Characterizations of Majoritarianism A Uni…ed Approach
M. Remzi Sanver
Istanbul Bilgi University
December, 2006
Abstract
De…ning regularity as the conjunction of unanimity, anonymity,
neutrality and monotonicity, we show that every regular social choice
rule can be expressed as a function of vote di¤erences and the number
of cancelling couples. This equivalence not only identi…es the class of
regular social choice rules but also allows a uni…ed conception of many
previous majority characterizations.
Corresponding author: [email protected]
1
1
Introduction
In a society confronted to make a collective choice from a given pair of alternatives, two distinct conceptions of majoritarianism arise. For one of these,
it is the relative support of the alternatives which matters, while for the
other, the collective decision depends on whether the alternatives receive
some absolute support. We refer to the former approach as relative and to
the latter as absolute majoritarianism. Relative majoritarianism dates back
to the seminal majority characterization of May (1952) where a majority
winner is de…ned as an alternative whose support exceeds the support of its
rival.1 Requiring that this excess goes beyond some given treshold leads to
the class of relative majority rules which have been considered by Fishburn
(1973), Saari (1990), García-Lapresta and Llamazares (2001), and characterized by Llamazares (2006) and Houy (2006).2 As for absolute majority rules,
a majority winner is an alternative that receives the support of a (possibly
quali…ed) majority. An early characterization of this class of rules is made
by Fishburn (1973), which has been followed by Austen-Smith and Banks
(1999), Yi (2005), Asan and Sanver (2006).
Relativism and absolutism are essentially incompatible conceptions of
majoritarianism.3 On the other hand, they share the following four central
axioms of social choice theory as common denominator: Unanimity which requires that a unanimously supported alternative must be the social outcome;
anonymity which requires the equal treatment of voters; neutrality which
requires the equal treatment of alternatives and monotonicity which requires
that a unidirectional shift in voters’opinions should not harm the alternative
towards which this shift occurs4 . Both relative and absolute majoritarianism
satisfy these four axioms which explicitly appear in various majority characterizations.5 As a result, we use them as a common basis to present a
1
Asan and Sanver (2002), Woeginger (2003), Miroiu (2004) and Woeginger (2005) characterize the same rule in societies of variable size.
2
A part of this literature calls relative majority rules as “majority rules based on
di¤erence of votes”. Our choice of nomenclature re‡ects our preference to emphasize the
contrast between relative and absolute majoritarianism.
3
In fact, Sanver (2006) shows that relative majoritarianism fails to be Nash implementable and admits absolute majoritarianism as its minimal Nash implementable extension. On the other hand, when indi¤erences are ruled out and an odd number of agents is
assumed, absolute and relative majoritarianism coincide. For other explorations of majoritarianism in various restricted frameworks where the two concepts converge to each other,
one can see Maskin (1995), Dasgupta and Maskin (1998), Campbell and Kelly (2000).
4
The monotonicity condition to which we refer is the weakest among those existing in
the literature. We give the formal de…nition of all four conditions in the next section.
5
To give a few examples, May (1952) uses anonymity, neutrality and positive responsiveness which is a strenghtening of monotonicity and whose conjunction with neutrality im-
2
uni…ed exposition of the separate characterizations of absolute and relative
majoritarianism.
Refering to the conjunction of unanimity, anonymity, neutrality and monotonicity as regularity, we identify the class of regular social choice rules in terms of
elementary social choice rules which depend on two parameters of the prevailing preference pro…le: The vote di¤erence6 and the number of cancelling
couples 7 . This family contains but is not restricted to absolute and relative
majoritarianism. Hence it can be used as a basis for the uni…ed exposition
we are after. In Section 2, we establish the equivalence between regular
and elementary social choice rules. In Section 3, we show how relative and
absolute majoritarianism can be seen as particular cases of regularity. In
Section 4 we conclude by noting that the equivalence established in Section
2 not only identi…es the class of regular social choice rules but also allows a
uni…ed perspective of majority characterizations.
