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Chapter 5
Fuzzy Arrow’s Theorem
5.1 Dictatorial Fuzzy Preference Aggregation
Rules
Arrow’s Theorem [1] is one of the most important discoveries in social science
theory. Arrow asks if there are any methods for aggregating the preferences
of individuals that meet several reasonable conditions: universal admissibility,
transitivity, unanimity (weak Paretianism), and independence from irrelevant
alternatives. Through a series of formal proofs he concludes that the only
methods for achieving all four of these conditions are dictatorial, that is, the
social choice is perfectly aligned with the preferences of one individual.
A voting system is a function that maps the voting preferences of the voters
for the candidates to a ranking of the candidates. We list certain reasonable
assumptions one might think a perfect voting system should satisfy:
(i) The voting system preserves rationality, that is, the output of the voting
system is a total ordering.
(ii) The output of the voting system is determined only by the ranking
preferences of the voters; no other factors are allowed.
(iii) If all voters prefer one candidate to another, then the output of the
voting system is the favored candidate.
(iv) All candidates are treated equally.
(v) The ranking of two candidates in the output of the voting system is
independent of the voters’ preferences for a third candidate.
(vi) There is no dictator.
Arrow’s theorem says that these six axioms are inconsistent.
In the situation where fuzzy preference aggregation rules are allowed, there
are many types of transitivity, independence of irrelevant alternatives, strict
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138 5. Fuzzy Arrow’s Theorem
fuzzy preference relations, and so on. We examine how various combinations
of these concepts yield impossibility theorems.
We begin by considering a certain set of deﬁnitions for Arrow’s conditions.
This set of deﬁnitions does not lead to a diﬀerent conclusion than that reached
by Arrow under crisp logic. We next consider applications of the fuzzy ver
sion of Arrow’s Theorem involving representation rules, oligarchies, and veto
players. We also consider an approach that uses the notion of ﬁlters to further
strengthen the fuzzy version of Arrow’s Theorem. In the last three sections,
we consider approaches by Dutta, Banerjee, and Richardson.
Deﬁnition 5.1.1 Let ρ be a fuzzy binary relation on X. Then ρ is called a
fuzzy weak order on X if it is reﬂexive, complete, and partially transitive.
Deﬁnition 5.1.2 Let ρ be a fuzzy binary relation on X. Deﬁne the fuzzy
subset ι of X × X by ∀x, y ∈ X, ι(x, y) = ρ(x, y) ∧ ρ(y, x).
Assume that each individual i has a fuzzy weak order ρ
i
on X. Let FWR
denote the set of all fuzzy weak orders on X. When there is no indiﬀerence,
i.e., ι
i
(x, y) = 0 for all x, y ∈ X, x 6= y, individual i’s preferences π
i
are
said to be strict. Let FWP denote the set of all strict orders on X. A fuzzy
preference proﬁle on X is a ntuple of fuzzy weak orders ρ = (ρ
1
, . . . , ρ
n
)
describing the fuzzy preferences of all individuals. Let FWR
n
denote the set
of all fuzzy preference proﬁles. Recall P(X) denotes the power set of X. For
any ρ ∈ FWR
n
and for all S ∈ P(X), let ρe
S
= (ρ
1
e
S
, . . . , ρ
n
e
S
), where
ρ
i
e
S
= ρ
i

S×S
, i = 1, . . . , n. For all ρ ∈ FWR
n
and ∀x, y ∈ X, let
P (x, y; ρ) = {i ∈ N  π
i
(x, y) > 0} and R(x, y; ρ) = {i ∈ N  ρ
i
(x, y) > 0}.
Let FB denote the set of all reﬂexive and complete fuzzy binary relations on
X.
Deﬁnition 5.1.3 A function
e
f : FWR
n
→ FB is called a fuzzy preference
aggregation rule.
We sometimes suppress the underlying fuzzy preference aggregation rule
and write ρ(x, y) for
e
f(ρ)(x, y) and π(x, y) > 0 for
e
f(ρ)(x, y) > 0 and
e
f(ρ)(y, x) = 0.
Deﬁnition 5.1.4 Let
e
f be a fuzzy preference aggregation rule.
(1)
e
f is said to be a simple majority rule if ∀ρ ∈ FWR
n
, ∀x, y ∈ X,
e
f(ρ)(x, y) > 0 if and only if P (x, y; ρ) > n/2.
(2) Let strict preferences be regular.
e
f is said to be a Pareto extension
rule if ∀ρ ∈ FWR
n
, ∀x, y ∈ X, π(x, y) > 0 if and only if
b
R(x, y; ρ) = N and
P (x, y; ρ) 6= ∅, where
b
R(x, y; ρ) = {i ∈ N  ρ
i
(x, y) ≥ ρ
i
(y, x)}.
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