Threshold models for comparative probability on finite sets (Q1584803)

The paper considers comparative probability relations \(\succ\) defined on the power set \({\mathcal B}_S\) of a finite state space \(S\). For these relations the author proposes necessary and sufficient conditions for the existence of a probability \(P\) defined on \({\mathcal B}_S\) and a nonnegative relevant threshold function \(\Omega\) defined on \({\mathcal B}_S\times {\mathcal B}_S\) such that \[ A\succ B\Leftrightarrow P(A)>P(B)+\Omega(A,B). \] In particular the paper analyzes the problem when \(\Omega\) is supposed to be a bivariate function (considering both nonseparable and separable case) and when \(\Omega\) is a univariate function. All the conditions proposed in the paper are stated in terms of sums of indicator functions and then they are essentially algebraic conditions.