Skip to main content

Complexity of Manipulative Actions When Voting with Ties

Part of the Lecture Notes in Computer Science book series (LNAI,volume 9346)

Abstract

Most of the computational study of election problems has assumed that each voter’s preferences are, or should be extended to, a total order. However in practice voters may have preferences with ties. We study the complexity of manipulative actions on elections where voters can have ties, extending the definitions of the election systems (when necessary) to handle voters with ties. We show that for natural election systems allowing ties can both increase and decrease the complexity of manipulation and bribery, and we state a general result on the effect of voters with ties on the complexity of control.

Keywords

  • Electrical Problems
  • Total Order
  • Faliszewski
  • Narodytska
  • Preferred Candidate Wins

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 is a preview of subscription content, access via your institution.

Buying options

Chapter
USD   29.95
Price excludes VAT (USA)
  • DOI: 10.1007/978-3-319-23114-3_7
  • Chapter length: 17 pages
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
eBook
USD   79.99
Price excludes VAT (USA)
  • ISBN: 978-3-319-23114-3
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book
USD   99.99
Price excludes VAT (USA)

Notes

  1. 1.

    Here and elsewhere we write \(\sum \!A\) to denote \(\sum _{a \in A} a\).

  2. 2.

    A similar situation occurred in the proof of Proposition 5 in Narodytska and Walsh [25], where a (very different) specialized version of Subset Sum was constructed to prove that 3-candidate Borda CWCM (in the non-single-peaked case) for top orders using average remained NP-complete.

  3. 3.

    By triviality we mean a scoring rule with a scoring vector that gives each candidate the same score.

  4. 4.

    Menon and Larson independently proved the top order case of the following theorem [24].

References

  1. Bartholdi III, J., Tovey, C., Trick, M.: The computational difficulty of manipulating an election. Soc. Choice Welf. 6(3), 227–241 (1989)

    MathSciNet  CrossRef  Google Scholar 

  2. Bartholdi III, J., Tovey, C., Trick, M.: How hard is it to control an election? Math. Comput. Model. 16(8/9), 27–40 (1992)

    MathSciNet  CrossRef  Google Scholar 

  3. Baumeister, D., Faliszewski, P., Lang, J., Rothe, J.: Campaigns for lazy voters: truncated ballots. In: Proceedings of the 11th International Conference on Autonomous Agents and Multiagent Systems, pp. 577–584, June 2012

    Google Scholar 

  4. Black, D.: On the rationale of group decision-making. J. Polit. Econ. 56(1), 23–34 (1948)

    CrossRef  Google Scholar 

  5. Brandt, F., Harrenstein, P., Kardel, K., Seedig, H.: It only takes a few: on the hardness of voting with a constant number of agents. In: Proceedings of the 12th International Conference on Autonomous Agents and Multiagent Systems, pp. 375–382, May 2013

    Google Scholar 

  6. Conitzer, V., Sandholm, T., Lang, J.: When are elections with few candidates hard to manipulate? J. ACM 54(3), 1–33 (2007). Article 14

    MathSciNet  CrossRef  Google Scholar 

  7. Copeland, A.: A “reasonable” social welfare function. Mimeographed notes from a Seminar on Applications of Mathematics to the Social Sciences, University of Michigan (1951)

    Google Scholar 

  8. Emerson, P.: The original Borda count and partial voting. Soc. Choice Welf. 40(2), 352–358 (2013)

    MathSciNet  CrossRef  Google Scholar 

  9. Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.: How hard is bribery in elections? J. Artif. Intell. Res. 35, 485–532 (2009)

    MathSciNet  CrossRef  Google Scholar 

  10. Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.: Weighted electoral control. In: Proceedings of the 12th International Conference on Autonomous Agents and Multiagent Systems, pp. 367–374, May 2013

    Google Scholar 

  11. Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: Llull and Copeland voting computationally resist bribery and constructive control. J. Artif. Intell. Res. 35, 275–341 (2009)

    MathSciNet  CrossRef  Google Scholar 

  12. 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. Inf. Comput. 209(2), 89–107 (2011)

    MathSciNet  CrossRef  Google Scholar 

  13. Faliszewski, P., Hemaspaandra, E., Schnoor, H.: Copeland voting: ties matter. In: Proceedings of the 7th International Conference on Autonomous Agents and Multiagent Systems, pp. 983–990, May 2008

