Generating Social Welfare Functions over Restricted Domains for Two Individuals and Three Alternatives Using Prolog pp. 163-187
Authors: (Kenryo Indo, Department of Business Administration, Faculty of Economics, Kanto Gakuen University, Gunma, Japan)
Abstract: Kenneth J. Arrow‟s general impossibility theorem postulates a set of axioms for the aggregation of rankings of individuals into a society‟s rankings, known as the social welfare function (SWF), and shows that it cannot be non-dictatorial unless one of the axioms is abandoned, especially under the unrestricted (or full) domain condition. For the case of two individuals and three alternatives, although it is considered to be the base case for the inductive proof of impossibility, there has been little examination of the possibilities when the domain is restricted to subsets of profiles with no axioms other than when the full domain is abandoned. The results of this study demonstrate, by using a Prolog program, that for linear orderings of the base case there are exactly 18 near-full domains if a pair of profiles is carefully selected from the two minimal super Arrovian domains, which are specified by Fishburn and Kelly, and that there are 18 corresponding non-dictatorial SWFs that can be intuitively associated with a variety of graded compromises ranging from uncompromising to full concession across the individual‟s set of rankings and summarized as near-constant mappings. Throughout this paper, a recursive Prolog program for SWF is presented to demonstrate the abovementioned results instead of the standard proof.
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