Meeting SCW 2002
Cloning manipulation of the Borda rule
J´erˆome Serais
∗
GEMMA-CREME, University of Caen, France
Abstract
In an election contest, a losing candidate
a
can manipulate the election outcome
in his favor by introducing his clone
b
in the choice set, the clone
b
being defined as
an alternative which is ranked immediatly below
a
in the individual preferences. We
characterise the voting situations where this manipulation is efficient for the Borda rule
and express its vulnerability for a 3 alternative election.
Keywords
: Borda rule, Manipulation, Clone.
JEL Classification
: D72
1
Introduction
From the theorem of Gibbard-Satterthwaite (1973, 1975) asserting that all reasonable vo-
ting procedures are sensitive to manipulation by strategic voters, an important litterature
has been developped on the possible strategies of voters and their efficiencies. Recently,
a different way of acting upon the election result has araised. By changing the choice set
rather than the preferences, the final outcome might be radically different and favour the
manipulators. Saari proved (1989, 1990) that adding or removing a candidate may affect
seriously the ranking of scoring rules. Inspired by the theorem of Gibbard-Satterthwaite,
Dutta, Jackson and Le Breton (2001) assert that all reasonable voting procedures are sen-
sitive to strategic candidacy, i.e., a candidate can alter the election outcome by deciding
whether or not to enter the choice set.
In this paper, we will focus on a specific way of changing the choice set suggested by Dum-
mett (1998) that he called
agenda manipulation
. We will refer to this manipulation as
cloning manipulation
in order not to be confused with the agenda used in parliament to
choose amendment. A losing candidate may try to manipulate the election outcome by pro-
moting the candidature of a clone, i.e., another candidate who is considered similar to him,
by all the voters. Dummett noticed that the Borda rule may suffer from this manipulation
∗
Tel: (33) 2 31 56 66 29. Fax: (33) 2 31 56 55 13. email: serais@econ.unicaen.fr. I would like to thank
Dominique Lepelley and Vincent Merlin for their helpful comments.
1
and gave some examples as the one displayed in Example 1.
The Borda rule (see Borda 1781) belongs to the class of point ranking rules where points
are given to each candidate acccording to his rank in the preference of the voters. When
n
voters have to choose among
m
alternatives, each one assigns
m
−
r
points to the candidate
with the rank
r
in his individual preference. The Borda score of a candidate is the amount
of points that he collects and the chosen candidate is the one with the highest Borda score.
In Example 1, eleven voters have to choose among four alternatives
a
,
b
,
c
, and
d
. Each
column represents a preference ordering, with the number of individuals with this specific
preference, and the last column represents the number of points that a voter gives to a
candidate according to his rank.
Example 1
2 3 2 1 2 1
points
b c c a b b
3
d d a b a a
2
a a d d d c
1
c b b c c d
0
In Example 1,
a
is chosen with a Borda score of 18 (17 for
b
, 16 for
c
and 15 for
d
). Dummett
assumes that before the vote is taken, a fifth alternative,
e
, is introduced by
d
, whom every
voter rank immediately below
d
. The chosen candidate among this new choice set is
d
with
a Borda score of 26 (24 for
a
, 23 for
b
, 22 for
c
, and 15 for
e
).
As the reader can notice, this manipulation is sufficiently powerfull to make
d
chosen while
he has the lowest Borda score with the initial voting situation. In this example, cloning
manipulation can radically change the oucome in favour of the manipulator and it is natural
to raise the question of the theoretical frequency of occurence of this manipulation. The
procedure suggested by Gerhlein and Fishburn (1976) to obtain analytical representation of
the probability of an event is of great interest for our purpose. From a system of linear in-
equalities characterizing an event, their procedure is able to provide polynomials expressing
the likelihood of occurence of this event. The voting situations where cloning manipula-
tion occurs can be characterized, with some conditions, by a system of linear inequalities.
