Abstract
A strong Condorcet winner (SCW) is an alternative, x, that a majority of voters rank higher than z, for every other alternative, z. A weak Condorcet winner (WCW) is an alternative, y, that no majority of voters rank below any other alternative, z, but is not a SCW. There has been some confusion in the voting/social choice literature as to whether particular voting rules that are SCW-consistent are also WCW-consistent. The purpose of this paper is to revisit this issue, clear up the confusion that has developed, and determine whether three additional SCW-consistent voting rules—that as far as we know have not been investigated to date regarding their possible WCW consistency—are indeed WCW-consistent.
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Notes
If indifference in voters’ rankings is permitted then ties are possible with odd numbers as well as even numbers of voters.
The list of relevant publications is vast and we mention here only a relatively small sample. A reader interested in finding additional analyses is advised to look for the “Condorcet criterion for evaluating voting rules” using the search engine Google Scholar on the Internet.
As far as we know, there is no SCW-consistent voting rule that is employed in public elections.
Jean Charles de Borda suggested his election method to the French Royal Academy in 1770 in a paper entitled (in English translation) “On Elections by Ballot”. This paper was published in the annals of the Academy 14 years later. (cf. de Borda 1784). According to the method that Borda proposed, all voters submit ballots that rank all n candidates. Thereafter one assigns to every candidate n − 1 points for every ballot in which s/he is ranked first, n − 2 points for every ballot in which s/he is ranked second, and so on, and 0 points for every ballot in which s/he is ranked last. The Borda score of every candidate is the sum of points s/he received over all ballots (Cf. McLean and Urken 1995, pp. 83–89). .
Several authors, e.g., Schwartz (1986, p. 180), Nurmi (1987, p. 46; 1995, pp. 101–102), Saari (2000, p. 40) and Tideman (2006, p. 201), state erroneously that according to the Nanson method either one eliminates at the end of each round only the candidate with the lowest Borda score, or that one eliminates at the end of every round all candidates whose Borda score is lower than the average Borda score—rather than all candidates whose Borda score is equal to or less than the average Borda score. Although Fishburn (1977, p. 474) was aware of Nanson’s original method he nevertheless gave the name “Nanson” to a method which, he stated, “is similar in spirit to Nanson’s original method” whereby “At each stage in a sequential process we delete all candidates who have the lowest Borda count at that stage unless all have the same Borda count” (p. 473). This procedure, which was originally proposed by Baldwin (1926) (cf. also McLean 2002; http://en.wikipedia.org/wiki/Nanson’s_method; and http://en.wikipedia.org/wiki/Joseph_M._Baldwin), is similar to Nanson’s but, as shown by Niou (1987), may result, ceteris paribus, in a different outcome than the one obtained under Nanson’s method. Fishburn (1990) clarifies his understanding of the Nanson rule. Both the Baldwin rule and the Nanson rule are known in the literature not to be WCW-consistent.
Dodgson did not use the term “top cycle” but it is clear that this is what he meant in rule V of his proposed rules (see Black 1958, p. 225). Following Smith (1973), if no SCW exists, the top cycle is the smallest subset of the candidates such that every candidate in the subset beats every candidate not in the subset, in head-to-head comparisons.
This is not ungrammatical if the sentence is read as being in the present subjunctive mood, which, at the time that Dodgson wrote, was common in formal English for conditions that might or might not obtain.
When the candidate with the fewest votes is determined by analyzing the remaining candidates on the basis of voters’ rankings rather than from iterated plurality votes, this rule becomes the alternative vote.
Fishburn (1977, p. 472) calls this voting rule ‘Condorcet’s function’. Young (1977, p. 349) prefers to call this procedure ‘The Minimax function’. Other authors (e.g., Tideman 2006, p. 212) prefer to call this method ‘the maximin rule’ because the above citation can perhaps be interpreted to mean that when no majority candidate exists, the winner should be that alternative whose minimal support in the paired comparisons against each of the other alternatives is largest.
Tideman (2006, pp. 187–189) proposes two heuristic procedures that simplify the need to examine all m! rankings.
According to Kemeny (1959) the “distance” between two rankings, 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 > a > 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 ranking a versus d, b versus d, and c versus 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. Kemeny’s procedure selects the ranking, R, that minimizes the sum of distances of the voters’ rankings from R. Because this R has the properties of the median central measure in statistics, it is called the median ranking. The median ranking will be identical to the ranking W that maximizes the sum, over all paired comparisons implied by W, of the number of voters who agree with the order of the pair implied by W.
See also Black (1958, p. 21) where Black calls this rule ‘Procedure α’.
For x to become a SCW the fourth voter should change his or her ranking such that, ceteris paribus, x is moved to constitute this voter’s second ranking – a total of three transpositions. For z to become a SCW the first and fourth voters should move z to constitute their top rank – also a total of three transpositions. .
Similar to Black’s rule, Fishburn (1977, pp. 471–472) mis-stated also Dodgson’s rule by asserting that according to this rule the candidate, x, who ought to be elected is the one needing the “fewest inversions in [its] linear order that will make [him or her] tie or beat every other candidate … on the basis of simple majority.” Of course Fishburn’s replacement of the word “beat” by the phrase “tie or beat” causes this rule to become WCW-consistent. So the conflict between Richelson and Fishburn regarding the WCW consistency of Dodgson’s rule is only apparent because Richelson investigated Dodgson’s original rule (which Example 2 shows is not WCW-consistent) while Fishburn investigated a “revised” version of this rule (which is WCW-consistent). Tideman (2006, pp. 199–201) proposed the “Simplified Dodgson Rule,” (see Sect. 4.3) that is both SCW-consistent and WCW-consistent.
From our summary of Dodgson’s contributions in Sect. 2.3 it may seem that his ideas boil down to be the first hybrid rule. We continue not to regard Dodgson as the first hybrid rule because there is a straightforward statement of it (vis. count the number of reversals that are needed) that entails no hybridization; and there is also the fact that it was not proposed as a rule until after Black proposed his. .
Tideman (2006, pp. 232–235) mentions two additional hybrid SCW-consistent rules which he labels ‘The Alternative Smith’ and ‘The Alternative Schwartz’ rules which are iterative alternations between the Smith (or GOCHA) sets, and the Alternative Vote rule. These are examples of rules that are not WCW-consistent but can easily be altered to be WCW-consistent if one wants such rules. Fishburn (1977, pp. 475–476) mentions a hybrid SCW-consistent rule which he attributes to Hoag and Hallett (1926, pp. 503–505) that he describes as follows:
[C]ompute first the Schwartz choice set to obtain a reduced choice set, then use Copeland’s method iteratively until a stable limit is reached; if this limit choice set contains several candidates, use Borda’s method of marks on the Schwartz–Copeland limit set to obtain a reduced choice set; if several candidates remain at this point, choose among them according to most first-place votes; if several still remain, choose among them by most second-place votes, and so forth; if two or more candidates remain after all this, choose a winner from among them by lot.
A Condorcet loser is a candidate who is ranked lower than any other candidate by the majority of voters. The possible election of a Condorcet Loser is considered as a very serious pathology that may characterize a voting rule. For other very serious pathologies afflicting SCW-consistent rules, see Felsenthal (2012, Table 3.3, p. 33).
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Felsenthal, D.S., Tideman, N. Weak Condorcet winner(s) revisited. Public Choice 160, 313–326 (2014). https://doi.org/10.1007/s11127-014-0180-4
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DOI: https://doi.org/10.1007/s11127-014-0180-4