Individual Choice Behavior

A Theoretical Analysis

R. Duncan Luce

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486441368
• eBook ISBN: 9780486153391
• Topic: Mathematics
• Subtopic: Probability & Statistics
• Index: Mathematics

This influential treatise presents upper-level undergraduates and graduate students with a mathematical analysis of choice behavior. It begins with the statement of a general axiom upon which the rest of the book rests; the following three chapters, which may be read independently of each other, are devoted to applications of the theory to substantive problems: psychophysics, utility, and learning.
Applications to psychophysics include considerations of time- and space-order effects, the Fechnerian assumption, the power law and its relation to discrimination data, interaction of continua, discriminal processes, signal detectability theory, and ranking of stimuli. The next major theme, utility theory, features unusual results that suggest an experiment to test the theory. The final chapters explore learning-related topics, analyzing the stochastic theories of learning as the basic approach—with the exception that distributions of response strengths are assumed to be transformed rather than response probabilities. The author arrives at three classes of learning operators, both linear and nonlinear, and the text concludes with a useful series of appendixes.

chapter 1

THE BASIC THEORY

A. INTRODUCTION

One large portion of psychology—including at least the topics of sensation, motivation, simple selective learning, and reaction time—has a common theme: choice. To be sure, in the study of sensation the choices are among stimuli, in learning they are among responses, and in motivation, among alternatives having different preference evaluations; and some psychologists hold that these distinctions, at least the one between stimulus and response, are basic to an understanding of behavior. This book attempts a partial mathematical description of individual choice behavior in which the distinction is not made except in the language used in different interpretations of the theory. Thus the more neutral word “alternative” is used to include the several cases.

In essence, the approach taken—in this respect, by no means novel—is orthogonal to that of S-R psychology, but not at variance with it. Rather than search for lawfulness between stimuli and responses and attempt to formulate a theory to describe those relationships, we shall be concerned with possible lawfulness found among different, but related, choice situations, whether these are choices among stimuli or among responses. Possibly the simplest prototype of this type of theory is the frequently assumed rule of transitivity among choices: given that a person chooses a over b and that he chooses b over c, then he chooses a over c when a and c are offered. This assumption, were it true, would be a law relating a person’s choice in one situation to those in two others, not a law relating responses to stimuli. It is evident that a sufficiently rich set of relations of this sort, coupled with a few simple S-R connections, will allow one to derive many more, and possibly quite complicated, S-R connections.

Such an approach seems to merit careful consideration, since several decades of pure S-R psychology have not resulted in notably simple laws of behavior. However, there seems little point in trying to discuss in detail its merits and demerits now, except to mention it in order to avoid confusion later. The results that follow—which seem to afford some insight into, and some integration of, psychological and psychophysical scaling, utility theory, and learning theory—will implicitly serve as the argument for the course taken.

1. Probabilistic vs. Algebraic Theories

A basic presupposition of this book is that choice behavior is best described as a probabilistic, not an algebraic, phenomenon. That is to say, at any instant when a person reaches a decision between, say, a and b we will assume that there is a probability P(a, b) that the choice will be a rather than b. These probabilities will generally be different from 0 and 1, although these extreme (and important) cases will not be excluded. The alternative is to suppose that the probabilities are always 0 and 1 and that the observed choices tell us which it is; in this case the algebraic theory of relations seems to be the most appropriate mathematical tool.

The decision between these two approaches does not seem to be empirical in nature. Various sorts of data—intransitivities of choices and inconsistencies when the same choices are offered several times—suggest the probabilitistic model, but they are far from conclusive. Both of these phenomena can be explained within an algebraic framework provided that the choice pattern is allowed to change over time, either because of learning or because of other changes in the internal state of the organism. The presently unanswerable question is which approach will, in the long run, give a more parsimonious and complete explanation of the total range of phenomena.

The probabilistic philosophy is by now a commonplace in much of psychology, but it is a comparatively new and unproven point of view in utility theory. To be sure, economists when pressed will admit that the psychologist’s assumption is probably the more accurate, but they have argued that the resulting simplicity warrants an algebraic idealization. Ironically, some of the following results suggest that, on the contrary, the idealization may actually have made the utility problem artificially difficult.

2. Multiple Alternative Choices

Once choice behavior is assumed to be probabilistic, a problem arises which does not exist in the algebraic models. Complete data concerning the choices that a person makes from each possible pair of alternatives taken from a set of three or more alternatives do not appear to determine what choice he will make when the whole set is presented. Because they cannot escape multiple alternative choice proble...

Table of contents

• DOVER BOOKS ON MATHEMATICS
• Title Page
• Copyright Page
• Dedication
• PREFACE
• Table of Contents
• chapter 1 - THE BASIC THEORY
• chapter 2 - APPLICATIONS TO PSYCHOPHYSICS
• chapter 3 - APPLICATIONS TO UTILITY THEORY
• chapter 4 - APPLICATIONS TO LEARNING
• chapter 5 - SUMMARY AND CONCLUSIONS
• appendix 1 - ALTERNATIVE FORMS OF AXIOM 1
• appendix 2 - FORM OF LATENCY DISTRIBUTION
• appendix 3 - MAXIMUM LIKELIHOOD EQUATIONS FOR THE TWO-ALTERNATIVE, TWO-OUTCOME BETA LEARNING MODEL
• appendix 4 - OPEN PROBLEMS
• BIBLIOGRAPHY
• INDEX