Hi) 2 8
.M414
no. i7?2-
86
Dewey
v^,.^«?A»»-i.
/AN
EXPERIMENT IN APPROVAL VOTING
Peter C. Fishburn
AT&T Bell Laboratories, Murray Hill, NJ 0797A
John D. C. Little
Sloan School of Management, MIT, Cambridge, MA 02139
Mav 1986
WP #1782-86
'I
I
AUG 2 7
1986
j
ABSTRACT
In its 1985 election for council The Institute of Management Sciences
(TIMS) ran a test of approval voting by sending a non-binding approval ballot
In approval voting
to its members along with the regular plurality ballot.
many
candidates as desired, the winner
a person votes for (approves of) as
being the candidate with the most votes. Two of the TIMS contests had
three candidates for a single office and a third had five candidates for two
In such situations approval voting can produce winners who are
offices.
more generally acceptable to the electorate than standard plurality voting.
This is because the approval mechanism tends to prevent two candidates with
broad appeal from splitting a majority constituency and electing a minority
candidate.
Of these three TIMS contests two would have had different winners had
In the first election with candidates A, B,
approval voting been binding.
and C, C narrowly beat B in the regular election, but B was the approval
winner because considerably more of A's supporters approved of B than C.
In the second election approval and plurality voting agreed on the winner.
In the third election with two positions to be filled from a field of five
candidates. A, B, C, D, and E, B was a winner by either method, but the
official second winner. A, placed fourth in the approval vote behind C and D.
Close examination of ranking data shows that in the first election a
head to head contest between B and C would be a toss up. Nevertheless,
because of secondary support, B is a better choice by the criterion of broad
acceptance by the electorate. In the third election C and D are similarly
close but either would have broader support than the plurality choice A and
of the two C has somewhat broader general approval.
It should be pointed out that, as approval voting creates new winners,
A minority constituency has more difficulty
it must also create new losers.
taking advantage of a majority split to install their candidate.
The TIMS experiment demonstrated remarkably well the points made by
the proponents of approval voting, namely, that it is a simple and effective
way to do a better job than plurality voting at selecting the candidates who
have the broadest base of support among the electorate.
-z-
Introduction
1.
In the spring of
1985 The Institute of Management Sciences (TIMS) conducted a
of approval voting, an innovative
test
Approval voting
is
method of assessing the wishes of an
a simple procedure that differs in
regular plurality voting and has certain advantages over
received
an exfjcrimentaj ballot along with the regular
election of officers
The
make
it.
In the
official
TIMS
TIMS
test all
members
ballot during the annual
voters filled in the test ballot.
it
showed that approval
by example, how those differences can occur and what
a difference and,
In the
they signify.
electorate.
modest but important ways from
exf)eriment was extremely successful in the sense that
voting can
ballot
Most
and council.
field
election several
outcomes would have changed had the approval
Our
been binding and, equally as interesting, several others would not.
shows that the changes would have resulted
analysis
electing candidates that had a broader base
in
of popular support than those actually elected.
Voting plays a central role
to
hold
positions
mechanism
in the
of responsibility.
where
that, in cases
more widely desired by the
split
TIMS
Newton Garber, chairman
society
the
candidates
is
believe
makes any
it
more
provides a
difference, will select candidates
difficult
election
for
simple
who
are
By the same token,
a small organized
group
to
take
were informed of the pending experiment by H.
of the 1984-5 nominating committee, and agreed to participate.
grateful for their cooperation.
will
that approval voting
majority and elect a minority candidate.
All candidates in the
The
We
electorate than standard plurality voting.
however, approval voting makes
advantage of a
it
democratic process as a means of selecting individuals
To
preserve anonymity in so far as possible,
be identified only by code designations
elections will not be identified
by
office.
(A,B,...).
In
addition
the
-3-
Section
report
the
of
2
describes
approval
experimental design and the basis of the analyses.
three elections.
appear
in
The
a
list
When
one person
of candidates.
are asked to vote
voters
Additional supporting data
findings.
Approval Voting
TIMS,
including those of
elections,
plurality voting.
from
the
the Appendix.
2.
