How frequently do different voting rules encounter voting paradoxes in three-candidate elections?

See, for example, Merrill (1984), Chamberlin and Featherston (1986), Nurmi (1992, 1999), Stern (1993), and Tideman and Plassmann (2012).

For example, Gehrlein (2006, p. 104) writes: “[N]one of the studies referenced above have ever suggested that IAC, IC, DC or UC reflect reality in any particular situation” where IAC, IC, DC, and UC refer to models of vote-casting outcomes used in the vast majority of previous studies.

The spatial model can be viewed either as a model of voter behavior that explains why voters submit specific rankings or as a model of vote-casting outcomes with which to simulate voting situations. Previous research has focused on the spatial model as a model of voter behavior. In contrast, we use the spatial model only because of its ability to simulate voting situations with the same distribution as observed voting situations, not because we want to defend it as a model of voter behavior.

See Tideman (2006, pp. 57–74) for details.

Approval Voting was listed on only 15 of the 22 ballots. However, Approval Voting received 5 more votes than the voting rule with the second-most votes: the Alternative Vote. It is notable that, unlike the question of who should be mayor, the question of which voting rules deserve further study is inherently a question that invites a plural answer and is therefore more likely to be regarded as answered suitably through Approval Voting.

Nurmi (1999, p. 120) also classifies voting paradoxes into four categories. His and our first two categories are the same, but his and our last two categories are different.

In contrast, a weak Condorcet winner is a candidate who is not beaten by any other candidates in pairwise comparison, using majority rule.

For example, Felsenthal (2012, p. 33) writes: “Although assessing the severity of the various paradoxes is largely a subjective matter, there seems to be a wide consensus that a voting procedure which is susceptible to an especially serious paradox ... i.e., a voting procedure which may elect a pareto-dominated candidate, or elect a Condorcet (and absolute) loser, or display lack of monotonicity, or not elect an absolute winner, should be disqualified as a reasonable voting procedure regardless of the probability that these paradoxes may occur.” Felsenthal (2012, p. 21) writes: “...we hold that a Condorcet winner, if one exists, ought always to be elected.”

See, for example, Fishburn (1974a) and Saari (2001).

Suppose that a weak monotonicity paradox occurs when three voters change their ballots. Then consider the voting situation after two of those changes have been made. This voting situation provides an instance of the strong version of the paradox because the paradox occurs when the third voter changes his ballot. However, we should expect to observe occurrences of the weak versions of the monotonicity paradoxes more frequently than occurrences of the strong versions, because a weak version occurs every time when the strong version occurs, but not vice versa.

Some voting rules—for example, the Kemeny rule and the Coombs rule (see Sect. 3.2)—ignore information from ballots with a single candidate.

Consider an election with 11 voters and three candidates, \(A,\,B\), and \(C\), and voting situation ABC (3), ACB (2), CAB (3), CBA (1), BCA (2), and BAC (0). Candidate \(C\) wins under the Nanson rule (see Sect. 3.2). Candidate \(C\) still wins if one of the two voters with ranking ACB truncates his ballot, which we record as 3.5 votes for ABC and 1.5 votes for ACB. If the other voter also truncates his ballot so that there are 4 votes for ABC and 1 vote for ACB, then the Nanson rule declares \(A\) the winner. If the starting voting situation can contain only integers (that is, if we cannot start with truncated ballots), then this example describes a violation of the weak truncation paradox but not of the strong truncation paradox.

With our exclusion of truncated ballots from the election results that we simulate, we find that the Nanson rule does not encounter the strong truncation paradox at all, while the Copeland rule and the Alternative Schwartz rule do not encounter the strong truncation paradox when the number of voters is odd. However, all three rules encounter the weak truncation paradox for even and odd numbers of voters.

