The geometry of manipulation — A quantitative proof of the Gibbard-Satterthwaite theorem

Abstract

We prove a quantitative version of the Gibbard-Satterthwaite theorem. We show that a uniformly chosen voter profile for a neutral social choice function f of q ≥ 4 alternatives and n voters will be manipulable with probability at least 10−42 n −3 q −30, where is the minimal statistical distance between f and the family of dictator functions.

Our results extend those of [11], which were obtained for the case of 3 alternatives, and imply that the approach of masking manipulations behind computational hardness (as considered in [4,6,9,15,7]) cannot hide manipulations completely.

Our proof is geometric. More specifically it extends the method of canonical paths to show that the measure of the profiles that lie on the interface of 3 or more outcomes is large. To the best of our knowledge our result is the first isoperimetric result to establish interface of more than two bodies.

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

References

  1. [1]

    D. Aldous: Random walks on finite groups and rapidly mixing Markov chains, in: Seminar on probability, XVII, volume 986 of Lecture Notes in Math., 243–297. Springer, Berlin, 1983.

    Google Scholar 

  2. [2]

    K. Arrow: A difficulty in the theory of social welfare, J. of Political Economy 58 (1950), 328–346.

    Article  Google Scholar 

  3. [3]

    K. Arrow: Social choice and individual values. John Wiley and Sons, 1963.

  4. [4]

    J. Bartholdi, III and J. Orline: Single transferrable vote resists strategic voting, Soc. Choice Welf. 8(4) (1991), 341–354.

    Google Scholar 

  5. [5]

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

    MathSciNet  Article  Google Scholar 

  6. [6]

    V. Coniftzer and T. Sandholm: Universal voting protocol tweaks to make manipulation hard, in: Georg Gottlob and Toby Walsh, editors, IJCAI-03, Proceedings of the Eighteenth International Joint Conference on Artificial Intelligence, Acapulco, Mexico, August 9–15, 2003, 781–788. Morgan Kaufmann, 2003.

  7. [7]

    V. Conitzer and T. Sandholm: Nonexistence of voting rules that are usually hard to manipulate, in: AAAI, AAAI Press, 2006.

  8. [8]

    S. Dobzinski and A. D. Procaccia: Frequent manipulability of elections: The case of two voters, in: Christos H. Papadimitriou and Shuzhong Zhang, editors, Internet and Network Economics, 4th International Workshop, WINE 2008, Shanghai, China, December 17–20, 2008. Proceedings, volume 5385 of Lecture Notes in Computer Science, 653–664. Springer, 2008.

  9. [9]

    E. Elkind and H. Lipmaa: Hybrid voting protocols and hardness of manipulation, in: Xiaotie Deng and Ding-Zhu Du, editors, Algorithms and Computation, 16th International Symposium, ISAAC 2005, Sanya, Hainan, China, December 19–21, 2005, Proceedings, volume 3827 of Lecture Notes in Computer Science, 206–215. Springer, 2005.

  10. [10]

    P. Faliszewski and A. D. Procaccia: Ai’s war on manipulation: Are we winning? AI Magazine special issue on algorithmic game theory, to appear, 2010.

  11. [11]

    E. Friedgut, G. Kalai and N. Nisan: Elections can be manipulated often, in: Proceedings of the 49th Annual IEEE Symposium on Foundations of Computer Science (FOCS), 243–249, 2009.

  12. [12]

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

    MathSciNet  MATH  Article  Google Scholar 

  13. [13]

    M. Jerrum and A. Sinclair: Polynomial-time approximation algorithms for ising model (extended abstract), in: Automata, Languages and Programming, 462–475, 1990.

  14. [14]

    J. S. Kelly: Almost all social choice rules are highly manipulable, but a few aren’t, Social Choice and Welfare 10 (1993).

  15. [15]

    A. D. Procaccia and J. S. Rosenschein: Junta distributions and the average-case complexity of manipulating elections, in: Hideyuki Nakashima, Michael P. Wellman, Gerhard Weiss, and Peter Stone, editors, 5th International Joint Conference on Autonomous Agents and Multiagent Systems (AAMAS 2006), Hakodate, Japan, May 8–12, 2006, 497–504. ACM, 2006.

  16. [16]

    M. A. Satterthwaite: Strategy-proofness and Arrow’s Conditions: Existence and Correspondence Theorems for Voting Procedures and Social Welfare Functions, J. of Economic Theory 10 (1975), 187–217.

    MathSciNet  MATH  Article  Google Scholar 

  17. [17]

    D. B. Wilson: Mixing times of lozenge tiling and card shuffling markov chains, Ann. Appl. Probab. 14(1) (2004).

  18. [18]

    L. Xia and V. Conitzer: A sufficient condition for voting rules to be frequently manipulable, in: Lance Fortnow, John Riedl, and Tuomas Sandholm, editors, Proceedings 9th ACM Conference on Electronic Commerce (EC-2008), Chicago, IL, USA, June 8–12, 2008, 99–108. ACM, 2008.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Marcus Isaksson.

Additional information

Supported by the Israel Science Foundation and by the Binational Science Foundation.

Supported by DMS 0548249 (CAREER) award, by ISF grant 1300/08, by a Minerva Foundation grant and by an ERC Marie Curie Grant 2008 239317.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Isaksson, M., Kindler, G. & Mossel, E. The geometry of manipulation — A quantitative proof of the Gibbard-Satterthwaite theorem. Combinatorica 32, 221–250 (2012). https://doi.org/10.1007/s00493-012-2704-1

Download citation

Mathematics Subject Classification (2000)

  • 05A05