Theory of Computing Systems

, Volume 53, Issue 4, pp 507–531 | Cite as

Normalized Range Voting Broadly Resists Control

  • Curtis Menton


We study the behavior of Range Voting and Normalized Range Voting with respect to electoral control. Electoral control encompasses attempts from an election chair to alter the participation or structure of an election in order to change the outcome. We show that a voting system resists a case of control by proving that performing that case of control is computationally hard. Range Voting is a natural extension of approval voting, and Normalized Range Voting is a simple variant which alters each vote to maximize the potential impact of each voter. We show that Normalized Range Voting has among the largest known number of control resistances among natural voting systems.


Computational social choice Electoral control Computational complexity Range voting 



For helpful comments and suggestions, I am grateful to Edith Hemaspaandra, who advised my M.S. thesis in which an earlier version of part of this work appeared, Lane A. Hemaspaandra, Preetjot Singh, Andrew Lin, and the anonymous ToCS referees.


  1. 1.
    Arrow, K.: A difficulty in the concept of social welfare. J. Polit. Econ. 58(4), 328–346 (1950) CrossRefGoogle Scholar
  2. 2.
    Baumeister, D., Erdélyi, G., Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: Computational aspects of approval voting. In: Laslier, J., Sanver, R. (eds.) Handbook of Approval Voting, pp. 199–251. Springer, Berlin (2010) CrossRefGoogle Scholar
  3. 3.
    Brelsford, E., Faliszewski, P., Hemaspaandra, E., Schnoor, H., Schnoor, I.: Approximability of manipulating elections. In: Proceedings of the 23rd AAAI Conference on Artificial Intelligence, July 2008, pp. 44–49 (2008) Google Scholar
  4. 4.
    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, pp. 55–60 (2011) Google Scholar
  5. 5.
    Bartholdi, J. III, Tovey, C., Trick, M.: The computational difficulty of manipulating an election. Soc. Choice Welf. 6(3), 227–241 (1989) MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Bartholdi, J. III, Tovey, C., Trick, M.: How hard is it to control an election? Math. Comput. Model. 16(8/9), 27–40 (1992) MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Betzler, N., Uhlmann, J.: Parameterized complexity of candidate control in elections and related digraph problems. In: Proceedings of the 2nd Annual International Conference on Combinatorial Optimization and Applications. Lecture Notes in Computer Science, vol. 5156, pp. 43–53. Springer, Berlin (2008) CrossRefGoogle Scholar
  8. 8.
    Conitzer, V., Sandholm, T.: Complexity of manipulating elections with few candidates. In: Proceedings of the 18th National Conference on Artificial Intelligence, July/August 2002, pp. 314–319 (2002) Google Scholar
  9. 9.
    Conitzer, V., Sandholm, T.: Nonexistence of voting rules that are usually hard to manipulate. In: Proceedings of the 21st National Conference on Artificial Intelligence, pp. 627–634. AAAI Press, Menlo Park (2006) Google Scholar
  10. 10.
    Davies, J., Katsirelos, G., Narodytska, N., Walsh, T.: Complexity of and algorithms for Borda manipulation. In: Proceedings of the 25th AAAI Conference on Artificial Intelligence, August (2011) Google Scholar
  11. 11.
    Duggan, J., Schwartz, T.: Strategic manipulability without resoluteness or shared beliefs: Gibbard–Satterthwaite generalized. Soc. Choice Welf. 17(1), 85–93 (2000) MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Erdélyi, G., Fellows, M.: Parameterized control complexity in bucklin voting and in fallback voting. In: Proceedings of the 3rd International Workshop on Computational Social Choice, pp. 163–174 (2010) Google Scholar
  13. 13.
    Erdélyi, G., Hemaspaandra, L., Rothe, J., Spakowski, H.: Generalized juntas and NP-hard sets. Theor. Comput. Sci. 410(38–40), 3995–4000 (2009) zbMATHCrossRefGoogle Scholar
  14. 14.
    Erdélyi, G., Nowak, M., Rothe, J.: Sincere-strategy preference-based approval voting fully resists constructive control and broadly resists destructive control. Math. Log. Q. 55(4), 425–443 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Erdélyi, G., Piras, L., Rothe, J.: The complexity of voter partition in Bucklin and fallback voting: solving three open problems. In: Proceedings of the 10th International Conference on Autonomous Agents and Multiagent Systems, pp. 837–844 (2011) Google Scholar
  16. 16.
