Advertisement

Combinatorica

, Volume 35, Issue 3, pp 317–387 | Cite as

A quantitative Gibbard-Satterthwaite theorem without neutrality

  • Elchanan Mossel
  • Miklós Z. Rácz
Original Paper
  • 102 Downloads

Abstract

Recently, quantitative versions of the Gibbard-Satterthwaite theorem were proven for k=3 alternatives by Friedgut, Kalai, Keller and Nisan and for neutral functions on k ≥ 4 alternatives by Isaksson, Kindler and Mossel.

We prove a quantitative version of the Gibbard-Satterthwaite theorem for general social choice functions for any number k ≥ 3 of alternatives. In particular we show that for a social choice function f on k ≥ 3 alternatives and n voters, which is ε-far from the family of nonmanipulable functions, a uniformly chosen voter profile is manipulable with probability at least inverse polynomial in n, k, and ε −1.

Ours is a unified proof which in particular covers all previous cases established before. The proof crucially uses reverse hypercontractivity in addition to several ideas from the two previous proofs. Much of the work is devoted to understanding functions of a single voter, and in particular we also prove a quantitative Gibbard-Satterthwaite theorem for one voter.

Mathematics Subject Classification (2000)

05A05 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    K. Arrow: A Difficulty in the Concept of Social Welfare, Journal of Political Economy 58 (1950), 328–346.CrossRefzbMATHGoogle Scholar
  2. [2]
    K. Arrow: Social Choice and Individual Values, Yale University Press, 2nd edition, 19–3.Google Scholar
  3. [3]
    J. J. Bartholdi, C. A. Tovey and M. A. Trick: The Computational Difficulty of Manipulating an Election, Social Choice and Welfare 6 (1989), 227–241.MathSciNetCrossRefzbMATHGoogle Scholar
  4. [4]
    C. Borell: Positivity improving operators and hypercontractivity, Mathematische Zeitschrift 180 (1982), 225–234.MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    V. Conitzer and T. Sandholm: Nonexistence of Voting Rules That Are Usually Hard to Manipulate, in: Proceedings of the 21st National Conference on Artificial Intelligence, volume 21, 627–634, 2006.Google Scholar
  6. [6]
    S. Dobzinski and A. D. Procaccia: Frequent Manipulability of Elections: The Case of Two Voters, in: Proceedings of the 4th International Workshop on Internet and Network Economics, 653–664. Springer, 2008.CrossRefGoogle Scholar
  7. [7]
    P. Faliszewski and A. D. Procaccia: AI’sWar on Manipulation: AreWe Winning? AI Magazine 31 (2010), 53–64.Google Scholar
  8. [8]
    E. Friedgut, G. Kalai, N. Keller and N. Nisan: A Quantitative Version of the Gibbard-Satterthwaite Theorem for Three Alternatives, SIAM J. Comput. 40 (2011), 934–952.MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    E. Friedgut, G. Kalai and N. Nisan: Elections can be manipulated often, in: Proceedings of the 49th Annual Symposium on Foundations of Computer Science, 243–249. IEEE, 2008.Google Scholar
  10. [10]
    A. Gibbard: Manipulation of Voting Schemes: A General Result, Econometrica: Journal of the Econometric Society, 587–601, 1993.Google Scholar
  11. [11]
    O. Goldreich, S. Goldwasser, E. Lehman, D. Ron and A. Samorodnitsky: Testing Monotonicity, Combinatorica 20 (2000), 301–337.MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    L. H. Harper: Global Methods for Combinatorial Isoperimetric Problems, Cambridge University Press, 2004.CrossRefzbMATHGoogle Scholar
  13. [13]
    M. Isaksson, G. Kindler and E. Mossel: The Geometry of Manipulation: A Quantitative Proof of the Gibbard-Satterthwaite Theorem, Combinatorica 32 (2012), 221–250.MathSciNetCrossRefzbMATHGoogle Scholar
  14. [14]
    J. S. Kelly: Almost all social choice rules are highly manipulable, but a few aren’t, Social Choice and Welfare 10 (1993), 161–175.MathSciNetCrossRefGoogle Scholar
  15. [15]
    J. H. Lindsey: Assignment of Numbers to Vertices, American Mathematical Monthly, 508–516, 1994.Google Scholar
  16. [16]
    E. Mossel: A quantitative Arrow theorem, Probability Theory and Related Fields 154 (2012), 49–88.MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    E. Mossel, R. O’Donnell, O. Regev, J. E. Steif and B. Sudakov: Noninteractive correlation distillation, inhomogeneous Markov chains, and the reverse Bonami-Beckner inequality, Israel Journal of Mathematics 154 (2006), 299–336.MathSciNetCrossRefGoogle Scholar
  18. [18]
    E. Mossel and M. Z. Rácz: A quantitative Gibbard-Satterthwaite theorem without neutrality, in: Proceedings of the 44th ACM Symposium on Theory of Computing (STOC), 1041–1060. ACM, 2002.Google Scholar
  19. [19]
    H. Moulin: The Strategy of Social Choice, North-Holland, 1993.Google Scholar
  20. [20]
    A. D. Procaccia and J. S. Rosenschein: Junta Distributions and the Average-case Complexity of Manipulating Elections, Journal of Artificial Intelligence Research, 28 (2007), 157–181.MathSciNetzbMATHGoogle Scholar
  21. [21]
    M. A. Satterthwaite: Strategy-proofness and Arrow’s Conditions: Existence and Correspondence Theorems for Voting Procedures and SocialWelfare Functions, Journal of Economic Theory 10 (1975), 187–217.MathSciNetCrossRefzbMATHGoogle Scholar
  22. [22]
    L. Xia and V. Conitzer: A Sufficient Condition for Voting Rules to be Frequently Manipulable, in: Proceedings of the 9th ACM Conference on Electronic Commerce, 99–108. ACM, 2008.Google Scholar

Copyright information

© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  1. 1.University of PennsylvaniaPhiladelphiaUSA
  2. 2.University of CaliforniaBerkeleyUSA

Personalised recommendations