Approval voting and Shapley ranking


Approval voting allows electors to list any number of candidates and their final scores are obtained by summing the votes cast in their favor. Equal-and-even cumulative voting instead follows the One-person-one-vote principle by endowing each elector with a single vote that may be distributed evenly among several candidates. It corresponds to satisfaction approval voting, introduced by Brams and Kilgour (in: Fara et al (eds) Voting power and procedures. Essays in honor of Dan Fesenthal and Moshé Machover, Springer, Heidelberg, 2014) as an extension of approval voting to a multiwinner election. It also corresponds to the concept of Shapley ranking, introduced by Ginsburgh and Zang (J Wine Econ 7:169–180, 2012) as the Shapley value of a cooperative game with transferable utility. In the present paper, we provide an axiomatic foundation for Shapley ranking and analyze the properties of the resulting social welfare function.

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  1. 1.

    See also Brams and Fishburn (1983/2007, 2005), Weber (1995), Brams (2008) and Laslier and Remzi Sanver (2010).

  2. 2.

    Convention: we use "she" for voters and "he" for candidates.

  3. 3.

    See the ensuing discussion in the issue of Public Choice where their paper was published.

  4. 4.

    See Brams and Fishburn (2005). Recently, the electoral system in the city of Fargo, North Dakota, was changed from plurality voting to approval voting. See

  5. 5.

    Ranking is cardinal and, in some contexts such as wine competitions, rankings matter.

  6. 6.

    For an overall analysis of multiwinner elections based on approval balloting, see Brams et al. (2019).

  7. 7.

    Alcantud and Laruelle (2014) study and characterize a voting rule that allows voters to divide candidates into three classes, approved, disapproved and indifferent, thereby allowing for incomplete preferences.

  8. 8.

    The reference is Arrow’s (1951) famous book. The 1963 edition reproduces the first edition and adds a chapter reviewing the developments in social choice theory since 1951.

  9. 9.

    Notation The cardinality of a finite set A is denoted |A|. Upper-case letters are used to denote finite sets and subsets, and the corresponding lower-case letters are used to denote the numbers of their elements: n = |N|, s = |S|, and so on.

  10. 10.

    Braces are omitted in the absence of ambiguity.

  11. 11.

    The terminology used by Bogolmania et al. (2005).

  12. 12.

    That candidate is therefore also the unique Condorcet winner (see Sect. 4).

  13. 13.

    Positive games form a particular subclass of convex games on which the set of asymmetric values obtained by considering all distributions of dividends (the "Harsanyi set") coincides with the set of weighted Shapley values and the core. See Dehez (2017) for details.

  14. 14.

    It is assumed implicitly that the sets of voters are disjoint.

  15. 15.

    Acknowledged by Brams and Fishburn (2007, p. 137) and confirmed by Mongin and Maniquet (2015).

  16. 16.

    Theorem 3.1 in Brams and Fishburn (2007, p. 38).

  17. 17.

    Cycles are possible if more than two grades are employed, giving rise to what Brams and Potthoff (2015) call the "paradox of grading systems", which is comparable to the Condorcet paradox.


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The authors are grateful to Steve Brams, Claude d’Aspremont, Herrade Igersheim, Annick Laruelle, Israel Zang and two referees for helpful comments and suggestions.

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Correspondence to Pierre Dehez.

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Dehez, P., Ginsburgh, V. Approval voting and Shapley ranking. Public Choice 184, 415–428 (2020).

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  • Approval voting
  • Equal-and-even cumulative voting
  • Ranking game
  • Shapley value

JEL Classification

  • D71
  • C71