Abstract
This note supplements the recent work of McCune and Wilson (Public Choice 196(1–2):19–50, 2023) by providing a complete analysis of spoiler effects under both plurality voting and Ranked-Choice Voting in the case of three (potential) candidates. The trick for definitively identifying all spoiler possibilities under both voting rules in the three-candidate case is to partition the set of all three-candidate preference profiles into eight types by cross-classifying the candidates in terms of their plurality status and Condorcet relationships. The resulting typology allows us to identify the winners in all possible two-candidate and three-candidate elections under both voting rules and therefore suffices to identify all spoiler effects. It implies, among other things, that the set of profiles that are vulnerable to spoilers under Ranked-Choice Voting is a proper subset of those vulnerable to spoilers under plurality rule.
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Notes
This supplement might be incorporated into Sect. 3 of McCune and Wilson’s paper as it elaborates on their Remark 1.
We follow McCune and Wilson, as well as Graham-Squire and McCune (2023) and many others, by ignoring the possibility of ties—a defensible assumption given a large (electorate-sized) number of voters. Essentially the same definitions and results apply if ties are allowed but all statements become considerably more complicated. Moreover, there is no standard tie-breaking rule for elections. We also assume that there are no ties in individual preference orderings.
However, the cyclical profiles (4a) and (4b) evidently occur infrequently in empirical elections.
In contrast, type (1a) represents a ‘Buchanan non-effect’ situation. (Buchanan won about 35,000 votes in Florida in 2000, presumably coming largely at Bush’s expense, but Bush remained the Plurality Winner even with Buchanan in the race.).
The section on spoilers was dropped before publication to reduce the paper to appropriate journal article length. The analysis in was restricted to English constituencies because virtually all were effectively three-candidate (Labour, Liberal Democrat, Conservative) contests, which was not the case in Wales, Scotland, or Northern Ireland. Since the resulting 2642 constituency elections were conducted under plurality voting, the electoral data provided only (what were taken to be) first preferences. Second preferences in each constituency were allocated in proportion to second preferences nationwide, as determined by surveys for each of the five elections. These surveys indicated that English voter preferences were ‘partially single-peaked’—that is, most but not all Labour (‘left-of-center’) voters had the Liberal Democrats (the ‘centrist’ party) as their second preference as did most but not all Conservative (‘right-of-center’) voters, while Liberal Democrat voters had more evenly divided second preferences.
References
Graham-Squire, A., & McCune, D. (2022). A mathematical analysis of the 2022 Alaska special election for U.S. House. arXiv preprint, arXiv:2209.04764.
Graham-Squire, A., & McCune, D. (2023). An examination of ranked-choice voting in the United States, 2004–2022. Representation. https://doi.org/10.1080/00344893.2023.2221689
McCune, D., & Wilson, J. (2023). Ranked-choice voting and the spoiler effect. Public Choice, 196(1–2), 19–50.
Miller, N. R. (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(3–4), 399–425.
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Acknowledgements
I thank David McCune for helpful comments and corrections, particularly pertaining to the RCV elections in Alaska.
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Miller, N.R. Ranked-choice voting and the spoiler effect: a supplementary note. Public Choice 198, 153–159 (2024). https://doi.org/10.1007/s11127-023-01116-2
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DOI: https://doi.org/10.1007/s11127-023-01116-2