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Winner Determination and Manipulation in Minisum and Minimax Committee Elections

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Algorithmic Decision Theory (ADT 2015)

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

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Abstract

In a committee election, a set of candidates has to be determined as winner of the election. Baumeister and Dennisen [2] proposed to extend the minisum and minimax approach, initially defined for approval votes, to other forms of votes. They define minisum and minimax committee election rules for trichotomous votes, incomplete linear orders and complete linear orders, by choosing a winning committee that minimizes the dissatisfaction of the voters. Minisum election rules minimize the voter dissatisfaction by choosing a winning committee with minimum sum of the disagreement values for all individual votes, whereas in a minimax winning committee the maximum disagreement value for an individual vote is minimized. In this paper, we investigate the computational complexity of winner determination in these voting rules. We show that winner determination is possible in polynomial time for all minisum rules we consider, whereas it is \(\mathrm{NP}\)-complete for three of the minimax rules. Furthermore, we study different forms of manipulation for these committee election rules.

The first author is supported in part by a grant for gender-sensitive universities. The middle author is supported by the Deutsche Forschungsgemeinschaft under Grant GRK 1931 – SocialCars: Cooperative (De-)Centralized Traffic Management.

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Notes

  1. 1.

    In an m-candidate Borda election (see [5]) each candidate gets points according to her position in the votes, where a first position gives \(m-1\) points, a second \(m-2\), and so on.

  2. 2.

    Note, that this problem does not equal coalitional manipulation which is proved to be \(\mathrm{NP}\)-hard even for two manipulators and three voters by Betzler et al. [3].

References

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

    Article  MathSciNet  Google Scholar 

  2. Baumeister, D., Dennisen, S.: Voter dissatisfaction in committee elections. In: Proceedings of the 14th International Joint Conference on Autonomous Agents and Multiagent Systems, IFAAMAS. Extended abstract (2015)

    Google Scholar 

  3. Betzler, N., Niedermeier, R., Woeginger, G.: Unweighted coalitional manipulation under the borda rule is NP-hard. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence, IJCAI, pp. 55–60 (2011)

    Google Scholar 

  4. Betzler, N., Slinko, A., Uhlmann, J.: On the computation of fully proportional representation. J. Artif. Intell. Res. 47, 475–519 (2013)

    Article  MathSciNet  Google Scholar 

  5. Borda, J.: Mémoire sur les élections au scrutin. Histoire de L’Académie Royale des Sciences, Paris. English translation appears in [19] (1781)

    Google Scholar 

  6. Boutilier, C., Rosenschein, J.: Incomplete information and communication in voting. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A. (eds.) Handbook of Computational Social Choice, Chapter 10. Cambridge University Press, Cambridge (2015)

    Google Scholar 

  7. Brams, S., Kilgour, D., Sanver, R.: A minimax procedure for negotiating multilateral treaties. In: Wiberg, M. (ed.) Reasoned Choices: Essays in Honor of Hannu Nurmi. Finnish Political Science Association, Turku (2004)

    Google Scholar 

  8. Brams, S., Kilgour, D., Sanver, R.: A minimax procedure for electing committees. Public Choice 132, 401–420 (2007)

    Article  Google Scholar 

  9. Brams, S., Kilgour, D., Sanver, R.: A minimax procedure for negotiating multilateral treaties. In: Avenhus, R., Zartmann, I. (eds.) Diplomacy Games: Formal Models and International Negotiations. Springer, Heidelberg (2007)

    Google Scholar 

  10. Byrka, J., Sornat, K.: PTAS for minimax approval voting. In: Liu, T.-Y., Qi, Q., Ye, Y. (eds.) WINE 2014. LNCS, vol. 8877, pp. 203–217. Springer, Heidelberg (2014)

    Chapter  Google Scholar 

  11. Caragiannis, I., Hemaspaandra, E., Hemaspaandra, L.: Dodgson’s rule and Young’s rule. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A. (eds.) Handbook of Computational Social Choice, Chapter 5. Cambridge University Press, Cambridge (2015)

    Google Scholar 

  12. Caragiannis, I., Kalaitzis, D., Markakis, E.: Approximation algorithms and mechanism design for minimax approval voting. In: Proceedings of the 24th AAAI Conference on Artificial Intelligence, pp. 737–742, AAAI Press (2010)

    Google Scholar 

  13. Chamberlin, J., Courant, P.: Representative deliberations and representative decisions: proportional representation and the Borda rule. Am. Polit. Sci. Rev. 77(3), 718–733 (1983)

    Article  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  15. Conitzer, V., Walsh, T.: Barriers to manipulation in voting. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A. (eds.) Handbook of Computational Social Choice, Chapter 6. Cambridge University Press, Cambridge (2015)

    Google Scholar 

  16. Faliszewski, P., Rothe, J.: Control and bribery in voting. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A. (eds.) Handbook of Computational Social Choice, Chapter 7. Cambridge University Press, Cambridge (2015)