It would nevertheless be in the spirit by which other anomalies are labeled as voting paradoxes to count a tie as a “paradox.” One purpose of voting is to determine a winning candidate, and the voting rule, “paradoxically,” fails to achieve that purpose when there is a tie. This is arguably as paradoxical as the situation in which a voting rule fails to elect a Condorcet winner. Another voting event in this category occurs when an electorate is committed to majority rule, but there are more than two candidates and no candidate receives a majority of the votes.

The Alternative Vote can be described as an application of the Single Transferable Vote to the situation in which only a single candidate is to be elected. The Single Transferable Vote was proposed independently by Carl George Andrae in 1855 and by Thomas Hare in 1857.

For example, for the ranking \(A > B >C\), count (1) the number of ballots on which \(A\) is ranked ahead of \(B\), (2) the number of ballots on which \(A\) is ranked ahead of \(C\), and (3) the number of ballots on which \(B\) is ranked ahead of \(C\). The score of the ranking \(A> B > C\) is determined as (1) + (2) + (3).

See Tideman (2006, Ch. 13) for descriptions of these voting rules.

Tideman (2006, p. 237), Felsenthal (2012), and Nurmi (2012) report similar tables that include additional voting rules and additional paradoxes, and they provide proofs for the vulnerability of each voting rule to the paradoxes in Table 1. Nurmi (1999) establishes the vulnerability of various voting rules to different paradoxes without summarizing his results in a table. These three authors also provide examples of the vulnerability of each rule to the paradoxes listed in Table 1. Tideman (2006, Ch. 13) provides explanations for why different rules are not vulnerable to certain paradoxes.

For each simulated election, we choose randomly one of the six strict rankings as a tie-breaking ranking, and we resolve all ties between any two candidates according to the order in which these candidates are listed in the tie-breaking ranking. This procedure ensures that for any given voting situation we resolve all ties that we encounter during the evaluation of our 14 voting rules in the same way.

We determine the frequencies of the reinforcement paradox by sampling ballots for two separate elections, which we then combine to determine whether the paradox occurred for any voting rule. In contrast, we determine the frequencies of all other paradoxes from a single sample of ballots. There is no obvious way in which to divide the ballots from a single election into two distinct sets that would yield the appropriate distribution of rankings in the two new sets and would permit us to examine reinforcement. Similarly, we might not achieve the appropriate distribution of rankings if we routinely sample two separate sets of rankings that we then combine into a single set and which would permit us to examine the frequency of all anomalies besides reinforcement. However, the frequencies with which the reinforcement paradox occurs in elections with 20 and more voters (see Table 12) are so low that their omission has no more than a minor effect on the frequencies in Table 2.

For example, Article 1 of the US Constitution specifies that, in his function as president of the US Senate, the Vice President of the United States does not vote, except to break a tie.

For example, the 12th Amendment to the United States Constitution specifies that, if the Electoral College does not reach a majority decision (which may happen because of a tie) to elect the president of the United States, then “from the persons having the highest numbers not exceeding three on the list of those voted for as President, the House of Representatives shall choose immediately, by ballot, the President.”

For example, in elections for the French National Assembly in which no candidate receives a majority of the votes in the first round, each member of the electorate is asked to cast a vote for one of the candidates who received at least 12.5 % of the votes, and the winner is the candidate who receives a plurality of the votes.

For example, rather than declaring a mistrial, judges often send deadlocked juries back to the jury room for further deliberations.

One could also resolve the tie by assessing the Copeland scores [which are 0 \((A)\), 1 \((B)\), and -1 \((C)\)], the Dodgson scores [which are 2 \((A)\), 1 \((B)\), and 6 \((C)\)], or the Kemeny scores of the six rankings [which are 84 (ABC), 84 (ACB), 64 (CAB), 66 (CBA), 66 (BCA) and 86 (BAC)]. Candidate \(B\) would win the election in each case.

Avoiding ties by using a different voting rule is equivalent to the common solution to the “anomaly” that there might be no majority winner when an attempt is made to use majority rule to elect a winner from more than two candidates. Thus in elections with more than two candidates, most people advocate replacing majority rule with a different rule.

