Utility and Means in the 1930s

Abstract

This paper reviews the early axiomatic treatments of quasi-linear means developed in the late 1920s and the 1930s. 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 computing 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 focusses 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.