Algebraic and Stochastic Models of Structures Arising in Utility Theory and Psychophysics

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The research concerns the modeling of choices between, and judgments about, uncertain and risky options. It involves both algebraic modeling of a sort that has been actively studied for many years, but where important improvements are still possible, and probabilistic modeling to accommodate the somewhat inconsistent behavior apparent in such choices and judgments. Four of the algebraic problems are: (1) Provide a fundamental behavioral model (axiomatization) of the general linear weighted utility representation. Although many models are of this type, a general understanding of the behavioral properties that give rise to it still is lacking. For instance, a behavioral formulation of Birnbaum's Transfer of Attention Exchange model is needed. (2) Understand at a deeper level the assumption that two gambles are independently realized, which is often invoked in these theories. (3) Although often mentioned, little has been done to understand the utility or disutility of uncertainly itself. Following non-axiomatic work of Meginniss (1976) in the context of risk, the investigator in collaboration with others has been working on an axiomatization that yields a linear weighted utility plus a constant times a standard measure of entropy. This needs further clarification and development. (4) Develop more fully models of mixed gains and losses, which often are postponed in favor of the simpler cases of either all gains or all losses. Turning to probabilistic models, the investigators' approach is designed to supplement and improve the existing literature that focuses mainly on the random analogue of one-dimensional measurement. The crux of the problem is to deal with interlocked structures, such as gambles and their joint receipt, and, in particular, to gain what amounts to a random version of the (algebraic distribution) laws that interlock two structures. Attempts to resolve this type of problem have not been very satisfactory, except for the recent sophisticated axiomatization of a probabilistic version of expected utility.This research will impact the sciences, including economics, management science, psychology, and statistics, that examine individual decision making under uncertainty. Sensory psychophysicists also should benefit because the formal results of utility theory can be applied, sometimes with modifications, to the measurement of subjective intensity. The social importance of the developments in utility theory lies largely in the advice that is given by decision analysts to clients in areas such as investment and medicine. Up to now, such advice has typically been based on rational theories such as subjective expected utility. With a deeper understanding of the ways in which traditional theories have failed to capture people's goals, we can expect considerably improved advising.