Some further results on logrolling in committees
Louis Aimé Fonoa , Boniface Mbihb , Aristide Valeub
a
b
Faculté des Sciences, Université de Douala, Cameroun
CREM UMR CNRS 6211, Université de Caen, 14032 Caen, France
Abstract
The aim of this paper is to study the circumstances at which vote trading is
susceptible to arise when individual votes are dichotomous. We provide an answer to some questions about the size of the society, the majority quotas, some
domain restriction of individual votes and the likelihood of such a phenomenon.
Conclusions are also obtained for logrolling and Anscombe’s paradox.
Keywords: logrolling, Anscombe paradox,
majority, impartial anonymous
culture.
JEL Classi…cation: D71
1
Introduction
One usual way to vote on motions in committees consists in yes-no voting. A motion is accepted if more than a (possibly quali…ed) majority of individuals vote yes;
otherwise, it is rejected. Further, it often happens that a committee has to vote on
several independent motions, in the sense that the social decision on some motion
does not have any in‡uence on the decision on some other motion: all motions can be
accepted or all of them can be rejected. For example, without any budget constraint,
a city council may have to decide whether to (a) build a bridge, (b) build a school
and/or (c) build a retirement house. Majority votes are then organized successively
on these di¤erent motions. Dummett (1984, Chap. 1) extensively studies this process
and speci…cally, he points out that after all successive votes, the con…guration of the
votes may lead to a somewhat paradoxical situation:
Corresponding author: Tel: +33(0)231565826; e-mail: [email protected];
1
“Suppose that the composition of a committee is …xed for a certain period,
during which it has to take a number of decisions. The following proposition,
at …rst glance surprising, belongs to the dynamic theory of voting: a majority
of the committee members may be in the minority on a majority of occasions.”
Situations exhibiting such a feature are known to describe Anscombe’s paradox
(Anscombe 1976). It also appears that this paradox is closely related to the possibility
for some group of voters to secure a better outcome for each of its members: with
three motions (or more), it is easy to construct an example showing that “if they
form a party, and agree always to vote for the outcome preferred by the majority
within the party... each of them obtains the outcome he prefers in two out of three
occasions...”(Dummett 1984, pp 15-16). More precisely, such a behavior amounts to
vote trading, or to logrolling, that is a behavior by which voters - usually committee
or legislative members - act in order to obtain passage of motions of interest for each
of them (see precisions in Section 2).
Vote trading and logrolling have been widely studied in modern public choice
literature (see for example Wilson 1969, Riker and Brams 1973, Stratmann 1997
or Miller 2001), especially according to their relationships with voting paradoxes;
Miller (1975, 1977) studies the relation with the Condorcet’s paradox with reference
to parliamentary contexts, while Lagerspetz (1996) also examines the relation with
other paradoxes, such as the Ostrogorski’s paradox and the Anscombe’s paradox.
More recently, empirical works examine the evidence of vote trading agreements in
Congressional voting (e.g. Stratman 1992, or Wiseman 2004).
The aim of this paper is to provide an answer to some unsolved questions related
to vote trading, logrolling and Anscombe’s paradox under dichotomous votes, that
is yes-no successive (possibly quali…ed) majority voting. More precisely, we start by
determining, depending on the number of motions, the minimum size of a committee
for vote those phenomena to be possible. We then tackle questions about what
quali…ed majority is necessary to avoid vote trading opportunities, how large can be
coalitions involved in vote trading behavior, can we …nd a way to restrict the domain
of individual voting vectors in order to avoid vote trading, and how likely are vote
trading opportunities?
The paper is organized as follows: Section 2 introduces the notation and basic
de…nitions; Section 3 is concerned with preliminary results; then Section 4 studies the
characterization of vote trading situations and other types of behavior are analyzed
2
in Section 5, Section 6 provides results about how frequent vote trading opportunities
are. Finally Section 7 concludes the paper.
2
Notations and de…nitions
We consider a …nite set of n voters N = f1; 2; :::; ng, and a …nite set A of m motions
(not necessarily alternative motions). Every individual i 2 N reports a vector v i
of votes, by accepting (+1) or rejecting ( 1) a motion; for simplicity, we assume
that no voter abstains. Note however that in some cases, for example voting on
the budget in french universities, abstaining amounts to voting “no”. For example,
given three motions: v i = (v1i ; v2i ; v3i ) = (+1; 1; +1) means that individual i accepts
motions 1 and 3, but rejects motion 2. Similarly, a social (decision) vector describes
the decision (+1) or ( 1) of the whole society. A pro…le of individual votes is an
n tuple v N = (v 1 ; :::; v n ) of individual vectors, one vector for each individual. We
also denote by vkN = (vk1 ; :::; vkn ) the n tuple of the k th individuals components of
v N ; k = 1; :::; m. A social decision rule (SDR) is a mapping the domain of which is
the set of all possible pro…les whose range is the set of all possible social vectors.
In this paper, we are concerned with a particular class of SDRs f ,
1
2
<
1,
where for each possible pro…le v N , and all k = 1; :::; m,
N
v N = 8f v1N ; :::; f vm
and
i=n
>
1
< 1 if P v i > 2n(
)
k
2
N
(ii) f (vk ) =
i=1
>
: 1
otherwise
We call these SDRs the successive
majority votes.
(i) f
Property (i) describes some kind of homogeneity between votes on di¤erent motions, in the sense that the same rule is used for all motions; but also, in the context of
successive votes, the vote on motions coming farther does not depend on the outcome
(acceptation or rejection) of previous votes.
