Introduction to voting methods

James Green-Armytage

 

            What is democracy? Basically, the extent of democracy is the extent to which people are able to participate on an equal basis in collective decision-making.

            In the context of a modern state, there are several elements that are necessary for a healthy democracy. For example, it is necessary to have protections for civil liberties. Also, it is necessary not only for the media to be free from government censorship, but for distribution of media production and consumption to be relatively broad. That is, neither control of media production nor access to information should be concentrated in the hands of a small, elite group. It is necessary that all people should have access to a good education, and not only one which trains students to obey or to efficiently process information, but one which nurtures students’ ability to think critically and creatively. Perhaps some measure of economic equality is necessary for democracy as well.

            The list of elements needed for a healthy democracy can go on, and can also be the subject of lively discussion. However, the element that I am concerned with here is perhaps the most straightforward and apparently obvious element of democracy: voting.

            It is difficult to argue that democracy can work fairly and effectively without some form of voting. Indeed, if most Americans were asked to name our most fundamental democratic institution, they would probably say it is the fact that political leaders are subject to popular election.

            There is a fair amount of discourse that surrounds the institution of voting. For example, discussion of election fraud, of faulty voting equipment, of campaign finance reforms, of the electoral college system. However, the basic voting system is rarely questioned in the United States. “The winner of an election is the person with the most votes.” This rule, called “plurality,” seems so simple and axiomatic that most people look elsewhere in trying to explain our severely limited level of democracy. This rule is the most fundamental pillar of what we call democracy, but people rarely examine it to see how sturdy it is.

            However, the issue of voting systems is not as simple as it is often taken to be.

            Although the plurality rule is simple, it is anything but completely effective, and its deficiencies can create very serious barriers to authentic democracy. Fortunately, more effective systems do exist, and the material that follows serves as an introduction to some of these systems.

            It might be useful to note from the beginning that the ‘perfect’ voting system is a highly elusive, perhaps impossible, object. Let’s begin with the concept of majority rule: that the outcome of an election should be that option preferred by the majority. Finding this option is the most logical goal of a single-winner voting method.

            Plurality is not the same as majority rule. For example, imagine a case where one candidate receives 20% of the vote, and eight other candidates each receive somewhere around 10% of the vote. Although 20% is of course not a majority of the vote, it is in this case a plurality of the vote, and the candidate with 20% will win if the plurality system is used.

            Other voting rules make more of an effort to establish a majority winner. The most common single-winner method aside from plurality is the two round runoff, in which a second election is held between the two candidates with the highest vote totals if no candidate receives a majority of the vote in the first election. Thus, the winner of the election is at least preferred by a majority over the other candidate who makes it into the runoff, although it is possible that one of the candidates eliminated after the first round might have been preferred by a majority to the actual winner.

            The instant runoff voting system, which uses ranked ballots to in effect simulate a multiple-round runoff, has a similar effect. It guarantees that the winner will at least be preferred by a majority over the other candidate who makes it to the final round, but not that this candidate will be preferred by a majority over all candidates.

            The most satisfying definition of majority rule for multicandidate elections is known as the Condorcet principle. It was formulated by the Marie Jean Antoine Nicolas Caritat, the Marquis de Condorcet, in 1785. A Condorcet winner is defined as a candidate who is preferred by a majority over all other candidates in the election. That is, if there was a series of two-candidate elections between every candidate and every other candidate in the race, a Condorcet winner is a candidate who would win all of the elections they were placed in.

            If voters are allowed to express their preferences using a ranked ballot (that is, indicating their first choice, second choice, and so on), then it is possible to use these ballots to simulate such a series of two-candidate elections. This is known as the method of pairwise comparison.

In a pairwise contest between candidate A and candidate B, a ballot counts as one vote for candidate A if he is ranked above B on that ballot. Also, if candidate A is ranked, and candidate B is not ranked at all on that ballot, then it counts as one vote for candidate A. The position of the other candidates is irrelevant.

