Introduction

In 1770, Jean-Charles de Borda raised objection to the opinion, then generally held, that "in an election by ballot the plurality of voices indicates the will of the electors." He argued that this opinion, "true in the case where the election is conducted between two candidates only, may lead to error in all other cases." He provided an example in which several candidates espousing similar positions might split the votes of the majority, permitting an opposing minority candidate to receive a plurality of votes and win the election. History has borne out Borda's concern. For example, in the 1912 U.S. presidential election, Roosevelt and Taft (former and incumbent Republican presidents) split a majority of the popular vote, allowing Wilson to win.

To address this problem, Borda proposed "election by order of merit," now known as Borda's rule. Under this method, each voter ranks the candidates in order, and each candidate is awarded a number of votes (from that voter) equal to the number of other candidates ranked below him; the candidate receiving the greatest total number of votes wins the election. However, Borda presented no argument that his proposal was the only method which would "solve" the perceived problem.

In the state of New York in 1970, a seat in the U.S. Senate was at stake. The Democratic and Liberal-Republican nominees split the liberal vote, and the Conservative candidate was elected with only 39 percent of the vote. The following year, in a mayoral election in Ithaca, New York, the Democratic nominee drew 29.1 percent of the vote and edged out the Republican nominee, who drew 28.9 percent; an independent Democratic candidate received 10.2 percent, and two independent Republican candidates received a combined total of 31.8 percent. This author, then a graduate student at Cornell (and a supporter of losing candidates in both elections), was prompted by these two results to propose an alternative to the plurality rule called "approval voting."

The idea is a simple one: Each voter is allowed to cast a single vote for each of as many candidates as he or she wishes -- that is, the voter votes for all candidates of whom the voter "approves." The candidate receiving the greatest total number of votes is declared the winner.

A particular attraction of this method was that most voting machines require no special modification for approval voting to be implemented. In most jurisdictions, there are elections where voters choose several candidates from a longer list: perhaps multiple judgeships in a single district, or seats on public commissions. In such elections, the machines can be set to impose an upper bound on the number of single votes a voter can cast; for example, if there were only three seats on a commission, the voter could cast single votes for at most three candidates. By setting the upper bound equal to the total number of candidates in the race, approval votes could easily be cast and tallied.

From the start, approval voting appeared to be an idea whose time had come. Fishburn and Gehrlein had independently analyzed the results of the 1968 U.S. Presidential race, in which George Wallace ran as a third-party candidate, using much the same idea. In the mid-1970s, Brams studied properties of a voting system under which voters could cast either a single positive, or a single negative, vote: In the three-candidate case, this system is equivalent to approval voting (since casting approval votes for two candidates has the same effect on relative vote totals as casting a negative vote for the third).

Brams became an apostle of approval voting. Through a book (Brams and Fishburn, 1983) and a number of public presentations, he built support for the idea. Today, professional organizations such as the Institute of Management Sciences, the IEEE, the Mathematical Association of America, and the American Statistical Association, with total membership in excess of 400,000, use approval voting to elect their officers.

In the former Soviet Union, many elections involved the presentation of a list of candidates to the voters, and voters were allowed only to cross names off the list: this system is equivalent to allowing the casting of approval votes (for the candidates not crossed off). A bill permitting the use of approval voting in public elections has been passed by the North Dakota Senate, and in 1991 Oregon conducted a public referendum involving five alternatives using approval voting. Clearly, approval voting is a viable and practical alternative to the plurality rule.

How Voters Vote

The political arena is complex. Parties form, merge, split, and dissolve. Individual candidates stake out positions, sometimes as a matter of principle, and sometimes to increase their chances of being elected. Voters at some times have little information, and at other times are inundated with poll results and political advertising. The voting system in use can affect all of these dimensions of complexity. The following sections will discuss some of the effects of approval voting, and offer comparisons with plurality rule and Borda's rule. This section will set a general stage for a broader discussion of the ultimate choice problem facing individual voters.

Assume that there are k candidates, numbered 1, 2, ..., k. A scoring rule for an election is a collection of vote-sets, where each vote-set consists of k numbers. A voter selects a vote-set, and assigns the numbers within that set to the candidates. The candidate assigned the greatest total across all voters wins the election. Plurality rule offers to each voter a single vote-set, {1, 0, ..., 0}. Borda's rule also offers a single vote-set, {k-1, k-2, ..., 0}. Approval voting offers a collection of vote-sets, {1, 0, ..., 0}, {1, 1, 0, ..., 0}, ..., {1, 1, ..., 1, 0}.

