A simple sufficient condition for the unique representability of a finite qualitative probability by a probability measure (Q1116146)

Let \(X=\{x_ 1,x_ 2,...,x_ n\}\) be a finite set and \(\Omega\) be an algebra of subsets of X called events. Let \(\gtrsim\) be a qualitative probability relation on \(\Omega\) \(\times \Omega\). A probability measure p is said to uniquely agree with \(\gtrsim\) if, for all A and B in \(\Omega\), \(A\gtrsim B\) if and only if p(A)\(\geq p(B)\), and p is the only probability measure with that property. We give a sufficient condition for the existence of a uniquely agreeing probability measure which is significantly simpler and more general than \textit{R. D. Luce}'s [Ann. Math. Stat. 38, 780-786 (1967; Zbl 0158.165)] condition.