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Are Condorcet procedures so bad according to the reinforcement axiom?

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Abstract

A Condorcet social choice procedure elects the candidate that beats every other candidate under simple majority when such a candidate exists. The reinforcement axiom roughly states that given two groups of individuals, if these two groups select the same alternative, then this alternative must also be selected by their union. Condorcet social choice procedures are known to violate this axiom. Our goal in this paper is to put this important voting theory result into perspective. We then proceed by evaluating how frequently this phenomenon is susceptible to occur.

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Notes

  1. \(C_{1}\) was introduced by Black (1958), \(C_{2}\) by Copeland (1951), \(C_{3}\) by Dodgson (1876), \(C_{4}\) and \(C_{5}\) by Young (1975), \(C_{6}\) by Kemeny (1959) and Levenglick (1975), \(C_{7}\) by Schwartz (1972), and \(C_{8}\) by Fishburn (1970).

  2. with a complete enumeration procedure of voting situations.

  3. Note that in the tables, the frequencies are in percentages.

  4. The main difficulty encountered is the evaluation of 10-dimensional volumes over more than 40 parameterized polytopes.

  5. Note that the minor differences between Tables 3 and 4 are a consequence of sampling only 200,000 profiles.

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Acknowledgments

This research has been developed within the center of excellence MME-DII (ANR-11-LBX-0023-01). We are grateful to two anonymous referees and one associate editor for very valuable comments and suggestions.

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Authors and Affiliations

  1. THEMA, UMR CNRS 8184, University of Cergy-Pontoise, Cergy-Pontoise Cedex, France

    Sébastien Courtin

  2. CREM, UMR CNRS 6211, Université de Caen Basse-Normandie, Caen, France

    Boniface Mbih

  3. Department of Mathematics, Advanced Teachers’ Training College, University of Yaounde I, Yaoundé, Cameroon

    Issofa Moyouwou

Authors
  1. Sébastien Courtin
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  2. Boniface Mbih
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  3. Issofa Moyouwou
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Correspondence to Sébastien Courtin.

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Courtin, S., Mbih, B. & Moyouwou, I. Are Condorcet procedures so bad according to the reinforcement axiom?. Soc Choice Welf 42, 927–940 (2014). https://doi.org/10.1007/s00355-013-0758-7

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