Each voter can be assumed to hold two relevant pieces of information. First, the voter have personal preferences over the possible outcomes of the election¹: We will assume that voters are expected utility maximizers. In addition, the voters will hold subjective beliefs about the relative likelihood that any particular choice of voting action will change the result of the election in some way. We represent these beliefs by assuming that each voter holds in his mind a set of "close-race" probabilities; p_ij will indicate the probability, given that the election is a near-tie between some two candidates, that the near-tie will be between candidates i and j.² In this case, a voter's decision problem is to maximize his expected utility payoff, based on the utility that voter would receive from various candidates being elected, the probability that a close race will occur, and the chance that a change in voting will tip the race.

Where do the perceptions of the close-race probabilities come from? We will consider two cases in the following sections. In one case, each voter will be assumed to have no specific knowledge concerning the intended voting behavior of the others. We represent this by taking all of the close-race probabilities to be equal.³ Voters who select their vote vectors in accordance with the resulting objective function are said to be voting "sincerely," since they choose based on their own preferences, without direct strategic consideration of the voting intentions of others. Under the plurality rule, sincere voters vote for their most-preferred candidate. Under Borda's rule, sincere voters assign the highest number of votes to their most-preferred candidate, second-highest to the second-most-preferred, and so on. Under approval voting, sincere voters cast their approval votes for all candidates who are "above average" in the field.

However, most modern elections of public officials take place in settings where pre-election polls are conducted, and the results are published. In such a case, the perceived close-race probabilities can differ across the various candidate pairs. As a result, voters may not select their most-preferred if that candidate is perceived to have little chance of being in a close race for victory.

Indeed, under plurality rule, any candidate other than the voter's least-preferred could potentially draw the vote, depending on which close contests are considered most likely.⁴ Under Borda's rule, even the least-preferred candidate might draw a positive number of votes. Consider a three-candidate race, and a voter who would derive utilities (u₁, u₂, u₃) = (10, 5, 0) from the election of each of the three candidates respectively. If the close-race probabilities are perceived to be (p₁₂, p₁₃, p₂₃) = (0.7, 0.2, 0.1), then the high likelihood of a close race involving the first two candidates will lead the voter to cast two votes for the first and none for the second, "discarding" his middle vote on the third (i.e., the prospective ranking of candidate 3 (-2.5) exceeds that of candidate 2 (-3.0), and the voter will cast the vote vector (2, 0, 1)).

Under approval voting, a voter will cast approval votes for all candidates with positive prospective rankings, and will not vote for any candidate with a negative ranking. It is an easy exercise to show that, in any race, a voter will have a positive prospective ranking for his most-preferred candidate, and a negative ranking for his least-preferred. So, in a three-candidate race, the voter will either vote for his most-preferred candidate alone, or for the two candidates who are both preferred to the third.

In races involving more than three candidates, it is possible to construct close-race probabilities which will lead to non-monotonicity in a voter's casting of approval votes. For example, if a voter assigns any utilities u₁ > u₂ > u₃ > u₄ to the election of each of four candidates respectively, and perceives the close-race probabilities to be (p₁₂, p₁₃, p₁₄, p₂₃, p₂₄, p₃₄) = (0.5, 0, 0, 0, 0, 0.5), then he will prefer to vote for candidates 1 and 3, but not for candidates 2 and 4. Note, however, that the close-race perceptions indicate that 2 is relatively more likely to be in a close race with 1 than with 4, while 3 is relatively more likely to be in a close race with 4 than with 1.

If we make the not-unreasonable assumption that for all candidates i, j, k, and l, p_ik/p_jk = p_il/p_jl (i.e., that the relative chances of i or j being in contention with any other candidate are the same for all other candidates), then a voter will only cast an approval vote for a particular candidate if he also casts approval votes for all candidates preferred to that one.⁵ Under this assumption, approval voting poses to voters a particularly simple strategic choice problem: All a voter must decide is where to draw the line between preferred candidates (who receive approval votes) and not-so-preferred candidates (who don't).

Sincere Comparisons

One use of voting systems is to select from among a number of alternatives, in settings where the voters have little access to information concerning either the preferences of the other voters or the intended voting behavior of the others. In these settings a voter can be presumed to vote sincerely, since the lack of information about other voters means there is no basis for voting in some clever strategic way. A typical example of such a setting would be the election of an officer for a professional society. A nominating committee selects a group of candidates, all of whom are considered eminently qualified for the post. The list of nominees is presented to the society's membership, and individuals form their preferences based on a wide variety of personal factors.

