Skip to main content
SpringerLink
Log in

Guarantees for the Success Frequency of an Algorithm for Finding Dodgson-Election Winners

  • Conference paper
Mathematical Foundations of Computer Science 2006 (MFCS 2006)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 4162))

Abstract

Dodgson’s election system elegantly satisfies the Condorcet criterion. However, determining the winner of a Dodgson election is known to be \({\mathrm{\Theta}^{\mathit{p}}_2}\)-complete ([1], see also [2]), which implies that unless P = NP no polynomial-time solution to this problem exists, and unless the polynomial hierarchy collapses to NP the problem is not even in NP. Nonetheless, we prove that when the number of voters is much greater than the number of candidates (although the number of voters may still be polynomial in the number of candidates), a simple greedy algorithm very frequently finds the Dodgson winners in such a way that it “knows” that it has found them, and furthermore the algorithm never incorrectly declares a nonwinner to be a winner.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 84.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. Hemaspaandra, E., Hemaspaandra, L., Rothe, J.: Exact analysis of Dodgson elections: Lewis Carroll’s 1876 voting system is complete for parallel access to NP. Journal of the ACM 44(6), 806–825 (1997)

    Article  MATH  MathSciNet  Google Scholar 

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

    Article  MATH  MathSciNet  Google Scholar 

  3. Condorcet, M.: Essai sur l’Application de L’Analyse à la Probabilité des Décisions Rendues à la Pluralité des Voix (1785); Facsimile reprint of original published in Paris, The Imprimerie Royale (1972)

    Google Scholar 

  4. McLean, I., Urken, A.: Classics of Social Choice. University of Michigan Press, Ann Arbor (1995)

    Google Scholar 

  5. Dodgson, C.: A method of taking votes on more than two issues. Clarendon Press, Oxford, pamphet (1876)

    Google Scholar 

  6. Black, D.: The Theory of Committees and Elections. Cambridge University Press, Cambridge (1958)

    MATH  Google Scholar 

  7. Nanson, E.: Methods of election. Transactions and Proceedings of the Royal Society of Victoria 19, 197–240 (1882)

    Google Scholar 

  8. Borda, J.C.d.: Mémoire sur les élections au scrutin. Histoire de L’Académie Royale des Sciences Année 1781 (1784)

    Google Scholar 

  9. Papadimitriou, C., Zachos, S.: Two remarks on the power of counting. In: Cremers, A.B., Kriegel, H.-P. (eds.) GI-TCS 1983. LNCS, vol. 145, pp. 269–276. Springer, Heidelberg (1982)

    Chapter  Google Scholar 

  10. Hemaspaandra, E., Spakowski, H., Vogel, J.: The complexity of Kemeny elections. Theoretical Computer Science 349(3), 382–391 (2005)

    MATH  MathSciNet  Google Scholar 

  11. Rothe, J., Spakowski, H., Vogel, J.: Exact complexity of the winner problem for Young elections. Theory of Computing Systems 36(4), 375–386 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  12. Cormen, T., Leiserson, C., Rivest, R., Stein, C.: Introduction to Algorithms, 2nd edn. MIT Press/McGraw Hill (2001)

    Google Scholar 

  13. Ausiello, G., Crescenzi, P., Protasi, M.: Approximate solution of NP optimization problems. Theoretical Computer Science 150(1), 1–55 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  14. Kaporis, A.C., Kirousis, L.M., Lalas, E.G.: The probabilistic analysis of a greedy satisfiability algorithm. In: Möhring, R.H., Raman, R. (eds.) ESA 2002. LNCS, vol. 2461, pp. 574–585. Springer, Heidelberg (2002)

    Chapter  Google Scholar 

  15. Chang, L., Korsh, J.: Canonical coin changing and greedy solutions. Journal of the ACM 23(3), 418–422 (1976)

    Article  MATH  MathSciNet  Google Scholar 

  16. Protasi, M., Talamo, M.: A new probabilistic model for the study of algorithmic properties of random graph problems. In: Karpinski, M. (ed.) FCT 1983. LNCS, vol. 158, pp. 360–367. Springer, Heidelberg (1983)

    Google Scholar 

  17. Slavik, P.: A tight analysis of the greedy algorithm for set cover. Journal of Algorithms 25(2), 237–254 (1997)

    Article  MATH  MathSciNet  Google Scholar 

  18. Brown, D.: A probabilistic analysis of a greedy algorithm arising from computational biology. In: Proceedings of the 12th Annual ACM-SIAM Symposium on Discrete Algorithms, pp. 206–207. ACM Press, New York (2001)

