., Tohmé, F.: Single-crossing, strategic voting and the median choice rule. Social Choice and Welfare 26(2), 363–383 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Skowron, P., Faliszewski, P., Slinko, A.: Achieving fully proportional representation is easy in practice. In: Proceedings of the 12th International Conference on Autonomous Agents and Multiagent Systems, pp. 399–406 (2013)Google Scholar
  16. 16.
    Skowron, P., Faliszewski, P., Slinko, A.: Fully proportional representation as resource allocation: Approximability results. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence, pp. 353–359 (2013)Google Scholar
  17. 17.
    Skowron, P., Yu, L., Faliszewski, P., Elkind, E.: The complexity of fully proportional representation for single-crossing electorates. Technical Report arXiv:1307.1252 [cs.GT], (2013)Google Scholar
  18. 18.
    Yu, L., Chan, H., Elkind, E.: Multiwinner elections under preferences that are single-peaked on a tree. In: Proceedings of the 23rd International Joint Conference on Artificial Intelligence, pp. 425–431 (2013)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Piotr Skowron
    • 1
  • Lan Yu
    • 2
  • Piotr Faliszewski
    • 3
  • Edith Elkind
    • 2
  1. 1.University of WarsawPoland
  2. 2.Nanyang Technological UniversitySingapore
  3. 3.AGH University of Science and TechnologyPoland

Personalised recommendations