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Unique subjective probability on finite sets. (English) Zbl 0678.60013

Summary: Let \({\mathcal A}_ n\) be the family of all subsets of \(\{\) 1,2,...,n\(\}\) and p a probability measure on \({\mathcal A}_ n\). We say that p uniquely agrees with a comparative probability relation \(>\) on \({\mathcal A}_ n\) if, for all A and B in \({\mathcal A}_ n\), \(A>B\) 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 \({\mathcal A}_ n\) 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.

MSC:

60C05 Combinatorial probability
05A05 Permutations, words, matrices
11B99 Sequences and sets
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