Abstract
This paper expands the illustration and analysis regarding the susceptibility of nine voting procedures to two types of what are generally known as No-Show paradoxes. Following the article by Felsenthal and Tideman (Theory and Decision 75:59–77, 2013), the two paradoxes are denoted as P-TOP and P-BOT paradoxes. According to the P-TOP paradox it is possible that if candidate x has been elected by a given electorate then, ceteris paribus, another candidate, y, may be elected if additional voters join the electorate who rank x at the top of their preference ordering. Similarly, according to the P-BOT paradox it is possible that if candidate y has not been elected by a given electorate then, ceteris paribus, y may be elected if additional voters join the electorate who rank y at the bottom of their preference ordering. Voting procedures that are susceptible to these paradoxes are considered to be afflicted with a particularly serious defect because instead of encouraging voters to participate in an election and vote according to their true preference orderings, they may inhibit voters from participating in an election and thereby undermine the rationale for conducting elections.
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Notes
Following Moulin (1988, p. 56) we define a voting procedure to be Condorcet-consistent if it elects the Condorcet winner when one exists. This definition differs from Fishburn’s (1977, p. 482) definitions of moderate and weak Condorcet-consistency. A Condorcet winner—named after the Frenchman Marquis de Condorcet, who called this candidate ‘the majority candidate’ —is the candidate whom the majority of the voters prefer over each of the other alternatives in pairwise comparisons. Condorcet thought that if such a candidate exists s/he ought to be elected.
The Twin paradox (Moulin 1988, p. 59) is a special version of the No-Show paradox. Two voters having the same preference ordering may obtain a preferable outcome if, ceteris paribus, one of them decides not to participate in the election while the other votes sincerely according to his preference ordering. The Twin paradox implies the No-Show paradox.
The five voting methods investigated by Felsenthal and Tideman (2013) were the Plurality with Runoff, Alternative Vote, Coombs, Nanson, and Dodgson methods. The first three methods are non-Condorcet-consistent and the last two are Condorcet-consistent. “Moving x up” means that all candidates other than x are in the same order after the change as before: all candidates initially below x remain below x after the change, and one or more candidates that were initially above x are below x after the change.
In the sequel we use a slightly revised definition for illustrating the P-TOP and P-BOT paradoxes under Schwartz’s procedure where multiple winners exist either in the original electorate or in the enlarged electorate.
However, it is unclear how a tie between two candidates, say a and b, ought to be broken under Bucklin’s procedure when both a and b are supported in the same counting round by the same number of voters and this number constitutes a majority of the voters. If one tries to break the tie between a and b in such an eventuality by performing the next counting round in which all other candidates are also allowed to participate, then it is possible that the number of (cumulated) votes of another candidate, c, will exceed that of a and b.
To see this, consider the following simple example. Suppose there are 18 voters who must elect one candidate under Bucklin’s procedure and whose preference orderings among four candidates, a, b, c, d, are as follows: seven voters with preference ordering a > b > c > d, eight voters with preference ordering b > a > c > d, one voter with preference ordering d > c > a > b, and two voters with preference ordering d > c > b > a. None of the candidates constitutes the top preference of a majority of the voters. However, both a and b constitute the top or second preference of a majority of voters (15). If one tries to break the tie between a and b by performing the next (third) counting round in which c and d also are allowed to participate, then c will be elected (with 18 votes), but if only a and b are allowed to participate in this counting round then b will be elected (with 17 votes).
So which candidate ought to be elected in this example under Bucklin’s procedure? As far as we know, Bucklin did not supply an answer to this question.
Young (1977, p. 349) prefers to call this procedure ‘The Minimax function’. It is also sometimes called in the literature ‘the max–min method’.
Borda’s Count (Borda 1784; Black 1958) is a voting procedure that was proposed by Jean Charles de Borda in a paper he delivered in 1770 before the French Royal Academy of Sciences entitled ‘Memorandum on election by ballot’ (‘Mémoire sur les élections au scrutin’). According to Borda’s procedure, each candidate x gets no points for each voter who ranks x last in his preference ordering, 1 point for each voter who ranks x second-to-last in his preference order, and so on, and m-1 points for each voter who ranks x first in his preference order (where m is the number of candidates). Thus, if all n voters have linear preference orderings among the m candidates then the total number of points obtained by all candidates is equal to the number of voters multiplied by the number of paired comparisons, i.e., nm(m-1)/2.
Tideman (2006, pp. 187–189) proposes two heuristic procedures that simplify the need to examine all m! preference orderings.
According to Kemeny (1959), the distance between two (individual) preference orderings, R and R′, is the number of pairs of candidates (alternatives) on which they differ. For example, if R = a > b > c > d and R′ = d > a > b > c, then the distance between R and R’ is 3, because they agree on three pairs [(a > b), (a > c), (b > c)], but differ on the remaining three pairs, i.e., on the preference ordering between a and d, b and d, and between c and d. Similarly, if R′′ is c > d > a > b, then the distance between R and R′′ is 4 and the distance between R’ and R′′ is 3. According to Kemeny’s procedure, the most likely social preference ordering is that R such that the sum of distances of the voters’ preference orderings from R is minimized. Because this R has the properties of the median central tendency in statistics it is called the median preference ordering. The median preference ordering (but not the mean preference ordering, which is that R which minimizes the sum of the squared differences between R and the voters’ preference orderings) will be identical to the possible (transitive) social preference ordering W which maximizes the sum of voters that agree with all paired comparisons implied by W.
This statement is inaccurate. Felsenthal and Tideman (2013) have demonstrated that two of the well-known Condorcet-consistent procedures that they studied (Dodgson’s and Nanson’s), are vulnerable to the P-TOP paradox but not also to the P-BOT paradox.
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Acknowledgments
The authors are grateful to two anonymous reviewers of this article for their positive feedback as well as for suggesting to them some examples and explanations regarding the vulnerability or invulnerability of some of the investigated voting methods to the P-TOP or P-BOT paradoxes. They are also grateful to Associate Editor G. Vanberg for helpful comments, and to the Editor-in-Chief, W.F. Shughart II, for his excellent copy-editing of the article.
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Felsenthal, D.S., Nurmi, H. Two types of participation failure under nine voting methods in variable electorates. Public Choice 168, 115–135 (2016). https://doi.org/10.1007/s11127-016-0352-5
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DOI: https://doi.org/10.1007/s11127-016-0352-5