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WORKING PAPER
SLOAN SCHOOL OF MANAGEMENT
Single Transferable Vote
Resists Strategic Voting
John
J.
Bartholdi,
III
and James B. Orlin
Working Paper No. 3221-90-MS
November, 1990
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
50 MEMORIAL DRIVE
CAMBRIDGE, MASSACHUSETTS 02139
Single Transferable Vote
Resists Strategic Voting
John
J.
Bartholdi,
III
and James B. Orlin
Working Paper No. 3221-90-MS
November, 1990
Single Transferable Vote
Resists Strategic Voting
John
J.
Bartholdi, III
School of Industrial and Systems Engineering
Georgia Institute of Technology, Atlanta, GA 30332
James B. Orlin
Sloan School of Management
Massachusetts Institute of Technology, Cambridge,
November
13,
MA
02139
1990
Abstract
We
give evidence that Single Tranferable Vote
ally resistant to
manipulation:
It is
NP-complete
(STV)
to
is
computation-
determine whether
there exists a (possibly insincere) preference that will elect a favored candidate, even in an election for a single seat.
STV
is
qualitatively
more
difficult
Thus
strategic voting under
than under other commonly-used vot-
ing schemes. Furthermore, this resistance to manipulation
STV
is
inherent to
and does not depend on hopeful extraneous assumptions
presumed
like the
difficulty of learning the preferences of the other voters.
when an STV
elec-
tion violates monotonicity. This suggests that non-monotonicity in
STV
We
also prove that
it is
NP-complete
to recognize
elections might be perceived as less threatening since
den" and hard to exploit
for strategic
advantage.
it
is in
effect "hid-
Strategic voting
1
For strategic voting the fundamental problem for any would-be manipulator
to decide
what preference
be impractically
difficult
to claim.
We
will
show that
this
even
this difficulty pertains
which the manipulator knows the preferences of
in
the ideal situation in
other voters and knows that
all
they will vote their complete and sincere preferences. Thus
among
voting schemes
resistant to manipulation.
It
the integrity of social choice:
mean
excessive, this might
modest task can
under the voting scheme known as Single Transferable
Vote (STV). Furthermore
unique
is
actual use today
in
might be that
If
in
that
STV
it is
this resistance
apparently
is
computationally
can help protect
the work to construct a strategic preference
that strategic voting
is
is
not practical, even though
theoretically possible.
Following the conventions of
[11]
we formalize the fundamental problem
of
strategic voting as the following "yes/no" question.
EFFECTIVE PREFERENCE
GIVEN: A
set
C
of candidates; a distinguished candidate
sincere, transitive, strict,
QUESTION:
Is
preferences of
V
A
will
and complete preferences of the
there a preference ordering on
will
C
that
c;
and the
set
V
of
voters.
when
tallied
with the
ensure the election of c?
polynomial-time algorithm
for
EFFECTIVE PREFERENCE
is
one that
always answer the question correctly within a number of steps bounded
above by a polynomial
in the size of a description of the election
(which
is
0(| V||C|)). Polynomial-time algorithms are considered fast because the work to
answer the question does not increase too rapidly as a function of the
problem; similarly, problems
are considered "easy"
.
for
size of the
which there exist polynomial time algorithms
In contrast, an algorithm which requires exponential time
quickly becomes impractical as the size of the problem increases; accordingly,
such problems are considered "hard".
This theoretical distinction
is
widely
borne out
The
in practical
experience
[11].
celebrated theorems of Gibbard, Satterthwaite, and Gardenfors show
that any voting scheme that
gic voting; that
minimally
is
fair is in principle
there exist instances in which
is,
misrepresent his true preferences [10,12,21].
if
there exists an algorithm that
ERENCE
This paper
STV,
in
in
may be
common
is
it
EFFECTIVE PREF-
more
it
effort to vote strategically.
can be
show a
nipulation. In
Kemeny had
[3]
real voting
it
we prove that
qualitatively different from the others
is
difficult to
Results of this type were anticipated in
possible to
which proved that most voting
[3],
use are vulnerable to strategic voting. Here
encourage sincere voting, since
vulnerable;
is
resistant.
read as a companion to
requires distinctly
incentive to
For a susceptible voting scheme,
guaranteed to answer
a voting scheme in practical use,
that
some voter has
within polynomial time, then we say the voting scheme
otherwise the voting scheme
schemes
is
susceptible to strate-
scheme that
[3],
is
Thus
STV
might
do otherwise.
but until now
has not been
it
computationally resistant to ma-
was observed that voting schemes due to Dodgson and
the undesirable property that
particular candidate has
won
it
is
NP-hard
to
whether any
to tell
the election. Such schemes are hard to manipulate
for the uninteresting reason that
they are hard to operate.
