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The Condorcet Efficiency Advantage that Voter Indifference Gives to Approval Voting Over Some Other Voting Rules

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  • William V. Gehrlein

    (University of Delaware)

  • Dominique Lepelley

    (CEMOI, University of La Réunion)

Abstract

Approval Voting is known to possess many good properties when voters have dichotomous preferences. But, when attention was restricted to the limiting case for large electorates with three candidates in an early study, Approval Voting was found to have the same Condorcet Efficiency as both Plurality Rule and Negative Plurality rule when no voter indifference is allowed in voters’ preferences with the assumption of the impartial culture condition (IC). However, a later study by Diss et al. (Handbook on approval voting, 2010) shows that the introduction of any degree of indifference in an extended impartial culture condition leads to a dominance of Approval Voting over both Plurality Rule and Negative Plurality Rule on the basis of Condorcet Efficiency. Scenarios were also found for which Approval Voting had greater Condorcet Efficiency than Borda Rule. The assumptions of that study are analyzed here, and an arguably more reasonable set of assumptions leads to the conclusion that Borda Rule will dominate Approval Voting on the basis of Condorcet Efficiency for all degrees of voter indifference, except for the case of completely dichotomous preferences. The same outcome is found to result in the current study for an extended version of the Impartial Anonymous Culture Condition.

Suggested Citation

  • William V. Gehrlein & Dominique Lepelley, 2015. "The Condorcet Efficiency Advantage that Voter Indifference Gives to Approval Voting Over Some Other Voting Rules," Group Decision and Negotiation, Springer, vol. 24(2), pages 243-269, March.
    • Handle: RePEc:spr:grdene:v:24:y:2015:i:2:d:10.1007_s10726-014-9388-4
    DOI: 10.1007/s10726-014-9388-4
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    1. Steven J. Brams & William V. Gehrlein & Fred S. Roberts (ed.), 2009. "The Mathematics of Preference, Choice and Order," Studies in Choice and Welfare, Springer, number 978-3-540-79128-7, June.
    2. Dominique Lepelley & Ahmed Louichi & Hatem Smaoui, 2008. "On Ehrhart polynomials and probability calculations in voting theory," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 30(3), pages 363-383, April.
    3. Wilson, Mark C. & Pritchard, Geoffrey, 2007. "Probability calculations under the IAC hypothesis," Mathematical Social Sciences, Elsevier, vol. 54(3), pages 244-256, December.
    4. Fabrice Valognes & William V. Gehrlein, 2001. "Condorcet efficiency: A preference for indifference," Social Choice and Welfare, Springer;The Society for Social Choice and Welfare, vol. 18(1), pages 193-205.
    5. Sven Berg, 1985. "Paradox of voting under an urn model: The effect of homogeneity," Public Choice, Springer, vol. 47(2), pages 377-387, January.
    6. Steven J. Brams & M. Remzi Sanver, 2009. "Voting Systems that Combine Approval and Preference," Studies in Choice and Welfare, in: Steven J. Brams & William V. Gehrlein & Fred S. Roberts (ed.), The Mathematics of Preference, Choice and Order, pages 215-237, Springer.
    7. Young, H. P., 1974. "An axiomatization of Borda's rule," Journal of Economic Theory, Elsevier, vol. 9(1), pages 43-52, September.
    8. William Gehrlein & Fabrice Valognes, 2001. "Condorcet efficiency: A preference for indifference," Post-Print halshs-02173169, HAL.
    9. Mostapha Diss & Vincent Merlin & Fabrice Valognes, 2010. "On the Condorcet Efficiency of Approval Voting and Extended Scoring Rules for Three Alternatives," Studies in Choice and Welfare, in: Jean-François Laslier & M. Remzi Sanver (ed.), Handbook on Approval Voting, chapter 0, pages 255-283, Springer.
    10. Gehrlein, William V., 1982. "Condorcet efficiency and constant scoring rules," Mathematical Social Sciences, Elsevier, vol. 2(2), pages 123-130, March.
    11. Gehrlein, William V. & Lepelley, Dominique, 1998. "The Condorcet efficiency of approval voting and the probability of electing the Condorcet loser," Journal of Mathematical Economics, Elsevier, vol. 29(3), pages 271-283, April.
    12. Gehrlein, William V. & Fishburn, Peter C., 1978. "Probabilities of election outcomes for large electorates," Journal of Economic Theory, Elsevier, vol. 19(1), pages 38-49, October.
    13. Jean-François Laslier & M. Remzi Sanver (ed.), 2010. "Handbook on Approval Voting," Studies in Choice and Welfare, Springer, number 978-3-642-02839-7, June.
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    Cited by:

    1. Mostapha Diss & Ahmed Doghmi, 2016. "Multi-winner scoring election methods: Condorcet consistency and paradoxes," Public Choice, Springer, vol. 169(1), pages 97-116, October.
    2. Durand, François & Macé, Antonin & Núñez, Matías, 2024. "Voter coordination in elections: A case for approval voting," Games and Economic Behavior, Elsevier, vol. 146(C), pages 1-34.
    3. Erik Friese & William V. Gehrlein & Dominique Lepelley & Achill Schürmann, 2017. "The impact of dependence among voters’ preferences with partial indifference," Quality & Quantity: International Journal of Methodology, Springer, vol. 51(6), pages 2793-2812, November.
    4. Eric Kamwa, 2023. "On two voting systems that combine approval and preferences: fallback voting and preference approval voting," Public Choice, Springer, vol. 196(1), pages 169-205, July.
    5. Begoña Subiza & Josep E. Peris, 2017. "A Representative Committee by Approval Balloting," Group Decision and Negotiation, Springer, vol. 26(5), pages 1029-1040, September.
    6. Eric Kamwa, 2018. "On the Condorcet Efficiency of the Preference Approval Voting and the Probability of Selecting the Condorcet Loser," Working Papers hal-01786121, HAL.
    7. Gehrlein, William V. & Lepelley, Dominique & Moyouwou, Issofa, 2016. "A note on Approval Voting and electing the Condorcet loser," Mathematical Social Sciences, Elsevier, vol. 80(C), pages 115-122.
    8. Eric Kamwa, 2023. "On Two Voting systems that combine approval and preferences: Fallback Voting and Preference Approval Voting," Working Papers hal-03614585, HAL.
    9. William V. Gehrlein & Dominique Lepelley & Florenz Plassmann, 2018. "An Evaluation of the Benefit of Using Two-Stage Election Procedures," Homo Oeconomicus: Journal of Behavioral and Institutional Economics, Springer, vol. 35(1), pages 53-79, June.
    10. Eric Kamwa, 2019. "Condorcet efficiency of the preference approval voting and the probability of selecting the Condorcet loser," Theory and Decision, Springer, vol. 87(3), pages 299-320, October.
    11. Abdelhalim El Ouafdi & Dominique Lepelley & Jérôme Serais & Hatem Smaoui, 2022. "Comparing the manipulability of approval, evaluative and plurality voting with trichotomous preferences," SN Business & Economics, Springer, vol. 2(8), pages 1-22, August.
    12. Abdelhalim El Ouafdi & Dominique Lepelley & Hatem Smaoui, 2020. "On the Condorcet efficiency of evaluative voting (and other voting rules) with trichotomous preferences," Annals of Operations Research, Springer, vol. 289(2), pages 227-241, June.

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