Property (ii) explains how the individual votes are aggregated into a social decision: a motion is accepted by the committee if it is supported by a majority of more
than n voters, with
2 [ 21 ; 1[: Otherwise, it is rejected. For example, with
= 1=2,
for a motion to be socially accepted, it must obtain the support of more than half
of the voters; in other words, the number of voters accepting it minus the number of
voters rejecting if must be strictly positive; and with
3
= 2=3; this di¤erence must
be strictly greater than 13 n.
Note that the class of SDRs de…ned above is su¢ ciently general to include the
possibility for voters to abstain (by voting 0) on some motions. If this arises, the
social decision on each motion is based on the votes of those voters who did not
abstain.
It is also worth noting that in our SDR f , if everybody in the society abstains on
a motion, that motion is rejected. One consequence of such a hypothesis is that the
society never abstains. Anyway, for the sake of simplicity we assume in what follows
below that voters never abstain.
Throughout the paper, for a real number u, the ‡oor of u is the largest integer,
denoted by buc; less than or equal to u: buc = maxfn 2 Z; n
3
ug:
Characterization of majorities in a minority
3.1
Preliminary observations
With independent successive votes, it may arise that a majority of voters feel unsatis…ed after all votes because although all decisions have been taken on the basis of
majority votes, they are in minority on a majority of votes. To illustrate, consider
the examples below, where
= 21 .
Example 1 Let N = f1; 2; 3; 4; 5g and A = fa1 ; a2 ; a3 g.
motions voter 1 voter 2 voter 3 voter 4 voter 5 social decision
a1
+1
1
1
+1
+1
+1
a2
1
1
+1
+1
+1
+1
a3
1
+1
1
+1
+1
+1
Every motion has been passed, but voters 1, 2 and 3, that is 3 voters out of 5 (a
majority), have voted with the majority only one time out of three.
It also appears that in some of such cases, those voters can form an alliance and
“trade their votes” in order for each of them to secure a greater number of social
outcomes in accordance with their votes. Then, voters 1, 2 and 3 can change their
votes as illustrated in the next example.
Example 2 Let N = f1; 2; 3; 4; 5g and A = fa1 ; a2 ; a3 g.
4
motions voter 1 voter 2 voter 3 voter 4 voter 5 social decision
a1
1
1
1
+1
+1
1
a2
1
1
1
+1
+1
1
a3
1
1
1
+1
+1
1
Now, every motion has been rejected, but every individual in coalition f1; 2; 3g is
better-o¤ since a greater number (two out of three, that is a majority indeed) of social
decisions are in accordance with her true votes as reported in Example 1.
Let us now consider the …rst question we raised in the introduction of this paper:
what is the smallest number of voters for which an alliance for logrolling is susceptible
to emerge? In other words, given the number of motions, what is the minimum size
of committee for logrolling to be possible?
The statement below provides an answer to that question.
Proposition 1 Suppose f1=2 is the SDR and let m be the number of proposals. Then,
the minimum number r of voters for which logrolling is susceptible to arise is
(
m + 2 if m is odd
r=
:
m + 3 if m is even
Proof. We distinguish three steps.
Step 1. Suppose n is odd. Since logrolling is susceptible to arise, that is, most of
the voters are in minority on most motions, then
in minority on
b m2 c
n+1
2
voters (S = fi1 ; :::; i n+1 g) are
2
+ 1 motions. We …rst establish that: “if a voter ij of S is in a
minority on a motion aj , then there exists another voter ik 2 S
fij g which is not in
minority on aj ”. Suppose on the contrary that ij is in minority on a motion aj and,
for all t 2 S
fij g; t is in minority on aj . Thus all the voters in S are in minority on
aj and some of the rest of
n 1
2
n 1
2
voters are in accordance on aj . To say that some of
voters are in accordance on aj ; means that the number c of these voters, which
is smaller than or equal to
n 1
,
2
is greater that
n+1
,
2
a contradiction. It then follows
that
n+1
m
m
(b c + 1) + 1 = b c + 2
(1)
2
2
2
; and, in this case, the inequality (1)
Hence, for m odd, we have b m2 c + 1 = m+1
2
becomes
n+1
2
m+1
2
+ 1; i.e., n
m + 2. And for m even, we have b m2 c + 1 =
5
m
2
+ 1;
n+1
2
and, in this case, the inequality (1) becomes
m
2
+ 2; i.e., n
m + 3.
Step 2. Assume that n is even. Since logrolling is susceptible to arise, that is, most
of the voters are in minority on most motions, then
n
2
+ 1 voters (S = fi1 ; :::; i n2 +1 g)
are in minority on b m2 c + 1 motions. We …rst establish that “if a voter ij of S is
in minority on a motion aj , then there exists at least two voters ik ; it 2 S
fij g
which are not in minority on aj ". We distinguish two cases: (i) First suppose on the
contrary that ij is in minority on a motion aj and, for all t 2 S
fij g, t is in minority
on aj . Thus all the voters of S are in minority on aj and some of the rest of
voters are in accordance on aj . To say that some of
n
2
n
2
n
2
1,
+ 1, a contradiction. (ii) Now suppose on the contrary that ij is in
minority on a motion aj and there exists a unique voter t 2 S
on aj . Thus all the voters of S
n
2
1
1 voters are in accordance on
aj ; means that the number c of these voters, which is smaller than or equal to
is greater that
n
2
fij g, t is in minority
ftg are in minority on aj and some of the rest of
voters are in accordance on aj . To say that some of
n
2
voters are in accordance on
aj , means that the number c of these voters, which is smaller than or equal to n2 , is
greater that
n
2
+ 1, a contradiction. Hence,
n
2
Therefore, for m odd, we have b m2 c =
comes
n
2
m 1
2
+ 2; i.e., n
the inequality (2) becomes
n
2
b
n
2
+1
m
c+2
2
m 1
;
2
(b m2 c + 1) + 2 = b m2 c + 3, that is,
and, in this case, the inequality (2) be-
m + 3. For m even, we have b m2 c =
m
2
(2)
+ 2; i.e., n
m
;
2
and, in this case,
m + 4.