What
wanted
is
is
a voting scheme that supports quick computation for authorized use, such as
determining winners, but erects computational barriers to abuse. In
was displayed a scheme that computes winners quickly but
However
it is
resists
[3]
there
manipulation.
a rather contrived scheme whose only use, as far as we know, has
been as a tie-breaking rule by the International Federation of Chess. This paper, taken with the results of
[3]
suggest that
STV
is
the only voting scheme in
actual use that computes winners quickly (in polynomial time) but
is
inherently
resistant to strategic voting.
Note that we are not arguing
faults
documented elsewhere
for
the adoption of
[4,7,9,13,16]
— rather we
STV — it
has troubling
are suggesting that
com-
among
putational properties ought to be
voting scheme
evaluated.
is
possibly helpful
the criteria by which a prospective
STV
In this regard
differs in
an interesting and
way from other common voting schemes.
2
Single Transferable Vote
Under
STV
each voter submits a total order of the candidates.
STV
tallies
votes
by reallocating support from "weaker" candidates to "stronger" candidates and
excess support from elected candidates to remaining contenders. In comparison
with other voting schemes in practical use,
despite
complexity,
its relative
STV
or
it is
rather complicated. However,
variants are used in elections for the
its
parliaments of the Republic of Ireland, Tasmania, and Malta; for the senates
of Northern Ireland, Australia, and South Africa; for
Republic of Ireland and some
dom
for
many
in
election of its city council
Australia and Canada; and in the United King-
it
is
used by Cambridge, Massachusetts for the
and school committee and by New York City
for the
John Stuart
advan-
election of its district school boards [22].
tages as being
".
.
.
such and so numerous
.
.
.
that, in
[STV] among the greatest improvements yet made
of
government"
[15].
what, since STV,
of
its
weaknesses
like
is
commend
it,
Electoral
We
my
in
that
it is
the theory and practice
this
most notably
its
[7].
Nevertheless,
in Figures 2, 3,
STV
and
4.
Most troubling
from a winner
STV
has
much
tendency to guarantee proportional represen-
STV
is
championed by such organizations
Reform Society of Great Britain and Ireland
formalize
enthusiasm some-
possible for a candidate to change
more support
its
conviction, they place
any voting system, must be imperfect.
tation. Largely for this property,
The
Mill praised
Subsequent analysis has tempered
to a loser as a result of gaining
to
local authorities in the
public and professional institutions, trade unions, and voluntary
In the U. S. A.
societies [17].
all
as an algorithm in Figure
All variants
we know
1,
differ
as
[17].
with supporting procedures
only in
how they
reallocate
The
excess support from an elected candidate to the remaining contenders.
"pure" form of
STV
uses the procedure given in Figure
For comparison,
4.
Figure 5 gives the procedure used by the city of Cambridge, Massachusetts.
Our complexity
how excess
STV, independently
results will hold for all such variants of
votes are reallocated.
For a picturesque rather than algorithmic description of
an
official
guide to one version of
STV
in
of
in [22].
STV may
A good summary
be found
Cambridge Election, 1941,
STV
J.
Brams,
in private
to the first author, conjectured that strategic voting
cult
is
under an
communication
computationally
diffi-
under STV. In addition, Chamberlin found that by several empirical mea-
sures strategic voting under
schemes he tested
in contrast to
STV
seemed more
(Plurality, Borda,
the other schemes,
the NP-completeness of
there
is
difficult
and Coombs)
". ..
[6].
usually has a
election-specific manipulation strategy". This
is
than under other voting
He observed
that
consistent with our results since
EFFECTIVE PREFERENCE
under
STV means
that
no simple pattern, structure, or rule to simplify the construction of
tive preference
large set
STV,
much more complex and
an effective preference. Thus, for each strategic voter, the search for an
—even
to achieve a
is
For
Counting
difficulty of strategic voting
Most immediately, Steven
election.