2
An Equivalence
Picking a natural number n 2, we consider a set of voters N = f1; :::; ng
confronting a set of alternatives A = fx; yg. Each i 2 N , has a preference i 2 f 1; 0; 1g over A.8 A preference pro…le is an n-tuple
=
N
( 1 ; :::; n ) 2 f 1; 0; 1g of individual preferences. A social choice rule is
a mapping : f 1; 0; 1gN ! f 1; 0; 1g.9 A social choice rule is unanimous
i¤ (1; :::; 1) = 1 and ( 1; :::; 1) = 1. A social choice rule is anonymous
i¤ ( 1 ; :::; n ) = ( (1) ; :::; (n) ) for any ( 1 ; :::; n ) 2 f 1; 0; 1gN and any
permutation : N ! N of the voters. A social choice rule is neutral i¤
( )=
( ) for all 2 f 1; 0; 1gN . A social choice rule is monotonic i¤
0
( )
( 0 ) for all ; 0 2 f 1; 0; 1gN with i
be the
i 8i 2 N . Let
plies unanimity; Asan and Sanver (2006) use unanimity, anonymity, neutrality and Maskin
(1999) monotonicity which is also a strenghtening of monotonicity though it is logically
independent of positive responsiveness; Llamazares (2006) uses unanimity, anonymity,
neutrality, monotonicity and cancellation.
6
By the vote di¤erence, we mean the absolute value of the di¤erence between the
number of voters who support one alternative and the number of voters who support the
other.
7
By a cancelling couple, we mean any two voters who have opposite views. Remark
that if c is the number of cancelling couples and d is the vote di¤erence, then the total
number of voters equals 2c + d + ! where ! is the number of indi¤erent voters.
8
We interpret, as usual, i = 1 as i preferring x to y; i = 1 as i preferring y to x
and i = 0 as i being indi¤erent between x and y.
9
At each 2 f 1; 0; 1gN , f ( ) = 1 means x being socially preferred to y; f ( ) = 1
means y being socially preferred to x and f ( ) = 0 re‡ects social indi¤erence between x
and y.
3
set of all unanimous, anonymous, neutral and monotonic social choice rules.
We qualify any 2 as regular.
Writing Z+ for the set of non-negative integers, we de…ne the set D =
f(c; d) 2 Z+ Z+ : 2c + d
ng. To interpret the elements of D, we need
some additional notation and de…nitions: Take any 2 f 1; 0; 1gN . Let
n+ ( ) = #fi 2 N : i = 1g and n ( ) = #fi 2 N : i = 1g. We call d( )
= jn+ ( ) n ( )j the vote di¤erence.10 Any distinct i; j 2 N with i = 1
and i = 1 is a cancelling couple. Writing !( ) = #fi 2 N : i = 0g for
the number of indi¤erent voters, we have c( ) = n d( 2) !( ) as the number of
cancelling couples. Now, for each (c; d) 2 D, we interpret c as the number of
cancelling couples and d as the vote di¤erence.11 An elementary social choice
rule is a mapping f : D ! f0; 1g such that
(i) f (0; n) = 1
(ii) f (c; 0) = 0 for all (c; 0) 2 D
(iii) f (c; d)
f (c0 ; d0 ) for all (c; d); (c0 ; d0 ) 2 D with d
d0 , c
c0 and
d d0 c0 c.12
Let F be the set of all elementary social choice rules.
Any f 2 F induces a mapping f : f 1; 0; 1gN ! f 1; 0; 1g such that at
each 2 f 1; 0; 1gN
f (n ( ); d( )) if n+ ( ) n ( ) 0
(
)
=
f
f (n+ ( ); d( )) if n+ ( ) n ( ) < 0
Theorem 2.1
= [f 2F f
f g.
