Robustness Among Multiwinner Voting Rules

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 10504)


We investigate how robust are results of committee elections to small changes in the input preference orders, depending on the voting rules used. We find that for typical rules the effect of making a single swap of adjacent candidates in a single preference order is either that (1) at most one committee member can be replaced, or (2) it is possible that the whole committee can be replaced. We also show that the problem of computing the smallest number of swaps that lead to changing the election outcome is typically \({\mathrm {NP}}\)-hard, but there are natural \({\mathrm {FPT}}\) algorithms. Finally, for a number of rules we assess experimentally the average number of random swaps necessary to change the election result.



We are grateful to anonymous SAGT reviewers for their useful comments. R. Bredereck was supported by the DFG fellowship BR 5207/2. P. Faliszewski was supported by the NCN, Poland, under project 2016/21/B/ST6/01509. A. Kaczmarczyk was supported by the DFG project AFFA (BR 5207/1 and NI 369/15). P. Skowron was supported by a Humboldt Fellowship. N. Talmon was supported by an I-CORE ALGO fellowship.


  1. 1.
    Aziz, H., Elkind, E., Faliszewski, P., Lackner, M., Skowron, P.: The Condorcet principle for multiwinner elections: from shortlisting to proportionality. arXiv preprint arXiv:1701.08023 (2017)
  2. 2.
    Barberà, S., Coelho, D.: How to choose a non-controversial list with \(k\) names. Soc. Choice Welf. 31(1), 79–96 (2008)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Betzler, N., Slinko, A., Uhlmann, J.: On the computation of fully proportional representation. J. Artif. Intell. Res. 47, 475–519 (2013)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Blom, M., Stuckey, P., Teague, V.: Towards computing victory margins in STV elections. arXiv preprint arXiv:1703.03511 (2017)
  5. 5.
    Bredereck, R., Faliszewski, P., Kaczmarczyk, A., Niedermeier, R., Skowron, P., Talmon, N.: Robustness among multiwinner voting rules. arXiv preprint arXiv:1707.01417 (2017)
  6. 6.
    Caragiannis, I., Hemaspaandra, E., Hemaspaandra, L.: Dodgson’s rule and Young’s rule. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A.D. (eds.) Handbook of Computational Social Choice. Cambridge University Press, Cambridge (2016)Google Scholar
  7. 7.
    Cary, D.: Estimating the margin of victory for instant-runoff voting. Presented at EVT/WOTE-2011, August 2011Google Scholar
  8. 8.
    Chamberlin, B., Courant, P.: Representative deliberations and representative decisions: proportional representation and the Borda rule. Am. Polit. Sci. Rev. 77(3), 718–733 (1983)CrossRefGoogle Scholar
  9. 9.
    Coelho, D.: Understanding, evaluating and selecting voting rules through games and axioms. Ph.D. thesis, Universitat Autònoma de Barcelona (2004)Google Scholar
  10. 10.
    Conitzer, V., Rognlie, M., Xia, L.: Preference functions that score rankings and maximum likelihood estimation. In: Proceedings of IJCAI-2009, pp. 109–115, July 2009Google Scholar
  11. 11.
    Cygan, M., Fomin, F.V., Kowalik, Ł., Lokshtanov, D., Marx, D., Pilipczuk, M., Pilipczuk, M., Saurabh, S.: Parameterized Algorithms. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  12. 12.
    Elkind, E., Faliszewski, P., Skowron, P., Slinko, A.: Properties of multiwinner voting rules. Social Choice Welf. 48(3), 599–632 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Elkind, E., Faliszewski, P., Slinko, A.: Swap bribery. In: Mavronicolas, M., Papadopoulou, V.G. (eds.) SAGT 2009. LNCS, vol. 5814, pp. 299–310. Springer, Heidelberg (2009). doi: 10.1007/978-3-642-04645-2_27CrossRefGoogle Scholar
  14. 14.
    Faliszewski, P., Skowron, P., Slinko, A., Talmon, N.: Committee scoring rules: axiomatic classification and hierarchy. In: Proceedings of IJCAI-2016, pp. 250–256 (2016)Google Scholar
  15. 15.
    Gehrlein, W.: The Condorcet criterion and committee selection. Math. Soc. Sci. 10(3), 199–209 (1985)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Kaczmarczyk, A., Faliszewski, P.: Algorithms for destructive shift bribery. In: Proceedings of AAMAS-2016, pp. 305–313 (2016)Google Scholar
  17. 17.
    Kamwa, E.: On stable voting rules for selecting committees. J. Math. Econ. 70, 36–44 (2017)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Lenstra Jr., H.: Integer programming with a fixed number of variables. Math. Oper. Res. 8(4), 538–548 (1983)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Lu, T., Boutilier, C.: Budgeted social choice: from consensus to personalized decision making. In: Proceedings of IJCAI-2011, pp. 280–286 (2011)Google Scholar
  20. 20.
    Magrino, T., Rivest, R., Shen, E., Wagner, D.: Computing the margin of victory in IRV elections. Presented at EVT/WOTE-2011, August 2011Google Scholar
  21. 21.
    Mattei, N., Walsh, T.: Preflib: a library for preferences. In: Proceedings of the 3rd International Conference on Algorithmic Decision Theory, pp. 259–270 (2013)CrossRefGoogle Scholar
  22. 22.
    McGarvey, D.: A theorem on the construction of voting paradoxes. Econometrica 21(4), 608–610 (1953)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Procaccia, A., Rosenschein, J., Zohar, A.: On the complexity of achieving proportional representation. Soc. Choice Welf. 30(3), 353–362 (2008)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Sekar, S. Sikdar., Xia, L.: Condorcet consistent bundling with social choice. In: Proceedings of AAMAS-2017, May 2017Google Scholar
  25. 25.
    Shiryaev, D., Yu, L., Elkind, E.: On elections with robust winners. In: Proceedings of AAMAS-2013, pp. 415–422 (2013)Google Scholar
  26. 26.
    Xia, L.: Computing the margin of victory for various voting rules. In: Proceedings of EC-2012, pp. 982–999, June 2012Google Scholar

Copyright information

© Springer International Publishing AG 2017

Authors and Affiliations

  1. 1.University of OxfordOxfordUK
  2. 2.AGH UniversityKrakowPoland
  3. 3.TU BerlinBerlinGermany
  4. 4.Weizmann Institute of ScienceRehovotIsrael

Personalised recommendations