[8].
in [5]
Others have remarked on the apparent
STV
see
assessment of the strengths and weaknesses
Strategic voting under
3
STV,
practice, see Rules for
Ballots under Proportional Representation, City of
reproduced
of
must be
essentially enumerative search over an exponentially
when the
common
effec-
strategic voters are trying to coordinate their votes
goal.
In contrast, for Plurality, Borda,
and Coombs
it
always the case that a voter can, within polynomial time, either construct a
strategic preference or else conclude that none exists
[3].
procedure STV;
begin;
InitializeScoresQ;
fg +
quota :=
1]
;
repeat
if any candidate has score > quota then
status of candidate := elected;
ReallocateVotesOfWinner( candidate
);
else
with candidate of
lowest score
do
status := defeated;
Reallocate VotesOfLoser( candidate
);
end;
end;
until total of elected candidates and viable candidates
for each remaining viable candidate
=
k;
do
his status := Elected;
end;
end;
Figure
1:
STV,
from a set
is
C
written in pseu do- Pascal. This
based on the preferences of a
set
tallies
V
votes to elect k candidates
of voters.
A
viable candidate
one who has not yet been defeated or elected.
procedure
InitializeScores();
begin;
for each candidate
do
his score :— 1.0;
his status := viable;
end;
for each ballot
its
weight :=
add
end;
end;
its
do
1.0;
weight to the score of the
Figure
2:
Procedure to
first
viable candidate on ballot;
initialize scores for the
STV
algorithm
procedure ReallocateVotesOfLoser( candidate
);
begin;
for each ballot
if
candidate
do
is
the
first
viable candidate on the ballot
then
Increase score of next viable candidate on ballot by weight of ballot;
end;
end;
end;
Figure
3:
Procedure Reallocate VotesOfLoser
procedure Reallocate VotesOfWinner( candidate
);
begin;
excess votes for candidate := score of candidate - quota;
for each ballot
if
candidate
do
is
the
first
viable candidate on the ballot
then
Set weight of this ballot :— (weight) (excess/score);
Increase score of next viable candidate on ballot by weight of ballot;
end;
end;
end;
Figure
Procedure Reallocate VotesOfWinner
4:
for the
procedure Reallocate VotesOfWinner( candidate
"pure" form of
STV
);
begin;
excess votes for candidate := score of candidate - quota;
for
:=
j
1
Draw
to excess
do
a ballot randomly from
first
among
those on which candidate
is
the
viable candidate;
Increase score of next viable candidate on ballot by weight of ballot;
end;
end;
Figure
5:
Procedure ReallocateVotesOfWinner
used in Cambridge, Massachusetts
for the variant
of
STV
that
is
We
STV
case of
cial
EFFECTIVE PREFERENCE
prove that
will
which
in
all
is
hard, even for the spe-
candidate vie for a single seat.
In this case the
voting scheme works by successively eliminating a candidate with the fewest
votes and reallocating his support.
This type of voting scheme
The
alternative voting or successive elimination.
cal
evidence that there
PREFERENCE
Theorem
no dependably
is
fast
is
also called
strong theoreti-
EFFECTIVE
algorithm to solve
under Successive Elimination.
EFFECTIVE PREFERENCE
1
following
is
is
NP-complete under Successive
Elimination.
First observe that
ination
is
EFFECTIVE PREFERENCE
Proof
NP, since the
in
under Successive Elim-
effectiveness of a preference can be verified in poly-
nomial time by tallying the election and checking whether c wins.
Now we
nation
is
EFFECTIVE PREFERENCE
prove that
"as hard as" the following
problem that
is
under Successive Elimi-
known
NP-complete
to be
[11].
3-COVER
GIVEN:
i
=
A
set
S
with
\S\
=
n; subsets
Si,...,Sm
C
=
S, with \S,\
3 for
l,...,m.
QUESTION: Does
Now
STV
let
{S\,
.
.
there exist an index set / with
.
|/|
=
n/3 and U, e/
,Sm } denote an instance of 3-COVER.
election for which there exists an effective preference
Our
exists a 3-cover for S.
from among
5m + + 3
if
and only
1.
"possible winners" c and w;
2.
"first losers" aj,
3.