In this setting, the goal of the election is to select a candidate who well-represents the preferences of the voters. To choose among alternative voting systems, it is necessary to specify how the "representativeness" of the selected candidate will be measured. Take a specific voter. Prior to presentation of the list of nominees, the voter will not know what the structure of his preferences will be like; for example, he will not know whether he will like some of the nominees, and dislike others, or will like them all, but slightly prefer some over the others. The voter would most prefer (and would be best served by) the use of a voting system which maximizes the expected utility to himself of the eventually-elected candidate.

Consider an election involving only two candidates, in which just over half of the voters slightly prefer the first of the two candidates, while the rest strongly prefer the second.⁶ In this situation, a nominal voter, not knowing in advance of the election whether he will be one of the voters with slight or strong preferences, would like to see in place a voting system which would lead to the election of the second candidate (the one strongly preferred by the minority).

It is possible to construct voting systems with this property. For example, each voter could be given a limited number of votes to use in a series of elections. Or voting could be made costly. Under either of these systems, voters with only slight preferences in a particular election might choose to abstain. In effect, both systems force the voters to make budget-allocation decisions across multiple elections.

However, if we wish to use a voting system for a single election, under which the actions of the voters have no carry-over effect, then we must accept the fact that the candidate slightly preferred by the majority will win (when each voter, at the time of the election, votes in his own best interest).

We will restrict our further attention to single elections. Assume that the utilities of the nominated candidates to each voter are independent, identically-distributed draws from a fixed probability distribution. Before the presentation of the list of nominees, a voter would most prefer the use of a voting system which captures his eventual intensity of preference by selecting the candidate with maximal mean utility across the electorate (since the voter does not know, before the nominees are announced, which position he will hold in the electorate).

Although the earlier discussion shows that no such voting system can be constructed, it can still be useful to consider such a voting system as a benchmark for comparison. In this spirit, let B represent the expected utility of the selected candidate to a voter, did such a voting system exist. Let A be the expected utility to a voter of the elected candidate under some particular voting system, when all voters vote sincerely. And let C be the expected utility of a randomly-selected candidate to a voter. Then a measure of the "effectiveness" of the voting system in representing the preferences of the voters is (A-C)/(B-C). The difference between A and C represents the gain from using the voting system over choosing a candidate randomly. The difference between B and C represents the maximum possible gain from any voting system over random selection. Thus, the ratio offers a measure of what share of the ideal gain is captured by a particular voting system.

Weber (1977) explored this measure of effectiveness in some detail. Here, we cite several results for the case in which the underlying distribution of the utilities of the candidates to the voters is uniform, and the electorate is large.

For two-candidate elections, all voting systems are essentially equivalent, and have an effectiveness of 6/3 = 81.65 percent. For three-candidate elections, the plurality rule has an effectiveness of 75 percent, Borda's rule has an effectiveness of 3/2 = 86.6 percent, and approval voting has an effectiveness of 87.5 percent: Approval voting is the most effective of these three commonly-discussed voting systems in the three-candidate case.

Of course, there's more to this story. Other voting systems are even more effective in the three-candidate case. The best known result is for the voting system which allows voters to cast 4 votes for one candidate, either 3 or 1 votes for another, and 0 votes for the third; this system has an effectiveness of 13/4 = 90.14 percent. In the k-candidate case, the effectiveness of the plurality rule is (3k)/(k+1), and the effectiveness of Borda's rule is (k/(k+1)). In other words, as the number of candidates becomes large, the plurality rule is little better than random selection of a winner, while Borda's rule approaches 100 percent effectiveness. However, the effectiveness of approval voting is not known for k > 3.

Strategic Comparisons

In many political settings, voters have access to substantial information, typically gleaned from public opinion polls, concerning the expressed preferences and voting intentions of others. This information can affect each voter's perception of the relative chances of the various candidates being in contention for victory, which in turn can affect the how voters cast their ballots.

Just as prices both summarize consumer demand and generate that same demand in a competitive equilibrium, so one might expect that, after a series of polls are reported, voters might eventually hold perceptions of the candidate's relative chances of contending for victory which both summarize the electorate's voting intentions, and generate vote totals which justify the voters' perceptions. From this perspective, Myerson and Weber (1993) developed the notion of a "voting equilibrium." A voting equilibrium arises in an "electoral situation" consisting of a set of candidates, a distribution of voters (indicating the proportions of the electorate holding different types of preferences), and a voting rule. The equilibrium itself consists of a set of relative probabilities of the election ending in a close race between any pair of candidates, and a specification of voting behavior for the various types of voters. At equilibrium, each voter is specified to vote in a manner which maximizes that voter's expected utility from the outcome of the election (given the perceived close-race probabilities), and the close-race probabilities are consistent with the candidate vote totals resulting from the specified behavior.

