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.
This is a preview of subscription content, access via your institution.
Buy single article
Instant access to the full article PDF.
Tax calculation will be finalised during checkout.
Subscribe to journal
Immediate online access to all issues from 2019. Subscription will auto renew annually.
Tax calculation will be finalised during checkout.
Convention: we use "she" for voters and "he" for candidates.
See the ensuing discussion in the issue of Public Choice where their paper was published.
Ranking is cardinal and, in some contexts such as wine competitions, rankings matter.
For an overall analysis of multiwinner elections based on approval balloting, see Brams et al. (2019).
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.
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.
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.
Braces are omitted in the absence of ambiguity.
The terminology used by Bogolmania et al. (2005).
That candidate is therefore also the unique Condorcet winner (see Sect. 4).
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.
It is assumed implicitly that the sets of voters are disjoint.
Theorem 3.1 in Brams and Fishburn (2007, p. 38).
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.
Alcantud, J. C. R., & Laruelle, A. (2014). Dis&approval voting: A characterization. Social Choice and Welfare,43, 1–10.
Arrow, K. (1951/1963). Social choice and individual values (2nd ed.). New York: Wiley.
Baujard, A., & Igersheim, H. (2010). Framed field experiments on approval voting: Lessons from the 2002 and 2007 French presidential elections. In J. F. Laslier & M. R. Remzi Sanver (Eds.), Handbook on approval voting (pp. 357–395). Berlin: Springer-Verlag.
Black, D. (1958). The theory of committees and elections. Cambridge: Cambridge University Press.
Bogolmania, A., Moulin, H., & Stong, R. (2005). Collective choice under dichotomous preferences. Journal of Economic Theory,122, 165–184.
Borda, J. C. (1781). Mémoire sur les élections au scrutin, Mémoires de l’Académie des Sciences, Paris, 657–664.
Brams, S. J. (2008). The mathematics and democracy. Designing better voting and fair-division procedures. Princeton: Princeton University Press.
Brams, S. J., & Fishburn, P. C. (1978). Approval voting. American Political Science Review,72, 831–847.
Brams, S. J., & Fishburn, P. C. (1983/2007). Approval voting (2nd ed.). Berlin: Springer.
Brams, S. J., & Fishburn, P. C. (2005). Going from theory to practice: The mixed success of approval voting. Social Choice and Welfare,25, 457–474. (Reproduced in Laslier J.-F. and M.R. Remzi Sanver (Eds. 2010), 19–37).
Brams, S. J., & Kilgour, D. M. (2014). Satisfaction approval voting. In R. Fara, et al. (Eds.), Voting power and procedures. Essays in honor of Dan Fesenthal and Moshé Machover (pp. 323–346). Heidelberg: Springer.
Brams, S. J., Kilgour, D. M., & Potthoff, R. F. (2019). Multiwinner approval voting: an apportionment approach. Public Choice,178, 67–93.
Brams, S. J., & Potthoff, R. F. (2015). The paradox of grading systems. Public Choice,165, 193–210.
Condorcet, N. (1785). Essai sur l’application de l’analyse à la probabilité des décisions rendues à la pluralité des voix. Imprimerie Royale, Paris.
Dehez, P. (2017). On Harsanyi dividends and asymmetric values. International Game Theory Review,19(3), 1750012.
Ginsburgh, V., & Zang, I. (2012). Shapley ranking of wines. Journal of Wine Economics,7, 169–180.
Harsanyi, J. C. (1959). A bargaining model for cooperative n-person games. In A. W. Tucker & R. D. Luce (Eds.), Contributions to the theory of games IV (pp. 325–355). Princeton: Princeton University Press.
Laslier, J.-F., & Remzi Sanver, M. R. (2010). Handbook on approval voting. Berlin: Springer-Verlag.
Mongin, Ph., & Maniquet, F.(2015). Approval voting and Arrow’s impossibility theorem. Social Choice and Welfare,44, 519–532.
Roth, A. (Ed.). (1988). The Shapley value. Essays in honor of Lloyd Shapley. Cambridge: Cambridge University Press.
Saari, D. G., & van Newenhizen, J. (1988). The problem of indeterminacy in approval, multiple and truncated voting systems. Public Choice,59, 101–120.
Shapley, L. S. (1953). A value for n-person games. In H. Kuhn & A. W. Tucker (Eds.), Contributions to the theory of games II. Annals of Mathematics Studies (Vol. 24, pp. 307–317). Princeton: Princeton University Press. (Reproduced in A. Roth (Ed. 1988), 31–40).
Weber, R. J. (1977). Comparison of voting systems, mimeo. (Reproduced in Comparison of public choice systems,Cowles Foundation Discussion Paper498, 1979, Yale University)
Weber, R. J. (1995). Approval voting. Journal of Economic Perspectives,9, 39–49.
The authors are grateful to Steve Brams, Claude d’Aspremont, Herrade Igersheim, Annick Laruelle, Israel Zang and two referees for helpful comments and suggestions.
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
About this article
Cite this article
Dehez, P., Ginsburgh, V. Approval voting and Shapley ranking. Public Choice 184, 415–428 (2020). https://doi.org/10.1007/s11127-019-00729-w
- Approval voting
- Equal-and-even cumulative voting
- Ranking game
- Shapley value