The stochastic 2-binary choice problem

The classic (to date unsolved) stochastic binary choice problem asks under what conditions a given stochastic choice function defined on pairs of alternatives derives from a random ranking. We propose a solution to the problem for the case in which at most two rankings are assigned positive probability. This case is psychologically motivated and interesting for applications. It is structurally different from the general case in that the choice functions that are derived from a random ranking do not necessarily form a convex polytope, hence they are not even in principle described by a set of linear inequalities.