Abstract
Plurality rule selects whichever alternative is most preferred by the greatest number of voters. The majoritarian principle states that if a simple majority of voters agree on the most preferred alternative, then it must be selected uniquely. Lepelley (RAIRO-Recherche Opérationnelle 26: 361–365, 1992) adopts a proof by contradiction approach to show that plurality is the only scoring rule satisfying the majoritarian principle. We make use of the relationship that majoritarianism implies faithfulness to present a new proof allowing us to derive limits on the size of the group for which a particular scoring rule will satisfy majoritarianism without restricting voter preferences. We then determine the limits for three specific faithful scoring rules where voters rank the alternatives: positive/negative voting, wherein one point is awarded to a voter’s top preference and one point is subtracted from a voter’s bottom preference; Borda, in which an equal increase in points is awarded to each successively higher rank; and Dowdall, for which rank points entail an harmonic sequence. Comparing these rules by the sizes of group and alternative set combinations for which they are majoritarian we find that Borda is dominated by positive/negative voting, and both are dominated by Dowdall. We also derive the relative point gaps between certain pairs of rankings beyond which a scoring rule will not be majoritarian for any group of more than two voters.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
Availability of data and material
not applicable
Change history
23 December 2020
The original online version of this article has been revised due to spacing and formatting error
Notes
Some of the social choice properties have multiple names. Our focus is on the property we refer to here as majoritarianism, which was called “weak Condorcet” by Lepelley (1992), the unified majority principle by Richelson (1978b), the majority criterion by Dougherty and Edward (2011), and majority consistent by Heckelman (2015), whereas violating the condition has been described as the absolute majority (winner) paradox (Felsenthal 2012; Felsenthal and Nurmi 2018). In addition, Lepelley (1992) referred to reinforcement as consistency, following Young’s (1974) original introduction of the property. Young would later switch to calling it reinforcement (e.g., Young 1995) when that label became prominent. Both terms are still in use (see, for example, Miller (forthcoming) and Heckelman and Ragan (2020)).
Similarly, a variety of names have been adopted to describe the voting rules discussed herein and defined below. For example, Lepelley (1992) referred to plurality as simple majority. It also goes by the names relative majority (Golder 2005) and first-past-the-post (Miller 2014). The positive/negative rule is sometimes called best-worst (García-Lapresta et al. 2010). A more formal label for Borda is the method of marks (Kilgour et al. 2020). Dowdall is known as harmonic voting in the computational literature (Filos-Ratsikas et al. 2019).
Lepelley’s paper also is entirely in French, which may make it less accessible to some readers.
Negative voting is defined by \(s_{m}=-1, s_{r}=0\text { for }r<m.\)
Borda generates weights identical to positive/negative when m = 3 if c = 2 and k =1.
Convergence is achieved when the integer value of the group size cap stabilizes, which occurs around five to six alternatives.
Under this condition, every distribution of voters on the m! linear orderings of A is equally likely to occur.
Their analytic formula yields probability = 0.5 for n = 4 and probability > 0.5 for all other \(n\ge 3\).
Non-positional scoring rules are not constrained by \(0<\lambda \le 1\).
In this section, we ignore the trivial cases of fewer than three persons or three alternatives because all faithful scoring rules will be majoritarian under those conditions.
To be precise, no restrictions are needed beyond the standard assumptions that voter preferences are complete, asymmetric, and transitive.
However, because Plassmann and Tideman’s analysis is limited to three alternatives, their Borda simulations also would apply to positive/negative voting.
The reverse, of course, is not true.
References
Ching, S. (1996). A simple characterization of plurality rule. Journal of Economic Theory, 71, 298–302.
Diss, M., Kamwa, E., & Tlidi, A. (2018). A note on the likelihood of the absolute majority paradoxes. Economic Bulletin, 38, 1727–1736.
Dougherty, K. L., & Edward, J. (2011). The calculus of consent and constitutional design. New York: Springer.
El Ouafdi, A., Lepelley, D., & Smaoui, H. (2020). Probabilities of electoral outcomes: From three-candidate to four-candidate elections. Theory & Decision, 88, 205–229.
