Nth axiomatization of the majority rule
Antonio Quesada†
Departament d’Economia, Universitat Rovira i Virgili, Avinguda de la Universitat 1, 43204 Reus, Spain
16th January 2009
159.1
Abstract
For the case in which there are only two alternatives, the majority rule is characterized in terms of axioms
of unanimity, reduction and cancellation.
Keywords: Social welfare function, majority rule, axiomatic characterization, two alternatives.
JEL Classification: D71
†
E-mail address: [email protected]. Financial support from the Spanish Ministerio de Educación y Ciencia,
under research project SEJ2007-67580-C02-01, is gratefully acknowledged.
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1. Introduction
If someone is asked to choose a mechanism that aggregates individual preferences over
two alternatives into a summarizing (or collective) preference, it is very likely that the
chosen mechanism will be the majority rule: if the are more individuals preferring one
of the alternatives to the other then the first alternative is preferred to the second in the
collective preference; otherwise, the result is indifference between the alternatives.
There are several axiomatic characterizations of the majority rule; see, for instance, May
(1952, p. 682), Fishburn (1973, p. 58), Aşan and Sanver (2002, p. 411) and Woeginger
(2003, p. 91). Neutrality is assumed in all these four characterizations: if the ranking
between two alternatives is reversed in each preference then the ranking between them
is also reversed in the collective preference. Two other axioms are postulated in two of
these characterizations. One is anonymity: the collective preference is not modified
when two individuals exchange their preferences. The second one is Pareto optimality:
if some individual i prefers one alternative to the other and no individual j ≠ i has the
opposite preference then the collective preference coincides with i’s.
This note suggests another characterization of the majority rule in which neither
neutrality nor anonymity nor Pareto optimality is postulated. The characterization is
based on three axioms; see A1, A2 and A3 in Section 2. A1 is unanimity: if all the
individuals have the same preference then that preference is the collective preference.
A2 is a reducibility axiom, since it states conditions under which an aggregation
problem has the same solution as a simpler aggregation problem. In this respect, A2 is
related to the axiom of weak path independence by Aşan and Sanver (2002) and to the
axiom of reducibility to subsocieties by Woeginger (2003). A3 expresses a cancellation
property: if the collective preference is the same in two cases and if a certain individual
has the same preference in both cases then the removal of that individual does not alter
the fact that the collective preference remains the same in the two resulting cases.
Though it is Proposition 3.2 that characterizes the majority rule in terms of A1, A2 and
A3, the arguably most interesting result of the note is Lemma 3.1, because it shows that
not much is needed for an aggregation rule satisfying A1 and A2 to become the majority
rule.
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2. Definitions and assumptions
Let ℕ be the set of natural numbers. Members of ℕ are names for individuals. The set
of alternatives is {α, β}, with α ≠ β. A preference over {α, β} is represented by a
number from the set {−1, 0, 1}. If the number is 1, α is preferred to β; if −1, β is
preferred to α; if 0, α is indifferent to β. A preference profile for a finite non-empty
subset I of ℕ is a function xI : I → {−1, 0, 1} assigning a preference over {α, β} to each
member of I. For preference profile xI and i ∈ I, xIi abbreviates xI(i). For n ∈ ℕ, Xn is the
set of all preference profiles xI : I → {−1, 0, 1} such that I has n elements. The set X is
the set of all preference profiles xI : I → {−1, 0, 1} such that I is a finite non-empty
subset of ℕ.
Definition 2.1. A social welfare function is a mapping f : X → {−1, 0, 1}.
A social welfare function transforms the preferences over {α, β} of all the members of
any given non-empty finite subset I of individuals into a collective preference over {α,
β}. In particular, for xI ∈ X: (i) f(xI) = 1 means that, in the preference f(xI), the collective
I prefers α to β; (ii) f(xI) = −1, that I prefers β to α; and (iii) f(xI) = 0, that I is indifferent
between α and β.
Definition 2.2. The majority rule is the social welfare function µ : X → {−1, 0, 1} such
that, for all xI ∈ X: (i) if ∑i∈I xIi > 0 then µ(xI) = 1; (ii) if ∑i∈I xIi < 0 then µ(xI) = −1; and
(iii) if ∑i∈I xIi = 0 then µ(xI) = 0.
A1. For all xI ∈ X, if there is a ∈ {−1, 0, 1} such that, for all i ∈ I, xIi = a then, f(xI) = a.
A1 is the unanimity principle: if all the individuals in a group I have the same
preference then that preference defines the collective preference attributed to I.
For xI ∈ X and yJ ∈ X such that I ∩ J = ∅, (xI, yJ) designates the member zK of X such
that: (i) K = I ∪ J; and (ii) for all i ∈ I, zKi = xIi; and (iii) for all i ∈ J, zKi = yJi. Therefore,
(xI, yJ) is obtained by joining xI with yJ.
A2. For all xI ∈ X and yJ ∈ X such that I ∩ J = ∅, if f(yJ) = 0 then f(xI, yJ) = f(xI).
A2 can be viewed as an absorption property: if f(yJ) yields indifference, the adding yJ to
any other preference profile xI is innocuous, so it is as if the preferences of the members
of J were absorbed by the rest of the individuals. By A2, if the preferences of a
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subgroup J imply indifference then those preferences are irrelevant to define the
preference of the whole group. Hence, if f(yJ) = 0 then the aggregation of (xI, yJ) can be
reduced to the aggregation of f(xI).
For xI ∈ X and non-empty J ⊂ I, xIJ designates the member yJ of X such that, for all i ∈
J, yJi = xIi. Hence, xIJ is the restriction of xI to the set J.
A3. For all xI ∈ X, yI ∈ X and i ∈ I, if I has at least two members, f(xI) = f(yI) and xIi = yIi
then f(xII\{i}) = f(yII\{i}).