Step 3. From the two previous steps, we have: for m odd, r = m + 2; and, for m even,
r = m + 3.
In the sequel of the paper we will focus on the three-motion case - with m >
3, computations are much more complex - and we will be interested in a series of
problems: (i) does the phenomenon illustrated in the examples above occur for any
majority quota
that is with
? (ii) if the social decision function is the simple majority rule =
1
2
- what is the maximum proportion of voters in a minority on
a majority of occasions? or alternatively, what is the maximum size of a coalition
whose members can pro…tably engage in logrolling? (iii) does this still happen if all
individuals have single-plateaued (see Subsection 4.1 below) voting vectors? And (iv)
how frequent opportunities of logrolling are?
6
Problems (i) and (ii) are successively dealt with in this section, while problems
(iii) and (iv) are the topics of Section 4.
3.2
Majority quotas and logrolling for the three-motion case
In this subsection we determine values of quota
under which there exist majority
coalitions susceptible to form an alliance in order to trade their votes.
Successive votes are taken on three alternatives and every voter i accepts or rejects motions. Then the set of all possible individual voting vectors is described and
numbered as follows:
v1 : (+1; +1; +1)
v2 : (+1; +1; 1)
v3 : (+1; 1; +1)
v4 : (+1; 1; 1)
v5 : ( 1; +1; +1) v6 : ( 1; +1; 1) v7 : ( 1; 1; +1) v8 : ( 1; 1; 1)
The following notations will be useful:
x1 is the number of voters who are in a minority on a2 ; a3 and not on a1 ,
x2 is the number of voters who are in a minority on a1 ; a3 and not on a2 ,
x3 is the number of voters who are in a minority on a1 ; a2 and not on a3 ,
x4 is the number of voters who are in a minority on a1 ; a2 and a3 ,
y1 is the number of voters who are in a minority only on a1 ,
y2 is the number of voters who are in a minority only on a2 ,
y3 is the number of voters who are in a minority only on a3 and
y4 is the number of voters who are not in a minority on any motion.
It follows that the number of voters who are in a minority on a majority of times is
s = x1 + x2 + x3 + x4 and n = s + y1 + y2 + y3 + y4 .
Let us give some further explanation about these variables. In order to do that,
let us choose some possible social decision, say (+1; 1; +1). Then, x1 is the number
of voters who report (+1; +1; 1) (they accept a1 and a2 , and reject a3 ), x2 is the
number of voters who report ( 1; 1; 1) (they reject all motions), x3 is the number
of voters who report ( 1; +1; +1) (they accept a2 and a3 , and reject a1 ), x4 is the
number of voters who report ( 1; +1; 1) (they accept a2 , and reject a1 and a3 ), y1
is the number of voters who report ( 1; 1; +1) (they accept a3 , and reject a1 and
a2 ), y2 is the number of voters who report (+1; +1; +1) (they accept all motions), y3
is the number of voters who report (+1; 1; 1) (they accept a1 , and reject a2 and
a3 ), and y4 is the number of voters who report (+1; 1; +1) (they accept a1 and a3 ,
7
and reject a2 ).
The next result determines the majority quotas
under which logrolling is possi-
ble.
Proposition 2 1) Suppose n is even. Then there exists some voting pro…les at which
logrolling
is possible if
8
n
> + 1 0 mod 3, n
>
< 2
>
>
:
n
2
n
2
+1
1 mod 3, n
2n 1
,
3 n
< 13 2nn 3 ,
< 23 nn 2 .
1
2
and 12
and 12
10, and
12,
<
+ 1 2 mod 3, n 14,
2) Suppose n is odd. Then there exists some voting pro…les at which logrolling is
possible
8 if
n+1
>
>
< 2
>
>
:
n+1
2
n+1
2
1
2
and 12
and 21
0 mod 3, n
5, and
1 mod 3, n
7,
2 mod 3, n
9,
1 2n 1
,
3 n
< 23 nn 1 ,
< 13 2nn 3 .
<
Proof. Since x1 + y2 + y3 + y4 (resp. x2 + y1 + y3 + y4 , resp. x3 + y1 + y2 + y4 ) is
the number of voters who are in accordance on a1 (resp. a2 ; resp. a3 ), then we have
the following system:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
n < x1 + y 2 + y 3 + y 4
n < x2 + y 1 + y 3 + y 4
n < x3 + y 1 + y 2 + y 4
n
2
(3)
:
< x 1 + x2 + x3 + x4 = s
>
>
>
>
x1 + x2 + x3 + x4 + y1 + y2 + y3 + y4 = n
>
>
>
>
>
x1 ; x2 ; x3 ; x4 2 f0; 1; :::; ng; y1 ; y2 ; y3 ; y4 2 f0; 1; :::; b n 2 1 cg
>
>
>
>
:
2 [ 21 ; 1[
We …nd the upper bound of the values of
for which (3) has a solution. We distinguish
three steps.