Most
summarizes
3
Sections 4, 5 and 6 analyze the data for
summarizes the
section
final
Section
voting.
as
method of
one
to be elected, voters are instructed to vote for
is
When more
for
are conducted by the familiar
than one position
many
is
from
to be filled
a single
list,
filled.
The
people as there are positions to be
candidates receiving the most votes are elected.
Approval
voting
restricting the
differs
number
from
voting
plurality
of candidates a person
allows each person to vote for any number.
may
in
one
vote.
The candidates
This gives a person the opportunity to vote for
the rest.
of,
or to discriminate between a
Each candidate
superior to plurality and other simple election procedures
candidates to choose from.
elected
way
Their basic thesis
to assess the overall support for
candidates
Analyses of past
selected receives a full
receiving the most votes are elected.
Brams and Fishburn (1983) have analyzed approval voting
effective
of
Instead
vote for in an election, approval voting
every candidate he or she finds acceptable or approves
more preferred subset of candidates and
way.
important
should
elections
be
those
who
is
in
when
depth and argue that
is
an
efficient
and
each candidate within the electorate, and that
are
most widely supported or approved
conducted essentially
by plurality voting,
including
For example, each of the 1970 and 1980
of.
multi-
candidate contests for the United States Senate and Presidency, clearly show that this
not always so under plurality voting.
is
more
there are three or
that approval voting
it
is
New York
-4-
Senaiorial elections had three main candidates on the ballot.
candidate,
i.e.,
contests.
Yet
some of
their
Each
election
one who would have defeated each of the other two
in
both cases the majority candidates
support
was diverted
minority candidates to win.
If
to
rather
simple pairwise
the plurality elections because
lost
candidates,
similar
in
had a majority
thereby
permitting
approval voting had been used, the majority candidates
would have been elected.
A
Brams and Fishburn (1983).
further introduction to approval voting appears in
practice, approval voting
would use the same type of
ballot as plurality uses
More complex methods,
only slightly harder to tabulate.
at
mailed
to
of the election.
TIMS member. The
The experimental
In these elections
names.
is
not clear that such
Experiment
TIMS members
in
John D. C.
OR/MS TODAY. On
official plurality ballot
Little's presidential
April 21, two ballots were
determined the binding results
ballot assessed the effects of approval voting in the three
races with at least three candidates.
names on
to
the February 1985 issue of
each
TIMS
The
The experiment was announced
in
it
uncovering the candidates most favored by the electorate.
3.
column
and would be
including ones that ask voters to
rank order candidates, require substantially more processing, and
methods would do better
In
We
shall designate these three elections as:
El
an election of one out of three candidates;
E2
E3
another election of one out of three;
an election of two out of
five
candidates.
each voter was asked to check approved candidates to the right of the
the experimental ballot and to rank the candidates (1,2,...)
to the left of the
Although rankings are not part of approval voting, they help to determine majority
comparisons.
Since
majority candidates
(which
might not
exist
if
there are cyclical
-5-
majorities) are often claimed to be most desirable,
it
was
fell
important to include this
Of
these,
information.
The
returned the
by
the
ballot
official
test ballot.
members and
exceptionally small
Almost
1,851
members.
1,579
(85%)
also
This remarkably high rate of return shows widespread cooperation
lends
number
credence
the
to
results.
of inconsistencies: see the
the test ballots had
all
candidates.
was returned by
Further
Appendix
support
arises
in
the
for details.
some approvals checked, but
a few did not rank the
The counts by response category were:
Rl.
Only
R2.
Test ballot also returned, but without rankings:
R3.
Test ballot also returned with some rankings:
returned:
official ballot
272
(14.7%)
68
(3.7%)
1511
(81.6%)
1851
numbers of
Slightly different
voters participated in each election, as will be reflected in the
ensuing analyses.
We
of each
discuss each of the three elections in turn in the following sections.
are
presented:
official
plurality
counts,
approval
voting
counts,
Three aspects
and majority
comparisons.
4.
In election
have
changed
official election
HI, a three-way race with candidates A, B, and C, approval voting would
the
winner
from
C
B would have won by 170 (6.1%)
I
to
B.