A3 is a cancellation property: for any group I having at least two individuals, if the
collective preference for profiles xI and yI is the same then the collective preference is
also the same for the profiles xII\{i} and yII\{i} obtained, respectively, from xI and yI by
removing the preference of an individual having the same preference in both profiles.
By A3, if individual i has the same preference in two profiles and the collective
preference is the same in the two profiles then the cancellation of i’s preference
preserves the equality between the resulting collective preferences.
3. Results
Lemma 3.1. A social welfare function f : X → {−1, 0, 1} that satisfies A1, A2 and (1) is
the majority rule.
For all i ∈ ℕ and j ∈ ℕ\{i}, f(1i, −1j) = f(1j, −1i) = 0.
(1)
Proof. Let f satisfy the required conditions. It will be shown that, for all n ∈ ℕ, f agrees
with µ on Xn. Case 1: n = 1. By A1, f agrees with µ on X1. Case 2: n = 2. Choose I ⊂ ℕ
having two members, i and j. Let (ai, bj) represent the member xI of X2 such that xIi = a
and xIj = b. Choose xI = (ai, bj) ∈ X2. Case 2a: a = b. By A1, f(xI) = a = µ(xI). Case 2b: a
≠ b and {a, b} = {1, −1}. By (1), µ(xI) = 0. Case 2c: a ≠ b and {a, b} ≠ {1, −1}. This
means that 0 ∈ {a, b}. Without loss of generality, suppose that a = 0, so b ∈ {1, −1}.
By A1, f(bj) = b and f(ai) = a = 0. Given this, by A2, f(bj, ai) = f(bj) = b = µ(bj, ai). Case
3: n ≥ 3. By cases 1 and 2, choose k ≥ 3 and, arguing inductively, suppose that, for all r
∈ {1, … , k − 1}, f agrees with µ on Xr. To show that f agrees with µ on Xk, choose I ⊂
ℕ having k members and xI ∈ Xk. Case 3a: for some i ∈ I, xIi = 0. By A1, f(xIi) = 0. By
A2, f(xI) = f(xII\{i}, xIi) = f(xII\{i}). By the induction hypothesis, f(xII\{i}) = µ(xII\{i}). Since µ
is the majority rule, µ(xII\{i}) = µ(xI). Case 3b: for all i ∈ I, xIi ≠ 0. Case 3b1: there are i
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∈ ℕ and j ∈ ℕ\{i} such that xIi = 1 and xIj = −1. By (1), f(xI{i,j}) = 0. By A2, f(xI) =
f(xII\{i,j}, xI{i,j}) = f(xII\{i,j}). By the induction hypothesis, f(xII\{i,j}) = µ(xII\{i,j}). And being µ
the majority rule, µ(xII\{i,j}) = µ(xI). Case 3b2: for no i ∈ ℕ and no j ∈ ℕ\{i}, xIi = 1 and
xIj = −1. This means that all the components of xI are 1 or all the components are −1. In
both cases, by A1, f(xI) = µ(xI).
Proposition 3.2. A social welfare function f : X → {−1, 0, 1} satisfies A1, A2 and A3
if, and only if, f is the majority rule.
Proof. “⇐” Let f be the majority rule. It is not difficult to verify that f satisfies A1, A2
and A3. “⇒” Let f satisfy A1, A2 and A3. By Lemma 3.1, it is enough to show that (1)
holds. To this end, choose i ∈ ℕ and j ∈ ℕ\{i}. Step 1: f(1i, −1j) = 0. Suppose not. Case
1: f(1i, −1j) = 1. By A1, f(1i, 1j) = 1. By A3, f(1i, −1j) = f(1i, 1j) implies f(−1j) = f(1j). By
A1, −1 = f(−1j) ≠ f(1j) = 1: contradiction. Case 2: f(1i, −1j) = −1. By A1, f(−1i, −1j) = −1.
By A3, f(1i, −1j) = f(−1i, −1j) implies f(1i) = f(−1i). By A1, 1 = f(1i) ≠ f(−1i) = −1:
contradiction. Step 2: f(1j, −1i) = 0. Rewrite step 1 replacing “i” with “j” and “j” with
“i”.
Remark 3.3. The social welfare function f such that, for all xI ∈ X, f(x) = 0 is not the
majority rule, does not satisfy A1 and satisfies both A2 and A3.
Remark 3.4. The following social welfare function f is not the majority rule, does not
satisfy A2 and satisfies both A1 and A3: there is a linear order → on ℕ that, for all xI ∈
X, f(x) = xIi if, and only if, for all j ∈ I\{i}, i → j.
Remark 3.5. Let f be the social welfare function such that, for all xI ∈ X: (i) if, for some
i ∈ I, xIi = 1 then f(x) = 1; (ii) if, for all i ∈ I, xIi ≠ 1 and, for some i ∈ I, xIi = −1 then f(x)
= −1; and (iii) if, for all i ∈ I, xIi ≠ 1 and xIi ≠ −1 then f(x) = 0. Then f is not the majority
rule, does not satisfy A3 (since f(−1i, 1j) = f(1i, 1j) but f(−1i) ≠ f(1i)) and satisfies both
A1 and A2.
References
Aşan, G. and Sanver, M. R. (2002): “Another characterization of the majority rule”,
Economics Letters 75, 409−413.
Fishburn, P. C. (1973): The Theory of Social Choice, Princeton University Press,
Princeton, New Jersey.
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May, K. O. (1952): “A set of independent, necessary and sufficient conditions for
simple majority decision”, Econometrica 20, 680−684.
Woeginger, G. J. (2003): “A new characterization of the majority rule”, Economics
Letters 81, 89−94.
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