Step 1. The …rst step of the proof is to show that the upper bound of the values
of
would be obtained on solutions of (3) of the forms x = (x1 ; x2 ; x3 ; 0; 0; 0; 0; y4 ).
If x = (x1 ; x2 ; x3 ; x4 ; y1 ; y2 ; y3 ; y4 ) is a solution of (3) satisfying (y1 > 0; y2 > 0
or y3 > 0), then there exists p 2 N, and k 2 f0; 1; 2g, such that x4 = 3p +
k. Therefore, a = (x1 + p; x2 + p; x3 + p + k; 0; 0; 0; 0; y1 + y2 + y3 + y4 ) is also
a solution of (3). More importantly, for the solution x (resp. a) we have
[ 12 ;
x[
where
x
3 +y4 x2 +y1 +y3 +y4 x3 +y1 +y2 +y4
= min( x1 +y2 +y
;
;
) (resp.
n
n
n
8
2 [ 21 ;
a[
2
where
a
2 +y3 +y4 x2 +p+y1 +y2 +y3 +y4 x3 +p+k+y1 +y2 +y3 +y4
= min( x1 +p+y1 +y
;
;
)). Since
n
n
n
then
[ 12 ;
x[
[ 12 ;
a[
a;
x
Thus the result.
With this step, (3) becomes:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
n < x1 + y 4
n < x2 + y 4
n < x3 + y 4
n
2
:
< x 1 + x2 + x3 = s
>
>
>
>
x1 + x2 + x3 + y4 = n
>
>
>
>
>
x1 ; x2 ; x3 2 f0; 1; :::; ng; y4 2 f0; 1; :::; b n 2 1 cg
>
>
>
>
:
2 [ 1 ; 1[
(4)
2
In the rest of the proof, we …nd the upper bound of the values of
for which (4) has
a solution.
Step 2. In this step we show that the upper bound would be obtained on solutions
on (4) of the forms x = (x1 ; x2 ; x3 ; b n 2 1 c). It su¢ ces to show that if there exists
a = (a1 ; a2 ; a3 ; a4 ) a solution of (4) satisfying a4 < b n 2 1 c; then there exists a a
solution z = (z1 ; z2 ; z3 ; b n 2 1 c) such that [ 21 ;
With this step, (4) becomes:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
>
<
a[
[ 12 ;
z [:
n < x1 + b n 2 1 c
n < x2 + b n 2 1 c
n < x3 + b n 2 1 c
n
2
< x 1 + x2 + x3
>
x1 + x2 + x3 = s
>
>
>
>
>
>
x1 + x2 + x3 + b n 2 1 c = n
>
>
>
>
>
x1 ; x2 ; x3 2 f0; 1; :::; b n2 c + 1g
>
>
>
>
:
2 [ 1 ; 1[
(5)
:
2
Step 3. In the remainder of the proof, we …nd the upper bound of the values of
for
which (5) has a solution. In order to do that, we distinguish two cases: n is either
even or odd. (i) If n is even, then b n 2 1 c =
9
n
2
1 and n
b n 2 1 c = b n2 c + 1 =
n
2
+ 1;
and (5) becomes:
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
n < x1 +
n < x2 +
n < x3 +
n
2
n
2
n
2
1
1
1
:
x1 + x2 + x3 = n2 + 1
>
>
>
n
>
x1 ; x2 ; x3 2 f0; 1; :::; 2 + 1g
>
>
>
>
>
2 [ 21 ; 1[
>
>
>
>
:
n is even
We distinguish three subcases: (i 1 ) If
that
n
2
n
2
+ 1 = 3p; then for x1 = x2 = x3 =
+1
n+2
,
6
0 mod 3, i.e., there exists p 2 N such
the upper bound of
It is easy to show that for all solutions x = (x1 ; x2 ; x3 ) of (6),
we have
1
2
<
2n 1
,
3 n
that is, n > 4. It follows that n
there exists p 2 N such that
n
2
(6)
x
10. (i 2 ) If
<
n
2
2
3
2n 1
.
3 n
is
2
3n
=
Since
2n 1
.
3 n
1
,
2
1 mod 3, i.e.,
+1
+ 1 = 3p + 1; then for x1 = x2 = x3 =
n
,
6
the upper
2
1
= 13 2nn 3 . It is easy to show that for all solutions x = (x1 ; x2 ; x3 )
3
n
1
of (6), x < 13 2nn 3 . Since
, then we have 12 < 13 2nn 3 , that is, n > 6: Thus
2
n
12. (i 3 ) If n2 + 1
2 mod 3, i.e., there exists p 2 N such that n2 + 1 = 3p + 2,
4
then x1 = x2 = x3 = n 6 2 , the upper bound of is 32
= 23 nn 2 . It is easy to
3n
1
, then we
show that for all solutions x = (x1 ; x2 ; x3 ) of (6), x < 32 nn 2 . Since
2
have 21 < 23 nn 2 , that is, n > 8. Thus n 14. (ii) If n is odd, then b n 2 1 c = n 2 1 and
n b n 2 1 c = b n2 c + 1 = n+1
. Thus (5) becomes:
2
bound of
is
8
>
>
>
>
>
>
>
>
>
>
>
>
>
<
>
>
>
>
>
>
>
>
>
>
>
>
>
:
n 1
2
n 1
n < x2 + 2
n < x3 + n 2 1
x1 + x2 + x3 = n+1
2
x1 ; x2 ; x3 2 f0; 1; :::; n+1
g
2
1
2 [ 2 ; 1[
n < x1 +
(7)
n is odd
n+1
0 mod 3, i.e., there exists p
2
n+1
. And the upper bound of is 32
6
We distinguish three subcases: (ii 1 ) If
n+1
2
:
2 N such that
1
= 13 2nn 1 .
