Joumal of Economic Perspectives—Volume 9, Number 1—Winter 1995—Pages 39-49
Approval Voting
Robert J. Weber
I
n 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 that
would "solve" the perceived problem.
'A more detailed description of Borda scores, and comparisons with other voting methods, can be
found in the article by Levin and Nalebuff in this issue.
• Robert J. Weber is Professor of Managerial Economics and Decision Sciences, J. L.
Kellogg Graduate School of Management, Northwestern University, Evanston, Illinois.
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Joumal of Economic Perspectives
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, tbe 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 Ceorge 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 Bve alternatives using ap-
Robert J. Weber
41
proval 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 \,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 — I, k — 2,... ,0). Approval voting
offers a collection of vote-sets, {1, 0 , . . . , 0},{l, 1, 0,... , 0 } , . . . ,{1, 1,..., 1,0}.
Each voter can be assumed to hold two relevant pieces of information.
First, tbe voter will bave personal preferences over the possible outcomes of the
election, and 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; pjj will indicate the probability, given
that tbe election is a near tie between some two candidates, that the near tie will
be between candidates i and j . 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 tbat 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
^We will 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.
^More precisely, we assume that the probability that a change of Ar; = t;, — Vj > 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 At;, and that the close-race probabilities are a common
rescaling of the constants of proportionality. With these assumptions, a voter's decision problem is
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Joumal of Economic Perspectives
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.'' 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 Held.
However, most modern elections of public officials take place in settings
where preelection 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. 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 iu^,U2,u^) = (10,5,0) from the
election of each of the three candidates respectively. If the close-race probabilities are perceived to be (pi2>P\3> 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 (that is, 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
to choose the vote vector v = (i;,, t^j,. • •, v*) which maximizes
this expression simplifies to a constant multiple of
E ", E/>,;(",-«/)•
1
j
In the terms used in the previous note, the objective function then simplifies to a constant multiple
of L,U|(u, — u), where u = Hu^/k is the mean of the utilities of all candidates to the voter.
"'Call T,jpijiUj — Uj) a voter's "prospective ranking" of candidate i. Under the plurality rule, a
(strategic) voter will vote for the candidate with the greatest prospective ranking.
Approval Voting
43
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 tbe two candidates
who are both preferred to the third.
In races involving more than three candidates, it is possible to construct
close-race probabilities that will lead to non-monotonicity in a voter's casting of
approval votes. For example, if a voter assigns any utilities M, > M2 > W3 > u^
to the election of each of four candidates respectively and perceives the
close-race probabilities to be (/),2, ^13,/'M,/'23'/'24'i'34) = (C*-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 /, pik/pjk — Pit/Pit (tbat is, that the relative chances of i or j being in
contention witb 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.® 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 tbis setting, tbe 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 tbe
list of nominees, tbe 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
Indeed, this monotonicity property characterizes approval voting.
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Journal of Economic Perspectives
of a voting system that maximizes the expected utility to himself of the
eventually elected candidate.
Consider an election involving only two candidates, in which just over half
of tbe voters slightly prefer tbe first of tbe two candidates, while tbe rest
strongly prefer tbe second.^ In tbis situation, perhaps a nominal voter, not
knowing in advance of tbe 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 tbis 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 carryover effect, then we must accept
the fact that tbe candidate slightly preferred by the majority will win (when
each voter, at the time of the election, votes in bis own best interest).
We will restrict our further attention to single elections. Assume that the
utilities of tbe 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
that captures his eventual intensity of preference by selecting the candidate
with maximal mean utility across tbe electorate (since the voter does not know,
before the nominees are announced, which position he will hold in tbe
electorate).
Although tbe 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 tbe expected utility of
tbe 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 tbe voting system in representing the preferences of the voters is
{A - G)/{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.
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.
Robert J. Weber
45
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 tbe electorate is large.
