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Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Negative voting is tantamount to allowing the voters to support either a single candidate or all but one candidate (taking a point away from a candidate \(C\) is equivalent to giving one point to all candidates except \(C\)). That is, the voters are asked to choose a set of candidates that they support, where the choice is between sets consisting of a single candidate or sets consisting of all except one candidate. The next voting method generalizes this idea by allowing voters to choose any subset of candidates:

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Approval Voting: Each voter selects a subset of the candidates (where the empty set means the voter abstains) and the candidate(s) with selected by the most voters wins.

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

If a candidate \(X\) is in the set of candidates selected by a voter, we say that the voter approves of candidate \(X\). Then, the approval winner is the candidate with the most approvals. Approval voting has been extensively discussed by Steven Brams and Peter Fishburn (Brams and Fishburn 2007; Brams 2008). See, also, the recent collection of articles devoted to approval voting (Laslier and Sanver 2010).

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Approval voting forces voters to think about the decision problem differently: They are asked to determine which candidates they approve of rather than selecting a single candidate to voter for or determining the relative ranking of the candidates. That is, the voters are asked which candidates are above a certain “threshold of acceptance”. Ranking a set of candidates and selecting the candidates that are approved are two different aspects of a voters overall opinion about the candidates. They are related but cannot be derived from each other. See Brams and Sanver 2009, for examples of voting methods that ask voters to both select a set of candidates that they approve and to (linearly) rank the candidates.

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Approval voting is a very flexible method. Recall the election scenario illustrating the \(k\)-Approval Voting methods:

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

In this election scenario, \(k\)-Approval for \(k=1,2,3\) cannot guarantee that the Condorcet winner \(A\) is elected. The Approval ballot \((\{A\},\{B\}, \{A, C\})\) does elect the Condorcet winner. In fact, Brams (2008, Chapter 2) proves that if there is a unique Condorcet winner, then that candidate may be elected under approval voting (assuming that all voters vote sincerely: see Brams 2008, Chapter 2, for a discussion). Note that approval voting may also elect other candidates (perhaps even the Condorcet loser). Whether this flexibility of Approval Voting should be seen as a virtue or a vice is debated in Brams, Fishburn and Merrill 1988a, 1988b and Saari and van Newenhizen 1988a, 1988b.

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Approval Voting asks voters to express something about their intensity of preference for the candidates by assigning one of two grades: "Approve" or "Don’t Approve". Expanding on this idea, some voting methods assume that there is a fixed set of grades, or a grading language, that voters can assign to each candidate. See Chapters 7 and 8 from Balinksi and Laraki 2010 for examples and a discussion of grading languages (cf. Morreau 2016).

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

There are different ways to determine the winner(s) given a profile of ballots that assign grades to each candidate. The main approach is to calculate a "group" grade for each candidate, then select the candidate with the best overall group grade. In order to calculate a group grade for each candidate, it is convenient to use numbers for the grading language. Then, there are two natural ways to determine the group grade for a candidate: calculating the mean, or average, of the grades or calculating the median of the grades.

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Cumulative Voting: Each voter is asked to distribute a fixed number of points, say ten, among the candidates in any way they please. The candidate(s) with the most total points wins the election.

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Score Voting (also called Range Voting): The grades are a finite set of numbers. The ballots are an assignment of grades to the candidates. The candidate(s) with the largest average grade is declared the winner(s).

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Cumulative Voting and Score Voting are similar. The important difference is that Cumulative Voting requires that the sum of the grades assigned to the candidates by each voter is the same. The next procedure, proposed by Balinski and Laraki 2010 (cf. Bassett and Persky 1999 and the discussion of this method at rangevoting.org), selects the candidate(s) with the largest median grade rather than the largest mean grade.

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Majority Judgement: The grades are a finite set of numbers (cf. discussion of common grading languages). The ballots are an assignment of grades to the candidates. The candidate(s) with the largest median grade is(are) declared the winner(s). See Balinski and Laraki 2007 and 2010 for further refinements of this voting method that use different methods for breaking ties when there are multiple candidates with the largest median grade.

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

I conclude this section with an example that illustrates Score Voting and Majority Judgement. Suppose that there are 3 candidates \(\{A, B, C\}\), 5 grades \(\{0,1,2,3,4\}\) (with the assumption that the larger the number, the higher the grade), and 5 voters. The table below describes an election scenario. The candidates are listed in the first row. Each row describes an assignment of grades to a candidate by a set of voters.

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The bottom two rows give the mean and median grade for each candidate. Candidate \(A\) is the score voting winner with the greatest mean grade, and candidate \(B\) is the majority judgement winner with the greatest median grade.

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

There are two types of debates about the voting methods introduced in this section. The first concerns the choice of the grading language that voters use to evaluate the candidates. Consult Balinski and Laraki 2010 amd Morreau 2016 for an extensive discussion of the types of considerations that influence the choice of a grading language. Brams and Potthoff 2015 argue that two grades, as in Approval Voting, is best to avoid certain paradoxical outcomes. To illustrate, note that, in the above example, if the candidates are ranked by the voters according to the grades that are assigned, then candidate \(C\) is the Condorcet winner (since 3 voters assign higher grades to \(C\) than to \(A\) or \(B\)). However, neither Score Voting nor Majority Judgement selects candidate \(C\).

