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

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  • John Pratt

Abstract

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

Suggested Citation

  • John Pratt, 2005. "How Many Balance Functions Does it Take to Determine a Utility Function?," Journal of Risk and Uncertainty, Springer, vol. 31(2), pages 109-127, September.
    • Handle: RePEc:kap:jrisku:v:31:y:2005:i:2:p:109-127
    DOI: 10.1007/s11166-005-3551-x
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    1. Partha Dasgupta & Douglas Gale & Oliver Hart & Eric Maskin (ed.), 1992. "Economic Analysis of Markets and Games: Essays in Honor of Frank Hahn," MIT Press Books, The MIT Press, edition 1, volume 1, number 0262541599, December.
    2. Gollier, Christian & Pratt, John W, 1996. "Risk Vulnerability and the Tempering Effect of Background Risk," Econometrica, Econometric Society, vol. 64(5), pages 1109-1123, September.
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