How Many Balance Functions Does it Take to Determine a Utility Function?

Call b your balance function at wealth W if you are indifferent between W and a 50–50 lottery with outcomes x and b(x). Given one b, u is arbitrary on one side of W but then determined on the other. Given two b‘s, u is arbitrary between the two W′ s but then determined elsewhere. Additional properties of u restrict the b’s but do not ordinarily make u unique. Contradictions can occur. Given three b′ s, an algorithm is developed using minimal domains of definition that determines the relative utility of the W’s. If it is irrational, then the set S generated by applying all combinations of b’s to W′ s is dense and u is determined. If finitely many b’s are rationally related, then S is discrete, a further algorithm determines it, the values of u on S are equally spaced, and u is arbitrary between any two adjacent points of S but then determined elsewhere. Infinitely many balance functions determine u unless they are rationally related in a uniform way. Copyright Springer Science + Business Media, Inc. 2005