Voting Methods

2. Examples of Voting Methods

2.2 Voting by Grading

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The second type of debate concerns the method used to calculate the group grade for each candidate (i.e., whether to use the mean as in Score Voting or the median as in Majority Judgement). One important issue is whether voters have an incentive to misrepresent their evaluations of the candidates. Consider the voter in the middle column that assigns the grade of 2 to \(A\), 0 to \(B\), and 3 to \(C\). Suppose that these grades represents the voter’s true evaluations of the candidates. If this voter increases the grade for \(C\) to 4 and decreases the grade for \(A\) to 1 (and the other voters do not change their grades), then the average grade for \(A\) becomes 2.4 and the average grade for \(C\) becomes 2.6, which better reflects the voter’s true evaluations of the candidates (and results in \(C\) being elected according to Score Voting). Thus, this voter has an incentive to misrepresent her grades. Note that the median grades for the candidates do not change after this voter changes her grades. Indeed, Balinski and Laraki 2010, chapter 10, argue that using the median to assign group grades to candidates encourages voters to submit grades that reflect their true evaluations of the candidates. The key idea of their argument is as follows: If a voter’s true grade matches the median grade for a candidate, then the voter does not have an incentive to assign a different grade. If a voter’s true grade is greater than the median grade for a candidate, then raising the grade will not change the candidate’s grade and lowering the voter’s grade may result in the candidate receiving a grade that is lowering than the voter’s true evaluation. Similarly, if a voter’s true grade is lower than the median grade for a candidate, then lowering the grade will not change the candidate’s grade and raising the voter’s grade may result in the candidate receiving a grade that is higher than the voter’s true evaluation. Thus, if voters are focused on ensuring that the group grades for the candidates best reflects their true evaluations of the candidates, then voters do not have an incentive to misrepresent their grades. However, as pointed out in Felsenthal and Machover 2008 (Example 3.3), voters can manipulate the outcome of an election using Majority Judgement to ensure a preferred candidate is elected (cf. the discussion of strategic voting in Section 4.1 and Section 3.3 of List 2013). Suppose that the voter in the middle column assigns the grade of 4 to candidate \(A\), 0 to candidate \(B\) and 3 to candidate \(C\). Assuming the other voters do not change their grades, the majority judgement winner is now \(A\), which the voter ranks higher than the original majority judgement winner \(B.\) Consult Balinski and Laraki 2010, 2014 and Edelman 2012b for arguments in favor of electing candidates with the greatest median grade; and Felsenthal and Machover 2008, Gehrlein and Lepelley 2003, and Laslier 2011 for arguments against electing candidates with the greatest median grade.

Voting Methods

2. Examples of Voting Methods

2.3 Quadratic Voting and Liquid Democracy

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

In this section, I briefly discuss two new approaches to voting that do not fit nicely into the categories of voting methods introduced in the previous sections. While both of these methods can be used to select representatives, such as a president, the primary application is a group of people voting directly on propositions, or referendums.

Voting Methods

2. Examples of Voting Methods

2.3 Quadratic Voting and Liquid Democracy

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Quadratic Voting: When more than 50% of the voters support an alternative, most voting methods will select that alternative. Indeed, when there are only two alternatives, such as when voting for or against a proposition, there are many arguments that identify majority rule as the best and most stable group decision method (May 1952; Maskin 1995). One well-known problem with always selecting the majority winner is the so-called tyranny of the majority. A complete discussion of this issue is beyond the scope of this article. The main problem from the point of view of the analysis of voting methods is that there may be situations in which a majority of the voters weakly support a proposition while there is a sizable minority of voters that have a strong preference against the proposition.

Voting Methods

2. Examples of Voting Methods

2.3 Quadratic Voting and Liquid Democracy

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

One way of dealing with this problem is to increase the quota required to accept a proposition. However, this gives too much power to a small group of voters. For instance, with Unanimity Rule a single voter can block a proposal from being accepted. Arguably, a better solution is to use ballots that allow voters to express something about their intensity of preference for the alternatives. Setting aside issues about interpersonal comparisons of utility (see, for instance, Hausman 1995), this is the benefit of using the voting methods discussed in Section 2.2, such as Score Voting or Majority Judgement. These voting methods assume that there is a fixed set of grades that the voters use to express their intensity of preference. One challenge is finding an appropriate set of grades for a population of voters. Too few grades makes it harder for a sizable minority with strong preferences to override the majority opinion, but too many grades makes it easy for a vocal minority to overrule the majority opinion.

Voting Methods

2. Examples of Voting Methods

2.3 Quadratic Voting and Liquid Democracy

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Using ideas from mechanism design (Groves and Ledyard 1977; Hylland and Zeckhauser 1980), the economist E. Glen Weyl developed a voting method called Quadratic Voting that mitigates some of the above issues (Lalley and Weyl 2018a). The idea is to think of an election as a market (Posner and Weyl, 2018, Chapter 2). Each voter can purchase votes at a costs that is quadratic in the number of votes. For instance, a voter must pay $25 for 5 votes (either in favor or against a proposition). After the election, the money collected is distributed on a pro rata basis to the voters. There are a variety of economic arguments that justify why voters should pay \(v^2\) to purchase \(v\) votes (Lalley and Weyl 2018b; Goeree and Zhang 2017). See Posner and Weyl 2015 and 2017 for further discussion and a vigorous defense of the use of Quadratic Voting in national elections. Consult Laurence and Sher 2017 for two arguments against the use of Quadratic Voting. Both arguments are derived from the presence of wealth inequality. The first argument is that it is ambiguous whether the Quadratic Voting decision really outperforms a decision using majority rule from the perspective of utilitarianism (see Driver 2014 and Sinnott-Armstrong 2019 for overviews of utilitarianism). The second argument is that any vote-buying mechanism will have a hard time meeting a legitimacy requirement, familiar from the theory of democratic institutions (cf. Fabienne 2017).