See
Table
1.
Candidate
C won
by 8 votes (0.4% of the votes cast) but, had approval voting been
B would have won by 130 (5.5% of the
In Table
Election El
in
the
in effect,
votes cast) using the actual returned approval votes.
extrapolated totals.
the extrapolated totals take account of the f)eople
who submitted an
official
-6-
TABLE
Official
1
and approval votes for election El show that C won
B would have won under approval voting
the official election but
Official
Actual
Extrapolated
plurality
approval
approval
vote
vote
vote
417
1038**
486
1224**
A
166
B
C
827
835**
908
1828
(
1
1054
2363
567 voters)
2764
(1828 voters)
largest vote
ballot but not the test ballot.
behaved
in
the
This
done by assuming that the non-responders would have
is
manner of the responders
— number
N^
to
both ballots.
of non-responders
who
Let
voted for candidate
X
on the
official ballot;
P{Y\X) -
fraction of responders approving candidate
they voted for
Then our estimate
on the
Y
given that
official ballot.
of additional approval votes attributable to non-responders
added approval votes
We
X
see from Table
1
for
is:
Y - P{Y\a)N^ + P{Y\B)>1b + P{Y\C)Nc
•
that the extrapolation does not change the results in any fundamental
way.
Next we look
who would win
here
is
results.
a
little
for a majority candidate.
in
As
stated earlier, a majority candidate
pairwise (plurality) elections with
all
other candidates.
more complicated and makes use of the rankings.
Table
The
is
one
analysis
2 presents the
-7-
TABLE
2
For election El majority comparisons show that in hypothetical two-way
B and C would defeat A but that B against C would be a virtual tie
,
races both
Inferred votes for:
Revealed
first
choices from
official ballot
counts:
ABAC
A
Contest:
vs.
B
A
vs.
C
B
B
vs.
C
-8-
Thus
B versus
in a
C
comparison the 70 are credited
This leaves 27 who voted for
A
2
estimates
how
27(66/139)
C.
We
A
voters
and
B
between
-
12.8.
C.
The
The
+ 66 +
3)
who
calculation
or
make
The
few further notes on El:
for
B
approval:
Brams and Fishburn (1983) observe
election as not voting.
row
14.2
a preference
C
is
B and 913.8
for
and
for
between the two candidates.
number of approvals among
the 1435
(5%).
i.e.,
two candidates
who approved
Three approvals have the same
effect
of one or two candidates
on the outcome of the
However, three approvals do indicate that
the results from election El,
in a three-
have the same chance of affecting the
acceptable to the voter as people to perform the functions of the
To summarize
-
to last
If
(46%)
(49%)
660
706
69
In this particular test election, the voters
split.
and also expressed
that votes for either one or
candidate election are equally efficacious,
evenly
preference.
they adhered to the
if
decimals, are 914.2 for
distribution of
3 approvals:
fairly
A
C
The next
for C.
27(73/139)
is
versus
B and C
a valid distinction
2 approvals:
were
voted for
B
3 to B.
more candidates was:
1
outcome.
B and 901
for
final totals, carried into
declare ourselves unable to
who approved one
900
the 27 would split between
pattern of the 139 voters (70
66 to C, and the
but gave no indication of a
the 27 are ignored, the revealed choices are
of Table
to B, the
C
wins the
all
the candidates are
office.
official
plurality election by
0.5%, B would have won an approval vote by 6.1% and a head-to-head election would be
too close for us to call.
little bit
larger than B's.
to
23%)
C
(27%).
The
picture emerges that
C
has a loyal following that
However, among A's followers, more approve of B than
is
just a
C (36%
Furthermore, more of C's followers approve of B (43%) than B's followers do of
Hence B wins the most approval
votes.
What
can we conclude?
Although we
_9-
cannot definitively say
that
B has
who would win
a head-to-head election
between B and C.
it
appears
a broader acceptance in the electorate.
5.
Election
E2
is
Election
E2
another three-way race to elect one person, but here plurality and
approval results agree.
Since once again the extrapolated approval votes do not change the
fundamental outcome, we omit them for
this case.