3n
It is easy to show that for all solutions x = (x1 ; x2 ; x3 ) of (7), x < 31 2nn 1 . Since
1
, then we have 21 < 13 2nn 1 , that is, n > 2. Thus n 5. (ii2 ) If n+1
1 mod 3,
2
2
i.e., there exists p 2 N such that n+1
= 3p + 1, then for x1 = n 6 1 or x2 = n 6 1 or
2
2
x3 = n 6 1 , the upper bound of is 23
= 32 nn 1 . It is easy to show that for all
3n
= 3p; then for x1 = x2 = x3 =
10
solutions x = (x1 ; x2 ; x3 ) of (7),
that is, n > 4. Thus n
<
7. (ii3 ) If
that n+1
= 3p + 2, then for x1
2
is 23 n1 = 31 2nn 1 . It is easy
1 2n 1
1
, then
x < 3 n . Since
2
3.3
x
=
n 3
6
2n 1
.
3 n
n+1
2
1
,
2
Since
or x2 =
then we have
1
2
<
2n 1
,
3 n
2 mod 3, i.e., there exists p 2 N such
n 3
6
or x3 =
n 3
,
6
the upper bound of
to show that for all solutions x = (x1 ; x2 ; x3 ) of (7),
we have
1
2
<
1 2n 1
,
3 n
that is, n > 6. Thus n
9.
The size of logrolling coalitions for the three-motion case
This subsection is devoted to the determination of the maximum number of voters
smax who can form an alliance and trade their votes, under the simple majority rule
= 21 .The following result gives the maximum value of
on all motions, that is when
s. Its proof outlines the distribution of n voters in order to obtain smax .
Proposition 3 Let f1=2 be the SDR. Then the maximum size smax of coalitions susceptible to 8
trade their
3n
>
2
>
>
4
>
>
< 3n 3
4
smax =
3n 6
>
>
4
>
>
>
: 3n 5
4
votes is
if n 0 mod 4 and n 62 f0; 4; 8g
if
if
if
n
n
n
1 mod 4 and n 6= 1
2 mod 4 and n 62 f2; 6g
3 mod 4 and n 6= 3
Proof. With the simple majority rule, (3) becomes the following system:
8
n
>
< x 1 + y2 + y3 + y4
>
2
>
>
>
n
>
< x 2 + y1 + y3 + y4
>
>
2
>
>
<
n
< x 3 + y1 + y2 + y4
2
:
n
>
< x 1 + x2 + x3 + x4 = s
>
2
>
>
>
>
>
x1 + x2 + x3 + x4 + y1 + y2 + y3 + y4 = n
>
>
>
:
x1 ; x2 ; x3 ; x4 2 f0; 1; :::; ng; y1 ; y2 ; y3 ; y4 2 f0; 1; :::; b n 2 1 cg
(8)
We …nd the maximum value of s. We will show that maximum would be obtained on
certain solutions of (8). In order to do that we distinguish two steps:
Step1. We show that the maximum value of s would be obtained on solutions of the
form x = (x1 ; x2 ; x3 ; 0; 0; 0; 0; y4 ).
If x = (x1 ; x2 ; x3 ; x4 ; y1 ; y2 ; y3 ; y4 ) is a solution of (8) satisfying (y1 > 0; y2 > 0 or
y3 > 0), then there exist p 2 N and k 2 f0; 1; 2g such that x4 = 3p + k. Therefore,
a = (x1 + p; x2 + p; x3 + p + k; 0; 0; 0; 0; y1 + y2 + y3 + y4 ) is also a solution of (8).
11
More importantly, the sum of the four …rst coordinates of x and a are equal. This
step shows that when y1 = y2 = y3 = x4 = 0, (8) becomes:
8
n
>
< x 1 + y4
>
2
>
>
>
n
>
< x 2 + y4
>
>
2
>
>
<
n
< x 3 + y4
2
:
n
>
< x 1 + x2 + x3 = s
>
2
>
>
>
>
>
x1 + x2 + x3 + y4 = n
>
>
>
:
x1 ; x2 ; x3 2 f0; 1; :::; ng; y4 2 f1; :::; b n 2 1 cg
(9)
Step 2. In the rest of the proof, we will determine the maximum value of s. We distinguish four cases: i) If n
0 mod 4 and n 62 f0; 4; 8g; i.e., there exists p 2 N
such that n = 4p; then z = (p
1=
n
4
1; p
1=
n
4
1; p = n4 ; p + 2 =
solution of (9). The sum of the three …rst coordinates of z is sz =
show that for all solutions x = (x1 ; x2 ; x3 ; y4 ) of (9), sx
3n
4
n
4
f0; 1; 2g
+ 2) is a
2. We now
sz , where sx is the sum
of the three …rst coordinates of x. Let x = (x1 ; x2 ; x3 ; y4 ) be a solution of (9). We
distinguish two subcases:
- If y4
sx =
p+2 =
3n
4
2
3n
4
+ 2, then there exists t 2 N
f0g such that y4 =
n
4
+ 2 + t. Thus,
+ 2, then there exists t 2 N
f0g such that y4 =
n
4
+2
s0 .
t
- If y4 < p + 2 =
sx =
n
4
2+t
n
4
3n
4
t. Thus,
2 = sz . In this subcase, there exist t1 ; t2 ; t3 2 Z, such that
t1 +t2 +t3 = t. Thus x1 = p 1+t1 , x2 = p 1+t2 ; x3 = p+t3 . x1 +y4 = n2 +1 t+t1 ;
n
2
x2 + y4 =
+1
t + t2 and x3 + y4 =
For t1 < t, then x1 + y4 =
n
2
+1
n
2
+2
t + t1 <
n
2
t + t3 .