    Google Scholar 

  19. Goldberg, A.V., Marchetti-Spaccamela, A.: On finding the exact solution of a zero-one knapsack problem. In: Proceedings of the 16th ACM Symposium on Theory of Computing, pp. 359–368 (1984)

    Google Scholar 

  20. Downey, R., Fellows, M.: Parameterized complexity. Springer, Heidelberg (1999)

    Google Scholar 

  21. Levin, L.: Average case complete problems. SIAM Journal on Computing (1986)

    Google Scholar 

  22. Spielman, D., Teng, S.: Smoothed analysis: Why the simplex algorithm usually takes polynomial time. Journal of the ACM 51(3), 385–463 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hemaspaandra, E., Hemaspaandra, L.A.: Computational politics: Electoral systems. In: Nielsen, M., Rovan, B. (eds.) MFCS 2000. LNCS, vol. 1893, pp. 64–83. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  24. Spakowski, H., Vogel, J.: Θ2 p-completeness: A classical approach for new results. In: Kapoor, S., Prasad, S. (eds.) FST TCS 2000. LNCS, vol. 1974, pp. 348–360. Springer, Heidelberg (2000)

    Chapter  Google Scholar 

  25. Spakowski, H., Vogel, J.: The complexity of Kemeny’s voting system. In: Proceedings of the Workshop Argentino de Informática Teórica. Anales Jornadas Argentinas de Informática e Investigación Operativa, vol. 30, pp. 157–168. SADIO (2001)

    Google Scholar 

  26. Hemaspaandra, E., Spakowski, H., Vogel, J.: The complexity of Kemeny elections. Technical Report Math/Inf/14/03, Institut für Informatik, Friedrich-Schiller-Universität, Jena, Germany (2003)

    Google Scholar 

  27. Wagner, K.: More complicated questions about maxima and minima, and some closures of NP. Theoretical Computer Science 51(1–2), 53–80 (1987)

    Article  MATH  MathSciNet  Google Scholar 

  28. Cai, J., Gundermann, T., Hartmanis, J., Hemachandra, L., Sewelson, V., Wagner, K., Wechsung, G.: The boolean hierarchy I: Structural properties. SIAM Journal on Computing 17(6), 1232–1252 (1988)

    Article  MATH  MathSciNet  Google Scholar 

  29. Kadin, J.: The polynomial time hierarchy collapses if the boolean hierarchy collapses. SIAM Journal on Computing 17(6), 1263–1282 (1988); Erratum appears in the same journal, 20(2), 404

    Google Scholar 

  30. Raffaelli, G., Marsili, M.: Statistical mechanics model for the emergence of consensus. Physical Review E 72(1), 016114 (2005)

    Google Scholar 

  31. Chernoff, H.: A measure of the asymptotic efficiency for tests of a hypothesis based on the sum of observations. Annals of Mathematical Statistics 23, 493–509 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  32. Alon, N., Spencer, J.: The Probabilistic Method, 2nd edn. Wiley–Interscience, Chichester (2000)

    Book  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Rochester Institute of Technology, Rochester, NY, 14623, USA

    Christopher M. Homan

  2. University of Rochester, Rochester, NY, 14627, USA

    Lane A. Hemaspaandra

Authors
  1. Christopher M. Homan
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Lane A. Hemaspaandra
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Department of Computer Science, Comenius University, Mlynská dolina, 84248, Bratislava, Slovakia

    Rastislav Královič

  2. Institute of Informatics, University of Warsaw, Poland

    Paweł Urzyczyn

Rights and permissions

Reprints and permissions

Copyright information

© 2006 Springer-Verlag Berlin Heidelberg

About this paper

Cite this paper

Homan, C.M., Hemaspaandra, L.A. (2006). Guarantees for the Success Frequency of an Algorithm for Finding Dodgson-Election Winners. In: Královič, R., Urzyczyn, P. (eds) Mathematical Foundations of Computer Science 2006. MFCS 2006. Lecture Notes in Computer Science, vol 4162. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11821069_46

Download citation

  • DOI: https://doi.org/10.1007/11821069_46

  • Publisher Name: Springer, Berlin, Heidelberg

  • Print ISBN: 978-3-540-37791-7

  • Online ISBN: 978-3-540-37793-1

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics

Search

Navigation