Unfortunately, when the coefficient of the variables of the linear inequalities are fraction of
integer, which is the case for cloning manipulation, it becomes extremely difficult to give
an analytical solution. Huang and Chua (2000)
1
provide a useful algorithm derived from
the method of Gehrlein-Fishburn. We use the Huang-Chua technique which permits us to
obtain polynomials expressing the likelihood of cloning manipulation for the Borda rule.
Basic notions and assumptions are introduced in Section 2. We characterize the voting situ-
ations where cloning manipulation can be successful, present the method of Huang and Chua
and give the vulnerability of the Borda rule to cloning manipulation in a three candidate
election in Section 3. We discuss our results in Section 4.
1
see also Gehrlein, 2002.
2
2
Notations
Let
X
be the set of possible alternatives and
A
a finite subset of
X
, with
|
A
|
=
m
.
2
The
set of individuals who choose among the candidates is
I
=
{
1
, . . . , i, . . . , n
}
with
|
I
|
=
n
.
We assume that the individuals are able to rank all the alternatives without ties in
X
. The
preferences on a finite subset
A
are the restriction of
P
i
on
A
. The six possible preference
orderings over
A
will be numbered as follows when
A
=
{
a, b, c
}
:
Table 1
n
1
n
2
n
3
n
4
n
5
n
6
a
a
c
c
b
b
b
c
a
b
c
a
c
b
b
a
a
c
A
voting situation
is a vector
s
= (
n
1
, n
2
, n
3
, n
4
, n
5
, n
6
), with
n
t
the number of type
t
voters
and
P
6
t
=1
n
t
=
n
, that gives the distribution of the
n
voters over the six possible preference
types.
S
n
is the set of all possible voting situations. For a context
A
, a
social choice
function
(SCF)
g
:
∪
n
i
=1
S
n
→
A
, assigns to each voting situation a nonempty subset of
A
,
g
(
A, s
). We shall assume throughout the paper that the social decision for the context
A
only depends upon the restriction of the preferences in the profile on
A
.
3
N
xy
is the number
of voters who prefer
x
to
y
,
S
x
B,s
is the Borda score of
x
for the voting situation
s
and
S
xy
B,s
is the difference of Borda score between
x
and
y
for the voting situation
s
. In case of a tie,
we use the lexicographic order to choose the winner.
The definition of a clone we give express the idea of a candidate creating another candidate,
always ranked after him.
4
Definition 1
A candidate
y
is a clone of
x
for a voting situation
s
if and only if:
∀
z
∈
X
\ {
x, y
}
,
∀
i
∈
I, xP
i
z
â‡â‡’
yP
i
z
and
∀
i
∈
I, xP
i
y
Given a voting situation
s
, we say that a SCF is vulnerable to cloning manipulation at
s
if
the outcome of a vote is better for a candidate when his clone is introduced in the context.
The vulnerability of the Borda rule will be the number of voting situations where this rule
is vulnerable to cloning manipulation at
s
when compared with the number of all possible
voting situations.
3
Vulnerability to cloning manipulation
We characterise the voting situations at which the Borda rule is senstive to cloning mani-
pulation. Lemma 1 describes the case where only one losing candidate manipulate while
the other losing candidate doesn’t react.
2
The cardinality of
A
will vary when an attempt of cloning manipulation takes place.
3
This is not the condition of Independance of Irrelevant Alternatives of Arrow(1963): in a context
A
, the
social preference between
x
and
y
may depend upon all the preferences on
A
, not only on the pair
{
x, y
}
.
But the preference on
{
x, y
}
is not influenced by
z
, which is not in the menu
A
.
4
For diffferent definitions of a clone, see Laslier (1999), Tideman (1987) and Zavist and Tideman (1989).
3
Lemma 1
Suppose
m
= 3
and consider a voting situation
s
= (
n
1
, n
2
, n
3
, n
4
, n
5
, n
6
)
.
