Handbook on Approval Voting

Approval voting (AV) is a voting procedure in which voters can vote for, or approve of, as many candidates as they like in multicandidate elections (i.e., those with more than two candidates). Each candidate approved of receives one vote, and the candidate with the most votes wins.

Individual preferences often take simple structures in some restricted environments. The so-called universal domain assumption in the three impossibility results by Arrow (1951), Sen (1970a,b), and Gibbard (1973) and Satterthwaite (1975) have been scrutinized and (partially) abandoned in numerous later studies, which do not intend to identify well-behaved social welfare functions that “would be universal in the sense that it would be applicable to any community” (Arrow 1951, p. 24). Important breakthroughs have been made in this line of research: Gaertner (2002) provides a comprehensive survey of the literature on domain restrictions.

There has been a number of axiomatic studies of approval voting since its introduction by Brams and Fishburn (1978). The axiomatic characterizations of approval voting have given researchers a better understanding of the structure of approval voting, and have made the pros and cons of approval voting much sharper. In this article, we present a survey of various axiomatic characterizations of approval voting that are there in the literature.

Approval voting is a well-known voting procedure for single-winner elections. Voters approve of as many candidates as they like, and the candidate with the most approvals wins (Brams and Fishburn 1978, 1983, 2005). But Merrill and Nagel (1987) point out that there are many ways to aggregate approval votes to determine a winner, justifying a distinction between approval balloting, in which each voter submits a ballot that identifies the candidates the voter approves of, and approval voting, the procedure of ranking the candidates according to their total numbers of approvals.

An approval ballot is a voting ballot where voters indicate the candidates they approve among finitely many ones running for elections. We review below some recent studies of procedures that select groups of candidates, or committees, from approval ballots. Many examples can be found of collective decision-making situations where a committee, rather than a single candidate, has to be chosen: deciding about who among a class of students are the ones to be awarded, selecting a board of trustees, appointing new members of an academy, or new professors in a faculty department are all cases where a group of candidates has to be chosen by an electorate. Another example is provided by multiple referendum, where several issues are presented to the voters, who are asked issue-wise to answer by either yes or no.

There is a vast literature which conceives Approval Voting as a mechanism where the approval of voters is a mere strategic action with no intrinsic meaning. As usual, a group of voters who have preferences over a set candidates is considered. Every voter announces the list of candidates which he approves of and the winners are the candidates which receive the highest number of approvals. Assuming that voters take simultaneous and strategic actions, we are confronted to a normal form game whose analysis dates back to Brams and Fishburn (1983). This chapter surveys the main results of this literature.

The strategic analysis of voting rules has given some insight into the understanding of their properties. However, one can assert that these analyses are “too rich” in the sense that they show that a plethora of equilibria can arise under most voting rules. In particular, there is a controversy over Approval voting or AV, a voting rule which has been called “the electoral reform of the twentieth century.” This voting rule allows the voter to vote for as many candidates as he wishes and the candidate who gets the most votes wins the election. Its detractors claim that this kind of method enhances strategic voting when compared for instance to Plurality voting (henceforth PV), whereas its proponents consider that it has several advantages as far as strategic voting is concerned. For an extensive discussion on this controversy over AV, the reader can refer to Brams (2008) and Weber (1995).

“Yes, we can!” – Barack Obama’s campaign slogan inspired enough of his supporters to go to the polls and give him their “yes” votes that he won the 2008 U.S. presidential election. And this happened notwithstanding the fact that many other voters said “no” when pollsters asked if they viewed Barack Obama as qualified for the office. “Yes” and “no” are perhaps the most basic ways for us, as voters, to express our preferences about candidates, and “yes” and “no” are what approval voting is all about.

The results presented in this chapter belong to the long tradition of evaluating the voting rules on their propensity to select the Condorcet winner. This tradition dates back to de Borda (1781), who first noticed that just voting for one name and then selecting as a winner the candidate with more votes, could lead to the selection of an option that is beaten by all the other candidates in pairwise comparisons. In order to replace the rule that is now known as the Plurality rule, he suggested a new voting mechanism, which now bares his name, the Borda Count (BC). When m candidates are in competition, Borda suggested that each voter could give m — 1 points to his first choice, m — 2 points to his second choice, and so on down to one point for his next to the last alternative and zero point for the candidate he ranks last. Next, the candidate who receives the highest number of points is declared as the winner.

Approval voting (AV) has been defended and criticized from many different viewpoints. In this paper, I will concentrate on two topics: preference intensities and strategic behavior. A voter is usually defined as voting sincerely under AV if he or she gives a vote to all candidates standing higher in his or her ranking than the lowest-ranking candidate for whom he or she gives a vote. There are no ‘holes’ in a voter’s approval set.¹ Since this kind of behavior is extremely rare, it has been claimed that approval voting makes strategic voting unnecessary (Brams and Fishburn 1978). On the other hand, Niemi (1984) has argued (see also van Newenhizen and Saari 1988a,b), that even though strategic voting may be rare under AV, even incere voting may require a considerable amount of strategic thinking under this rule. If strategic voting is defined by the fact that a voter gives his or her vote to a candidate who is lower in his or her ranking than some candidate for whom he or she does not vote (see, e.g., Brams and Sanver 2006), I will be studying strategic behavior but not strategic voting under AV here.

