On the likelihood of Condorcet's profiles

Consider a group of individuals who have to collectively choose an outcome from a finite set of feasible alternatives. A scoring or positional rule is an aggregation procedure where each voter awards a given number of points, w _j, to the alternative she ranks in j ^th position in her preference ordering; The outcome chosen is then the alternative that receives the highest number of points. A Condorcet or majority winner is a candidate who obtains more votes than her opponents in any pairwise comparison. Condorcet [4] showed that all positional rules fail to satisfy the majority criterion. Furthermore, he supplied a famous example where all the positional rules select simultaneously the same winner while the majority rule picks another one. Let P ^* be the probability of such events in three-candidate elections. We apply the techniques of Merlin et al. [17] to evaluate P ^* for a large population under the Impartial Culture condition. With these assumptions, such a paradox occurs in 1.808% of the cases.

Authors and Affiliations

1. GEMMA-CREME and CNRS, MRSH–SH230, Université de Caen, Esplanade de la Paix, F-14032 Caen Cedex, France (e-mail: merlin@econ.unicaen.fr), , , , , , FR

   Vincent Merlin

2. Department of Mathematics, Northwestern University, 2033 Sheridan Road, Evanston, IL 60208-2730 USA (e-mail: maria@math.nwu.edu), , , , , , US

   Monica Tataru

3. Department of Economics, The University of Namur, Rempart de la Vierge 8, B-5000 Namur, Belgium (e-mail: fabrice.valognes@fundp.ac.be), , , , , , BE

   Fabrice Valognes

Received: 30 April 1999/Accepted: 14 September 2000

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