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

By John N. Mordeson

Fuzzy social selection conception comes in handy for modeling the uncertainty and imprecision wide-spread in social existence 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 research of fuzzy social selection theory.

The ebook explains the idea that of a fuzzy maximal subset of a suite of 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 models of Arrow’s theorem, and Black’s median voter theorem for fuzzy personal tastes. It examines how unambiguous and distinct offerings are generated via fuzzy personal tastes and no matter if specific offerings prompted via fuzzy personal tastes fulfill convinced believable rationality family members. 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 members. the ultimate bankruptcy discusses Georgescu’s measure of similarity of 2 fuzzy selection functions.

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Social choice theorists initially expected on the basis of revealed preference theory that collective choice was based on collective preferences. In the previous chapter, we demonstrated that a maximal set may not exist under all conditions. This conclusion has led scholars to question the relationship between revealed social preferences and social choice. In particular, they have asked whether the choices made by collective actors are consistent with their collective social preference. In other words, if we assume that the preferences of a set of individuals are not cyclic, we would like to know if their collective choices are rationalizable.

Proof. Since C = C, it suffices to show for all µ, ν ∈ B and ∀x ∈ X that the following inequality holds I(µ, ν) ∗ µ(x) ∗ C(ν)(x) ≤ C(µ)(x). 3. 1(2), it follows for all u ∈ X that µ(u) ∗ (µ(u) → ν(u)) ∗ (ν(u) → ρ(x, u)) = µ(u) ∧ ν(u) ∗ (ν(u) → ρ(x, u) = µ(u) ∧ ν(u) ∧ ρ(x, u) ≤ ρ(x, u). 1(1), with a = (µ(u) → ν(u)) ∗ (ν(u) → ρ(x, u)), b = µ(u), c = ρ(x, u)), we have (µ(u) → ν(u)) ∗ (ν(u) → ρ(x, u)) ≤ µ(u) → ρ(x, u). Thus I(µ, ν) ∗ µ(x) ∗ C(ν)(x) = ∧{µ(u) → ν(u) | u ∈ X} ∗ µ(x) ∗ ν(x) ∗ ∧{ν(u) → ρ(x, u) | u ∈ X} ≤ µ(x) ∗ ∧{(µ(u) → ν(u)) ∗ (ν(u) → ρ(x, u))} | u ∈ X} ≤ µ(x) ∗ ∧{µ(u) → ρ(x, u))} | u ∈ X} = C(µ)(x).

Proof. 34 and the fact that if T = ∅, then ρ(x, x) > 0 ∀x ∈ X else if ρ(x, x) = 0 for some x ∈ X, MM (ρ, 1{x} ) = 1T , where T = ∅. Note also that the assumption x, y ∈ X (not necessarily distinct) such that 0 < ρ(x, y) < 1 implies ρ(x, x) > 0 ∀x ∈ X. 3 Exercises 1. Let X be a finite set and a R relation on X. Prove that a necessary condition for the maximal set M (R, S) to be nonempty for all subsets S of X is that R be reflexive and complete. Prove also that transitivity of R together with completeness and reflexivity are sufficient for M (R, S) to be nonempty for all subsets S of X.

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