"second line"
.
.
.
o m and 5i
,
.
.
,
b
if
an
there
contrived election will be to select a single winner
into the following five groups.
.
= 5?
will create
candidates. In this election the candidates
71
b\,
We
Si
m and
6i
,
.
.
.
,
.
.
.
8
,
,
5m
6
m
;
;
fall
naturally
4.
"w-bloc" do,di,
5.
"garbage collectors" gi
The
election
Property
ai,
1:
..
.
Property
Property
Of
the
if
.
,d n
;
,
.
=
{i
and only
:
if
,
.
candidates to be eliminated,
eliminated prior to
ence which, when tallied with
is
eliminated prior to
a< if
C
all
as follows:
€
if i
them
are
a, }.
Then
c will
win the
{1,
.
,
.
.
m}, there
exists a prefer-
the other preferences, guarantees that a*
and only
if
a
£
can be constructed by allocating the
list
of
a 3-cover.
is
For any specified index set 7
3:
2m
.
a* is
7
;
have the following properties.
will
3m
first
gm
.
and ai,...,am
Let I
election
.
we construct
,a m
2:
.
Furthermore, such a preference
I-
first
I then place a, in the
m
positions on the preference
z'th
position; otherwise place a,
in the ith position.
Properties 1-3 together imply that a strategic voter can guarantee victory
and only
for c in this election if
Now we
follows.
he can solve the 3-cover problem.
if
describe the details of our contrived election.
The
voters are as
(Wherever we leave preferences unspecified, they can be
•
12m
•
12m —
•
10m + (2n/3)
•
For each
i
=
1,
.
.
• For each
i
=
1,
.
.
• For each
i
=
1,
.
voters with preference
1
(c,
.
.
.);
voters with preference (u\
c,
.
.);
voters with preference (do,
.
,n,
,
.
.
,
.
12m —
"', c,
)',
2 voters with preference (di,w,c,
m, 12m voters with preference
m:
6m +
arbitrary.)
Ai
—
(gi,
w,
c,
.
.
.);
.);
5 voters with preference (6,-,(,-,w,
c,
.
.)
and, for each of the three j such that j £ S,, 2 voters with preference
(b,,dj,w,c,
• For each
i
.. .);
=
1,
.
.
.
,m:
6m +
Ai
—
and 2 voters with preference (b,,d
voters with preference (b,,bi,w,
1
,
w,c,
.
.);
c,
.
..)
• For each
=
i
1,
.
.
.
6m + 4z —
m:
,
3 voters with preference (a,,
voter with preference (a,-,6t-,u;,c,
1
.
<?,,
u/,c,
.
.
.);
and 2 voters with preferences
..);
(a,,a,,u;,c,...);
• For each
=
i
1,
.
.
.
6m + 4i —
m:
,
voter with preference
1
(ai,ai,w,c,
Now we
(a,,6,-,ii;,c,
c
initial
supporters of c,
cannot get another vote until
w
votes, then
12m
w
The w-bloc. Candidate d
12m —
d, receives
second choice
w, so that
is
is
c.
Thus, to be
2 additional votes,
votes.
w
if
d, is
and
c
Except
a strategic
effective,
10m + 2n/3
whose
Thus
gets another 2
eliminated from the election before
initially receives
any
12m— 1
eliminated. Moreover,
is
2 votes. For every voter
if
has
u>
voters rank c immediately below w.
all
w
votes and
cannot be eliminated before
preference must ensure that
more than
and 2 voters with preferences
..);
consider the election.
12m
the other
.
...);
Votes for c and w. Initially c has
for the
3 voters with preference (ai,g,,w,c, ...);
c.
votes and each of
first
choice
w
eliminated prior to w, then
the
is d,-,
will
obtain
cannot win. Thus the strategic voter must
We
ensure that each candidate d achieves a score of at least 12m.
t
will contrive
the remainder of the election so that this can be accomplished only
if
certain of
the "second line" candidates are eliminated early and their votes reallocated to
candidates in the w-bloc.
The second
Candidate
line.
initially receives
6,
6m +
4»
eliminated, his votes are reallocated as follows: for each j
dj
and the remainder go to
votes initially. If
6^
is
6;.