    Erdélyi, G., Rothe, J.: Control complexity in fallback voting. In: Proceedings of Computing: The 16th Australasian Theory Symposium, pp. 39–48 (2010) Google Scholar
  17. 17.
    Faliszewski, P., Hemaspaandra, E., Hemaspaandra, L.: The complexity of bribery in elections. J. Artif. Intell. Res. 35, 485–532 (2009) MathSciNetzbMATHGoogle Scholar
  18. 18.
    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) MathSciNetzbMATHGoogle Scholar
  19. 19.
    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) MathSciNetzbMATHCrossRefGoogle Scholar
  20. 20.
    Friedgut, E., Kalai, G., Nisan, N.: Elections can be manipulated often. In: Proceedings of the 49th IEEE Symposium on Foundations of Computer Science, October 2008, pp. 243–249 (2008) Google Scholar
  21. 21.
    Gibbard, A.: Manipulation of voting schemes. Econometrica 41(4), 587–601 (1973) MathSciNetzbMATHCrossRefGoogle Scholar
  22. 22.
    Garey, M., Johnson, D.: Computers and Intractability: A Guide to the Theory of NP-Completeness. Freeman, New York (1979) zbMATHGoogle Scholar
  23. 23.
    Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: Anyone but him: The complexity of precluding an alternative. Artif. Intell. 171(5–6), 255–285 (2007) MathSciNetzbMATHCrossRefGoogle Scholar
  24. 24.
    Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: Hybrid elections broaden complexity-theoretic resistance to control. Math. Log. Q. 55(4), 397–424 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  25. 25.
    Hillinger, C.: The case for utilitarian voting. Discussion Papers in Economics 653, University of Munich, Department of Economics, May (2005) Google Scholar
  26. 26.
    Isaksson, M., Kindler, G., Mossel, E.: The geometry of manipulation—a quantitative proof of the Gibbard-Satterthwaite theorem. Combinatorica 32(2), 221–250 (2012) MathSciNetzbMATHCrossRefGoogle Scholar
  27. 27.
    Liu, H., Feng, H., Zhu, D., Luan, J.: Parameterized computational complexity of control problems in voting systems. Theor. Comput. Sci. 410(27–29), 2746–2753 (2009) MathSciNetzbMATHCrossRefGoogle Scholar
  28. 28.
    Liu, H., Zhu, D.: Parameterized complexity of control problems in maximin election. Inf. Process. Lett. 110(10), 383–388 (2010) zbMATHCrossRefGoogle Scholar
  29. 29.
    Mossel, E., Procaccia, A., Rácz, M.: A smooth transition from powerlessness to absolute power. Technical Report (2012). arXiv:1205.2074 [cs.GT]
  30. 30.
    Mossel, E., Rácz, M.: A quantitative Gibbard-Satterthwaite theorem without neutrality. In: Proceedings of the 44th ACM Symposium on Theory of Computing, May 2012, pp. 1041–1060 (2012) Google Scholar
  31. 31.
    Procaccia, A., Rosenschein, J.: Average-case tractability of manipulation in voting via the fraction of manipulators. In: Proceedings of the 6th International Joint Conference on Autonomous Agents and Multiagent Systems, pp. 718–720 (2007) Google Scholar
  32. 32.
    Procaccia, A., Rosenschein, J.: Junta distributions and the average-case complexity of manipulating elections. J. Artif. Intell. Res. 28, 157–181 (2007) MathSciNetzbMATHGoogle Scholar
  33. 33.
    Parkes, D., Xia, L.: A complexity-of-strategic-behavior comparison between Schulze’s rule and ranked pairs. In: Proceedings of the 26th AAAI Conference on Artificial Intelligence, pp. 1429–1435 (2012) Google Scholar
  34. 34.
    Satterthwaite, M.: Strategy-proofness and Arrow’s conditions: existence and correspondence theorems for voting procedures and social welfare functions. J. Econ. Theory 10(2), 187–217 (1975) MathSciNetzbMATHCrossRefGoogle Scholar
  35. 35.
  36. 36.
    Walsh, T.: Where are the hard manipulation problems? J. Artif. Intell. Res. 42, 1–29 (2011) zbMATHGoogle Scholar
  37. 37.
    Xia, L., Conitzer, V.: Generalized scoring rules and the frequency of coalitional manipulability. In: Proceedings of the 9th ACM Conference on Electronic Commerce, July 2008, pp. 109–118 (2008) Google Scholar
  38. 38.
    Xia, L.: How many vote operations are needed to manipulate a voting system? In: Proceedings of the 4th International Workshop on Computational Social Choice, September (2012) Google Scholar
  39. 39.
    Zuckerman, M., Procaccia, A., Rosenschein, J.: Algorithms for the coalitional manipulation problem. Artif. Intell. 173(2), 392–412 (2009) MathSciNetzbMATHCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  1. 1.Department of Computer ScienceUniversity of RochesterRochesterUSA

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