The Borda rule is sensitive to single cloning manipulation at
s
if and only if
−
S
ab
B,s
≥
0
and S
ac
B,s
≥
0
and
((
N
ba
> S
ab
B,s
and N
bc
≥
S
cb
B,s
)
or
(
N
ca
> S
ac
B,s
and N
cb
> S
bc
B,s
))
−
S
ba
B,s
>
0
and S
bc
B,s
≥
0
and
((
N
ab
≥
S
ba
B,s
and N
ac
≥
S
ca
B,s
)
or
(
N
cb
> S
bc
B,s
and N
ca
> S
ac
B,s
))
−
S
ca
B,s
>
0
and S
cb
B,s
>
0
and
((
N
ac
≥
S
ca
B,s
and N
ab
≥
S
ba
B,s
)
or
(
N
bc
≥
S
cb
B,s
and N
ba
> S
ab
B,s
))
Proof
.
•
We assume that the Borda rule is sensitive to single cloning manipulation and
a
chosen
initially:
S
ab
B,s
≥
0 and
S
ac
B,s
≥
0.
The candidate
b
creates
d
. We obtain the following voting situation
s
0
:
n
1
n
2
n
3
n
4
n
5
n
6
points
a
a
c
c
b
b
3
b
c
a
b
d
d
2
d
b
b
d
c
a
1
c
d
d
a
a
c
0
The new Borda scores are :
S
a
B,s
0
=
S
a
B,s
+
N
ab
,
S
b
B,s
0
=
S
b
B,s
+
n
,
S
c
B,s
0
=
S
c
B,s
+
N
cb
and
S
d
B,s
0
=
S
b
B,s
b
is chosen if he beats the other candidates. As
d
is a clone of
b
,
d
is always beaten by
b
.
b
beats
a
if the difference between their new score is strictly positive which is equivalent to
write
N
ba
> S
ab
B,s
and
b
beats
c
if the difference between their new score is positive which is
equivalent to write
N
bc
≥
S
cb
B,s
.
•
We assume
S
ab
B,s
≥
0
, S
ac
B,s
≥
0
, N
ba
> S
ab
B,s
and N
bc
≥
S
cb
B,s
.
S
ab
B,s
≥
0
and S
ac
B,s
≥
0 implies
a
chosen.
N
ba
> S
ab
B,s
â‡â‡’
S
b
B,s
+
n > S
a
B,s
+
N
ab
â‡â‡’
3(
n
5
+
n
6
) + 2(
n
1
+
n
4
) +
n
2
+
n
3
>
3(
n
1
+
n
2
) + 2
n
3
+
n
6
(1)
N
bc
≥
S
cb
B,s
â‡â‡’
S
b
B,s
+
n
≥
S
c
B,s
+
N
cb
â‡â‡’
3(
n
5
+
n
6
) + 2(
n
1
+
n
4
) +
n
2
+
n
3
≥
3(
n
3
+
n
4
) + 2
n
2
+
n
5
(2).
The conditions (1) and (2) correspond to the fact of having yet 6 preference types and 4
candidates as Borda gives now 3 points for a top candidate. One property of the Borda
rule is that each voter of type
n
t
gives
m
(
m
−
1) points to the candidates. As we know that
there is 4 candidates and the number of points given by each voter of type
n
t
for
a
,
b
and
c
we find that the new candidate
d
is always ranked after
b
in the individual preferences. We
conclude that
d
is a clone of
b
. From the conditions (1) and (2) we know that
b
is chosen:
the Borda rule is sensitive to single cloning manipulation. The other cases are similar and
we omitt them. Q.E.D.
Lemma 2 describes the situations where the two losing candidates create simultaneously a
clone. They act upon the choice set at the same time but with a different aim.