    Google Scholar 

  14. Faliszewski, P., Hemaspaandra, E., Schnoor, H.: Manipulation of Copeland elections. In: Proceedings of the 9th International Conference on Autonomous Agents and Multiagent Systems, pp. 367–374, May 2010

    Google Scholar 

  15. Faliszewski, P., Hemaspaandra, E., Schnoor, H.: Weighted manipulation for four-candidate Llull is easy. In: Proceedings of the 20th European Conference on Artificial Intelligence, pp. 318–323, August 2012

    Google Scholar 

  16. Gibbard, A.: Manipulation of voting schemes. Econometrica 41(4), 587–601 (1973)

    MathSciNet  CrossRef  Google Scholar 

  17. Hägele, G., Pukelsheim, F.: The electoral writings of Ramon Llull. Stud. Lulliana 41(97), 3–38 (2001)

    Google Scholar 

  18. Hemaspaandra, E., Hemaspaandra, L.: Dichotomy for voting systems. J. Comput. Syst. Sci. 73(1), 73–83 (2007)

    MathSciNet  CrossRef  Google Scholar 

  19. Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: Anyone but him: the complexity of precluding an alternative. Artif. Intell. 171(5–6), 255–285 (2007)

    MathSciNet  CrossRef  Google Scholar 

  20. Kemeny, J.: Mathematics without numbers. Daedalus 88, 577–591 (1959)

    Google Scholar 

  21. Konczak, K., Lang, J.: Voting procedures with incomplete preferences. In: Proceedings of the IJCAI-05 Multidisciplinary Workshop on Advances in Preference Handling, pp. 124–129, July/August 2005

    Google Scholar 

  22. Lackner, M.: Incomplete preferences in single-peaked electorates. In: Proceedings of the 28th AAAI Conference on Artificial Intelligence, pp. 742–748, July 2014

    Google Scholar 

  23. Mattei, N., Walsh, T.: PrefLib: a library for preferences. In: Perny, P., Pirlot, M., Tsoukiàs, A. (eds.) ADT 2013. LNCS, vol. 8176, pp. 259–270. Springer, Heidelberg (2013)

    CrossRef  Google Scholar 

  24. Menon, V., Larson, K.: Complexity of manipulation in elections with partial votes. Technical report. arXiv:1505.05900 [cs.GT], arXiv.org, May 2015

  25. Narodytska, N., Walsh, T.: The computational impact of partial votes on strategic voting. In: Proceedings of the 21st European Conference on Artificial Intelligence, pp. 657–662, August 2014

    Google Scholar 

  26. Satterthwaite, M.: Strategy-proofness and arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. J. Econ. Theor. 10(2), 187–217 (1975)

    MathSciNet  CrossRef  Google Scholar 

  27. Schulze, M.: A new monotonic and clone-independent, reversal symmetric, and condorcet-consistent single-winner election method. Soc. Choice Welf. 36(2), 267–303 (2011)

    MathSciNet  CrossRef  Google Scholar 

  28. Walsh, T.: Uncertainty in preference elicitation and aggregation. In: Proceedings of the 22nd AAAI Conference on Artificial Intelligence, pp. 3–8, July 2007

    Google Scholar 

Download references

Acknowledgments

The authors thank Aditi Bhatt, Kimaya Kamat, Matthew Le, David Narváez, Amol Patil, Ashpak Shaikh, and the anonymous referees for their helpful comments. This work was supported in part by NSF grant no. CCF-1101452 and a National Science Foundation Graduate Research Fellowship under NSF grant no. DGE-1102937.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Zack Fitzsimmons .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and Permissions

Copyright information

© 2015 Springer International Publishing Switzerland

About this paper

Cite this paper

Fitzsimmons, Z., Hemaspaandra, E. (2015). Complexity of Manipulative Actions When Voting with Ties. In: Walsh, T. (eds) Algorithmic Decision Theory. ADT 2015. Lecture Notes in Computer Science(), vol 9346. Springer, Cham. https://doi.org/10.1007/978-3-319-23114-3_7

Download citation

  • DOI: https://doi.org/10.1007/978-3-319-23114-3_7

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-23113-6

  • Online ISBN: 978-3-319-23114-3

  • eBook Packages: Computer ScienceComputer Science (R0)