Unbounded Utility for Savage's "Foundations of Statistics, " and Other Models

A general procedure for extending finite-dimensional "additive-like" representations for binary relations to infinite-dimensional "integral-like" representations is developed by means of a condition called truncation-continuity. The restriction of boundedness of utility, met throughout the literature, can now be dispensed with, and for instance normal distributions, or any other distribution with finite first moment, can be incorporated. Classical representation results of expected utility, such as Savage 1954, von Neumann and Morgenstern 1944, Anscombe and Aumann 1963, de Finetti 1937, and many others, can now be extended. The results are generalized to Schmeidler's 1989 approach with nonadditive measures and Choquet integrals, and Quiggin's 1982 rank-dependent utility. The different approaches have been brought together in this paper to bring to the fore the unity in the extension process.

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