Voting Methods

2. Examples of Voting Methods

2.3 Quadratic Voting and Liquid Democracy

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Liquid Democracy: Using Quadratic Voting, the voters’ opinions may end up being weighted differently: Voters that purchase more of a voice have more influence over the election. There are other reasons why some voters’ opinions may have more weight than others when making a decision about some issue. For instance, a voter may have been elected to represent a constituency, or a voter may be recognized as an expert on the issue under consideration. An alternative approach to group decision making is direct democracy in which every citizen is asked to vote on every political issue. Asking the citizens to vote on every issue faces a number of challenges, nicely explained by Green-Armytage (2015, pg. 191):

Voting Methods

2. Examples of Voting Methods

2.3 Quadratic Voting and Liquid Democracy

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Direct democracy without any option for representation is problematic. Even if it were possible for every citizen to learn everything they could possibly know about every political issue, people who did this would be able to do little else, and massive amounts of time would be wasted in duplicated effort. Or, if every citizen voted but most people did not take the time to learn about the issues, the results would be highly random and/or highly sensitive to overly simplistic public relations campaigns. Or, if only a few citizens voted, particular demographic and ideological groups would likely be under-represented

Voting Methods

2. Examples of Voting Methods

2.3 Quadratic Voting and Liquid Democracy

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

One way to deal with some of the problems raised in the above quote is to use proxy voting, in which voters can delegate their vote on some issues (Miller 1969). Liquid Democracy is a form of proxy voting in which voters can delegate their votes to other voters (ideally, to voters that are well-informed about the issue under consideration). What distinguishes Liquid Democracy from proxy voting is that proxies may further delegate the votes entrusted to them. For example, suppose that there is a vote to accept or reject a proposition. Each voter is given the option to delegate their vote to another voter, called a proxy. The proxies, in turn, are given the option to delegate their votes to yet another voter. The voters that decide to not transfer their votes cast a vote weighted by the number of voters who entrusted them as a proxy, either directly or indirectly.

Voting Methods

2. Examples of Voting Methods

2.3 Quadratic Voting and Liquid Democracy

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

While there has been some discussion of proxy voting in the political science literature (Miller 1969; Alger 2006; Green-Armytage 2015), most studies of Liquid Democracy can be found in the computer science literature. A notable exception is Blum and Zuber 2016 that justifies Liquid Democracy, understood as a procedure for democratic decision-making, within normative democratic theory. An overview of the origins of Liquid Democracy and pointers to other online discussions can be found in Behrens 2017. Formal studies of Liquid Democracy have focused on: the possibility of delegation cycles and the relationship with the theory of judgement aggregation (Christoff and Grossi 2017); the rationality of delegating votes (Bloembergen, Grossi and Lackner 2018); the potential problems that arise when many voters delegate votes to only a few voters (Kang et al. 2018; Golz et al. 2018); and generalizations of Liquid Democracy beyond binary choices (Brill and Talmon 2018; Zhang and Zhou 2017).

Voting Methods

2. Examples of Voting Methods

2.4 Criteria for Comparing Voting Methods

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

This section introduced different methods for making a group decision. One striking fact about the voting methods discussed in this section is that they can identify different winners given the same collection of ballots. This raises an important question: How should we compare the different voting methods? Can we argue that some voting methods are better than others? There are a number of different criteria that can be used to compare and contrast different voting methods:

Voting Methods

3. Voting Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

In this section, I introduce and discuss a number of voting paradoxes — i.e., anomalies that highlight problems with different voting methods. Consult Saari 1995 and Nurmi 1999 for penetrating analyses that explain the underlying mathematics behind the different voting paradoxes.

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

A very common assumption is that a rational preference ordering must be transitive (i.e., if \(A\) is preferred to \(B\), and \(B\) is preferred to \(C\), then \(A\) must be preferred to \(C\)). See the entry on preferences (Hansson and Grüne-Yanoff 2009) for an extended discussion of the rationale behind this assumption. Indeed, if a voter’s preference ordering is not transitive, for instance, allowing for cycles (e.g., an ordering of \(A, B, C\) with \(A \succ B \succ C \succ A\), where \(X\succ Y\) means \(X\) is strictly preferred to \(Y\)), then there is no alternative that the voter can be said to actually support (for each alternative, there is another alternative that the voter strictly prefers). Many authors argue that voters with cyclic preference orderings have inconsistent opinions about the candidates and should be ignored by a voting method (in particular, Condorcet forcefully argued this point). A key observation of Condorcet (which has become known as the Condorcet Paradox) is that the majority ordering may have cycles (even when all the voters submit rankings of the alternatives).

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Condorcet’s original example was more complicated, but the following situation with three voters and three candidates illustrates the phenomenon:

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Note that we have:

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

That is, there is a majority cycle \(A>_M B >_M C >_M A\). This means that there is no Condorcet winner. This simple, but fundamental observation has been extensively studied (Gehrlein 2006; Schwartz 2018).

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The Condorcet Paradox shows that there may not always be a Condorcet winner in an election. However, one natural requirement for a voting method is that if there is a Condorcet winner, then that candidate should be elected. Voting methods that satisfy this property are called Condorcet consistent. Many of the methods introduced above are not Condorcet consistent. I already presented an example showing that plurality rule is not Condorcet consistent (in fact, plurality rule may even elect the Condorcet loser).

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The example from Section 1 shows that Borda Count is not Condorcet consistent. In fact, this is an instance of a general phenomenon that Fishburn (1974) called Condorcet’s other paradox. Consider the following voting situation with 81 voters and three candidates from Condorcet 1785.

