A necessary but insufficient condition for the stochastic binary choice problem
Résumé
The "stochastic binary choice problem" is the following: Let there be given n alternatives, to be denoted by N = {1, ..., n}. For each of the n! possible linear orderings {m}m = 1n of the alternatives, define a matrix Yn × n(m)(1 ≤ m ≤ n!) as follows: Given a real matrix Qn × n, when is Q in the convex hull of {Y(m)}m? In this paper some necessary conditions on Q--the "diagonal inequality"--are formulated and they are proved to generalize the Cohen-Falmagne conditions. A counterexample shows that the diagonal inequality is insufficient (as are hence, perforce, the Cohen-Falmagne conditions). The same example is used to show that Fishburn's conditions are also insufficient.