 Theorem 2 (von Neumann-Morgenstern)  Let \(\bO\) be a finite set of outcomes, \(\bL\) a set of corresponding lotteries that is closed under probability mixture and \(\preceq\) a weak preference relation on \(\bL\). Then \(\preceq\) satisfies axioms 1–4 if and only if there exists a function \(u\), from \(\bO\) into the set of real numbers, that is unique up to positive linear transformation, and relative to which \(\preceq\) can be represented as maximising expected utility.