Decision Making with Incomplete Information

The interpretation of complex structural models is facilitated by a variety of operations that can be carried out by a computer. Several of these involve partitions on a reachability matrix, which describes a contextual relation among the elements of the system under study. When partitioning of the reachability matrix in various ways does not permit adequate interpretation, cycles are usually the source of difficulty. A weighting matrix applied to the elements of a maximal cycle set permits a set of digraphs to be developed, which allows for more detailed interpretation of a cycle. When still further development is needed, the maximal cycle sets can be identified and placed in a hierarchy based on an inclusion relation. This hierarchy suggests a natural order of study of the system proceeding from the least complex maximal cycle sets to the most complex sets.

Toward Interpretation of Complex Structural Models

This paper investigates the problem of combining ordinal preferences, expressed as priority vectors, to form a consensus. An axiomatic structure relating to the concept of distance between rankings is developed, uniqueness of the distance measure is proven and its form derived. Adopting the median ranking as a form of consensus, it is shown that this ranking can be determined, in the case of complete rankings, by solving a certain assignment problem. The results are compared and contrasted to earlier work due to Arrow, Kendall, Kemeny and Snell, and Bogart.

Priority Ranking and Consensus Formation

The median procedure in cluster analysis and social choice theory

A preference aggregation method through the estimation of utility intervals

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 lp norm and an indicator function. Implications of an integer programming solution, including alternate optima and post optimality analysis, are also discussed.

V. J. Bowman

Papers

Majority Rule Under Transitivity Constraints