If any candidate wins all of their pairwise contests with other candidates, then they are a Condorcet winner. A method that always selects a Condorcet winner when one exists is generally called a Condorcet method, or a method that is perfectly Condorcet-efficient.

            However, a Condorcet winner does not always exist. Strangely, it is possible for a situation to exist where a majority prefers candidate A to candidate B, a majority prefers candidate B to candidate C, and a majority prefers candidate C to candidate A. This is the simplest example of what is called a majority rule cycle.

            The fact that cycles can exist in the preferences of the electorate means that in effect no voting method can always satisfy majority rule in the ideal sense. If there is a cycle, no matter which candidate is selected, there will be another candidate who is preferred over them by a majority.

            Since no methods can guarantee perfect majority rule, it seems that the best we can do is to use a method that comes as close as possible to majority rule. Plurality rates fairly low in this regard, two round runoff and instant runoff voting are intermediate, and Condorcet methods are probably the closest. Also, there are a variety of Condorcet methods which each define a candidate’s ‘closeness’ to being a Condorcet winner in different ways, some of which are more convincing than others.

 

            Much of the complexity in evaluating voting methods has to do with the possible use of strategy on the part of the voters. That is, it cannot be assumed that people will always vote sincerely; instead they might weigh the desirability of different outcomes in balance with the probability of different outcomes given different votes on their part, and vote so as to maximize the combination of these.

            For example, the plurality system used in America asks voters to cast one vote for President. But who do they cast this one vote for? Is it always their first choice; the candidate whom they would most like to see in office? If this person is anyone other than the Democratic or Republican nominee for the office, the probability of their winning is very near to zero, even if they are on the ballot. So people do not always vote for their first choice. Instead, people may vote for their first choice among the candidates who are considered to be frontrunners, and hence who have a good chance of winning. Rather than simply indicating the candidate whom they like best, voters are faced with an unclear decision in which they will have to weigh the strength of their preference for each candidate in combination with that candidate’s chance of winning. Thus, their vote is heavily laden with strategic concerns.

In short, many people who prefer an alternative candidate end up voting for a major-party nominee because the probability of anyone else winning is perceived as being so low. Thus it is extremely difficult for alternative candidates to seriously challenge the two major parties. This is a fatal problem for the competitiveness of political races, the accountability of politicians, and the range of political discourse.

            So, the plurality system serves as a barrier against government accountability to the people, and against social change. It is a subtle barrier, which many people do not notice, but it is also an extremely effective one. People are not compelled to vote for major-party nominees by any law, but they do so based on the belief that they don’t have any other viable options, combined with the offensiveness of the major-party nominee whom they least prefer. And yet, if they do vote this way, no record exists of the fact that they might have preferred an alternative candidate if they had thought that candidate had a chance. Hence plurality creates an illusion of popular support for the two party system, and delegitimizes alternative parties.

            The general strategy of placing a candidate you prefer less over a candidate you prefer more, because the candidate you prefer less has a greater chance of winning, is known as the “compromising” strategy. (It doesn't sound so bad, but it often means leaving downranking your favorite or leaving her off the ballot altogether in favor of a lesser-of-two-evils candidate.) Although incentives for the compromising strategy reduced in other methods, it cannot be removed entirely in methods that meet other standards of fairness. It is removed to some degree in two round runoff, to a slightly larger degree in instant runoff voting, and to the greatest possible extent in Condorcet methods. However, these methods can create incentives for other strategies which may be problematic as well.

            Essentially there is no voting rule which can be completely free from strategic incentives. The goal, then, is to find a method that most frequently chooses an appropriate winner (such as a Condorcet winner, if one exists) with respect to the sincere preferences, even though the sincere preferences are not directly known. That is, a method which is both highly responsive to voters’ preferences and highly resistant to strategic manipulation. The search for such methods is a rich and dynamic puzzle which I think has yet to be fully solved; it is an interesting challenge for the human intellect in the attempt to reach towards more effective organization and greater democracy.