TABLE
In election
E2
plurality
Table
3 presents the results.
3
and approval votes pick the same winner
-10-
B) in pairwisc races, but
C
and B arc
in a virtual tic
when run
against each other.
Such
a
closeness would not have been anticipated from Table 3 and arises because A's supporters
tend to prefer
B
to C.
TABLE
For election E2.
hypothetical two-way races
Contest:
4
C is not quite a majority candidate: in
B would defeat A, C would defeat A, but C
B is too close to call
versus
-11-
2 approvals:
421
(54%)
(33%)
3 approvals:
174
(13%).
1
summary
In
agree on C.
election
We
E2
691
approval:
presents an example in which the plurality and approval votes
cannot say that
C
would actually beat B
in
a two-way race, but, as in El,
the approval vote indicates a wider acceptance of C.
6.
A new
Election
E3
surprise awaits us in election E3, in which two people are to be elected from
among
5
oflRcial
plurality
candidates. A, B, C,
summary. As
election
in election
D
finished
and
E.
fourth
We
in
find
the
that the second place winner in the
approval
voting.
Table
5
shows the
E2, the extrapolated approval votes show the same pattern as the
actual and are omitted.
TABLE
In election
B
E3
5
with iwo positions to be filled, the first
under both methods but the second choice by plurality
voting, candidate A. drops to fourth place behind C and D in approval voting
choice
is
candidate
Candidate
-12-
The majority comparisons
shown
in
Table
of hypothetical pairwisc contests between the candidates are
6.
TABLE
6
Results of hypothetical painvise contests between the five
E3 show that approval winners B and C would win pairwise
candidates in
contests with all others
-13-
FIGURE
A
I.
graph of the hypothetical pairwisc
E3 with arrows toward winners.
contests within
two winners are
B,
The
then C, and since there are no
directed cycles, no cyclical majorities arc present.
Table 6 and Figure
ranking B
C D A
E.
1
show
that the majority comparisons for
The rankings from
approval:
The approval ranking
is
transitive with
the three types of analysis are:
plurality:
majority:
E3 are
BADGE
B
B
C D A E
C D A E.
identical to the majority ranking.
On
the other hand, as already
noted, the plurality ranking elevates the fourth place majority candidate to second place
-14-
C
and drops the majority runner-up
What
has happened
in
minority has voted for A.
their opinions
Similarly,
that
is
B
clearly has wide support.
However, when the electorate
23%
20%
C
of those not voting for
is
Beyond that
a cohesive
given an opportunity to express
many more approve
by taking off the two vote constraint,
For example,
A.
E3
to fourth place.
on the
official
of
ballot
C
and
D
than
approved of C.
But only 14% of those not voting for
A
from E3 the distribution of the 1380 voters who approved of
at
not voting for
D
approved of D.
approved of A.
In other information
least
one candidate
is
approvals:
494
4 approvals;
75
(8%)
(46%)
(36%)
(5%)
5 approvals:
54
(4%).
1
641
3
Thus most
voters approved of either two or three candidates.
7.
Our
116
approval:
2 approvals:
Conclusions
analysis indicates that approval voting captures the general wishes of the electorate
considerably better than plurality voting.
The approval method compares favorably with
rankings methods, as shown for example by majority comparisons, but
of the voter and tabulator and
is
also
much
less
a
determined minority constituency
majority to elect
its
may
candidate but instead
will
requires less effort
vulnerable to strategic misrepresentation
by voters than are ranking methods (Brams and Fishburn, 1983).
that
it
be
A
possible detraction
unable to take advantage of a
have
to seek
is
split
approval from the whole
electorate.
Our work
with the experimental data has led us to believe that approval voting would
be desirable for
TIMS.
At the same time, consideration should be given
to
partition
-15-
multiple-seat elections into multiple single-position elections
particular constituencies.
Council
1
We
to
This
is
when
it
is
desired to represent
presently done, in part, by separating U.S. and non-U.S.
seats.'
are deeply indebted to
Mary DeMelim and
the
Newton Garbcr
TIMS
for his central role in the
business office for tbeir help
in
TIMS
approval voting experiment and
designing and counting ballots.