Similarly, candidate
6,
or
fc,
is
votes. If
Si, 2 votes
also receives
fc,
is
go to
6m + 4i +
1
6,-.
The
third choice of
all initial
supporters
w.
Note that
12m.
1
eliminated, his votes are reallocated as follows: 2 votes
go to do and the remainder go to
of
6,
€
-f-
if
fcj
Similarly,
is
eliminated before
if 6,
is
6i,
then the revised score of
eliminated before
10
6,,
6,
will
then the revised score of
exceed
6,
will
exceed 12m. This means that at most one of
can be eliminated before
6,
6,-,
c
or w.
The
The First Losers.
which
one
is
as follows:
Similarly,
a^ is
fc,.
If a,
and the remainder goes
a,-;
and the remainder goes to
Each candidate
The Garbage Collectors.
Now we show how
6m +
is
4i,
eliminated, his votes are reallocated
is
2 votes go to
6<;
a*, a;
eliminated, his votes are reallocated as follows:
2 votes go to a,;
6,-;
and
6;
vote goes to
1
if
than
less
each candidate
initial score of
to jj.
vote goes to
1
g,.
initially
<7,
has
12m
votes.
the sequence in which candidates are eliminated in the
election can encode a 3-cover.
Lemma
1
Exactly one of
6,
hi,
will be
among
3m
the first
eliminated. Furthermore, candidate c will win the election if
J
=
j
{
:
bj
is
among
3m
the first
candidates to be
and only
if the set
candidates to be eliminated} is the index of a
set cover for the 3-cover problem.
Proof The
first
claim
must be among the ai,a
will
true because the
is
t
6;, 6;;
,
and
if
first
one of
3m
6,,
fc;
candidates to be eliminated
is
eliminated, then the other
have score exceeding 12m.
To
Then
establish the second claim, suppose that
\J\
=
n/3. Consider the election after the
eliminated.
Suppose
i
6
5,
and j £ J
.
Then
J
first
6,
is
(10m
+
2n/3)
After
all
the
of the
3m
6;
's
!
candidates have been
-:-e.<
6;
eli
<f,,
i
n/3)
=
J
,
b } for j
an
i
1,
.
.
,
.
has been eliminated for j £ J
is
m, has
.
Thus
at least
12m, which exceeds that of candidate w.
candidates have been eliminated there remain only
$.
=
.
12m, which
have been eliminated and the revised score of do
+ 2(m —
g,, bj for j
.
.
d{ is at least
the index of a set cover, each
a revised score of at least 12m. Also,
m — n/3
3m
ha-'
has received 2 of bj's votes. Thus the revised score of
exceeds that of w. Since J
the index of a set cover.
is
£
J,
all
of which excepting
11
w
c, u;, all
the
d,-,
have at least 12m
Candidate w, with 12m
votes.
will inherit his votes
Next the
and
-
1
his score will rise to at least
votes of each d, are reallocated to
(n
+
l)(12m)
Now no
remaining
=
+ 36)m -
(12n
or
c
1.
12m
votes.
(24m —
rises to at least
The
1)
+
1.
has more than
6{
whose score
c,
has more than then
<j,
6;
24m —
candidates are eliminated since each has only
d{
Candidate
votes, will be eliminated next.
24m + 8m = 32m
32m
Since each
votes.
eliminated will have his votes reallocated to
votes.
gi,
must be that
c, it
Similarly,
or
6,-,
c
is
that
6,
no
is
the eventual
winner of the election.
Conversely, suppose that J
cannot win the election.
c
If \J\
not the index of a set cover; we will show that
is
>
n/3, then fewer than
m— n/3 of the bj's have
been eliminated, and the revised score of do cannot exceed 12m
be eliminated prior to w, and
will
score than
c.
w
In such case
w
will receive his votes
—
2.
Thus d
and achieve a higher
cannot be eliminated before c and so
c
cannot win
the election.
If
|
J
|
< n/3 and
the score of
w
di
i
€ S
is
not covered by the corresponding set cover, then
has not increased and
will receive his votes
must be eliminated before w. Thus
d{
and achieve a higher score than
c.
Again
w
cannot be
eliminated before c and so c cannot win the election.
Now we show how
among
are
Lemma
the
first
2 Let I
the ith candidate
in
which the
first
to construct a preference that will control which of the
3m
candidates to be eliminated.
