Unique Subjective Probability on Finite Sets

Let A_n be the family of all subsets of {1,2,..,n} n and p a probability measure on A_n. We say that p uniquely agrees with a comparative probability relation > on A if, for all A and B in A_n, A > B if and only if p(A) > p(B), and p is the only measure with this representation. The set of probability measures that uniquely agree with some > on A_n is small for small n but grows rapidly and even for modest n has an amazing number and variety of members.

The problem translates naturally into systems of equations that have nonnegative integer solutions, and the derivations of the paper are conducted in this format.