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The majority ordering is \(A >_M B >_M C\), so \(A\) is the Condorcet winner. Using the Borda rule, we have:

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

So, candidate \(B\) is the Borda winner. Condorcet pointed out something more: The only way to elect candidate \(A\) using any scoring method is to assign more points to candidates ranked second than to candidates ranked first. Recall that a scoring method for 3 candidates fixes weights \(s_1\ge s_2\ge s_3\), where \(s_1\) points are assigned to candidates ranked 1st, \(s_2\) points are assigned to candidates ranked 2nd, and \(s_3\) points are assigned to candidates ranked last. To simplify the calculation, assume that candidates ranked last receive 0 points (i.e., \(s_3=0\)). Then, the scores assigned to candidates \(A\) and \(B\) are:

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

So, in order for \(Score(A) > Score(B)\), we must have \((s_1 \times 31 + s_2 \times 39) > (s_1 \times 39 + s_2 \times 31)\), which implies that \(s_2 > s_1\). But, of course, it is counterintuitive to give more points for being ranked second than for being ranked first. Peter Fishburn generalized this example as follows:

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Theorem (Fishburn 1974). For all \(m\ge 3\), there is some voting situation with a Condorcet winner such that every scoring rule will have at least \(m-2\) candidates with a greater score than the Condorcet winner.

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

So, no scoring rule is Condorcet consistent, but what about other methods? A number of voting methods were devised specifically to guarantee that a Condorcet winner will be elected, if one exists. The examples below give a flavor of different types of Condorcet consistent methods. (See Brams and Fishburn, 2002, and Fishburn, 1977, for more examples and a discussion of Condorcet consistent methods.)

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The last method was proposed by Charles Dodgson (better known by the pseudonym Lewis Carroll). Interestingly, this is an example of a procedure in which it is computationally difficult to compute the winner (that is, the problem of calculating the winner is NP-complete). See Bartholdi et al. 1989 for a discussion.

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

These voting methods (and the other Condorcet consistent methods) guarantee that a Condorcet winner, if one exists, will be elected. But, should a Condorcet winner be elected? Many people argue that there is something amiss with a voting method that does not always elect a Condorcet winner (if one exists). The idea is that a Condorcet winner best reflects the overall group opinion and is stable in the sense that it will defeat any challenger in a one-on-one contest using Majority Rule. The most persuasive argument that the Condorcet winner should not always be elected comes from the work of Donald Saari (1995, 2001). Consider again Condorcet’s example of 81 voters.

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

This is another example that shows that Borda’s method need not elect the Condorcet winner. The majority ordering is

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

while the ranking given by the Borda score is

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

However, there is an argument that candidate \(B\) is the best choice for this electorate. Saari’s central observation is to note that the 81 voters can be divided into three groups:

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Groups 1 and 2 constitute majority cycles with the voters evenly distributed among the three possible rankings. Such profiles are called Condorcet components. These profiles form a perfect symmetry among the rankings. So, within each of these groups, it is natural to assume that the voters’ opinions cancel each other out; therefore, the decision should depend only on the voters in group 3. In group 3, candidate \(B\) is the clear winner.

Voting Methods

3. Voting Paradoxes

3.1 Condorcet’s Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Balinski and Laraki (2010, pgs. 74–83) have an interesting spin on Saari’s argument. Let \(V\) be a ranking voting method (i.e., a voting method that requires voters to rank the alternatives). Say that \(V\) cancels properly if for all profiles \(\bR\), if \(V\) selects \(A\) as a winner in \(\bP\), then \(V\) selects \(A\) as a winner in any profile \(\bP+\bC\), where \(\bC\) is a Condorcet component and \(\bP+\bC\) is the profile that contains all the rankings from \(\bP\) and \(\bC\). Balinski and Laraki (2010, pg. 77) prove that there is no Condorcet consistent voting method that cancels properly. (See the discussion of the multiple districts paradox in Section 3.3 for a proof of a closely related result.)

Voting Methods

3. Voting Paradoxes

3.2 Failures of Monotonicity

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

A voting method is monotonic provided that receiving more support from the voters is always better for a candidate. There are different ways to make this idea precise (see Fishburn, 1982, Sanver and Zwicker, 2012, and Felsenthal and Tideman, 2013). For instance, moving up in the rankings should not adversely affect a candidate’s chances to win an election. It is easy to see that Plurality Rule is monotonic in this sense: The more voters that rank a candidate first, the better chance the candidate has to win. Surprisingly, there are voting methods that do not satisfy this natural property. The most well-known example is Plurality with Runoff. Consider the two scenarios below. Note that the only difference between the them is the ranking of the fourth group of voters. This group of two voters ranks \(B\) above \(A\) above \(C\) in scenario 1 and swaps \(B\) and \(A\) in scenario 2 (so, \(A\) is now their top-ranked candidate; \(B\) is ranked second; and \(C\) is still ranked third).

Voting Methods

3. Voting Paradoxes

3.2 Failures of Monotonicity

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

In scenario 1, candidates \(A\) and \(B\) both have a plurality score of 6 while candidate \(C\) has a plurality score of 5. So, \(A\) and \(B\) move on to the runoff election. Assuming the voters do not change their rankings, the 5 voters that rank \(C\) transfer their support to candidate \(A\), giving her a total of 11 to win the runoff election. However, in scenario 2, even after moving up in the rankings of the fourth group (\(A\) is now ranked first by this group), candidate \(A\) does not win this election. In fact, by trying to give more support to the winner of the election in scenario 1, rather than solidifying \(A\)’s win, the last group’s least-preferred candidate ended up winning the election! The problem arises because in scenario 2, candidates \(A\) and \(B\) are swapped in the last group’s ranking. This means that \(A\)’s plurality score increases by 2 and \(B\)’s plurality score decreases by 2. As a consequence, \(A\) and \(C\) move on to the runoff election rather than \(A\) and \(B\). Candidate \(C\) wins the runoff election with 9 voters that rank \(C\) above \(A\) compared to 8 voters that rank \(A\) above \(C\).

