R Duncan Luce - US grants

The proposal outlines a valuable extension of research on expected utility theory. Forty years of experience has led to two conclusions. One, stressed by empirical researchers, is that the theory does not adequately describe behavior in many experimental and real-world situations. The other, stressed by decision analysts, is that the axioms of the theory are normatively appealing and difficult to dismiss. The gap between the descriptive inadequacy and normative appeal has widened in the past ten years. As a result, researchers have challenged not only the descriptive validity, but the normative basis of expected utility theory. The proposed research begins by identifying a common ground for descriptive and normative research in a newly developed rank and sign dependent (RSD) linear utility theory. The investigators propose (1) to extend the range of concepts included in RSD, (2) to test key assumptions of RSD, (3) to develop connections between RSD and traditional utility theory, and (4) to explore the normative-descriptive distinction in the context of RSD. Thus, the research combines theoretical development, experimental research, and applied experience in an effort to bridge the normative- descriptive gap in the theory of decision making under uncertainty.

This award provides funds for the establishment of a Research Training Group in Mathematical Behavioral Sciences. The faculty group is a mixture of outstanding senior and junior investigators who come from a diverse set of disciplinary backgrounds, but who share a common interest in the use of mathematical models in the study of cognition and related phenomena. The research programs in which trainees will participate are aimed at investigation of axiomatic and foundational issues, and at modeling behavioral phenomena from psychophysics, decision making, cognition processes and social networks. The funds will provide stipends for graduate students and postdoctoral fellows, will defray part of the cost of the trainees' research and will enable the trainees to attend scientific meetings. In addition, funds will be used to purchase specialized research equipment to be used by trainees, and to bring visiting faculty from other research and academic institutions to provide specialized seminars and coursework. Traditionally, the study of behavior has been undertaken by investigators in a number of distinct disciplines, often defined by the subjects with which the behavior deals. Thus there are important behavioral aspects to the study of economics, political science, sociology, anthropology, risk management and decision science, among others. Because of the considerable specialized knowledge required by each discipline and the differing arenas in which the behavior may occur, it has been difficult for investigators to identify underlying commonalities in behavior patterns. Increasingly, the use of mathematical models is providing an important approach for identifying such commonalities and for providing a means for applying principles recognized in one discipline to the problems of another. The training program funded by this award will produce researchers able to bridge disciplines and to bring the full power of mathematics to bear on a large variety of problems in the social and cognitive sciences.

9308959 Luce Almost forty years of experience with theories about rational decision making on the part of individuals confronting risk and uncertainty -- technically, subjective expected utility theories (SEU) -- has led to two firm conclusions. One, stressed primarily by psychologists, is that this class of models, while overall fairly descriptive, fails to describe certain crucial aspects of actual behavior in many experimental and real world situations. The other, stressed primarily by economists and decision analysts, is that the axioms of the SEU model are normatively very compelling and the model has proved useful in numerous applied situations. Recognizing both the descriptive failure and normative usefulness of the classical model, the PIs have explored the common ground held by both the experimentalists and the applied decision analysts, attempted to discover exactly where the two part company, and sought a prescriptive theory that satisfies both experimental concerns and applied expectations. Like SEU this new theory -- technically rank- and sign-dependent utility (RSDU) theory -- is based (axiomatically) on fairly simple, quite compelling, underlying behavioral principles, and also like SEU, it results in a numerical representation as a weighted average, but unlike SEU the weights are not probabilities and their values depend not only on the underlying chance event but on whether the associated outcome is a gain or a loss. Fundamental to the theory is the differential treatment of an event depending on whether it is associated to a gain or a loss relative to a status quo and the incorporation of the concept of joint receipt -- of getting two or more things at the same time -- that allows an explicit, simple characterization of the fundamental non-rationality of the theory. This project examines several theoretical, experimental, and applied issues that remain unresolved: On the theoretical side, while RSDU is a fairly good descriptive theory of choice behavior, it needs modification to accommodate the violations of its axioms which occur when judged certainty equivalents are used rather than choice procedures. On the empirical side, the project examines, among other things, Thaler's rule for the utility of the joint receipt of two or more things; the descriptive validity of the quasi-rational assumptions underlying RSDU; the construction of a reference level (local status quo) in terms of the choice set; and alternative combination rules for the aggregation of gains and losses around the status quo. Regarding applied issues, we are convinced that the status quo plays an important role in decision making and so it is highly desirable to develop prescriptive theories that depend explicitly on it. In addition, the project will investigate the limitations and potential pitfalls of sign dependent utility as a prescriptive theory. ***

