Products of Representations Characterize the Products of Dispersions and the Consistency of Beliefs

A "dispersion" specifies the relative probability between any two elements of a finite domain. It thereby partitions the domain into equivalence classes separated by infinite relative probability. The paper's novelty is to numerically represent not only the order of the equivalence classes, but also the "magnitude" of the gaps between them. The paper's main theorem is that the many products of two dispersions are characterized algebraically by varying the magnitudes of the gaps between each factor's equivalence classes. An immediate corollary is that the many beliefs consistent with two strategies are characterized by varying each player's "steadiness" in avoiding various zero-probability options.