Voting Methods

3. Voting Paradoxes

3.2 Failures of Monotonicity

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The above example is surprising since it shows that, when using Plurality with Runoff, it may not always be beneficial for a candidate to move up in some of the voter’s rankings. The other voting methods that violate monotonicity include Coombs Rule, Hare Rule, Dodgson’s Method and Nanson’s Method. See Felsenthal and Nurmi 2017 for further discussion of voting methods that are not monotonic.

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

In this section, I discuss two related paradoxes that involve changes to the population of voters.

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

No-Show Paradox: One way that a candidate may receive “more support” is to have more voters show up to an election that support them. Voting methods that do not satisfy this version of monotonicity are said to be susceptible to the no-show paradox (Fishburn and Brams 1983). Suppose that there are 3 candidates and 11 voters with the following rankings:

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

In the first round, candidates \(A\) and \(C\) are both ranked first by 4 voters while \(B\) is ranked first by only 3 voters. So, \(A\) and \(C\) move to the runoff round. In this round, the voters in the second column transfer their votes to candidate \(C\), so candidate \(C\) is the winner beating \(A\) 7-4. Suppose that 2 voters in the first group do not show up to the election:

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

In this election, candidate \(A\) has the lowest plurality score in the first round, so candidates \(B\) and \(C\) move to the runoff round. The first group’s votes are transferred to \(B\), so \(B\) is the winner beating \(C\) 5-4. Since the 2 voters that did not show up to this election rank \(B\) above \(C\), they prefer the outcome of the second election in which they did not participate!

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Plurality with Runoff is not the only voting method that is susceptible to the no-show paradox. The Coombs Rule, Hare Rule and Majority Judgement (using the tie-breaking mechanism from Balinski and Laraki 2010) are all susceptible to the no-show paradox. It turns out that always electing a Condorcet winner, if one exists, makes a voting method susceptible to the above failure of monotonicity.

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Theorem (Moulin 1988). If there are four or more candidates, then every Condorcet consistent voting method is susceptible to the no-show paradox.

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

See Perez 2001, Campbell and Kelly 2002, Jimeno et al. 2009, Duddy 2014, Brandt et al. 2017, 2019, and Nunez and Sanver 2017 for further discussions and generalizations of this result.

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Multiple Districts Paradox: Suppose that a population is divided into districts. If a candidate wins each of the districts, one would expect that candidate to win the election over the entire population of voters (assuming that the two districts divide the set of voters into disjoint sets). This is certainly true for Plurality Rule: If a candidate is ranked first by the most voters in each of the districts, then that candidate will also be ranked first by a the most voters over the entire population. Interestingly, this is not true for all voting methods (Fishburn and Brams 1983). The example below illustrates the paradox for Coombs Rule.

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Candidate \(B\) wins both districts:

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Combining the two districts gives the following table:

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

There are 15 total voters in the combined districts. None of the candidates are ranked first by 8 or more of the voters. Candidate \(C\) receives the most last-place votes, so is eliminated in the first round. In the second round, candidate \(A\) is beats candidate \(B\) by 1 vote (8 voters rank \(A\) above \(B\) and 7 voters rank \(B\) above \(A\)), and so is declared the winner. Thus, even though \(B\) wins both districts, candidate \(A\) wins the election when the districts are combined.

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The other voting methods that are susceptible to the multiple-districts paradox include Plurality with Runoff, The Hare Rule, and Majority Judgement. Note that these methods are also susceptible to the no-show paradox. As is the case with the no-show paradox, every Condorcet consistent voting method is susceptible to the multiple districts paradox (see Zwicker, 2016, Proposition 2.5). I sketch the proof of this from Zwicker 2016 (pg. 40) since it adds to the discussion at the end of Section 3.1 about whether the Condorcet winner should be elected.

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Suppose that \(V\) is a voting method that always selects the Condorcet winner (if one exists) and that \(V\) is not susceptible to the multiple-districts paradox. This means that if a candidate \(X\) is among the winners according to \(V\) in each of two districts, then \(X\) must be among the winners according to \(V\) in the combined districts. Consider the following two districts.

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Note that in district 2 candidate \(B\) is the Condorcet winner, so must be the only winner according to \(V\). In district 1, there are no Condorcet winners. If candidate \(B\) is among the winners according to \(V\), then, in order to not be susceptible to the multiple districts paradox, \(B\) must be among the winners in the combined districts. In fact, since \(B\) is the only winner in district 2, \(B\) must be the only winner in the combined districts. However, in the combined districts, candidate \(A\) is the Condorcet winner, so must be the (unique) winner according to \(V\). This is a contradiction, so \(B\) cannot be among the winners according to \(V\) in district 1. A similar argument shows that neither \(A\) nor \(C\) can be among the winners according to \(V\) in district 1 by swapping \(A\) and \(B\) in the first case and \(B\) with \(C\) in the second case in the rankings of the voters in district 2. Since \(V\) must assign at least one winner to every profile, this is a contradiction; and so, \(V\) is susceptible to the multiple districts paradox.