Lemma 2
Suppose
m
= 3
and consider a voting situation
s
= (
n
1
, n
2
, n
3
, n
4
, n
5
, n
6
)
The Borda rule is sensitive to simultaneous cloning manipulation at
s
if and only if
−
S
ab
B,s
≥
0
and S
ac
B,s
≥
0
and
((
S
b
B,s
> S
ab
B,s
+
N
ac
and S
bc
B,s
≥
N
cb
−
N
bc
)
or
(
S
c
B,s
> S
ac
B,s
+
N
ab
and S
cb
B,s
> N
bc
−
N
cb
))
−
S
ba
B,s
>
0
and S
bc
B,s
≥
0
4
and
((
S
a
B,s
≥
S
ba
B,s
+
N
bc
and S
ac
B,s
≥
N
ca
−
N
ac
)
or
(
S
c
B,s
> S
bc
B,s
+
N
ba
and S
ca
B,s
> N
ac
−
N
ca
))
−
S
ca
B,s
>
0
and S
cb
B,s
>
0
and
((
S
a
B,s
≥
S
ca
B,s
+
N
cb
and S
ab
B,s
≥
N
ba
−
N
ab
)
or
(
S
b
B,s
≥
S
cb
B,s
+
N
ca
and S
ba
B,s
> N
ab
−
N
ba
))
Proof
.
See Annex.
The lemmas 1 and 2 give the complete characterization of the voting situations where
the manipulators are able to win after cloning manipulation (single or simultaneous). Let
CM
(
n
) be the set of these voting situations. The conditions that characterize
CM
(
n
) can
be translated in terms of the
n
t
’s. We assume each voting situation to be equally likely to
occur ( this is the Impartial Anonymous Culture condition used by Gehrlein and Fishburn
(1976)). Under this assumption, it is possible to obtain an analytical representation for the
number of elements in
CM
(
n
) as a function of
n
and write the polynomial as following:
|
CM
(
n
)
|
=
x
5
n
5
+
x
4
n
4
+
x
3
n
3
+
x
2
n
2
+
x
1
n
+
x
0
for a sequence of
n
=
r
+
ej
,
j
= 0
,
1
,
2
, . . . ,
5,
where the term
e
is the periodicity of the integer sequence for which the representation is
valid. The procedure of Huang and Chua permits to find the values of the coefficients of
CM
(
n
)
5
. We use computer enumeration to evaluate exact values for the number of the
situations characterized by Lemma 1 and 2 for each
n
=
r
+
ej
,
j
= 0
,
1
, . . . ,
5 (see Table 2).
We set up 6 equations with 6 unknowns (
x
0
, x
1
, ..., x
5
) and solve the 6 simultaneous equa-
tions. The Huang-Chua algorithm permits us to find that the periodicity
e
is 24 (resp.15)
for single (resp. simultaneous) cloning manipulation. Finally, we divide the cardinality of
CM
(
n
) by the total number of situations, and obtain the following representations for the
vulnerability of the Borda rule to cloning manipulation.
V(
B
cl
,r(e)) (resp. V(
B
simcl
,r(e))) is the vulnerability of Borda rule to single (resp. simul-
taneous) cloning manipulation with n
≡
r(mod e). We express only the first polynomial.
Proposition 1
V
(
B
cl
,
1(24)) =
1909
n
5
+ 24420
n
4
+ 111750
n
3
+ 208580
n
2
+ 94221
n
−
440880
3072(
n
+ 1)(
n
+ 2)(
n
+ 3)(
n
+ 4)(
n
+ 5)
V
(
B
simcl
,
1(15)) =
1023
n
5
+ 12900
n
4
+ 58745
n
3
+ 109260
n
2
+ 32280
n
−
214208
1875(
n
+ 1)(
n
+ 2)(
n
+ 3)(
n
+ 4)(
n
+ 5)
4
Concluding remarks
The algoritm of Huang and Chua is of great usefull when we want to look at the likelihood
of an event characterized by a system of linear inequalities. They provide a usefull tool
which make easier the study of the likelihood of success of the manipulation by cloning
candidate. Our results (see Table 2) show the great vulnerability of the Borda rule to this
manipulation. We can compare it to the classical manipulation attempted by voters mis-
representing their individual preference. In an election contest of three candidates when
n
is large the vulnerability of the Borda rule to manipulation of coalition of strategic voters
5
See Huang and Chua (2000) for a general proof.