Proof. We …rst show
[f 2F f f g. Take any 2 . Now, we induce
some f : D ! f0; 1g through
as follows: At each (c; d) 2 D, we let
f (c; d) = ( ) where 2 f 1; 0; 1gN is such that n ( ) = c and n+ ( )
n ( ) = d. The existence of is ensured by 2c + d
n. Moreover, as
is anonymous, we have ( 0 ) = ( 00 ) for any 0 ; 00 2 f 1; 0; 1gN with
n ( 0 ) = n ( 00 ) and n+ ( 0 ) = n+ ( 00 ). Thus, f is well-de…ned. Now observe
that the anonymity and neutrality of implies ( ) = 0 8 2 f 1; 0; 1gN
with n+ ( ) = n ( ). Conjoining this observation with the monotonicity of
10
As usual, jrj stands for the absolute value of the real number r. Note that de…nition
of a vote di¤erence does not keep track of which alternatives receives a higher support.
11
As c cancelling couples means 2c voters, the condition 2c + d n follows. Note that
n (2c + d) is the number of indi¤erent voters.
12
These three conditions imposed over elementary social choice rules are related to the
regularity conditions for social choice rules. In particular, condition (i) -to which we refer
as consensus- is related to unanimity; condition (ii) -to which we refer as equal treatmentis related to the conjunction of anonymity and neutrality; condition (iii) -to which we
refer as elementary monotonicity is related to monotonicity.
4
, we have ( ) 2 f0; 1g 8 2 f 1; 0; 1gN with n+ ( )
n ( ). Thus, by
construction of f , we have f (c; d) 2 f0; 1g for all (c; d) 2 D. To see that
f 2 F , …rst note that f (0; n) = 1 holds by the unanimity of . Now take any
(c; 0) 2 D. Pick some 2 f 1; 0; 1gN with n ( ) = c and n+ ( ) n ( ) = 0.
As is anonymous and neutral, n+ ( ) = n ( ) implies ( ) = 0 = f (c; 0).
Finally take any (c; d); (c0 ; d0 ) 2 D with d d0 , c c0 and d d0 c0 c. Let ;
0
2 f 1; 0; 1gN be such that n ( ) = c, n ( 0 ) = c0 , n+ ( ) n ( ) = d and
n+ ( 0 ) n ( 0 ) = d0 . As c c0 , we have n ( ) n ( 0 ). Moreover, d d0
c0 c implies n+ ( ) n+ ( 0 ). Hence, the anonymity and monotonicity of
ensures ( )
( 0 ), thus f (c; d) f (c0 ; d0 ). Having established f 2 F; we
leave checking = f to the reader, which shows
[f 2F f f g.
To show [f 2F f f g
, take any f 2 F . If 2 f 1; 0; 1gN is such
that i = 1 8i 2 N , then n+ ( ) n ( ) 0. Thus f ( ) = f (n ( ); d( )).
Moreover, d( ) = n, so f ( ) = f (n ( ); n) which, by consensus, implies
1 when i = 1 8i 2 N ,
f ( ) = 1. Similar arguments establish f ( ) =
showing that f satis…es unanimity. To see that f is anonymous, take any
= ( 1 ; :::; n ) 2 f 1; 0; 1gN , any permutation : N ! N of the voters
and let 0 = ( (1) ; :::; (n) ). As n+ ( ) = n+ ( 0 ) and n ( ) = n ( 0 ), hence
d( ) = d( 0 ), we have f ( ) = f ( 0 ), showing the anonymity of f . To see
that f is neutral, take any 2 f 1; 0; 1gN . Note that n+ ( ) n ( ) 0 ,
n+ ( ) n ( )
0. Moreover, n+ ( ) = n ( ), n ( ) = n+ ( ) and
d( ) = d( ). Thus f ( ) =
f ( ), showing the neutrality of f . To see
0
that f is monotonic, take any ; 0 2 f 1; 0; 1gN with i
i 8i 2 N . So
+
+ 0
0
n ( ) n ( ) and n ( ) n ( ). Hence we have d( ) d( 0 ) as well as
d( ) d( 0 ) n ( 0 ) n ( ) which, by the elementary monotonicity of f ,
0
implies f ( )
f ( ), establishing the monotonicity thus the regularity of
f 2 .
