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How many voters are needed for paradoxes?


  • James S. Weber

    () (Department of Information and Decision Sciences, University of Illinois at Chicago, Chicago, IL 60607-7124, USA)


This paper presents a general procedure for finding profiles with the minimum number of voters required for many important paradoxes. Borda's and Condorcet's classic examples are revisited as well as generalizations. Using Saari's procedure line, we obtain an upper bound for the minimum number of voters needed for a profile for which the Condorcet winner is not strictly top ranked for all $w_{\rm s}^{3}$ weighted positional procedures. Also we give a simple upper bound on the minimum number of voters needed for a set of prescribed voting outcomes. In contrast to situations wherein small numbers of voters are needed, we consider paradoxes requiring arbitrarily large numbers of voters as well as large numbers of alternatives. Finally we indicate connections with statistical rank based tests.

Suggested Citation

  • James S. Weber, 2002. "How many voters are needed for paradoxes?," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 20(2), pages 341-355.
  • Handle: RePEc:spr:joecth:v:20:y:2002:i:2:p:341-355
    Note: Received: April 18, 2001; revised version: May 25, 2001

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    Cited by:

    1. Lee Gibson & Robert Powers, 2012. "An extension of McGarvey’s theorem from the perspective of the plurality collective choice mechanism," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 38(1), pages 101-108, January.
    2. Munda, Giuseppe, 2009. "A conflict analysis approach for illuminating distributional issues in sustainability policy," European Journal of Operational Research, Elsevier, vol. 194(1), pages 307-322, April.
    3. Giuseppe Munda, 2012. "Choosing Aggregation Rules for Composite Indicators," Social Indicators Research: An International and Interdisciplinary Journal for Quality-of-Life Measurement, Springer, vol. 109(3), pages 337-354, December.

    More about this item


    Voting paradox; Minimum number of voters; Condorcet pairwise procedure; Borda Count; Plurality; $w_{rm s}^3$ procedure; Procedure line; Committees; Kruskal-Wallis Test.;

    JEL classification:

    • D71 - Microeconomics - - Analysis of Collective Decision-Making - - - Social Choice; Clubs; Committees; Associations


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