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Scoring rules over subsets of alternatives: Consistency and paradoxes

Listed author(s):
  • Eric Kamwa

    (Université de Caen Normandie, CREM - Centre de Recherche en Economie et Management - UR1 - Université de Rennes 1 - Université de Caen Basse-Normandie - CNRS - Centre National de la Recherche Scientifique)

  • Vincent Merlin

    (CREM - Centre de Recherche en Economie et Management - UR1 - Université de Rennes 1 - Université de Caen Basse-Normandie - CNRS - Centre National de la Recherche Scientifique)

We know since the works of Gehrlein and Fishburn (1980, 1981), Fishburn (1981) and Saari (1987, 1988, 1990) that, the collective rankings of scoring rules are not stable when some alternatives are dropped from the set of alternatives. However, in the literature, attention has been mainly devoted to the relationship between pairwise majority vote and scoring rules rankings. In this paper, we focus on the relationships between four-candidate and three-candidate rankings. More precisely, given a collective ranking over a set of four candidates, we determine under the impartial culture condition, the probability of each of the six possible rankings to occur when one candidate is dropped. As a consequence, we derive from our computations, the likelihood of two paradoxes of committee elections, the Leaving Member Paradox (Staring, 1986) and the Prior Successor Paradox which occur when an elected candidate steps down from a two-member committee.

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Paper provided by HAL in its series Post-Print with number halshs-01238563.

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Date of creation: Dec 2015
Publication status: Published in Journal of Mathematical Economics, Elsevier, 2015, 61, pp.130-138. <10.1016/j.jmateco.2015.08.008>
Handle: RePEc:hal:journl:halshs-01238563
DOI: 10.1016/j.jmateco.2015.08.008
Note: View the original document on HAL open archive server: https://halshs.archives-ouvertes.fr/halshs-01238563
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  1. Tataru, Maria & Merlin, Vincent, 1997. "On the relationship of the Condorcet winner and positional voting rules," Mathematical Social Sciences, Elsevier, vol. 34(1), pages 81-90, August.
  2. Van Newenhizen, Jill, 1992. "The Borda Method Is Most Likely to Respect the Condorcet Principle," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 2(1), pages 69-83, January.
  3. Eric Kamwa, 2013. "The increasing committee size paradox with small number of candidates," Economics Bulletin, AccessEcon, vol. 33(2), pages 967-972.
  4. Merlin, Vincent & Valognes, Fabrice, 2004. "The impact of indifferent voters on the likelihood of some voting paradoxes," Mathematical Social Sciences, Elsevier, vol. 48(3), pages 343-361, November.
  5. Saari, Donald G., 1987. "The source of some paradoxes from social choice and probability," Journal of Economic Theory, Elsevier, vol. 41(1), pages 1-22, February.
  6. Mitchell, Douglas W & Trumbull, William N, 1992. "Frequency of Paradox in a Common n-Winner Voting Scheme," Public Choice, Springer, vol. 73(1), pages 55-69, January.
  7. Smith, John H, 1973. "Aggregation of Preferences with Variable Electorate," Econometrica, Econometric Society, vol. 41(6), pages 1027-1041, November.
  8. Mostapha Diss & Vincent Merlin & Fabrice Valognes, 2010. "On the Condorcet efficiency of approval voting and extended scoring rules for three alternatives," Post-Print halshs-00533124, HAL.
  9. Chevaleyre, Yann & Lang, Jérôme & Maudet, Nicolas & Monnot, Jérôme & Xia, Lirong, 2012. "New candidates welcome! Possible winners with respect to the addition of new candidates," Mathematical Social Sciences, Elsevier, vol. 64(1), pages 74-88.
  10. Dutta, Bhaskar & Jackson, Matthew O & Le Breton, Michel, 2001. "Strategic Candidacy and Voting Procedures," Econometrica, Econometric Society, vol. 69(4), pages 1013-1037, July.
  11. Davide Cervone & William Gehrlein & William Zwicker, 2005. "Which Scoring Rule Maximizes Condorcet Efficiency Under Iac?," Theory and Decision, Springer, vol. 58(2), pages 145-185, March.
  12. Donald G. Saari & Maria M. Tataru, 1999. "The likelihood of dubious election outcomes," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 13(2), pages 345-363.
  13. Merlin, V. & Tataru, M. & Valognes, F., 2000. "On the probability that all decision rules select the same winner," Journal of Mathematical Economics, Elsevier, vol. 33(2), pages 183-207, March.
  14. William Gehrlein & Peter Fishburn, 1981. "Constant scoring rules for choosing one among many alternatives," Quality & Quantity: International Journal of Methodology, Springer, vol. 15(2), pages 203-210, April.
  15. Donald G. Saari & Vincent R. Merlin, 1996. "The Copeland method (*)," Economic Theory, Springer;Society for the Advancement of Economic Theory (SAET), vol. 8(1), pages 51-76.
  16. Peter Fishburn & William Gehrlein, 1976. "Borda's rule, positional voting, and Condorcet's simple majority principle," Public Choice, Springer, vol. 28(1), pages 79-88, December.
  17. Gehrlein, William V. & Fishburn, Peter C., 1976. "The probability of the paradox of voting: A computable solution," Journal of Economic Theory, Elsevier, vol. 13(1), pages 14-25, August.
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