    Google Scholar 

  17. Felsenthal, D.: On combining approval with disapproval voting. Behav. Sci. 34(1), 53–60 (1989)

    Article  MathSciNet  Google Scholar 

  18. Gibbard, A.: Manipulation of voting schemes: a general result. Econometrica 41(4), 587–601 (1973)

    Article  MathSciNet  Google Scholar 

  19. Grazia, A.: Mathematical deviation of an election system. Isis 44, 41–51 (1953)

    Article  Google Scholar 

  20. Bartholdi III, J., Tovey, C., Trick, M.: Voting schemes for which it can be difficult to tell who won the election. Soc. Choice Welf. 6(2), 157–165 (1989)

    Article  MathSciNet  Google Scholar 

  21. Kilgour, D., Brams, S., Sanver, R.: How to elect a representative committee using approval balloting. In: Pukelsheim, F., Simeone, B. (eds.) Mathematics and Democracy: Recent Advances in Voting Systems and Collective Choice, pp. 83–95. Springer, Heidelberg (2006)

    Chapter  Google Scholar 

  22. Lang, J., Xia, L.: Voting in combinatorial domains. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A. (eds.) Handbook of Computational Social Choice, Chapter 9. Cambridge University Press, Cambridge (2015)

    Google Scholar 

  23. LeGrand, R.: Analysis of the minimax procedure. Technical report WUCSE-2004-67, Department of Computer Science and Engineering, Washington University, St. Louis, Missouri, November 2004

    Google Scholar 

  24. LeGrand, R., Markakis, E., Mehta, A.: Some results on approximating the minimax solution in approval voting. In: Proceedings of the 6th International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 1193–1195, ACM Press, May 2007

    Google Scholar 

  25. 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, AAAI Press (2011)

    Google Scholar 

  26. Meir, R., Procaccia, A., Rosenschein, J., Zohar, A.: Complexity of strategic behavior in multi-winner elections. J. Artif. Intell. Res. 33, 149–178 (2008)

    Article  MathSciNet  Google Scholar 

  27. Monroe, B.: Fully proportional representation. Am. Polit. Sci. Rev. 89(4), 925–940 (1995)

    Article  Google Scholar 

  28. Obraztsova, S., Elkind, E., Hazon, N.: Ties matter: complexity of voting manipulation revisited. In: Proceedings of the 10th International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 71–78, ACM Press (2011)

    Google Scholar 

  29. Obraztsova, S., Zick, Y., Elkind, E.: On manipulation in multiwinner elections based on scoring rules. In: Proceedings of the 12th International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 359–366, IFAAMAS (2013)

    Google Scholar 

  30. Papadimitriou, C.: Computational Complexity, 2nd edn. Addison-Wesley, Reading (1995). Reprinted with corrections

    MATH  Google Scholar 

  31. Procaccia, A., Rosenschein, J., Zohar, A.: Multi-winner elections: complexity of manipulation, control, and winner-determination. In: Proceedings of the 20th International Joint Conference on Artificial Intelligence, pp. 1476–1481, IJCAI (2007)

    Google Scholar 

  32. Procaccia, A., Rosenschein, J., Zohar, A.: On the complexity of achieving proportional representation. Soc. Choice Welf. 30, 353–362 (2008)

    Article  MathSciNet  Google Scholar 

  33. 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)

    Article  MathSciNet  Google Scholar 

  34. Skowron, P., Yu, L., Faliszewski, P., Elkind, E.: The complexity of fully proportional representation for single-crossing electorates. In: Vöcking, B. (ed.) SAGT 2013. LNCS, vol. 8146, pp. 1–12. Springer, Heidelberg (2013)

    Chapter  Google Scholar 

  35. Wilcoxon, F.: Individual comparisons by ranking methods. Biom. Bull. 1(6), 80–83 (1945)

    Article  Google Scholar 

  36. Zwicker, W.: Introduction to the theory of voting. In: Brandt, F., Conitzer, V., Endriss, U., Lang, J., Procaccia, A. (eds.) Handbook of Computational Social Choice, Chapter 2. Cambridge University Press, Cambridge (2015)

    Google Scholar 

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Author information

Authors and Affiliations

  1. Institut für Informatik, Heinrich-Heine-Universität Düsseldorf, 40225, Düsseldorf, Germany

    Dorothea Baumeister & Lisa Rey

  2. Institut für Informatik, Technische Universität Clausthal, 38678, Clausthal-Zellerfeld, Germany

    Sophie Dennisen

Authors
  1. Dorothea Baumeister
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  2. Sophie Dennisen
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  3. Lisa Rey
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Correspondence to Lisa Rey .

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Editors and Affiliations

  1. University of New South Wales, Kensington, Australia

    Toby Walsh

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Baumeister, D., Dennisen, S., Rey, L. (2015). Winner Determination and Manipulation in Minisum and Minimax Committee Elections. 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_28

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  • DOI: https://doi.org/10.1007/978-3-319-23114-3_28

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