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

By John N. Mordeson

Fuzzy social selection idea comes in handy for modeling the uncertainty and imprecision generic in social lifestyles but it's been scarcely utilized and studied within the social sciences. Filling this hole, Application of Fuzzy good judgment to Social selection Theory offers a complete learn of fuzzy social selection theory.

The booklet explains the concept that of a fuzzy maximal subset of a suite of possible 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 types of Arrow’s theorem, and Black’s median voter theorem for fuzzy personal tastes. It examines how unambiguous and distinct offerings are generated by means of fuzzy personal tastes and even if specified offerings brought on through fuzzy personal tastes fulfill definite believable rationality family. The authors additionally expand recognized Arrowian effects concerning fuzzy set idea to effects concerning intuitionistic fuzzy units in addition to the Gibbard–Satterthwaite theorem to the case of fuzzy vulnerable choice kin. the ultimate bankruptcy discusses Georgescu’s measure of similarity of 2 fuzzy selection functions.

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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. Recall that the operation ∧ is a continuous t-norm called the G¨ odel tnorm.

W ∈ Sy , let ty,w = ∨{t ∈ [0, 1] | ρ(w, y) ∗ t ≤ ρ(y, w)}. Then MG (ρ, 1S )(y) = {ty,w | w ∈ Sy }. Since Sy = ∅ and ρ(y, x) > 0 it follows that 0 < {ρ(y, w) | w ∈ Sy } < 1. However, this contradicts the hypothesis. Thus ρ(x, y) = ρ(y, x). Conversely, suppose that ∀x, y ∈ S, ρ(x, y) ∗ ρ(y, x) > 0 implies ρ(x, y) = ρ(y, x). Let T = {x ∈ S | w ∈ S, ρ(w, x) > ρ(x, w)}. Suppose x ∈ S\T. Then ∃w ∈ S such that ρ(w, x) > ρ(x, w). By our hypothesis, ρ(x, w) = 0. Thus MG (ρ, 1S )(x) = 0. Hence MG (ρ, 1S ) = 1T .

3 Let C be a fuzzy choice function on X such that ∀S ∈ P ∗ (X), C(1S ) = 1T for some nonempty T ⊆ S. Suppose that there exists ρ ∈ FR(X) such that C(1S ) = MG (ρ, 1S ) ∀S ∈ P ∗ (X). Then there exists a relation R on X such that ρ = 1R . Proof. Let S ∈ P ∗ (X) and let t ∈ (0, 1]. Then x ∈ MG (ρ, 1S )t ⇔ ∧{∨{u ∈ [0, 1] | ρ(w, x) ∧ u ≤ ρ(x, w)} | w ∈ S} ≥ t ⇔ ρ(x, w) ≥ ρ(w, x) ∧ t ∀w ∈ S and x ∈ MG (ρt , S) ⇔ MG (1ρt , 1S )(x) > 0 ⇔ ∧{∨{u ∈ [0, 1] | 1ρt (w, x) ∧ u ≤ 1ρt (x, w)} | w ∈ S} > 0 ⇔ ∀w ∈ S, [(w, x) ∈ ρt ⇒ (x, w) ∈ ρt or (w, x) ∈ / ρt ].

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