+ 1. Thus x1 + y4
n
.
2
This contradicts
the …rst inequality of (9).
For t1 = t, then t2 = t3 = 0 and x2 + y4 =
n
2
+1
t
n
2
+ 1. This contradicts the
second inequality of (9).
These two contradictions show that the second subcase is impossible.
ii) If n
1 mod 4 and n 6= 1, i.e., there exists p 2 N
f0g; n = 4p + 1, then
) is a solution of (9). The sum of the three …rst coordinates of
z = ( n 4 1 ; n 4 1 ; n 4 1 ; n+3
4
s is sz =
iii) If n
3n 3
.
4
Similarly as above, for all x = (x1 ; x2 ; x3 ; y4 ) solution of (9), sx
sz .
2 mod 4 and n 62 f2; 6g, i.e., there exists p 2 N f0; 1g such that n = 4p+2;
then z = ( n 4 2 ; n 4 2 ; n 4 2 ; n+6
) is a solution of (9). The sum of the three …rst coordinates
4
of s is sz =
iv) If n
3n 6
.
4
Similarly as above, for all solutions x = (x1 ; x2 ; x3 ; y4 ) of (9) sx
3 mod 4 and n 6= 3, i.e., there exists p 2 N
12
sz .
f0g such that n = 4p + 3, then
z = ( n 4 3 ; n 4 3 ; n+1
; n+5
) is a solution of (9). The sum of the three …rst coordinates of
4
4
s is sz =
3n 5
.
4
Similarly as above, for all solutions x = (x1 ; x2 ; x3 ; y4 ) of (9), sx
sz .
It is important to notice some interesting comments from the previous proof.
Remark 1 To obtain the maximum number of voters in minority most of the times,
no voter should be in minority on one motion and the number of voters in minority
on all motions should be 0, 1 or 2.
Furthermore, (i) if n
x2 =
n
4
0 mod 4 and n 62 f0; 4; 8g; we could have: y4 =
1 and x3 = n4 ; (ii) if n
x1 = x2 = x3 =
n 1
;
4
n 2
4
y4 =
n+5
; x1
4
= x2 =
n 3
4
+ 2; x1 =
n+3
4
and
2 mod 4 and n 62 f2; 6g; we could have: y4 =
n+6
4
1 mod 4 and n 6= 1; we could have: y4 =
(iii) if n
and x1 = x2 = x3 =
n
4
and, (iv) if n
and x3 =
3 mod 4 and n 6= 3; we could have:
n+1
:
4
For example if n = 87: Since n
3 mod 4, to obtain the maximum number of
voters in minority most of the times, we can distribute the n voters as follows: for
that we assume that social decision (SD) is (+1; 1; +1).
motions
x1 =21
x4 =0
y1 =0
y2 =0
y3 =0
y4 =23
SD
a1
+1
1
1
1
1
+1
+1
+1
+1
a2
+1
1
+1
+1
1
+1
1
1
1
a3
1
1
+1
1
+1
+1
1
+1
+1
x2 =21
x3 =22
Hence, using the fourth result of our proposition or the table, we obtain the maximum number of voters in minority most of times which is: smax =
3n 5
4
=
3 87 5
4
=
64:
We can also have the same smax if (x1 = 22; x2 = 21; x3 = 21; x4 = 0) or (x1 =
21; x2 = 22; x3 = 21; x4 = 0) or (x1 = 21; x2 = 21; x3 = 21; x4 = 1):
In the next section, we consider the case where the domain of individual voting vectors over the set of motions is restricted in some sense and, we evaluate the
frequency of logrolling opportunities.
13
4
A particular case and an evaluation
4.1
Single-plateaued individual voting vectors
In this subsection, we study the case where each voter has a single plateaued voting
vector. We show that in such a context, the whole society also has a single plateaued
voting vector. And we use this result to show that under the majority rule of quota
2 [ 12 ; 1[; there exists no “majority in minority on a majority of occasions”.
We begin with de…ning the notion of single-plateaued voting vectors.
Suppose a1 , a2 and a3 are three amounts of money proposed by three …rms as a
reply to a public call for tenders, with a1 < a2 < a3 and assume that these amounts
are successively voted on in that order (for example a voter in a city council); then
we can de…ne a kind of “single-plateaued” votes over the set of alternatives: vSP =
f(+1; +1; +1), (+1; +1; 1), (+1; 1; 1), ( 1; 1; 1)g. The intuition is as follows:
there is no rationale for a voter to reject a small amount, and then accept a bigger
one. Note that with such an intuition, not all possible single-plateaued votes are
considered here; for example, we exclude the two following vectors: ( 1; +1; +1)
and ( 1; 1; +1). Therefore, we have the following geometric representation of the
domain.
+1
+1
-1
-1
a1
a2
a3
+1
+1
-1
-1
a1
a2
a3
a1
a2
a3
a1
a2
a3
Then, in what follows, we show that, under the majority rule of quota
2 [ 21 ; 1[; in
the case of individual single-plateaued voting vectors: (i) if society rejects a motion,
14
them the next (if they exist) motions will be also rejected, (ii) social voting vectors
are single-plateaued and (iii) there is not exist majority in minority most of times.
