A necessary but insufficient condition for the stochastic binary choice problem

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 = 1}ⁿ of the alternatives, define a matrix Y_{n × n}^(m)(1 ≤ m ≤ n!) as follows: Y_ab^(m) = 1 0 a≻^mb otherwise. Given a real matrix Q_{n × 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.