C {l,...,m} and
is a,
3m
6;
if i
€ I and
a,
consider the strategic preference in which
Then
otherwise.
the following
is
candidates are eliminated.
•
The
3i
-
2 candidate to be eliminated
•
The
3i
-
1
•
The
3i candidate to be eliminated
candidate to be eliminated
is a;
a,
if
i
€
/,
and
is 6,
if
i
€
I,
andbi otherwise;
€
I,
is
if
12
i
and
ci{
a,
otherwise;
otherwise;
the order
Proof (By
Assume
induction)
that the
nated and that these candidates were,
one of each pair
Case
1
for
—3
each j
<
6m +
+
At
1
Candidate
votes.
eliminated, after which the revised score of
b{
6m +
is
Next, candidate
6,
4i
is
+
i
^
The next
/.
Candidate
is
of a,
is
1;
6m +
is
by
+
3.
all its
Lemmas
6, is
1
and
Corollary
dently of
1
how
6;
is
6m +
6m +
6m + 4i + 2;
6;
2 candidate c will
4j
a,-
is
and
4» votes
+
1;
is
6m +
who
a,,
is
the revised
+
Ai
3.
we have our main
a*.
is
6,
and the revised score
is
eliminated, followed by
is
as stated in the theorem.
win the election
if
By
a,-.
and only
if
/
is
O
since the single elimination voting
variants,
and exactly
next, after which the revised score of
Next, candidate
a solution to the set cover problem.
Now
Gj
viable candidate on the strategic preference
induction, the elimination of candidates
Now
and
as claimed.
the revised score of
4j
has
Hi
and the revised score of
2;
must be eliminated
a,
6m + 4i +
a,
eliminated, followed by a,. Thus, by induction, the
elimination of candidates
Case 2
both
i,
next viable candidate on the strategic preference
therefore has
score of
candidates have been elimi-
bj
,
/ The
€
j
bj
first 3i
scheme
is
a special case of
STV
in
result.
EFFECTIVE PREFERENCE
under
STV
is
NP-hard indepen-
the excess votes of a winning candidate are reallocated
among
the
remaining contenders.
4
Non-monotonicity
A
voting procedure
in
the preferences of voters; that
is
monotontc
if it
is,
does not respond negatively to changes
a candidate can never be hurt
13
if
some
move him up
voters
STV
Doron and Kronick have shown that
in their preferences.
non-monotonic: there exist elections
is
from an
STV
winner to a
which a candidate can change
simply as a result of gaining support
loser
it is
elections.
Thus
it
We
[7].
NP-complete to recognize when non-monotonicity occurs
show that
in
STV
can be impractical to exploit non-monotonicity for strategic
To some
purposes.
for
extent this weakens any argument against
STV
that
is
based on non-monotonicity. Since non-monotonicity can be hard to recognize,
voters are unlikely to be regaled with examples of
elections.
its
Furthermore, even
distortions, since hidden,
if
know
voters
It
pernicious effects in real
that non-monotonicity
might be perceived
unlikely to be exploited, benign.
its
as
possible,
"random" and even,
might be seen by voters
since
as similar in spirit
randomness occasionally designed into voting systems, as
to the
is
in [22].
Thus
non-monotonicity might be more easily tolerated than flaws that are easy to
When
find.
flaws are easy to find, then they are
more
likely to
be exploited by
manipulators and they are easier to display to an outraged electorate. Indeed,
when
flaws easy to find, manipulative behavior might be encouraged, while
if
flaws are hard to find, manipulative behavior might be uneconomic.
Doron and Kronick argue that any non-monotonic voting procedure
verse".
it is
They
vividly
make
their point
by imagining an
STV
is
"per-
election in which
observed that non-monotonicity affected the outcome.
Most voters would probably be alienated and outraged upon hearing
the hypothetical (but theoretically possible) election night report:
"Mr. O'Grady did not obtain a seat
in
today's election, but
him
in
second place instead of
of his supporters had voted for
place, he
if
5,000
first
would have won!"
However, because
it is
NP-complete to recognize an instance of non-monotonicity,
voters would almost certainly not hear any such news on election night. In fact
they would probably would never hear
because
it
would simply take too long
it
— not because the claim was
to discover.
14
false,
but
We
formalize the problem of recognizing non-monotonicity as follows.
