Likelihood relations and stochastic preferences

We define the concept of a qualitative, non-numerical relative likelihood relation, to capture the intuition that “it is at least as likely that a is preferred to b, as that c is preferred to d.” We provide necessary and sufficient conditions for this concept to be a basis for the numerical concept of a stochastic preference, which is a numerical probability measure on the set of deterministic preferences. As a qualitative comparison of comparisons, the concept has many interpretations and applications, including measurement of dissimilarity, characterization of non-transitive binary relations, and others.