Voting Methods

3. Voting Paradoxes

3.3 Variable Population Paradoxes

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

One last comment about this paradox: It is an example of a more general phenomenon known as Simpson’s Paradox (Malinas and Bigelow 2009). See Saari (2001, Section 4.2) for a discussion of Simpson’s Paradox in the context of voting theory.

Voting Methods

3. Voting Paradoxes

3.4 The Multiple Elections Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The paradox discussed in this section, first introduced by Brams, Kilgour and Zwicker (1998), has a somewhat different structure from the paradoxes discussed above. Voters are taking part in a referendum, where they are asked their opinion directly about various propositions (cf. the discussion of Quadratic Voting and Liquid Democracy in Section 2.3). So, voters must select either “yes” (Y) or “no” (N) for each proposition. Suppose that there are 13 voters who cast the following votes for the three propositions (so voters can cast one of eight possible votes):

Voting Methods

3. Voting Paradoxes

3.4 The Multiple Elections Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

When the votes are tallied for each proposition separately, the outcome is N for each proposition (N wins 7–6 for all three propositions). Putting this information together, this means that NNN is the outcome of this election. However, there is no support for this outcome in this population of voters. This raises an important question about what outcome reflects the group opinion: Viewing each proposition separately, there is clear support for N on each proposition; however, there is no support for the entire package of N for all propositions. Brams et al. (1998, pg. 234) nicely summarise the issue as follows:

Voting Methods

3. Voting Paradoxes

3.4 The Multiple Elections Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The paradox does not just highlight problems of aggregation and packaging, however, but strikes at the core of social choice—both what it means and how to uncover it. In our view, the paradox shows there may be a clash between two different meanings of social choice, leaving unsettled the best way to uncover what this elusive quantity is.

Voting Methods

3. Voting Paradoxes

3.4 The Multiple Elections Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

See Scarsini 1998, Lacy and Niou 2000, Xia et al. 2007, and Lang and Xia 2009 for further discussion of this paradox.

Voting Methods

3. Voting Paradoxes

3.4 The Multiple Elections Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

A similar issue is raised by Anscombe’s paradox (Anscombe 1976), in which:

Voting Methods

3. Voting Paradoxes

3.4 The Multiple Elections Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

It is possible for a majority of voters to be on the losing side of a majority of issues.

Voting Methods

3. Voting Paradoxes

3.4 The Multiple Elections Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

This phenomenon is illustrated by the following example with five voters voting on three different issues (the voters either vote ‘yes’ or ‘no’ on the different issues).

Voting Methods

3. Voting Paradoxes

3.4 The Multiple Elections Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

However, a majority of the voters (voters 1, 2 and 3) do not support the majority outcome on a majority of the issues (note that voter 1 does not support the majority outcome on issues 2 and 3; voter 2 does not support the majority outcome on issues 1 and 3; and voter 3 does not support the majority outcome on issues 1 and 2)!

Voting Methods

3. Voting Paradoxes

3.4 The Multiple Elections Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The issue is more interesting when the voters do not vote directly on the issues, but on candidates that take positions on the different issues. Suppose there are two candidates \(A\) and \(B\) who take the following positions on the three issues:

Voting Methods

3. Voting Paradoxes

3.4 The Multiple Elections Paradox

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Candidate \(A\) takes the majority position, agreeing with a majority of the voters on each issue, and candidate \(B\) takes the opposite, minority position. Under the natural assumption that voters will vote for the candidate who agrees with their position on a majority of the issues, candidate \(B\) will win the election (each of the voters 1, 2 and 3 agree with \(B\) on two of the three issues, so \(B\) wins the election 3–2)! This version of the paradox is known as Ostrogorski’s Paradox (Ostrogorski 1902). See Kelly 1989; Rae and Daudt 1976; Wagner 1983, 1984; and Saari 2001, Section 4.6, for analyses of this paradox, and Pigozzi 2005 for the relationship with the judgement aggregation literature (List 2013, Section 5).

Voting Methods

4. Topics in Voting Theory

4.1 Strategizing

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

In the discussion above, I have assumed that voters select ballots sincerely. That is, the voters are simply trying to communicate their opinions about the candidates under the constraints of the chosen voting method. However, in many contexts, it makes sense to assume that voters choose strategically. One need only look to recent U.S. elections to see concrete examples of strategic voting. The most often cited example is the 2000 U.S. election: Many voters who ranked third-party candidate Ralph Nader first voted for their second choice (typically Al Gore). A detailed overview of the literature on strategic voting is beyond the scope of this article (see Taylor 2005 and Section 3.3 of List 2013 for discussions and pointers to the relevant literature; also see Poundstone 2008 for an entertaining and informative discussion of the occurrence of this phenomenon in many actual elections). I will explain the main issues, focusing on specific voting rules.

Voting Methods

4. Topics in Voting Theory

4.1 Strategizing

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

There are two general types of manipulation that can be studied in the context of voting. The first is manipulation by a moderator or outside party that has the authority to set the agenda or select the voting method that will be used. So, the outcome of an election is not manipulated from within by unhappy voters, but, rather, it is controlled by an outside authority figure. To illustrate this type of control, consider a population with three voters whose rankings of four candidates are given in the table below:

Voting Methods

4. Topics in Voting Theory

4.1 Strategizing

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Note that everyone prefers candidate \(B\) over candidate \(D\). Nonetheless, a moderator can ask the right questions so that candidate \(D\) ends up being elected. The moderator proceeds as follows: First, ask the voters if they prefer candidate \(A\) or candidate \(B\). Since the voters prefer \(A\) to \(B\) by a margin of 2 to 1, the moderator declares that candidate \(B\) is no longer in the running. The moderator then asks voters to choose between candidate \(A\) and candidate \(C\). Candidate \(C\) wins this election 2–1, so candidate \(A\) is removed. Finally, in the last round the chairman asks voters to choose between candidates \(C\) and \(D\). Candidate \(D\) wins this election 2–1 and is declared the winner.