NON-MONOTONICITY
GIVEN: A
C
set
V
of candidates; the set
of sincere, transitive, strict, and
C
complete preferences of the voters; and a distinguished candidate c €
would be a
loser
QUESTION:
the election was tallied now.
if
V who
there a subset of voters in
Is
who
can change
c
from a
loser to
a winner by lowering c in their preferences?
Theorem
NON-MONOTONICITY
2
NP-complete for Successive Elimina-
is
tion.
Proof
Again
Our proof
{S\,
let
.
.
election (which
•
12m
•
12m —
•
12m
•
10m +
almost the same as
is
=
1,
.
.
• For each
i
=
1,
.
.
.
For each
i
=
1,
.
.
.
n,
,
.
12m —
,m:
6m
i
=
each
i
=
1,
.
preference (c,6,,
Candidate
before
c.
.
.);
.);
m:
,
.
.
.
.);
.
1
(<;,,
w,c,
.
voters with preference (bi,bi,w,c,
6m
l,...,m:
1).
2 voters with preference (di,w,c,
j
G
5,, 2 voters
.
.
.
.);
.);
.
.)
and, for
with preference (bi,dj,w,c,
voters with preference
voters with preference (6;,d ,u',c,
u;
.
.
,m, 12m voters with preference
each of the three; such that
• For
c,
(2n/3) voters with preference (do, w,c,.
each
details.
..);
voters with preference (u/, w,c,
t
• For
and so we omit
proof to Theorem
in the
voters with preference (w,
• For each
•
1
,Sm ] denote an instance of 3-COVER. Consider the following
.
voters with preference (c,w,.
1
Theorem
similar to that of
is
(6, ,6{,
w,c,
.
.)
.
and
.)
2
..);
voter with preference
(c, 6,,
.
.
.)
and
1
voter with
.);
c will lose this election since reallocated votes will
However the
last
always go to
group of voters, and only that group, can make
15
winner by moving him down on their preferences.
c a
appropriate preferences of the
last
gaining any (reallocated) votes until
votes; then
in
which
C
{i}
be eliminated and
{ 1,
.
.
.
m}
,
is
c in the
can be prevented from
remaining candidates have at least 12m
c will
win the election. The preferences
(c, 6j,
.
.
.)
and
a solution to the 3-cover problem.
NON-MONOTONICITY under STV is
Corollary 2
how
will
all
must be moved down are exactly those
c
which the
w
w
group of voters,
By lowering
(c,
6,-,
.
.
for
.)
D
NP-hard independently of
the excess votes of a winning candidate are reallocated
among
the remaining
contenders.
Finally,
voter in an
ing
him up
we
slightly strengthen the
STV
example of
election can change a candidate
in his preferences:
[7]
by observing even a single
from winner to
In the contrived election of
loser
Theorem
by movif
1,
the
manipulative voter constructs an effective preference then his favored candidate
c will
win; but
Thus
STV
is,
5
is
if
the voter moves
c to
the top of his preferences then c will
apparently as sensitive as
it
can be to changes
in preferences;
lose.
that
non-monotonicity can be realized by even a single voter.
Conclusions
There are several ways to measure the vulnerability of a voting procedure to
strategic voting (see, for example, [6,18,20]).
computational
effort.
is
have suggested measuring the
This complements the established approach, which
count the opportunities
bility are useful
We
for
A
weakness of the latter
assumption about the distribution of
to justify. Frequently
it is
to
manipulation. Both of these measures of manipula-
but imperfect. The primary weakness of the former
a worst-case measure.
is
societies
is
that
it
that
it
based on some
and that assumption
chosen for convenience of analysis.
16
is
is
is difficult
We
think that even the weakest reading of our theorem makes three con-
tributions.
First,
we have explained Chamberlin's empirical observation that
strategies to manipulate
results
an
STV
show how a would-be manipulator
STV
election: he
Also, our
[6].
constrained to plot his strategy in
is
cannot do significantly better than enumerative search
an effective preference. This
for
and instance-specific
are complex
is
different
from other standard voting procedures,
which a natural, "greedy" algorithm
ence or show that none exists
[3].
for
will either
produce an effective prefer-
Finally, our result refines the conclusions of
impossibility theorems such as those of Gibbard, Satterthwaite,
some
All voting procedures have flaws, but for
can be
it
and Gardenfors:
difficult to exploit
those
flaws.
