Utility and Means in the Thirties

Authors: 
Giovanni Parmigiani, Pietro Muliere
Universita di Pavia, Duke University

Jun 30 1993

This paper reviews the early axiomatic treatments of quasi-linear means developed in the late nineteen twenties and the nineteen thirties. These years mark the beginning of both axiomatic and subjectivist probability theory as we know them today. At the same time, Kolmogorov, de Finetti, and, in a sense, Ramsey, took part in a perhaps lesser known debate concerning the notions of mean and certainty equivalent. The results they developed offer interesting perspectives on computer data-summaries. They also anticipate key ideas in current normative theories of rational decision making.

This paper includes an extended and self-contained introduction discussing the main concepts in an informal way. the remainder focuses primarily on two early characterizations of quasi-linear means: the Nagumo-Kolmogorov theorem and de Finetti's extension of it. These results are then related to Ramsey's expected utility theory, to von Neumann and Morgenstern's, and to results on weighted means.

Keywords: 

Quasi-linear means, certainty equivalent, utility, subjective probability

Manuscript: 

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