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

By John N. Mordeson

Fuzzy social selection thought turns out to be useful for modeling the uncertainty and imprecision well-known 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 examine of fuzzy social selection theory.

The booklet explains the concept that of a fuzzy maximal subset of a suite of possible choices, fuzzy selection features, 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 specific offerings are generated through fuzzy personal tastes and even if certain offerings brought on by way of fuzzy personal tastes fulfill convinced believable rationality kin. The authors additionally expand recognized Arrowian effects concerning fuzzy set idea to effects related to intuitionistic fuzzy units in addition to the Gibbard–Satterthwaite theorem to the case of fuzzy susceptible choice kin. the ultimate bankruptcy discusses Georgescu’s measure of similarity of 2 fuzzy selection functions.

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Also, ρ(w, x)∗t ≤ ρ(x, w) implies t = 1 is the largest possible t if ρ(w, x) ≤ ρ(x, w) and ρ(w, x) ∗ t ≤ ρ(x, w) implies tx,w is the largest possible t if ρ(w, x) ≥ ρ(x, w). The desired result now follows since ρC (x, w) = tx,w if ρ(w, x) > ρ(x, w) and ρ(x, w) > ρ(w, x) ⇔ ρC (x, w) > ρC (w, x). 6 Let C be a fuzzy choice function on X and let ρ ∈ FR(X). Suppose ρ rationalizes C. Then ∀x ∈ X, ρC (x, x) = 1. Recall that in the crisp case, condition α requires that if an alternative x is chosen from a set T and S is a subset of T containing x, then x should still be chosen.

Since ρ is acyclic, it follows that (x1 , x2 ), (x2 , x3 ), . . , (xn−1 , xn ) ∈ πt = Supp(π) implies (x1 , xn ) ∈ ρt = Supp(ρ). 1. 3, it follows that Supp(ρ) = ρt is acyclic for all t ∈ (0, t∗ ]. Since (x, x) ∈ ρt ∀t ∈ [0, t∗ ] and ∀x ∈ X, ρt is reflexive ∀t ∈ [0, t∗ ]. Let s∗ = ∨{µ(x) | x ∈ X}. 1, p. 5]. 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∗ ].

Recall that Sy = {w ∈ X | ρ(w, y) > ρ(y, w)}. 2 Let ρ ∈ FR∗ (X) and let S be a subset of X. Suppose ∗ is a t-norm without zero divisors. Then MG (ρ, 1S ) = 1T for some T ⊆ S if and only if ∀x, y ∈ S, ρ(x, y) ∗ ρ(y, x) > 0 implies ρ(x, y) = ρ(y, x). Proof. Suppose MG (ρ, 1S ) = 1T for some T ⊆ S. Let x, y ∈ S. Suppose ρ(x, y) ∗ ρ(y, x) > 0. Then ρ(x, y) > 0 and ρ(y, x) > 0. Suppose ρ(x, y) > ρ(y, x) > 0. ∀w ∈ Sy , let ty,w = ∨{t ∈ [0, 1] | ρ(w, y) ∗ t ≤ ρ(y, w)}. Then MG (ρ, 1S )(y) = {ty,w | w ∈ Sy }.

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