The
strongest interpretation of our result
pect to vote strategically
in
an
STV
This probably overstates the case, for
is
that one cannot in practice ex-
election, even given perfect information.
like
some of the notions of
putational complexity does not perfectly model our concerns.
the caveats of
rize.
[3]
This means that,
PREFERENCE
under STV, some are
summa-
an asymptotic, worst-case measure
is
for large
com-
In particular,
hold here, the most important of which we briefly
Most immediately, NP-completeness
of complexity.
fairness,
enough instances of
difficult.
EFFECTIVE
This n.ight be insufficient to
provide practical protection against strategic voting. Certainly for small enough
elections one can simply generate each of the |C|! possible preference orders and
check each one to see whether
with
many
candidates are
it
elects the favored candidate.
in fact difficult to
manipulate
is
Whether
elections
an empirical question.
Naturally this will depend on the structure of a "typical" instance.
It is
worth
remarking that Chamberlin's observations on the complexity of manipulating
STV
were based on small elections (3 candidates and electorates of
1000); this suggests that the computational difficulty
size 21
and
becomes palpable rather
quickly as the size of the election increases.
We
close with
some additional comments on points that
17
are frequently mis-
understood. First,
it is
important to note that our assumptions of perfect
infor-
mation and sincere voting by most of the voters are intentionally unrealistic
we
are not describing voter behavior, but
show that even when the
difficult to
that
expects
it
unrealistically favorable
is
determine how to vote strategically.
difficult for
it is
situation
making a conservative assumption
it
can
still
to
be too
Furthermore, we have shown
even a single manipulator to determine how to vote. One
to be only harder, at least in an informal sense, for a group of people
to coordinate their votes to achieve a preferred outcome.
A
second possible misunderstanding
is
that one might be tempted to con-
clude, by the contrived nature of the election in the proof of Theorem
1,
that only
pathological instances are difficult.
This would be overinterpreting the proof.
The
is
contrived nature of our election
an artifact of
human
proof-building:
needed only to show that there exist some instances that are as
other hard problem; we constructed an instance that
say whether any particular election
We
sified
is
difficult to
difficult as an-
made our proof
as possible, but not necessarily representative. In short, our
We
as simple
theorem does not
manipulate.
also point out that the issues of computational difficulty are only inten-
when
the computational device
with pencil and paper. For example,
Summer Olympics was
is
a relatively slow one, such as a person
in [14] it
was reported that the
site of the
chosen by successive elimination, with six
cities
voted on by the 88 members of the International Olympic Committee.
Our
1996
results
imply that, even with perfect information, a strategic voter would have
had to perform enumerative search
possibilities.
Even
if
90%
for
an effective preference over
6!
=
840
of the possible preferences could have been eliminated
a priori, the prospect of evaluating the
remaining 84 elections by hand might
have been sufficiently discouraging to prevent strategic voting.
Finally,
we speculate on some broader suggestions
of our results. For
when
elections
we have
voting
possible and of recognizing instances of non-monotonicity.
is
established the formal difficulty of recognizing
18
STV
strategic
Most imme-
diately, this suggests that
of
its
weaknesses
perhaps
less
More
fail
STV
might
rise
somewhat
in
our estimation, since two
—susceptibility to strategic voting and non-monotonicity —are
threatening than previously thought.
generally,
we have
illustrated that even
to satisfy fairness criteria, nevertheless
it
when voting procedures can
can be hard to verify when
this
occurs. This might be interpreted as offering an explanation for an observation
made by Nurmi and
others of the
".
.
.the differences in difficulty of devising
counterexamples showing the violations of
some procedures and
criteria
criteria the productions of a
obvious while for other procedures
it
may
by various procedures. For
counterexample
be exceedingly
may
be rather
difficult" [19].
Acknowledgements
J.
Bartholdi was supported in part by a Presidential
Young
Investigator
Award
from the National Science Foundation (ECS-8351313) and by the Office of Naval
Research (N00014-85-K-0147).
tial
Young
Investigator
J.
Orlin was supported in part by a Presiden-
Award from
the National Science Foundation
(ECS-
8451517).
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21
2966 067
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MIT
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