Felsenthal, D. S. (2012). Review of paradoxes afflicting procedures for electing a single candidate. In D. S. Felsenthal & M. Machover (Eds.), Electoral systems: Paradoxes, assumptions, and procedures (pp. 19–91). Berlin: Springer.
Felsenthal, D. S., & Nurmi, H. (2018). Voting procedures for electing a single candidate: Proving their (In)vulnerability to various voting paradoxes. Switzerland: Springer.
Filos-Ratsikas, A., Micha, E., & Voudouris, A.A. (2019). The distortion of distributed voting. In Proceedings of the 12th international symposium on algorithmic game theory, pp. 312–325.
Fraenkel, J., & Grofman, B. (2014). The Borda count and its real-world alternatives: Comparing scoring rules in Nauru and Slovenia. Australian Journal of Political Science, 49, 186–205.
García-Lapresta, J. L., Marley, A. A. J., & Martínez-Panero, M. (2010). Characterizing best-worst voting systems in the scoring context. Social Choice & Welfare, 34, 487–496.
Gehrlein, W. V., & Lepelley, D. (2017). Elections, voting rules and paradoxical outcomes. Cham, Switzerland: Springer.
Golder, M. (2005). Democratic electoral systems around the world, 1946–2000. Electoral Studies, 24, 103–121.
Goodin, R. E., & List, C. (2006). A conditional defense of plurality rule: Generalizing May’s theorem in a restricted informational environment. American Journal of Political Science, 50, 940–949.
Heckelman, J.C. (2015). Properties and paradoxes of common voting rules. In J. C. Heckelman & N. R. Miller (Eds.), Handbook of social choice and voting. Cheltenham, UK: Edward Elgar Publishing Limited, pp. 263–283.
Heckelman, J. C., & Ragan, R. (2020). Symmetric scoring rules and a new characterization of the Borda count. Economic Inquiry,. https://doi.org/10.1111/ecin.12929.
Kilgour, D. M., Grégoire, J.-C., & Foley, A. M. (2020). The prevalence and consequences of ballot truncation in ranked-choice elections. Public Choice, 184, 197–218.
Lepelley, D. (1992). Une caractérisation du vote à la majorité simple. RAIRO-Recherche Opérationnelle, 26, 361–365.
Miller, A.D. (Forthcoming). Voting in corporations. Theoretical Economics.
Miller, N. (2014). The alternative vote and Coombs rule versus first-past-the-post: A social choice analysis of simulated data based on English elections, 1992–2010. Public Choice, 158, 399–425.
Plassmann, F., & Tideman, T. N. (2014). How frequently do different voting rules encounter voting paradoxes in three-candidate elections? Social Choice & Welfare, 42, 31–75.
Reilly, B. (2002). Social choice in the South Seas: Electoral innovation and the Borda Count in the Pacific Island countries. International Political Science Review, 23, 355–372.
Richelson, J. T. (1978a). A characterization result for the plurality rule. Journal of Economic Theory, 19, 548–550.
Richelson, J. T. (1978b). Comparative analysis of social choice functions, II. Behavioral Sciences, 23, 38–44.
Sekiguchi, Y. (2012). A characterization of the plurality rule. Economics Letters, 116, 330–332.
Yeh, C.-H. (2008). An efficiency characterization of plurality rule in collective choice problems. Economic Theory, 34, 575–583.
Young, H. P. (1974). An axiomatization of Borda’s rule. Journal of Economic Theory, 9, 43–52.
Young, H. P. (1975). Social choice scoring functions. SIAM Journal of Applied Mathematics, 28, 824–838.
Young, H. P. (1995). Optimal voting rules. Journal of Economic Perspectives, 9, 51–63.
Funding
None.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflicts of interest/Competing interests
none
Code availability
not applicable
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Rights and permissions
About this article
Cite this article
Heckelman, J.C. Characterizing plurality using the majoritarian condition: a new proof and implications for other scoring rules. Public Choice 189, 335–346 (2021). https://doi.org/10.1007/s11127-020-00845-y
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11127-020-00845-y