A linear inequality method of establishing certain social choice conjectures
✍ Scribed by J. R. Chamberlin; M. D. Cohen
- Publisher
- Springer US
- Year
- 1978
- Tongue
- English
- Weight
- 593 KB
- Volume
- 33
- Category
- Article
- ISSN
- 0048-5829
No coin nor oath required. For personal study only.
✦ Synopsis
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 t h e many available algorithms to solve the corresponding !inear 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.
I. Method
We begin with some definitions. When there are n alternatives and individuals may not be indifferent between any two of them, there are n! preference orderings that can possibly be held by members of the population that is to choose a winner or a social ordering of the alternatives. We will label the alternatives arbitrarily a 1 . . . a n and denote the set of alternatives