This chapter is devoted to computation-based simulations of voting. To perform such a simulation requires two things. On the one hand, one has to specify what might be called the “economic environment,” that is the number of voters, the number of alternatives, and the voter preferences (or tastes, values, utilities, opinions; : : :) over the alternatives. On the other hand, one has to specify the decision process, which is itself made of two ingredients: firstly the material decision procedure, for instance the formal voting rule, and secondly the individual behavior, that is how a voter decides to place in the urn one ballot rather than another, given her preferences and any other relevant information.

This chapter reviews the experimental work about Approval Voting which follows the now standard practice of Experimental Economics. The principle of Experimental Economics is to observe individual behavior in situations where the experimenter can control individual preferences. The classical way to induce and control preferences is to use money, that is to pay the subjects more or less, depending on what they do and, in group experiments, what the other subjects do. This methodology has slowly gained popularity among the economists and is summarized in several classical references (for instance Davis and Holt 1993; Kagel and Roth 1995).

Competitive elections are an essential feature of representative democracies; thus, the choice of voting method is partly constitutive of the form of the democracy. Clearly, this engenders fundamental debates on the properties that acceptable voting rules should and should not exhibit. These debates take place primarily in two spheres: the public and the scientific. Let us here consider an example from France. The President of the French Republic is elected by direct universal suffrage, on the basis of a two-round plurality vote. In other words, run-off voting ensures that the elected President always obtains a majority. On each round, each voter can vote for one and only one candidate. If no candidate receives a majority of votes in the first round of voting, there is a run-off between the two highest-scoring candidates. The winner of this latter round is the winner of the election. Hence, each round is determinant for the result and considered as an important source of information on citizens’ political preferences. The results of the first round of the 2002 French presidential election were a shock for a large part of the population: contrary to the predictions of the opinion polls, the candidate for the extreme Right, Jean-Marie Le Pen, and the sitting president, Jacques Chirac, were selected for the second round. This surprise has contributed to serious public debate on the mechanisms of the two-round single-name vote. This discussion focuses in particular on the tension between tactical and sincere voting, with many citizens pleading for the adoption of a voting method which would allow better expression of their true preferences.

The 2008 state elections in the German state of Hesse were expected to be extremely close. However, nobody expected that forming a new governmentwould reveal itself to be impossible and, after long months of unsuccessful attempts, new elections would have to be called for almost exactly 1 year later.

In large societies, collective decisions cannot be taken directly but have to be delegated to professional decision makers. In a democracy, these delegates are typically elected through a competitive mechanism. The simplest expression of such a mechanism is the now standard Downsian model of Politics (Downs 1951) in which a relatively small number of candidates face a relatively large number of voters, the candidates are purely office-motivated and the voters policy-motivated. For the purpose of winning the election, each candidate freely and independently proposes a policy from a fixed and common set of possible policies. Voters are only interested in policies and not in candidates per se. They trust that the elected candidate will implement the policy she is proposing.

Approval Voting is a voting procedure in which a voter can vote for as many candidates as she wishes, and the candidate who receives the most votes wins the election. Since the seminal contribution of Brams and Fishburn (1978), Approval Voting has received considerable attention. Of course, Approval Voting has been the subject of numerous scholarly papers, as this handbook testifies. But Approval Voting has also received considerable attention outside academics. This is best exemplified by the fact that there are now several professional associations that elect their officers by means of Approval Voting.¹ Also, the use of Approval Voting in political elections has been advocated relentlessly and, at one point, was even hailed as the electoral reform of the twentieth century (Brams 1980, p. 105).

An election is not only a mean to choose among options, candidates or representatives, but it also serves as a privileged occasion for voters to express publicly their opinions and to know the opinions of the others. These two goals — choice and expression — may be contradictory but they co-exist in the minds of the voters. The usual rationale for voting is to elect someone or to have a decision taken, that is collective choice, but voters also sometime declare that their rationale for voting is to “express themselves” or to “contribute to the political debate”. The choice and expression rationales also co-exist in practice since, for instance, on top of the identity of the elected candidate, the number of votes gathered by a candidate matters for deciding of her political fate, even if she is not elected.

The collective decision making problem can be conceived as the aggregation of a vector of utility functions whose informational content depends on the assumptions made about the cardinality and interpersonal comparability of individual preferences. To be more explicit, we consider a non-empty set N of individuals and a non-empty set A of alternatives. Letting U(A) be the set of real-valued “utility functions” defined over A, we model the problem through an aggregation function f : U(A) ^N → 2 ^A \{Ø} The assumptions about the cardinality and interpersonal comparability of individual preferences are formalized by partitioning U(A) into information sets, while requiring f to be invariant at any two vector of utility functions which belong to the same information set. At one extreme, one can assume the existence of an absolute scale over which the utilities of individuals are measured and compared. This assumption partitions U(A) into singleton information sets, hence imposing no invariance over f. At the other extreme, one can rule out any kind of cardinal information and interpersonal comparability, in which case an information set consists of the elements of U(A) which are ordinally equivalent, i.e., induce the same ordering of alternatives for every individual.¹ When cardinality and interpersonal comparability are ruled out, the problem can be modeled through an aggregation function f : W(A) ^N → 2 ^A \{Ø} where W(A) is the set of weak orders (i.e., complete and transitive binary relations) over A. We refer to this as the Arrovian model (Arrow 1950, 1951).