With partial support from the National Science Foundation, the Institute for Mathematical Behavioral Sciences at the University of California, Irvine will purchase a Digital Equipment Corporation AlphaServer 2100 4/275. The RISC chip in this server runs at 275 MHz and the operating system is the Open Software Foundation version of Unix, OSF. The server has 256 Mbytes of main memory with 8.3 Gbytes of disc space. This equipment will supplement and significantly upgrade the group's current system and will permit a broader range of analyses of large data sets. The Institute for Mathematical Behavioral Sciences is a multidisciplinary research center that draws its faculty from the Departments of Anthropology, Cognitive Sciences, Economics, Mathematics, Philosophy and Sociology and from the Graduate School of Management. The experience and expertise of the faculty cover much of the spectrum of formal and computational approaches to behavioral science. The equipment provided will further research in several areas that include cognitive sciences, economics, social networks and anthropology, and visual perception. This project has a strong educational component and the equipment will help graduate students in the Institute's M.A. and Ph.D. programs and graduate students in other Departments in the School of Social Sciences. The equipment will help with computer-aided learning for undergraduates in anthropology, economics and cognitive sciences.

The theory of measurement, which provides rigorous methods for assigning numbers to and developing various numerically based mathematical structures for empirical phenomenon, has generated in the last twenty years many new fundamental ideas that are applicable to science -- particularly experimental sciences that use sophisticated mathematical models. This research exploits and extends these recent developments by (1) directly applying them to areas of behavioral science that employ formal models of human judgment and decision, and (2) by developing new methods of measurement and new mathematical models that take into account empirical facts that contradict important, widely applied models of judgment and decision. Because of the wide use by the military, the government, and business of complicated decision strategies that have human judgments and the results of human decisions as inputs, the results of this research are expected to provide a basis for improving such decision strategies.

Luce Experimental studies and attempts by decision analysists to use extant models to predict people's behavior for decisions involving mixed consequences of both gains and losses have not been very successful. Examples where trouble has been seen is in medical decision making. Because of the great importance of such applications, it is urgent that better models be developed and tested. Recent theoretical work by the PI has shown why the troubles may arise. He has shown under very plausible, empirically testable conditions that the computations one should use considerably more complex expressions than the weighted averages of utilities of consequences that are commonly employed. Indeed, he derived two distinct mathematical forms from slightly different assumptions. One goal of the project is to pit these assumptions against one another directly and to see empirically which is more accurate. Also, attempts will be made to determine whether these new expressions actually provide improved predictions of observed behavior. Some numerical fitting will be based on existing data, but considerable additional data are needed for more thorough-going tests. The PI will continue to develop the area theoretically. In particular, he will focus considerable attention on the form of the weighting functions whose properties as far less specified than are those of the utility function.

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.

In studying sensory experience, a primary goal is to describe explicitly how sensations relate to the underlying physical stimuli. Although both physical stimuli and a person's behavioral responses are relatively easy to observe, a preeminent problem is to gain a handle on the unobservable intervening sensation. For example, if a patient judges a pain to be 4 on a 10-point scale, is that twice as much pain as a 2? Or, if two people judge a pain to be 4, how can we compare their experiences? For more than a century, these general issues have been studied intensely. Despite that long history and considerable progress, the problems are still far from completely solved, in part, because much past research has not been expressed in the form of testable foundational assumptions, formulated as explicit mathematical laws. The elaboration and testing of such is pivotal to progress in many areas of psychological science.

With support of the National Science Foundation, Drs. Duncan Luce, Louis Narens, and Ragnar Steingrimsson will continue their work in establishing a general, empirically-based, mathematically-expressed foundation of psychological measurement. In particular, the investigators will apply their general approach to the study of sensations. Experiments in both the auditory and visual domains will be used to evaluate the adequacy of behavioral axioms and, thus, of the numerical representations derived from them. The research extends their theoretical and empirical work to several topics including the effects of temporal or spatial order of stimulus presentation, sequential effects, and evaluating how observations obtained using different methodologies are related. An important aspect is the adequacy with which representations can accommodate individual differences without any statistical fitting of unspecified functions and free parameters. Successful progress on this program promises much unification of now separate theories, notably those involving intensity, frequency, and presentation order, and theories that treat those in a domain-specific fashion, and the relations between and evaluation of different methodologies.