After proving that voting equilibria exist in every electoral situation, Myerson and Weber (1993) determine the equilibria under the plurality rule, Borda's rule, and approval voting for a particular situation meant to represent the type of historical situations discussed early in this paper. Specifically, the situation has three candidates (1, 2, and 3), and three types of voters (A, B, and C). The utilities (u₁, u₂, u₃) derived by a voter of any type from the election of any of the candidates is given in Table 1. Notice that voters A and B together make up a majority of the electorate, and prefer either 1 or 2. However, they could quite possible split the vote, and hand the election to 3.

Under plurality rule, three voting equilibria exist. At one, all of the type-A and type-B voters cast their votes for candidate 1, and all of the type-C voters vote for candidate 3. The likelihood of candidates 1 and 3 being in a close race for victory is perceived by the voters to be much greater than the chance of any other pair of candidates being in a close race, and, since candidates 1 and 3 are the two highest vote-getters, the voters' perceptions are justified by the outcome. A similar equilibrium exists, wherein the type-A and type-B voters all vote for candidate 2.

However, there is a third voting equilibrium, at which all voters of each type vote for their most-favored candidate, and candidate 3 wins the election. The voters correctly perceive that close races between candidates 1 and 3, and between candidates 2 and 3, are of comparable likelihood and are much more likely than a close race between candidates 1 and 2 (the two lowest vote-getters), and these perceptions justify the voters' actions. This equilibrium appears to correspond to the outcome of the historical elections cited earlier.

Under Borda's rule, a family of voting equilibria exist. At all of these equilibria, all three candidates are expected to draw roughly equal vote totals, but a close race between candidates 1 and 2 is perceived by the voters to be somewhat more likely than between candidate 3 and either 1 or 2 (indeed, the first close race is perceived to be precisely 28 times as likely as each of the other two). At equilibrium, each voter casts his 2-vote for his most-favored candidate. However, some type-A or type-B voters give the 1-vote to their second-most-favored candidate, while others give the 1-vote to candidate 3. (The close-race perceptions justify this behavior by making the type-A and type-B voters indifferent between casting the 1-vote for either of the two less-favored candidates.)

Under approval voting, three voting equilibria exist. Two of the equilibria are similar in outcome to the first two under the plurality rule. One of candidates 1 or 2 draws approval votes from all of the type-A and type-B voters, the other draws approval votes only from the voters who most prefer him, and the type-C voters vote only for candidate 3. Since candidate 3 finishes with the second-highest vote total, the only justified perceptions are that a close race involving him and the likely winner are much more likely than any other close race. Yet, if some other close race were to develop, it is perceived to be much more likely to involve candidates 1 and 2 (the first- and third-place finishers) than candidate 3 and the third-place finisher. These perceptions in turn justify the voters' actions.

The third voting equilibrium resembles that found under Borda's rule. One-third of the type-A and type-B voters vote for both candidates 1 and 2, while everyone else votes only for his most-favored candidate. All three candidates are expected to draw roughly equal vote totals, but a close race between 1 and 2 is perceived to be 9 times as likely as the close races between one of them and candidate 3. (These perceptions make type-A and type-B voters indifferent between single- and double-voting.)

What can be made of all this? Only under approval voting do all of the equilibria involve every voter casting a ballot on which the votes for each candidate decrease monotonically with the utility derived by the voter from each candidate's election.

Approval voting is the only voting system among the three studied under which there are equilibria at which one of the first two candidates is the only likely winner, and at the same time there aren't any equilibria in which candidate 3 is the only likely winner. Borda's rule fails to have the first of these properties, and the plurality rule fails to have the second.

Under both plurality rule and approval voting, there remains room for the candidates to engage in political activities which seek to influence voter perceptions of their "viability," in order to lead to a particular equilibrium outcome. Much computational work remains to be done to provide a more-complete picture of how the sets of voting equilibria under these voting rules, and others, vary with the demographics of the electorate.

Candidate Positioning

One oft-cited claim is that the plurality rule induces candidates to take moderate positions. A typical justification for this claim comes from a variety of "median-voter" theorems. For example, in two-candidate races where the voters are distributed along a single dimension (in terms of the most-preferred position of each voter on a single issue), then the more-extreme candidate can increase his vote total by moving his stand closer to the median position; at equilibrium, both candidates will take this median position.

