Measurement-theoretic behavior of the category of social utility models is studied in terms of ordered sums and tensor products of mixture spaces. Necessary and sufficient conditions are given for the existence of individual utility functions and a social utility function (being a weighted average, and in another case a product, of the individual utilities) in terms of individual and social preference rankings.
1. Introduction
Utility models represent the formal properties of preference, choice, and decision behavior of individuals and social groups. They are usually defined in terms of concepts such as prospect, act, consequence, state of nature, decision, cost, utility, choice, preference, uncertainty, and satisfy certain conditions and axioms.
Our knowledge in this field has expanded prodigiously since yon Neumann and Morgenstern gave in 1944 the first mathematically satisfactory definition of a utility model in terms of ordered mixture spaces. Detailed information on this expansion can be gained, for example, from the survey and extensive bibliography in Lute and Suppes .
The yon Neumann-Morgenstern axiomatic system of utilities has been replaced several times subsequently by simpler axioms or conditions, which allowed to construct easier proofs of the so-called representation theorems than those known before (see, in particular, Marshak (1950) and ).
Parallel to the fast growth and evolution of utility, preference, and choice models, remarkable progress has been made in the area of (algebraic) measurement theory (see the survey in ). A number of measurement-theoretic results turned out to be essential for the foundations of utility models. As is well known, various empirical attributes, relations and operations, associated with the preference behavior of individuals and Theory andDecision 11 (1979) 375-399.