5
is of about 50% (see Favardin, Lepelley, Serais 2001) whereas it is of 62% (resp. 55%) for
single (resp. simultaneous) cloning manipulation. These results reinforce the intuition of
Dummett about the importance of this manipulation. However, this manipulation is a very
specific case of manipulation by changing the choice set. If one doesn’t impose specific
condition over the choice set, the Borda rule may be robuster than other positional rule.
Gehrlein and Fisburn (1980) looked at the likelihood of a positional rule to choose the same
candidate when one modified the choice set by eliminating one (resp. one or two) losing
candidate of the choice set in the case of a three (resp. four) candidate election. They
proved that, in each case, the Borda rule was the positional rule which maximise the prob-
ability of electing the same candidate after this modification of the choice set.
6
References
Arrow, K.J., 1963, Social choice and individual values (Wiley, New York).
de Borda, J. C., 1781, M´emoires sur les ´elections au scrutin, Histoire de l’Acad´emie Royale
des Sciences, Paris.
Dutta, B., Jackson, M.O., Le Breton, M., 2001, Strategic candidacy and voting procedures,
Econometrica 69, 1013-1037.
Dummett, M., 1998, The Borda count and agenda manipulation, Social Choice and Welfare
15, 289-296.
Favardin, P., Lepelley, D., Serais, J., 2001, Borda rule, Copeland method and strategic
manipulation, working paper, University of Caen.
Gehrlein, W.V., 2002, Obtaining representations for probabilities of voting outcomes with
effectively unlimited precision integer arithmetic, Social Choice and Welfare (forthcoming).
Gehrlein, W.V., Fishburn, P.C., 1976, Condorcet’s paradox and anonymous preference pro-
files, Public Choice 26, 1-18.
Gehrlein, W.V., Fishburn, P.C., 1980, Robustness of positional scoring over subsets ot al-
ternatives, Applied Mathematics and Optimization 6, 241-255.
Gibbard, A.F., 1973, Manipulation of voting schemes: a general result, Econometrica 41,
587-601.
Huang, H.C., Chua, V.C.H., 2000, Analytical representation of probabilities under the IAC
condition, Social Choice and Welfare 17, 143-155.
Laslier, J.-F., 2000, Aggregation of preferences with a variable set of alternatives, Social
Choice and Welfare 17, p 269-282.
Saari, D.G., 1989, A dictionary for voting paradoxes, Journal of Economic Theory 48, 443-
475.
Saari, D.G., 1990, The Borda dictionary, Social Choice and Welfare 7, 279-317.
Satterthwaite, M.A., 1975, Strategyproofness and Arrow’s conditions: existence and cor-
respondences for voting procedures and social welfare functions, Journal of Economic The-
ory 10, 187-217.
Tideman, T.N., 1987, Independence of clones as a criterion for voting rules, Social Choice
and Welfare 4, 185-206.
Zavist, T.M., Tideman, T.N., 1989, Complete independence of clones in ranked pairs rule,
Social Choice and Welfare 6, 167-173.
7
Table 2: Vulnerability of the Borda rule to cloning manipulation
(three candidate election)
Single
Simultaneous
n
n
1
0
1
0
2
0
.
4286
2
0
.
2857
3
0
.
3571
3
0
.
3571
4
0
.
4127
4
0
.
3333
5
0
.
4524
5
0
.
4087
6
0
.
4675
6
0
.
4026
7
0
.
4735
7
0
.
4090
8
0
.
5012
8
0
.
4312
9
0
.
5055
9
0
.
4380
10
0
.
5115 10
0
.
4469
11
0
.
5240 11
0
.
4560
12
0
.
5294 12
0
.
4584
13
0
.
5346 13
0
.
4628
14
0
.
5399 14
0
.
4711
15
0
.
5445 15
0
.
4753
16
0
.
5490 16
0
.
4776
17
0
.
5530 17
0
.
4807
18
0
.
5552 18
0
.
4836
19
0
.
5584 19
0
.
4868
20
0
.
5622 20
0
.
