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Public Choice

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22-10-2019

Approval voting and Shapley ranking

Authors: Pierre Dehez, Victor Ginsburgh

Published in: Public Choice | Issue 3-4/2020

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Abstract

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

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.

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 www.electionscience.org.

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.

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

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.CrossRef

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.CrossRef

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.CrossRef

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.CrossRef

Brams, S. J., & Fishburn, P. C. (1978). Approval voting. American Political Science Review,72, 831–847.CrossRef

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).CrossRef

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.CrossRef

Brams, S. J., & Potthoff, R. F. (2015). The paradox of grading systems. Public Choice,165, 193–210.CrossRef

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.CrossRef

Ginsburgh, V., & Zang, I. (2012). Shapley ranking of wines. Journal of Wine Economics,7, 169–180.CrossRef

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.CrossRef

Mongin, Ph., & Maniquet, F.(2015). Approval voting and Arrow’s impossibility theorem. Social Choice and Welfare,44, 519–532.CrossRef

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.CrossRef

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.CrossRef

Title: Approval voting and Shapley ranking
Authors: Pierre Dehez
Victor Ginsburgh
Publication date: 22-10-2019
Publisher: Springer US
Published in: Public Choice / Issue 3-4/2020
Print ISSN: 0048-5829
Electronic ISSN: 1573-7101
DOI: https://doi.org/10.1007/s11127-019-00729-w

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