Proposition 4 Suppose all individuals have single-plateaued voting vectors. Then
under the majority rule of quota
not vulnerable to logrolling.
2 [ 12 ; 1[; the method of successive majority votes is
This result is due to the following lemma.
Lemma 1 Suppose all individuals have single-plateaued voting vectors.
1) If a motion is rejected by the society, then all the next motions (if they exist) are
also rejected by the society.
2) Social voting vectors are single-plateaued.
Proof. 1) First suppose that the society rejects a1 . Thus there exists a coalition
S such that j S j> n
and any voter of S rejects a1 . Since individuals have single
plateaued vectors, then any voter of S rejects a2 and a3 . Since j S j> n , thus society
rejects a2 and a3 . Now assume that the society rejects a1 and a2 . Thus there exists a
coalition T such that j T j> n and any voter of T rejects a2 . Since individuals have
single-plateaued vectors, then any voter of T rejects a3 : Since j T j> n ; thus society
rejects a3 .
2) Let us show that the social voting vectors are single-plateaued. Since individuals
have single-plateaued voting vectors, the previous result implies that the four following
social voting vectors are not possible: (+1; 1; +1), ( 1; +1; +1), ( 1; +1; 1) and
( 1; 1; +1). Consequently the only possible social voting vectors are: (+1; +1; +1),
(+1; +1; 1), (+1; 1; 1) and ( 1; 1; 1). Hence the result.
Proof of Proposition 4 To show that the method of majority votes is not vulnerable to logrolling is to show that (3) do not have a solution.
The second result of the previous Lemma shows that social voting vector are (+1; +1; +1),
(+1; +1; 1), (+1; 1; 1) and ( 1; 1; 1). Thus we distinguish four cases:
i) If the social voting vector is (+1; +1; +1) : we have x2 = x3 = y1 = y2 = 0 since
15
individuals have single-plateaued voting vectors. And (3) becomes
8
>
n < x1 + y 2 + y 3 + y 4
>
>
>
>
>
>
n < y3 + y 4
>
>
>
>
>
n < y4
>
<
n
:
< x 1 + x4 = s
2
>
>
>
>
x1 + x4 + y3 + y4 = n
>
>
>
>
>
x1 ; x4 2 f0; 1; :::; ng; y3 ; y4 2 f0; 1; :::; b n 2 1 cg
>
>
>
>
:
2] 21 ; 1]:
The …fth and the second equations of (10) imply x1 + x4 = n y3
y4 < n(1
(10)
)
n
:
2
This contradicts the fourth inequality of (10).
ii) If the social voting vector is (+1; +1; 1) : we have x1 = x2 = x4 = y1 = 0 since
individuals have single-plateaued voting vectors. Thus, similarly as above, we show
that (3) has no solution.
iii) If the social voting vector is (+1; 1; 1) : we have x2 = x3 = x4 = y3 = 0 since
individuals have single-plateaued voting vectors. Thus, similarly as above, we show
that (3) has no solution.
iv) If the social voting vector is ( 1; 1; 1) : we have x1 = x2 = y2 = y3 = 0 since
individuals have single-plateaued voting vectors. Thus, similarly as above, we show
that (3) has no solution.
In the following section, we check our previous results with the evaluation of the
likelihood of logrolling under the impartial culture hypothesis.
4.2
Evaluation of the likelihood of logrolling for the threemotion case
In this subsection we evaluate, under the simply majority rule, the proportion of
voting pro…les at which there is an opportunity for some coalitions of individuals
to form an alliance and enter into a logrolling process. This evaluation is based on
the impartial anonymous culture, which assumes that voters are anonymous; in other
terms, the outcome of the voting process does not depend on the names of individuals
who vote in one or another way. Such an hypothesis is usual in the literature (see
for example Gehrlein and Fishburn 1976, Gehrlein 2006, Mbih, Moyouwou and Picot
16
2008). An extensive discussion of this hypothesis and some alternative hypotheses
can be found in Regenwetter M., Grofman B., Marley A. A. J.& Tselin I. M. (2006).
More precisely, with three motions and n voters, an anonymous voting pro…le can
be obtained from a voting pro…le in the sense de…ned in Section 2 by rewriting the
k=8
P
latter as ! = (! 1 ; ! 2 ; ! 3 ; ! 4 ; ! 5 ; ! 6 ; ! 7 ; ! 8 ),
! k = n, where for all k = 1; :::; 8, ! k is
k=1
the number of voters with voting vector vk . Then our aim is to compute the following
ratio:
number of anonymous voting pro…les at which logrolling is possible
:
total number of anonymous voting pro…les
We notice that the total number of voting situations (anonymous pro…les) ! is given
by
n+7
7
=
1
(n + 7) (n + 6) (n + 5) (n + 4) (n + 3) (n + 2) (n + 1) .