For two-candidate elections, all voting systems are essentially equivalent
and bave an effectiveness oi {% /2> = 81.65 percent. For three-candidate elections, the plurality rule has an effectiveness of 75 percent, Borda's rule bas an
effectiveness of \/3^/2 = 86.6 percent, and approval voting has an effectiveness
of 87.5 percent. Approval voting is tbe 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 tbe three-candidate case. The best known result is for tbe voting
system that allows voters to cast 4 votes for one candidate, either 3 or 1 votes
for another, and 0 votes for the third; this system bas an effectiveness of
/Y3"/4 = 90.14 percent. In the fe-candidate case, the effectiveness of tbe plurality rule is /ik /{k + 1), and the effectiveness of Borda's rule is yk/{k+l)
. In
other words, as the number of candidates becomes large, the plurality rule is
litde better tban random selection of a winner, while Borda's rule approaches
100 percent effectiveness. However, the effectiveness of approval voting is not
known for k > 4.
Strategic Comparisons
In many political settings, voters bave access to substantial information,
typically gleaned from public opinion polls, concerning tbe expressed preferences and voting intentions of others. Tbis information can affect each voter's
perception of the relative chances of tbe various candidates being in contention
for victory, which in turn can affect how voters cast their ballots.
Just as prices both summarize consumer demand and generate that same
demand in a competitive equilibrium, so one migbt expect tbat, after a series of
polls are reported, voters might eventually hold perceptions of tbe candidates'
relative cbances of contending for victory tbat both summarize the electorate's
voting intentions and generate vote totals tbat justify the voters' perceptions.
From tbis 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.
Tbe 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 bebavior for the various types of voters. At equilibrium, each voter is
specified to vote in a manner that maximizes that voter's expected utility from
tbe 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.
46
Joumal of Economic Perspectives
Table 1
Utilities Derived by a Voter
Voter Type
A
B
C
Utility Vector
Proportion of Population
(10,9,0)
(9,10,0)
(0,0,10)
30%
30%
40%
After proving that voting equilibria exist in every electoral situation,
Myerson and Weber (1993) determine tbe 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 (M,, Wg.Wj) 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 S.
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 whicb 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 likelibood and are mucb 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 tbe historical elections cited earlier.
Under Borda's rule, a family of voting equilibria exists. 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 tbe voters to be
somewhat more likely than between candidate 3 and eitber 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 bebavior by making the type-A and type-B voters
Approval Voting
47
indifferent between casting the 1-vote for either of the two less-favored candidates.)
Under approval voting, three voting equilibria exist. Two of tbe equilibria
are similar in outcome to tbe 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,
wbile 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 nine 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 tbe
equilibria involve every voter casting a ballot on wbich 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 wbich
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 tbe
candidates to engage in political activities that seek to influence voter perceptions of their "viability," to lead to a particular equilibrium outcome. Mucb
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
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Joumal of Economic Perspectives
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 tbe only candidates likely to be in
contention for victory, then under tbe 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 eitber extremist would lead supporters to expect all
otbers 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 do so; the
extremist then suffers from changing 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 tbe election. Subsequently,
Myerson and Weber (1993) proposed a formal definition of a "positional
equilibrium." Such an equilibrium specifies positions for tbe 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 wbich 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 tbat median
position.
Conclusion
Approval voting is a practical and attractive alternative to tbe 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 tbe tbree-candidate case) than either the plurality rule or Borda's rule in leading to an election
outcome that 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 tbe
Robert J. Weber
49
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 bas 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.
References
Borda, Jean-Charles de, Memoires sur les
Elections au Scrutin. Paris: Histoire de
TAcademie Royale des Sciences, 1781.
Brams, Steven J., and Peter C. Fishburn,
"Approval Voting," American Political Science
Review, September 1978, 72, 831-47.
Brams, Steven J., and Peter C. Fishburn,
Approval Voting. Boston: Birkhauser, 1983.
Cox, Gary W., "Electoral Equilibrium under Approval Voting," American Joumal of
Political Science, February 1985, 29, 112-18.
Merrill, Samuel, III, Making Multicandidate
Elections More Democratic. Princeton: Princeton
University Press, 1988.
Myerson, Roger B., and Robert J. Weber,
"A Theory of Voting Equilibria," American Political Science Review, March 1993, 87, 102-14.
Weber, Robert J., "Comparison of Voting
Systems." Cowles Foundation Discussion Paper No. 498, Yale Llniversity, 1977.