The Copeland rule encounters a tie whenever there is a voting cycle, and the spatial model of vote-casting outcomes implies that the frequency of voting cycles converges to about 0.085 % as the number of voters becomes large (see Plassmann and Tideman 2011).

This occurs most notably for the Anti-plurality rule and the Bucklin rule.

In elections with odd numbers of voters, the Bucklin rule is vulnerable to the weak version but not the strong version of the no-show paradox. In our simulations, all occurrences of the weak no-show paradox for the Bucklin rule involve the emergence of a strict majority rule winner when one or more voters do not vote. Because a strict majority rule winner can emerge only when the number of voters is reduced from an even to an odd number, the strong no-show paradox can occur only in elections with even numbers of voters. As an example, consider an election with 31 voters and voting situation ABC (10), ACB (5), CAB (4), CBA (5), BCA (4), and BAC (3). No candidate wins a majority of the votes (16), and candidates \(A\) and \(B\) are tied with equal Bucklin scores of 22. Candidate \(A\) will win outright with a strict majority of 15 votes if two of the voters with ranking CAB do not vote, because their abstentions reduce the number of voters to 29. The weak no-show paradox occurs because by abstaining these two voters can ensure a victory of their preferred candidate \(A\), instead of a tie. However, if only one voter with ranking CAB does not vote, then \(A\)’s 15 first-place votes are not sufficient to win a strict majority of the votes (16); instead \(A\)’s Bucklin score falls to 21 and \(B\) wins the election. Although the strong no-show paradox will occur when this voting situation is the starting situation and another voter with ranking CAB does not vote, the starting number of voters (30) is even in this case.

We can approximate the standard errors of estimate as follows: for 10,000 Bernoulli trials, the standard error of estimate of the probability of success \(p\) is \(\sqrt{p(1-p)}/100\), so it is 0.5 % when the probability is 50 %, 0.3 % when the probability is 10 %, and 0.1 % when the probability is 1 %. Thus, for example, for elections with 100 voters, the standard error of the Copeland rule is about 0.06 % given the frequency of 3.50 % with which the weak truncation paradox occurs.

Merrill (1984) also reports results from five-candidate elections simulated with a spatial model that was not calibrated to actual voting situations; Chamberlin and Cohen (1978) report results from a similar model with four candidates. Chamberlin and Featherston (1986) report results from four-candidate elections simulated with a model similar to IC but whose probabilities, constant across elections, for the six rankings differ from 1/6 and that were calibrated to observed voting situations from five presidential elections of the American Psychological Association. None of these results are directly comparable with ours because they apply to elections with more than three candidates.

Calibrating the spatial model to voting situations with four candidates requires numerical integration of non-central triangular wedges of a trivariate normal distribution, and we have not yet developed an algorithm to accomplish this task.

Owing to the difficulty of having to integrate wedges underneath multivariate normal distributions, we can currently evaluate the spatial model only for elections with three candidates which requires the (relative straightforward) integration of wedges underneath a bivariate normal distribution.

To match the variance of the angles that we determine from observed election data, we parameterize the Dirichlet distribution so that each of the three shares is multiplied by the common constant 73.5008. Dividing the three values in the text by this number yields three shares that sum to 1.

For example, if candidate \(A\) is the winner of voting situation \(R\), then the four pairs of rankings are (ABC, BAC), (ACB, CAB), (BAC, BCA), (CAB, CBA).

In three-candidate elections in which voters must submit full rankings, we treat a truncated ballot for candidate \(X\) as half a ballot for each of the two rankings with \(X \)first. This treatment permits us to examine truncated rankings with two rules, the Kemeny rule and the Coombs rule, that do not permit ballots listing only a single candidate.

The definition of SCC requires that the winner for \(R\) be the unique winner.