A linear inequality method of establishing certain social choice conjectures

Abstract

This paper demonstrates the equivalence between sets of linear inequalities and a variety of social choice problems, a fact which permits the solution of the problems using linear programming techniques. An important advantage of this approach is that the primal solution of the linear programming problem, if one exists, provides a specific example illustrating the social choice conjecture under consideration. Examples illustrating this approach to several standard types of social choice problems are presented.

Social choice theory relies heavily on demonstrations that various social choice functions can disagree with one another or that these functions are inconsistent with one or more fundamental conditions or axioms held to be desirable properties of a method of deriving a “fair and reasonable” social choice from a set of individual preference orderings. A large class of such conjectures can be shown to be equivalent to systems of linear inequalities. This equivalence makes it possible to establish the truth or falsehood of such social choice conjectures by testing the systems for consistency or by employing one of the many available algorithms to solve the corresponding linear programs. We will give below a number of examples of conjectures resolved in this fashion, but first we must show how the appropriate inequality systems and linear programs may be found.