-16-
Appendix
We
provide here various further data that support and enrich the analyses in the text,
and E3.
especially for elections El
Ballot transcription and discrepancies
The
TIMS
Business Office transcribed the
each voter who returned both
C
El for A, B and
For example, the
are 166, 827 and 835 respectively (Table
the transcriptions on the test ballots are
None
"official plurality vote").
a
in
the voting.
Nine
candidate ranked lower than one they did not approve
not approve of their official ballot choice, yet approved of
voters did the
same
in
E2.
votes
The same counts from
(see
Table
7
of the discrepancies affects our conclusions.
Several interesting inconsistencies occurred
approved of
1).
official plurality
826 and 835 respectively
167,
A
checks and to link the two ballots.
to provide consistency
few minor discrepancies arose from transcription.
in
votes onto the test ballot for
official ballot
In the five candidate
E3
voters in election
of.
Two
E3
El voters did
some other candidate.
election, 18 voters failed to
Seven
approve of
a candidate selected on their official ballot yet approved others not selected on the official
ballot.
Rankings
In election El,
ranked
all
95%
of ihe voters
three (1415 of 1484).
ranked the candidates, of
whom
choices, 189 ranked three,
their rankings.
In
who ranked one
E2
it
or
more candidates on the
was 88% (1161 of 1313).
46 ranked only their
and 975 (69%) ranked
first
all
choice,
five.
A
test ballot
In E3, 1420 voters
210 ranked
their first
few voters indicated
two
ties in
-17-
Approval voting, El
Table 7 shows the match-ups between approval votes and
other data used to obtain the approval vote summaries in Table
TABLE
official plurality
1.
The upper
7
Analysis of approval voting and plurality voting
for election El shows approval voting sets matched with
official plurality votes along with summaries and extrapolations
Candidates
in
approval set
(R2
+
R3)
along with
part of
-18-
ihe table includes
returned a
For example, of the 105 voters
ballot.
none voted
ballot,
official
approval vote" column
Table
who
1567 voters
all
for B,
Table
in
who approved
79 voted
for
of
official
A
on the
and C, 24 voted
for
The "actual
C, and two voted for nobody.
the upper part of
in
7.
The middle
of Table
part
7
summarizes the number of approval votes
candidate as a function of the candidate voted for on the
record
(A'^
A
El along with their
obtained from the totals column
is
1
test ballot for
-
the
-
21, A^5
134, A'c
who
nonresponders
test-ballot
-
1
11),
and
the
on
voted
plurality
total
The
official ballot.
the
next two lines
official
(counted
votes
each
for
ballot
from
transcriptions on the test ballots plus the nonresponders totals).
The bottom
described
in
section shows the calculation for extrapolated approval vote totals that
Section
Approval voting,
4.
E3
Table 8 includes the
official
E3
votes for election
for
matched against the subsets approved by these
subsets,
shows that 23
f)eople
who
A
p>eople
who
and C, 16
approved of A, B and
C
voted for
voted for
on the
A
and B on the
each of the ten two-candidate
voters.
official ballot,
For example, row
B and C, and one person who voted
test ballot.
of people voted for only one of the five candidates on that ballot.
counted
in
table.
For example, of the 255 voters
who
C
and E
all
official ballot, a
These votes were
8.
of the data in the main table are given at the bottom.
frequencies for approving of a candidate not in the
from the
for
Contrary to instruction on the
the official totals but are not reflected in Table
ABC
20 people who voted for
number
Summaries
was
official
voted for
Conditional relative
ballot pair
can be computed
B and C on
the
main
ballot.
-19-
TABLE
8
Approval votes for election E3 are matched wth
candidate pairs voted for on the
official ballot
29 approved of
in
Section
VsUe for
C
6.
A
14%
(11%). 57 approved of
of the voters
approved of C, and
who
D
(22%), and 53 approved of
did not vote for
20% who
A
did not vote for
E (21%). As
approved of A,
D
approved of D.
23% who
noted
did not
-21-
Reference
Brams,
S. J.
and
P.
Boston, 1983.
C. Fishburn. Approval Voting, Birkhauscr,
n3^
mB^
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