Social Choice and Welfare

, Volume 1, Issue 2, pp 149–158 | Cite as

Heights of representative systems: A proof of Fishburn's conjecture

  • H. Keiding
Article

Abstract

Representative systems with n-voters are hierarchical choice functions from {-1, 0, 1} n to {-1, 0, 1} constructed as iterations of weighted majority voting. The height of a representative system is the minimal number of iterations necessary for this construction.

In the paper we give an upper bound for μ(n), the maximal height of any n-voter representative system, and show that \(\frac{{\mu (n)}}{n}\) goes to zero as n goes to infinity, thus proving a conjecture made by Fishburn. Technically, the results are obtained by transferring the problem to the context of proper simple games, which have a similar hierarchical structure, and using known results on heights of simple games.

Keywords

Economic Theory Hierarchical Structure Maximal Height Majority Vote Choice Function 

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References

  1. Fishburn PC (1973) The theory of social choice. Princeton University Press, PrincetonGoogle Scholar
  2. Fishburn PC (1975) Three-valued representative systems. Math Systems Theory 9:265–280Google Scholar
  3. Fishburn PC (1979) Heights of representative systems. Discrete Appl Math 1:181–199Google Scholar
  4. Keiding H (1984) Heights of simple games. Int J Game Theory 13:15–26Google Scholar
  5. Murakami Y, (1966) Formal structure of majority decision. Econometrica 34:709–718Google Scholar
  6. Shapley LS (1962) Simple games: An outline of the descriptive theory. Behav Sci 7:59–66Google Scholar

Copyright information

© Springer-Verlag 1984

Authors and Affiliations

  • H. Keiding
    • 1
  1. 1.Institute of EconomicsUniversity of CopenhagenCopenhagenDenmark

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