Geometric and combinatorial properties of the polytope of binary choice probabilities (Q1184363)

A system of binary choice probabilities on a finite set is representable if the probabilities are compatible with a probability distribution over the family of linear orders of the set. In geometric terms representable binary choice probabilities form a polytope the vertices of which correspond to the permutations in a matrix indicating the dominant element in each pair. In the paper the geometrical aspect is developed reconciling two branches of research --- choice theory and optimization connected with the linear ordering problem --- which until recently developed independently. It is shown that most results of the choice literature are already known and have a geometric meaning, i.e. they describe facets of the polytope. In particular a few combinatorial results concerning this special kind of permutation matrices are proved, the geometric aspect is introduced and a few well-known results of the polytope under consideration are established. Then the known necessary conditions are analyzed concerning their facet-defining properties. Moreover, the diagonal inequality recently proved by \textit{I. Gilboa} [J. Math. Psychol. 34, 371-392 (1990)] is shown to contain facet-defining cases.