4905
..
.
..
.
..
.
..
.
..
.
..
.
91
0
.
5318
145
0
.
6121
..
.
..
.
..
.
..
.
..
.
..
.
∞
0
.
6214
∞
0
.
5456
8
5
Annex
Proof Lemma 2
•
We assume that the Borda rule is sensitive to simultaneous cloning manipulation and
a
chosen initially. We have
S
ab
B,s
≥
0 and
S
ac
B,s
≥
0
We assume that
b
is chosen after the partisans of
b
and
c
have acted upon the choice set by
creating clone
d
and
e
. We have the following profile
s
0
n
1
n
2
n
3
n
4
n
5
n
6
points
a
a
c
c
b
b
4
b
c
e
e
d
d
3
d
e
a
b
c
a
2
c
b
b
d
e
c
1
e
d
d
a
a
e
0
The new Borda scores are :
S
a
B,s
0
=
S
a
B,s
+
N
ab
+
N
ac
,
S
b
B,s
0
=
S
b
B,s
+
n
+
N
bc
,
S
c
B,s
0
=
S
c
B,s
+
n
+
N
cb
,
S
d
B,s
0
=
S
b
B,s
+
N
bc
and
S
e
B,s
0
=
S
c
B,s
+
N
cb
.
b
is chosen if he beats the other candidates. As
d
is the clone of
b
,
d
is always beaten by
b
:
b
beats
a
if the difference between their new score is strictly positive:
S
ba
B,s
0
>
0, which is
equivalent to write
S
b
B,s
> S
ab
B,s
+
N
ac
.
b
beats
c
if the difference between their new score is positive:
S
bc
B,s
0
≥
0, which is equivalent
to write
S
bc
B,s
≥
N
cb
−
N
bc
e
is the clone of
c
so we have always
S
ce
B,s
0
≥
0, as
S
bc
B,s
0
≥
0 it implies
S
be
B,s
0
≥
0.
•
We assume
S
ab
B,s
≥
0
, S
ac
B,s
≥
0
, S
b
B,s
> S
ab
B,s
+
N
ac
and S
bc
B,s
≥
N
cb
−
N
bc
.
S
ab
B,s
≥
0
and S
ac
B,s
≥
0 implies
a
chosen.
S
b
B,s
> S
ab
B,s
+
N
ac
â‡â‡’
2
S
b
B,s
> S
a
B,s
+
N
ac
â‡â‡’
S
b
B,s
+
N
ba
+
N
bc
> S
a
B,s
+
N
ac
â‡â‡’
S
b
B,s
+
N
ba
+
N
bc
+
N
ab
> S
a
B,s
+
N
ac
+
N
ab
â‡â‡’
S
b
B,s
+
n
+
N
bc
> S
a
B,s
+
N
ab
+
N
ac
â‡â‡’
4(
n
5
+
n
6
) + 3
n
1
+ 2
n
4
+
n
2
+
n
3
>
4(
n
1
+
n
2
) + 2(
n
3
+
n
6
) (1)
S
bc
B,s
≥
N
cb
−
N
bc
â‡â‡’
S
b
B,s
+
n
+
N
bc
≥
S
c
B,s
+
n
+
N
cb
â‡â‡’
4(
n
5
+
n
6
) + 3
n
1
+ 2
n
4
+
n
2
+
n
3
>
4(
n
3
+
n
4
) + 3
n
2
+ 2
n
5
+
n
1
+
n
6
(2)
The conditions (1) and (2) correspond to the fact of having yet 6 preference type but with
5 candidates as Borda gives now 4 points for a top candidate. We know that the number
of candidates and the number of points given by each voter of type
n
t
for
a
,
b
and
c
so we
find that the two new candidates
d
and
e
are always ranked respectively after
b
and
c
in
the individual preferences. We conclude that
d
and
e
are respectively the clone of
b
and
c
.
Borda rule is sensitive to simultaneous cloning manipulation. The other cases are similar
and we omitt them. Q.E.D.
9