Theorem 2.1 is our central result. It announces that regular (i.e., unanimous, anonymous, neutral and monotonic) social choice rules are those that
can be expressed through an elementary social choice rule f 2 F . So from
now on, we refer to F as the set of regular social choice rules. Moreover, we
exploit this equivalence to obtain a uni…ed exposition of majority characterizations.
3
Regular Social Choice Rules
Regular social choice rules are a function of the vote di¤erence and the
number of cancelling couples. Note that at some given vote di¤erence d,
the social choice is allowed to be a¤ected by variations of the number c of
5
cancelling couples.13 In more formal words, having f (c; d) 6= f (c0 ; d) for some
(c; d); (c0 ; d) 2 D does not contradict regularity. As a particular instance,
take f 2 F such that f (0; 0) = 0, f (0; d) = 1 8 (0; d) 2 D with d > 0 and
f (c; d) = 0 8 (c; d) 2 D with c > 0.14
Nevertheless, one can impose independence from the number of cancelling
couples as an additional axiom. We say that f 2 F is independent of cancelling couples (ICC) i¤ f (c; d) = f (c0 ; d) for all (c; d); (c0 ; d) 2 D.
Lemma 3.1 A regular social choice rule f 2 F satis…es ICC if and only
if there exists d 2 f0; 1; :::; n 1g such that for any (c; d) 2 D, we have
f (c; d) = 1 () d > d .
Proof. Take any regular f 2 F . To see the “if” part, assume the existence
of d 2 f0; 1; :::; n 1g such that for any (c; d) 2 D, we have f (c; d) = 1
() d > d . Take any (c; d); (c0 ; d) 2 D. If d > d , then f (c; d) = f (c0 ; d)
= 1 and if d
d , then f (c; d) = f (c0 ; d) = 0. Thus, f satis…es ICC. To
see the “only if” part, let f satisfy ICC. Let d 2 f0; 1; :::ng be such that
f (c; d ) = 1 for some (c; d ) 2 D while f (c; d) = 0 for all (c; d) 2 D with
d < d . The fact that f (0; n) = 1 ensures the existence of d and by ICC, we
have f (c0 ; d ) = 1 for all (c0 ; d ) 2 D. Now take any ( ; ) 2 D with > d .
As f ( ; d ) = 1, we have f ( ; ) = 1 by the elementary monotonicity of f ,
completing the proof.
Any f 2 F that admits some d 2 f0; 1; :::; n 1g such that f (c; d) = 1
() d > d 8(c; d) 2 D, is a relative majority rule as de…ned by Llamazares
(2006) and Houy (2006).15 Hence, Lemma 3.1 has the following immediate
corollary:
Proposition 3.1 A regular social choice rule f 2 F satis…es ICC if and
only if f is a relative majority rule.
13
All such variations are compatible with conditions (i); (ii) and (iii) that elementary
social choice rules do satisfy.
14
Remark that f is the Pareto rule which, at each preference pro…le, picks x if there is
at least one voter who prefers x to y and there are no voters who prefer y to x; picks y if
there is at least one voter who prefers y to x and there are no voters who prefer x to y;
and announces social indi¤erence if all voters are indi¤erent between x and y or there is
at least one cancelling couple.
15
A social choice rule : f 1; 0; 1gN ! f 1; 0; 1g is a relative majority rule i¤ there
exists t 2 f0; 1; :::;P
n 1g such that at each 2 f 1; 0; 1gN we have
1 if Pi2N i > t
1 if
t . Remark that the majority rule à la May (1952) is the
( )=
i2N i <
0 otherwise
particular case where t = 0.
6
Note that, ICC is equivalent to the cancellation axiom of Llamazares
(2006).16 As a result, Theorem 8 of Llamazares (2006) which characterizes
relative majoritarianism by the conjunction of regularity with cancellation
immediately follows from Proposition 3.1. We quote this result below:
Theorem 3.1 A social choice rule : f 1; 0; 1gN ! f 1; 0; 1g is regular
and satis…es cancellation i¤ is a relative majority rule.