5040
It remains to compute the numerator. We successively consider all possible social
voting vectors, and for each social voting vector vk , let Lk be the set of all situations
vulnerable to logrolling. This allows us to construct a partition of all possible cases:
S
L=
Lk and Li \Lj = ? for all distinct i and j in f1; 2; :::; 8g; and the numerator
k2f1;2;:::;8g
is then equal to the sum of the cardinalities of all subsets in the partition:
k=8
P
k=1
The table below illustrates all situations leading to logrolling:
jLk j.
social voting vector
individual votes of the majority in a minority
v1 = (+1; +1; +1)
f(+1; 1; 1) ; ( 1; +1; 1) ; ( 1; 1; +1) ; ( 1; 1; 1)g
v2 = (+1; +1; 1)
v3 = (+1; 1; +1)
v4 = (+1; 1; 1)
v5 = ( 1; +1; +1)
v6 = ( 1; +1; 1)
v7 = ( 1; 1; +1)
v8 = ( 1; 1; 1)
f(+1; 1; +1) ; ( 1; +1; +1) ; ( 1; 1; +1) ; ( 1; 1; 1)g
f(+1; +1; 1) ; ( 1; +1; +1) ; ( 1; +1; 1) ; ( 1; 1; 1)g
f(+1; +1; +1) ; ( 1; +1; +1) ; ( 1; +1; 1) ; ( 1; 1; +1)g
f(+1; +1; 1) ; (+1; 1; +1) ; (+1; 1; 1) ; ( 1; 1; 1)g
f(+1; +1; +1) ; (+1; 1; +1) ; (+1; 1; 1) ; ( 1; 1; +1)g
f(+1; +1; +1) ; (+1; +1; 1) ; (+1; 1; 1) ; ( 1; +1; 1)g
f(+1; +1; +1) ; (+1; 1; +1) ; (+1; +1; 1) ; ( 1; +1; +1)g
For example, for social voting vector v1 = (+1; +1; +1), logrolling is possible if
voters with the following individual voting vectors form a majority: (+1; 1; 1),
17
( 1; +1; 1),( 1; 1; +1) and ( 1; 1; 1). This is described by the set of inequalities below:
a1 is accepted
!1 + !2 + !3 + !4 > !5 + !6 + !7 + !8
a2 is accepted
!1 + !2 + !5 + !6 > !3 + !4 + !7 + !8
a3 is accepted
!1 + !3 + !5 + !7 > !2 + !4 + !6 + !8
Majority in a minority
!4 + !6 + !7 + !8 > !1 + !2 + !3 + !5
!1 + !2 + !3 + !4 + !5 + !6 + !7 + !8 = n
We then use a computer based program (available from the authors) giving formulae of sums of powers of integers, in order to obtain the result below.
Proposition 5 Let m = 3 and let n 2
= f3; 4; 6; 8g. Then under the social decision
rule f1=2 , the vulnerability of the method of successive majority votes to logrolling is
given by
8
>
>
>
>
>
<
1
vul(n; ) =
>
2
>
>
>
>
:
5
1 n(n 8)(n 4)( 32n n2 +n3 18)
if n
32 (n+7)(n+6)(n+5)(n+3)(n+2)(n+1)
1 (n 1)(n+11)( 11n+8n2 +n3 60)
if n
32 (n+6)(n+5)(n+4)(n+2)(n+1)
1 (n 6)(n 2)( 16n 7n2 +n3 +70)
if n
32 (n+5)(n+4)(n+3)(n+7)(n+1)
1 (n+9)(n 3)(53n+16n2 +n3 28)
if n
32 (n+5)(n+4)(n+3)(n+7)(n+1)
0 mod 4
1 mod 4
2 mod 4
3 mod 4
Concluding Remarks
In this paper, we were concerned with Anscombe’s paradox and its corollary, the
possibility of logrolling in committees where votes are taken successively. We have
examined technical conditions at which logrolling occurs in such committees and
most of our results are somewhat general in the sense that there consider any possible majority quota. Besides we provide frequencies at which such a phenomenon is
susceptible to occur in the three-motion case. One possible way to extend our some
of our results is to consider situations with more than three motions, and situations
where voters can abstain.
A
Tables
is the proportion of voters in a minority a majority of times.
18
Table 1. The vulnerability of successive majority votes to
logrolling with = 21 and = 12
n
3
4
5
6
7
8
9
12
frequencies
——
——
0:01010
——
0:01632
——
0:02028
0:00111
frequencies
n
15
18
21
24
27
30
33
36
0:02594
0:00161
0:02815
0:00690
0:02922
0:00946
0:02982
0:01161
n
frequencies
39
42
45
48
51
75
100
0:03019
1
0:03125
0:01342
0:03043
0:01494
0:03060
0:03094
0:02199
Table 2. Limit frequencies with respect to changes in
frequencies
0.50
0.51
0.52
0.53
0.54
0.55
0:03125
0:020 26
0:012 77
0:007 79
0:004 58
0:002 57
, with
frequencies
0.56
0.57
0.58
0.59
0.60
0.61
0:001 37
6: 899 9
10
4
10
4
1: 361 93
10
4
5:1 2
5
3: 212 72
10
1: 641 35
10
5
19
= 0:5.
frequencies
0.62
0.63
0.64
0.65
0.66
2/3
4: 216 54
10
6
7: 794 87
10
7
8: 388 61
10
3: 125
10
5: 12
10
0
8
9
12
Table 3. Limit frequencies with respect to changes in
frequencies
0.50
0.51
0.52
0.53
0.54
0.55
0.56
0.57
0.58
0:03125
0:023 48
0:017 43
0:012 77
0:009 22
0:006 55
0:004 58
0:003 13
0:002 10
frequencies
0.59
0.60
0.61
0.62
0.63
0.64
0.65
0.66
0.67
0:001 37
0:000 87
5: 397
10
4
3: 213
10
4
1: 835
10
4
10
5
9: 977
5:1 2
10
5
2: 449
10
5
1: 074
10
5
20
with
= 12
frequencies
0.68
0.69
0.70
0.71
0.72
0.73
0.74
0.75
—
4: 217
10
1: 433
10
4:0
10
10
1: 120
10
6: 4
10
10
0
6
7
8: 389
6: 553
6
8
8
10
13
References
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majority’s will, Analysis 36:4, 161-168.
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21