Application of fuzzy logic to social choice theory by John N. Mordeson

By John N. Mordeson

Fuzzy social selection concept turns out to be useful for modeling the uncertainty and imprecision primary in social lifestyles but it's been scarcely utilized and studied within the social sciences. Filling this hole, Application of Fuzzy common sense to Social selection Theory presents a entire learn of fuzzy social selection theory.

The publication explains the idea that of a fuzzy maximal subset of a suite of choices, fuzzy selection capabilities, the factorization of a fuzzy choice relation into the "union" (conorm) of a strict fuzzy relation and an indifference operator, fuzzy non-Arrowian effects, fuzzy models of Arrow’s theorem, and Black’s median voter theorem for fuzzy personal tastes. It examines how unambiguous and certain offerings are generated by way of fuzzy personal tastes and no matter if targeted offerings triggered through fuzzy personal tastes fulfill yes believable rationality kin. The authors additionally expand recognized Arrowian effects related to fuzzy set concept to effects concerning intuitionistic fuzzy units in addition to the Gibbard–Satterthwaite theorem to the case of fuzzy vulnerable choice family. the ultimate bankruptcy discusses Georgescu’s measure of similarity of 2 fuzzy selection functions.

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Yn }. y1 } (w) ∗ ρC (w, x) ∗ t ≤ ρC (x, w)})) = MG (ρC , 1S )(x). Hence C(1S ) ⊇ MG (ρC , 1S ). Suppose n ≥ 1 and C(1{x,y1 } ) ∗ . . ,yk } ) for 1 ≤ k < n, the induction hypothesis. ,yk+1 } ). Thus it follows by induction that ∗y∈S C(1{x,y} ) ⊆ C(1S ). Thus C(1S ) ⊇ MG (ρC , 1S ) and so C(1S ) = MG (ρC , 1S ). Conversely, suppose ρC rationalizes C for characteristic functions. Let x ∈ X. Let S, T ∈ P ∗ (X) be such that S ⊆ T. If (C(1T ) ∩ 1S )(x) = 0, then (C(1T ) ∩ 1S )(x) ≤ C(1S )(x). Suppose that (C(1T ) ∩ 1S )(x) > 0.

Thus {x ∈ X | x ∈ µt and (x, y) ∈ ρt ∀y ∈ Supp(µ)} = ∅ if t ∈ [0, t∗ ∧ s∗ ], where s∗ = ∧{µ(x) | x ∈ Supp(µ)}. Thus {x ∈ µt | ρ(x, w) ≥ t ∀w ∈ Supp(µ)} = ∅ ∀t ∈ (0, t∗ ∧ s∗ ]. Hence ∃x ∈ Supp(µ) such that ∀w ∈ X, ρ(x, w) ≥ tw , where tw ∈ (0, t∗ ∧ s∗ ]. ) Thus ρ(x, w) ≥ C(w, x) ∗ t ∀w ∈ X, where t = ∧{tw | w ∈ X}. Since X is finite, t > 0. Thus ∃x ∈ X such that M (ρ, µ)(x) > 0. 7 Let X = {x, y, z}. Suppose ∗ has no zero divisors. Define the fuzzy relation ρ on X as follows: ρ(x, x) = ρ(y, y) = ρ(z, z) = 1, ρ(x, y) = 1/2, ρ(x, z) = 1/2, ρ(y, z) = ρ(z, y) = 1/4, ρ(y, x) = ρ(z, x) = 0.

In this section, we present the concept of a weak form of fuzzy Tcongruence axiom, which was introduced in [17]. We characterize the full rationality of fuzzy choice functions in terms of this axiom and the fuzzy Chernoff axiom. We show that this form is insufficient to imply full rationality of fuzzy choice functions defined on base domain. However, the weak fuzzy T-congruence axiom together with the fuzzy Chernoff axiom characterizes full rationality. We also show that the fuzzy Arrow axiom alone does not imply full rationality of fuzzy choice functions defined on a base domain and also that G-rational fuzzy choice functions with transitive rationalizations characterize their full rationality if the fuzzy Chernoff axiom holds.

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