We now turn to absolute majoritarianism. We say that f 2 F is additive
i¤ f (c; d) = f (c0 ; d0 ) 8 (c; d); (c0 ; d0 ) 2 D with c + d = c0 + d0 .
Lemma 3.2 Given any additive f 2 F and any (c; d) 2 D, we have f (c; d) =
1 only if c + d > n2 .
Proof. Take any additive f 2 F , any (c; d) 2 D with c + d n2 and suppose
f (c; d) = 1. As c + d n2 , we have ((c + d); 0) 2 D as well and the additivity
of f implies f (c + d; 0) = 1, contradicting that f is an elementary social
choice rule.
Lemma 3.3 A regular social choice rule f 2 F is additive if and only if
there exists an integer k 2 (n=2; n] such that at each (c; d) 2 D, we have
f (c; d) = 1 () c + d k .
Proof. Take any regular f 2 F . Checking the “if”part is left to the reader.
To see the “only if” part, let f be additive. Let k 2 f0; 1; :::ng be the
integer such that f (c; d) = 1 for some (c; d) 2 D with c + d = k while
f (c; d) = 0 for all (c; d) 2 D with c + d < k . The fact that f (0; n) = 1
ensures the existence of k and by Lemma 3.1, k > n2 . Moreover, by the
additivity of f , we have f (c; d) = 1 for all (c; d) 2 D with c + d = k . Now
take any integer k 2 fk + 1; :::ng and any ( ; ) 2 D with + = k.
Suppose, for a contradiction, that f ( ; ) = 0. Writing r = k k , we have
+(
r) = k , hence f ( ;
r) = 1 while f ( ; ) = 0, contradicting the
elementary monotonicity of f .
Any f 2 F that admits an integer k 2 (n=2; n] such that f (c; d) = 1 ()
c + d k 8 (c; d) 2 D, is an absolute majority rule.17 Hence, Lemma 3.3
has the following immediate corollary:
16
which requires for : f 1; 0; 1gN ! f 1; 0; 1g that ( ) = ( 0 ) 8 ; 0 2 f 1; 0; 1gN
such that 9 distinct i; j 2 N with i = 1, j = 1, 0i = 0, 0j = 0 while k = 0k
8k 2 N nfi; jg. An equivalent version of this condition is used by Young (1974).
17
A social choice rule : f 1; 0; 1gN ! f 1; 0; 1g is an absolute majority rule i¤ there
exists an integer t 2 ( n2 ; n] such that at each 2 f 1; 0; 1gN we have
1 if #fi 2 N : i = 1g t
1 if #fi 2 N : i = 1g t .
( )=
0 otherwise
7
Proposition 3.2 A regular social choice rule f 2 F is additive if and only
if f is an absolute majority rule.
Additivity is equivalent to the decisiveness axiom of Austen Smith and
Banks (1999).18 So Proposition 3.2 can equivalently be seen as the characterization of absolute majoritarianism through the conjunction of regularity
with decisiveness, as we state below:
Theorem 3.2 A social choice rule : f 1; 0; 1gN ! f 1; 0; 1g is regular
and decisive i¤ is an absolute majority rule.
We wish to note that Theorem 3.2 is essentially equivalent to Theorem
3.7 of Austen Smith and Banks (1999) who characterize “q rules” in terms
of anonymity, neutrality, monotonicity and decisiveness.19
4
Concluding Remarks
Our main result is Theorem 2.1 which identi…es the class of regular social
choice rules. Given the centrality of the regularity axioms in social choice
theory, this identi…cation seems to have its own merit. Moreover, as regularity includes relative and absolute majoritarianism, Theorem 2.1 allows a
uni…ed exposition of many previous majority characterizations and clari…es
the role that the regularity axioms (i.e., unanimity, anonymity, neutrality
and monotonicity) as well as cancellation and additivity play in these.
5
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18
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19
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8
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9