In their note [Blin, J. M., A. B. Whinston. 1974. A note on majority rule under transitivity constraints. Management Sci. 20 (11) 1439–1440.], Blin and Whinston indicate that the linear integer programming formulation of the majority voting problem can also be formulated as a quadratic assignment problem. We wish to point out that both of these formulations are a result of the fact that majority decision functions can be linearized over the set of integer solutions to the linear program.

In this paper we are concerned with imposing constraints directly on the admissible majority decisions so as to insure transitivity without restricting individual preference orderings. We demonstrate that this corresponds to requiring that majority decisions be confined to the extreme points of a convex polyhedron. Thus, transitive majority decisions can be characterized as basic solutions of a set of linear inequalities. Through the use of a majority decision function (which is not restricted to be linear) it is shown that constrained majority rule is equivalent to an integer programming problem. Some special forms of majority decision functions are studied including the generalized l p norm and an indicator function. Implications of an integer programming solution, including alternate optima and post optimality analysis, are also discussed.

The objective of this paper is to provide a model to deal with multi-dimensional social goals on a regionalized (as well as national) basis and to relate these to each other in the context of a national economic system. Input-Output (Interindustry) Analysis is therefore regionalized and reformulated in a way that connects it to national economic policy variables in a Keynsian framework. Possible multiple objectives and controls are examined and their possible further developments are also sketched in a context of regionalized and social goals that differ from region to region and also from time to time. Various proposals for revenue sharing are exhibited and interpreted in the light of the possibilities this model admits. Special attention is given to the Nixon (President Nixon) and ACIR (Advisory Commission on Intergovernmental Relations) plans in contrast to a multi-goal dynamic and adaptive control-incentive scheme with “saturating” indices of goal attainment. The latter, suggested at the end of this paper, is designed so that it can permit variation between regions while maintaining minimum levels of comparable attainment on goals that are common to more than one region.

Abstract The article discusses the use of mathematical models across a variety of disciplines and practices. Mathematical models, naturally, are represented by means of mathematical relations. The mathematical models which will be of interest in this article generally proceed via the special kinds of mathematical relations called "functions," which are used to represent some or all of the relations. Double-entry accounting has been used as a basis for planning and control at both economy-wide and individual-enterprise levels. This is to say that double-entry accounting provides a tool of great utility which can be employed in a variety of ways and contexts. Beyond the convenience of moving back and forth between accounting and interindustry analyses, the mathematics associated with this modeling has permitted a variety of other uses and extensions. The examples in this article should make it clear, however, that this is not the end. Still more may be available from further research in model equivalences and related explorations. Indeed the "models" definition we introduced in the first section of the preceding paper is designed to underscore the potential value of such continuing explorations.