Concluding remarks
Logically satisfactory methods for narrowing the range of social choice can be designed. For example, the SDF which transforms (S, R) into P(S, R), that is, the Pareto rule, satisfies all the conditions imposed on C(·) in Theorem 9: The Pareto rule is normal. For if x ∈ P(S, R) and y ∈ S, then y cannot be Pareto superior to x; so x ∈ P({x, y}, R), and α2 is satisfied. If x ∈ S and x ∈ P({x, y}, R) for all y ∈ S, no y in S is Pareto superior to x, and therefore, x ∈ P(S, R); so γ2 is satisfied. The Pareto rule is clearly neutral and anonymous; it is unbiased among the alternatives and among the individuals. The Pareto rule is obviously non-imposed. The Pareto rule is nonmanipulable. For if xP iy, P({x, y}, R) must be either {x} or {x, y}. If P({x, y}, R) = {x, y}, there is an individual j ≠ i for whom yR jx. Consequently no misrepresentation by i can force y out of the set of optima, and therefore the rule is cheatproof.
The significance of this paper is that any rule which is logically satisfactory (in the sense of the conditions of Theorem 9) must be bracketed between the maximal and Pareto rules. So those rules are especially important: they provide lower and upper bounds for completely satisfactory multi-valued SDFs.
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Feldman, A. Nonmanipulable multi-valued social decision functions. Public Choice 34, 177–188 (1979). https://doi.org/10.1007/BF00129525
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DOI: https://doi.org/10.1007/BF00129525