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.