However, when three or more candidates are in the running, things change considerably. If two candidates, positioned at opposite ends of the political spectrum, are perceived to be the only candidates likely to be in contention for victory, then under the plurality rule voters will ignore more moderate candidates (that is, they will choose not to waste their votes), and instead will vote for their more-preferred of the two extremists. If motion towards the center by either extremist would lead supporters to expect all others to shift their votes to a more moderate candidate (rejecting the extremist as untrue to his ideals), then it is in fact rational for all of them to so; the extremist then suffers from changing his position, and prefers to remain at the extreme.

Cox (1985) argued that under approval voting, if the candidates are free to choose their positions in a three-candidate race, then at least one will choose the median position, and will be expected to win the election. Subsequently, Myerson and Weber (1993) proposed a formal definition of a "positional equilibrium." Such an equilibrium specifies positions for the candidates, and a voting equilibrium for the resulting electoral situation. Furthermore, for each candidate likely to contend for victory in the voting equilibrium, a shift in position will result in a new situation with either a voting equilibrium in which that candidate loses, or a voting equilibrium in which the set of likely winners is increased. Finally, for each candidate unlikely to contend for victory, a shift in position will not make that candidate a likely winner. Under this definition, they are able to generalize Cox's result to elections involving any number of candidates. They show that in an election involving any number of candidates, at any positional equilibrium, at least one candidate will take the median position, and the likely winner of the election will be a candidate at that median position.

Prospects

Approval voting is a practical and attractive alternative to the commonly-used plurality rule. Under plausible assumptions concerning the voters' perceptions of the relative chances of various pairs of candidates being in contention for victory, it never forces a voter to forgo voting for a more-preferred candidate in order to vote for a less-preferred candidate who is considered more likely to be in contention for victory. In the absence of polling data, when voters can be assumed to vote sincerely, it is more effective (in the three-candidate case) than either the plurality rule or Borda's rule in leading to an election outcome which well-represents the preferences of the electorate. In the classical three-candidate setting in which two similar candidates share the support of a majority of the voters, and a candidate preferred only by a minority of the electorate might win under the plurality rule, approval voting will not (at equilibrium) ever have the minority candidate emerge as the clear victor. Finally, the use of approval voting encourages candidates to take moderate positions.

Approval voting has been tried in a number of settings, and has proven to be successful, at least in the sense that it continues to be used. Given the well-known potential problems associated with use of the plurality rule when more than two candidates run for a single office, it seems inevitable that the use of approval voting will continue to spread.

Endnotes

1. We focus our attention on elections in which a single candidate must be elected. Our representation of voter preferences includes the tacit assumption that the voters care only about who wins, and not about the relative sizes of the candidates' vote totals.

2. More precisely, we assume that the probability that a change of dv = v_i-v_j > 0 in the relative vote totals of candidates i and j will change the winner of the election from candidate j to candidate i is perceived to vary linearly in dv, and that the close-race probabilities are a common rescaling of the constants of proportionality. With these assumptions, a voter's decision problem is to choose the vote vector v = (v₁, v₂, ..., v_k) which maximizes this expression simplifies to a constant multiple of

3. In the terms used in the previous note, the objective function then simplifies to a constant multiple of where  is the mean of the utilities of all candidates to the voter.

4. A voter's "prospective ranking" of candidate i is Under the plurality rule, a (strategic) voter will vote for the candidate with the greatest prospective ranking.

5. Indeed, this monotonicity property characterizes approval voting.

6. It is important to note that strength-of-preference as used here does not refer to interpersonal comparisons of utility. Rather, voters in the first group have a less-strong preference for one candidate over the other in this election than they might in other elections.

Bibliography

Borda, Jean-Charles de. 1781. Mémoires sur les Élections au Scrutin. Paris: Histoire de l'Academie Royale des Sciences.

Brams, Steven J. and Peter C. Fishburn. 1978. "Approval Voting." American Political Science Review. 72:113-134.

Brams, Steven J. and Peter C. Fishburn. 1983. Approval Voting. Birkhauser-Boston.

Cox, Gary W. 1985. "Electoral Equilibrium under Approval Voting." American Journal of Political Science. 29:112-118.

Merrill, Samuel III. 1988. Making Multicandidate Elections More Democratic. Princeton University Press.

Myerson, Roger B. and Robert J. Weber. 1993. "A Theory of Voting Equilibria." American Political Science Review. 87:102-114.

Weber, Robert J. 1977. "Comparison of Voting Systems." Cowles Foundation Discussion Paper No. 498. Yale University.

Robert J. Weber
J.L. Kellogg Graduate School of Management