Triadic Consensus

A Randomized Algorithm for Voting in a Crowd
  • Ashish Goel
  • David Lee
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7695)


Typical voting rules do not work well in settings with many candidates. If there are even several hundred candidates, then a simple task such as evaluating and choosing a top candidate becomes impractical. Motivated by the hope of developing group consensus mechanisms over the internet, where the numbers of candidates could easily number in the thousands, we study an urn-based voting rule where each participant acts as a voter and a candidate. We prove that when participants lie in a one-dimensional space, this voting protocol finds a \((1-\epsilon/\sqrt{n})\) approximation of the Condorcet winner with high probability while only requiring an expected \(O(\frac{1}{\epsilon^2}\log^2 \frac{n}{\epsilon^2})\) comparisons on average per voter. Moreover, this voting protocol is shown to have a quasi-truthful Nash equilibrium: namely, a Nash equilibrium exists which may not be truthful, but produces a winner with the same probability distribution as that of the truthful strategy.


Nash Equilibrium Communication Complexity Vote Rule Condorcet Winner Concave Utility Function 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Goel, A., Lee, D.: Triadic consensus: A randomized algorithm for voting in a crowd. arXiv:1210.0664 [cs.GT] (2012)Google Scholar
  2. 2.
    Brandt, F., Conitzer, V., Endriss, U.: Computational social choice. In: Multiagent Systems. MIT Press (2012)Google Scholar
  3. 3.
    Young, H.P.: Condorcet’s theory of voting. The American Political Science Review 82(4), 1231–1244 (1988)CrossRefGoogle Scholar
  4. 4.
    Arrow, K.: A difficulty in the concept of social welfare. Journal of Political Economy 58(4), 328–346 (1950)CrossRefGoogle Scholar
  5. 5.
    Pattanaik, P., Peleg, B.: Distribution of power under stochastic social choice rules. Econometrica 54(4), 909–921 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  6. 6.
    Gibbard, A.: Manipulation of voting schemes: A general result. Econometrica 41(4), 587–601 (1973)MathSciNetzbMATHCrossRefGoogle Scholar
  7. 7.
    Satterthwaite, M.: Strategy-proofness and arrow’s conditions: Existence and correspondence theorems for voting procedures and social welfare functions. Journal of Economic Theory 10, 187–217 (1975)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Gibbard, A.: Manipulation of schemes that mix voting with chance. Econometrica 45(3), 665–681 (1977)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Bartholdi, J., Tovey, C., Trick, M.: The computational difficulty of manipulating an election. Social Choice and Welfare 6, 227–241 (1989)MathSciNetzbMATHCrossRefGoogle Scholar
  10. 10.
    Faliszewski, P., Procaccia, A.: Ai’s war on manipulation: Are we winning? AI Magazine 31(4), 53–64 (2010)Google Scholar
  11. 11.
    Conitzer, V., Sandholm, T.: Nonexistence of voting rules that are usually hard to manipulate. In: Proceedings of the 21st AAAI Conference, pp. 627–634 (2006)Google Scholar
  12. 12.
    Procaccia, A.: Can approximation circumvent gibbard-satterthwaite? In: Proceedings of the 24th AAAI Conference on Artificial Intelligence, pp. 836–841 (2010)Google Scholar
  13. 13.
    Birrell, E., Pass, R.: Approximately strategy-proof voting. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence (2011)Google Scholar
  14. 14.
    Conitzer, V., Sandholm, T.: Communication complexity of common voting rules. In: Proceedings of the 6th ACM Conference on Electronic Commerce (ACM-EC), pp. 78–87 (2005)Google Scholar
  15. 15.
    Conitzer, V., Sandholm, T.: Vote elicitation: Complexity and strategy-proofness. In: Proceedings of the 17th AAAI Conference (2002)Google Scholar
  16. 16.
    Lu, T., Boutilier, C.: Robust approximation and incremental elicitation in voting protocols. In: Proceedings of the 22nd International Joint Conference on Artificial Intelligence (2011)Google Scholar
  17. 17.
    Black, D.: On the rationale of group decision-making. Journal of Political Economy 56(1), 23–34 (1948)CrossRefGoogle Scholar
  18. 18.
    Moulin, H.: On strategy-proofness and single peakedness. Public Choice 35(4), 437–455 (1980)CrossRefGoogle Scholar
  19. 19.
    Escoffier, B., Lang, J., Öztürk, M.: Single-peaked consistency and its complexity. In: Proceedings of the 18th European Conference on Artificial Intelligence, ECAI 2008 (2008)Google Scholar
  20. 20.
    Conitzer, V.: Eliciting single-peaked preferences using comparison queries. Journal of Artificial Intelligence Research 35, 161–191 (2009)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Lee, D., Bruck, J.: Modeling biological circuits with urn functions. In: Proceedings of the International Symposium on Information Theory, ISIT (2012)Google Scholar
  22. 22.
    Motwani, R., Raghavan, P.: Randomized Algorithms. Cambridge University Press (1995)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Ashish Goel
    • 1
  • David Lee
    • 1
  1. 1.Stanford UniversityUSA

Personalised recommendations