Foundations of self-progressive choice theories

Consider a population of agents whose choice behaviors are partially comparable according to given primitive orderings. The set of choice functions admissible in the population specifies a choice theory. A choice theory is self-progressive if any aggregate choice behavior consistent with the theory is uniquely representable as a probability distribution over admissible choice functions that are comparable. We establish an equivalence between self-progressive choice theories and (i) well-known algebraic structures called lattices; (ii) the maximizers of supermodular functions.