Voting Methods

4. Topics in Voting Theory

4.1 Strategizing

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

A second type of manipulation focuses on how the voters themselves can manipulate the outcome of an election by misrepresenting their preferences. Consider the following two election scenarios with 7 voters and 3 candidates:

Voting Methods

4. Topics in Voting Theory

4.1 Strategizing

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The only difference between the two election scenarios is that the third voter changed the ranking of the bottom three candidates. In election scenario 1, the third voter has candidate \(A\) ranked first, then \(C\) ranked second, \(B\) ranked third and \(D\) ranked last. In election scenario 2, this voter still has \(A\) ranked first, but ranks \(B\) second, \(D\) third and \(C\) last. In election scenario 1, candidate \(C\) is the Borda Count winner (the Borda scores are \(\BS(A)=9, \BS(B)=5, \BS(C)=10\), and \(\BS(D)=6\)). In the election scenario 2, candidate \(A\) is the Borda Count winner (the Borda scores are \(\BS(A)=9, \BS(B)=6, \BS(C)=8\), and \(\BS(D)=7\)). According to her ranking in election scenario 1, this voter prefers the outcome in election scenario 2 (candidate \(A\), the Borda winner in election scenario 2, is ranked above candidate \(C\), the Borda winner in election scenario 1). So, if we assume that election scenario 1 represents the “true” preferences of the electorate, it is in the interest of the third voter to misrepresent her preferences as in election scenario 2. This is an instance of a general result known as the Gibbard-Satterthwaite Theorem (Gibbard 1973; Satterthwaite 1975): Under natural assumptions, there is no voting method that guarantees that voters will choose their ballots sincerely (for a precise statement of this theorem see Theorem 3.1.2 from Taylor 2005 or Section 3.3 of List 2013).

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Much of the literature on voting theory (and, more generally, social choice theory) is focused on so-called axiomatic characterization results. The main goal is to characterize different voting methods in terms of abstract principles of collective decision making. See Pauly 2008 and Endriss 2011 for interesting discussions of axiomatic characterization results from a logician’s point-of-view.

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Consult List 2013 and Gaertner 2006 for introductions to the vast literature on axiomatic characterizations in social choice theory. In this article, I focus on a few key axioms and results and how they relate to the voting methods and paradoxes discussed above. I start with three core principles.

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

These properties ensure that the outcome of an election depends only on the voters’ ballots, with all the voters and candidates being treated equally. Other properties are intended to rule out some of the paradoxes and anomalies discussed above. In section 4.1, there is an example of a situation in which a candidate is elected, even though all the voters prefer a different candidate. The next principle rules out such situations:

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Unanimity (also called the Pareto Principle): If candidate \(A\) is ranked above candidate \(B\) by all voters, then candidate \(B\) should not win the election.

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

These are natural properties to impose on any voting method. A surprising consequence of these properties is that they rule out another natural property that one may want to impose: Say that a voting method is resolute if the method always selects one winner (i.e., there are no ties). Suppose that \(V\) is a voting method that requires voters to rank the candidates and that there are at least 3 candidates and enough voters to form a Condorcet component (a profile generating a majority cycle with voters evenly distributed among the different rankings). First, consider the situation when there are exactly 3 candidates (in this case, we do not need to assume Unanimity). Divide the set of voters into three groups of size \(n\) and consider the Condorcet component:

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

By Universal Domain and resoluteness, \(V\) must select exactly one of \(A\), \(B\), or \(C\) as the winner. Assume that \(V\) select \(A\) as the winner (the argument when \(V\) selects the other candidates is similar). Now, consider the profile in which every voter swaps candidate \(A\) and \(B\) in their rankings:

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

By Neutrality and Universal Domain, \(V\) must elect candidate \(B\) in this election scenario. Now, consider the profile in which every voter in the above election scenario swaps candidates \(B\) and \(C\):

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

By Neutrality and Universal Domain, \(V\) must elect candidate \(C\) in this election scenario. Notice that this last election scenario can be generated by permuting the voters in the first election scenario (to generate the last election scenario from the first election scenario, move the first group of voters to the 2nd position, the 2nd group of voters to the 3rd position and the 3rd group of voters to the first position). But this contradicts Anonymity since this requires \(V\) to elect the same candidate in the first and third election scenario. To extend this result to more than 3 candidates, consider a profile in which candidates \(A\), \(B\), and \(C\) are all ranked above any other candidate and the restriction to these three candidates forms a Condorcet component. If \(V\) satisfies Unanimity, then no candidate except \(A\), \(B\) or \(C\) can be elected. Then, the above argument shows that \(V\) cannot satisfy Resoluteness, Universal Domain, Neutrality, and Anonymity. That is, there are no Resolute voting methods that satisfy Universal Domain, Anonymity, Neutrality, and Unanimity for 3 or more candidates (note that I have assumed that the number of voters is a multiple of 3, see Moulin 1983 for the full proof).

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Section 3.2 discussed examples in which candidates end up losing an election as a result of more support from some of the voters. There are many ways to state properties that require a voting method to be monotonic. The following strong version (called Positive Responsiveness in the literature) is used to characterize majority rule when there are only two candidates:

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Positive Responsiveness: If candidate \(A\) is a winner or tied for the win and moves up in some of the voter’s rankings, then candidate \(A\) is the unique winner.