 

            So far, I have only mentioned situations where there is only one non-divisible issue to be decided, such as the election of a president or mayor. In these cases, majority rule is the best that we can really do. However, if we are electing a multi-seat legislature, it becomes possible and desirable to go beyond majority rule toward proportional representation.

Proportional representation provides representation such that the size of a group of voters divided by the total electorate is equivalent to the amount of representatives they elect divided by the total number of seats in the legislature.

For example, if there are 100 seats to be filled and 100,000 voters, then any group of 1,000 people voting together should be able to fill one of the seats with whomever they like. A group of 30,000 voters should be able to determine how 30 of the seats are filled. And so on.

            Proportional representation is an extremely common model of government around the world, and there is ample evidence that it leads to a greater number of political parties and a greater number of viable choices for each voter. It should also increase the fluidity with which changes in the balance of power can take place. That is, single winner systems work on an all-or-nothing basis which lends itself on the one hand to unresponsive governments which do not adapt to changing voter preferences at all, and on the other hand to over-drastic and chaotic shifts of ideology power, since in any given outcome one large minority or another is being ignored. A proportional system, by contrast, can respond to changes in collective preference more gradually and subtly, since support and representation for a party or political faction can decrease without suddenly disappearing from the map altogether.

            Also, compromising strategy incentives are far less intense and distorting, since the odds of a sincere favorite winning are much greater when there are many seats to be filled rather than only one. The single transferable vote principle, which makes up the most interesting family of proportional representation systems, is the most effective basis for proportional representation.

           

            In this paper, I usually describe and discuss voting methods in terms of a group of voters selecting elected representatives, either for a single-person executive office or a multi-seat legislature. However, I should emphasize here that selecting government representatives is certainly not the only important application of voting methods.

            For one thing, once a government legislature is formed, they will themselves have to use particular voting methods to come to collective decisions. Integration of Condorcet methods into legislative rules of order might be very attractive for the purpose of more effectively choosing between multiple versions of a bill or responses to a given circumstance. This could save time and curtail the practice of unpopular riders being added to otherwise popular bills, which can either kill popular bills or pass unpopular riders.

            Also, governments can use direct democracy, giving citizens a chance to vote directly on government policy, rather than limiting their input to the selection of representatives. Direct democracy systems can employ various voting methods, such as plurality, instant runoff voting, and Condorcet methods; in most cases Condorcet methods should be preferable.

Also, it would be attractive to have a direct democracy system which incorporates a proxy system, that is, which allows voters to delegate the weight of their vote to a proxy for the purpose of deciding a given issue. Different rules for proxy systems exist, but luckily the concept of proxy voting doesn’t offer any baffling paradoxes in and of itself.

            There are probably also situations where it is appropriate to use proportional representation principles for purposes other than selecting representatives. For example, filling slots on public television, or perhaps in setting certain kinds of budgets. The extent to which proportional representation is desirable in allocating public goods is a rich area for inquiry in the field of public choice.

            Also, the use of voting is by no means exclusive to governments. Voting is useful whenever a group is making a collective decision, which takes place in hundreds of spheres of human life. Alternative voting methods can be profitably adopted by unions, businesses, schools, churches, and so on, whether to elect boards, hire people, or make policy decisions. They can be used by informal hobby groups or even families and groups of friends.

            Better voting methods are tools for making group decisions with more effectiveness and less confusion. There may be many group decision situations where one option is a Condorcet winner but no one realizes it because it is not the first choice of many people. There may be situations where a proportional representation principle is appropriate but where instead the majority takes full determination, leaving the rest disgruntled. And so on.

            My opinion is that people should start using better voting methods wherever they can. In the long run, this might make voting changes possible on higher levels of power, and in the short run it will improve people’s ability to make clear collective decisions.

            So, although the search for the best voting methods is fraught with paradoxes, it is important to try our best. Although it is somewhat abstract, it is important, because any democracy is only as good as its voting methods.