Expected Utility:
Theory Assume that a utility index exists which conforms to the five axioms. The expected utility for the two-outcome lottery L = (P, A, B) is given by,
E [U (L)] = P U (A) +(1-P) U (B)
Consider the lotteries L_{1} = (P_{1}, A_{1}, A_{2}) and L_{2} = (P_{2}, A_{3}, A_{4}). An expected utility theorem states that if L_{1} is preferred to L_{2}, E [U (L_{1})] > E [U (L_{2})]. The significance of this theorem is that uncertain situations can be analysed in terms of the maximisation of expected utility.
The proof of the theorem is straightforward. Select outcomes such that B, the best, is preferred to all other outcomes under consideration, and W, the worst, is inferior to all other outcomes. By the continuity axiom, Qi's exist such that Ai is indifferent to (Qi, B, W) (i=1,...,4). Thus L_{1} and L_{2} are equivalent to, i.e., have the same expected utility as, the lotteries (Z1, B, W) and (Z2, B, W) respectively where Z1 = P1Q1 + (1-P_{1})Q_{2} and Z_{2} = P_{2}Q_{3} + (1-P_{2})Q_{4}. By assumption, L_{1} is preferred to L_{2} and it follows from the unequal probability axiom that Z_{1}>Z_{2}. Since origin and unit of measure are arbitrary for utility indexes, let U (B) = 1 and U (W) = 0. Then E [U (L_{1})] = Z_{1} and E [U (L_{2})] = Z , establish the theorem.