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

I can now state our first characterization result. Note that in all of the example discussed above, it is crucial that there are three or more candidates (for example, stating Condorcet’s paradox requires there to be three or more candidates). When there are only two candidates, or alternatives, Majority Rule (choose the alternative ranked first by more than 50% of the voters) can be singled out as “best”:

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Theorem (May 1952). A voting method for choosing between two candidates satisfies Neutrality, Anonymity, Unanimity and Positive Responsiveness if and only if the method is majority rule.

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

See May 1952 for a precise statement of this theorem and Asan and Sanver 2002, Maskin 1995, and Woeginger 2003 for alternative characterizations of majority rule.

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

A key assumption in the proof May’s theorem and subsequent results is the restriction to voting on two alternatives. When there are only two alternatives, the definition of a ballot can be simplified since a ranking of two alternatives boils down to selecting the alternative that is ranked first. The above characterizations of Majority Rule work in a more general setting since they also allow voters to abstain (which is ambiguous between not voting and being indifferent between the alternatives). So, if the alternatives are \(\{A,B\}\), then there are three possible ballots: selecting \(A\), selecting \(B\), or abstaining (which is treated as selecting both \(A\) and \(B\)). A natural question is whether there are May-style characterization theorems for more than two alternatives. A crucial issue is that rankings of more than two alternatives are much more informative than selecting an alternative or abstaining. By restricting the information required from a voter to selecting one of the alternatives or abstaining, Goodin and List 2006 prove that the axioms used in May’s Theorem characterize Plurality Rule when there are more than two alternatives. They also show that a minor modification of the axioms characterize Approval Voting when voters are allowed to select more than one alternative.

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Note that focusing on voting methods that limit the information required from the voters to selecting one or more of the alternatives hides all the interesting phenomena discussed in the previous sections, such as the existence of a Condorcet paradox. Returning to the study of voting methods that require voters to rank the alternatives, the most important characterization result is Ken Arrow’s celebrated impossibility theorem (1963). Arrow showed that there is no social welfare function (a social welfare function maps the voters’ rankings (possibly allowing ties) to a single social ranking) satisfying universal domain, unanimity, non-dictatorship (there is no voter \(d\) such that for all profiles, if \(d\) ranks \(A\) above \(B\) in the profile, then the social ordering ranks \(A\) above \(B\)) and the following key property:

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Independence of Irrelevant Alternatives: The social ranking (higher, lower, or indifferent) of two candidates \(A\) and \(B\) depends only on the relative rankings of \(A\) and \(B\) for each voter.

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

This means that if the voters’ rankings of two candidates \(A\) and \(B\) are the same in two different election scenarios, then the social rankings of \(A\) and \(B\) must be the same. This is a very strong property that has been extensively criticized (see Gaertner, 2006, for pointers to the relevant literature, and Cato, 2014, for a discussion of generalizations of this property). It is beyond the scope of this article to go into detail about the proof and the ramifications of Arrow’s theorem (see Morreau, 2014, for this discussion), but I note that many of the voting methods we have discussed do not satisfy the above property. A striking example of a voting method that does not satisfy Independence of Irrelevant Alternatives is Borda Count. Consider the following two election scenarios:

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Notice that the relative rankings of candidates \(A\), \(B\) and \(C\) are the same in both election scenarios. In the election scenario 2, the ranking of candidate \(X\), that is uniformly ranked in last place in election scenario 1, is changed. The ranking according to the Borda score of the candidates in election scenario 1 puts \(A\) first with 15 points, \(B\) second with 14 points, \(C\) third with 13 points, and \(X\) last with 0 points. In election scenario 2, the ranking of \(A\), \(B\) and \(C\) is reversed: Candidate \(C\) is first with 13 voters; candidate \(B\) is second with 12 points; candidate \(A\) is third with 11 points; and candidate \(X\) is last with 6 points. So, even though the relative rankings of candidates \(A\), \(B\) and \(C\) do not differ in the two election scenarios, the position of candidate \(X\) in the voters’ rankings reverses the Borda rankings of these candidates.

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

In Section 3.3, it was noted that a number of methods (including all Condorcet consistent methods) are susceptible to the multiple districts paradox. An example of a method that is not susceptible to the multiple districts paradox is Plurality Rule: If a candidate receives the most first place votes in two different districts, then that candidate must receive the most first place votes in the combined the districts. More generally, no scoring rule is susceptible to the multiple districts paradox. This property is called reinforcement:

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Reinforcement: Suppose that \(N_1\) and \(N_2\) are disjoint sets of voters facing the same set of candidates. Further, suppose that \(W_1\) is the set of winners for the population \(N_1\), and \(W_2\) is the set of winners for the population \(N_2\). If there is at least one candidate that wins both elections, then the winner(s) for the entire population (including voters from both \(N_1\) and \(N_2\)) is the set of candidates that are in both \(W_1\) and \(W_2\) (i.e., the winners for the entire population is \(W_1\cap W_2\)).

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

The reinforcement property explicitly rules out the multiple-districts paradox (so, candidates that win all sub-elections are guaranteed to win the full election). In order to characterize all scoring rules, one additional technical property is needed:

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

Continuity: Suppose that a group of voters \(N_1\) elects a candidate \(A\) and a disjoint group of voters \(N_2\) elects a different candidate \(B\). Then there must be some number \(m\) such that the population consisting of the subgroup \(N_2\) together with \(m\) copies of \(N_1\) will elect \(A\).

Voting Methods

4. Topics in Voting Theory

4.2 Characterization Results

voting-methods

First published Wed Aug 3, 2011; substantive revision Mon Jun 24, 2019

We then have: