Preference, Utility, and Subjective Probability

Preference, Utility, and Subjectiue Pro babilityl" R. Duncan Luce University of Pennsylvania Patrick Suppes Stanford Uniuersity I. The preparation of this chapter was supported in part by Natiorzal Science Foundation grants GI7637 and GB1462 to the University of Pennsylvania and by the Group Psychology Branch of the Ofice of Naval Research under contract NR 171-034 with Stanford University. 2. We very much appreciate the extensive comments and criticisms of drafts of various sections that were given by Ward Edwards, F. M7. Irwin, Jacob Marschak, and Dana Scott. Con tents 1. General Remarks on the Study of Preference 1.1 Origins of the mathematical theories of preference, 252 1.2 Preference, discrimination, and bias, 254 1.3 A classification of theories of preference, 255 1.4 Previous surveys of the literature, 258 2. General Algebraic Choice Theories 2.1 Ordinal utility functions, 259 2.2 Topological assumptions, 264 2.3 Additivity assumptions, 267 2.4 Higher ordered metrics, 272 2.5 JND assumptions, 279 3. Algebraic Choice Theories for Uncertain Outcomes 3.1 The expected utility hypothesis, 281 3.2 Axiom systems that assume numerical probability, 284 3.3 Axiom systems for subjective probability, 291 3.4 Other decision principles, 299 4. Experimental Tests of Algebraic Models 4.1 Higher ordered metrics, 307 4.2 Measurement of utility, 310 4.3 Measurement of subjective probability, 321 4.4 Experimental studies of variance-preference, utility-of- gambling, and other models, 327 5. General Probabilistic Choice Theories 5.1 Response probabilities, 331 5.2 Constant utility models, 332 5.3 Random utility models, 337 5.4 Observable properties, 339 5.5 Relations among the observable properties, 342 5.6 Relations between the models and observable properties, 346 6. General Probabilistic Ranking Theories 35 1 6.1 A function relating choice to ranking probabilities, 351 CONTENTS 251 6.2 A function relating ranking to choice probabilities, 353 6.3 An "impossibility" theorem, 356 7. Probabilistic Choice Theories for Uncertain Outcomes 359 7.1 Expected utility models, 359 7.2 Decomposition assumption, 362 7.3 Several theories applied to a special case, 367 8. Experimental Tests of Probabilistic Models 377 8.1 Some experimental issues, 377 8.2 Stochastic transitivity, 380 8.3 Plots of p, versus .rr, 390 8.4 Probabilistic expected utility models, 397 References Preference, Utilip, and Subjective Probability Of the major areas into which experimental psychology has been traditionally partitioned, motivation is the least well understood and systematized. This is true whether we consider theory, experimental paradigms, or experimental results. The psychology of motivation, compared with that of learning or of sensory and perceptual processes, is peculiarly retarded and confused. Much of what passes for discussion of motivation is pieced together out of fragments from physiological psychology, learning, personality, social psychology, and psycho- pathology. These patches often look violently out of place; worse still, they conceal the underlying cloth so well that one may doubt whether it exists at all. (Irwin, 1958, p. 152) Nevertheless, as Irwin goes on to point out, at least one motivational concept;preference, is just as basic to psychological theory as, for example, are the concepts of discrimination and instrumental conditioning that have arisen in the other more developed branches of psychology. More- over, of the various notions usually considered to be primarily motivational, preference is the only one that mathematical psychologists have attempted to analyze with any care: there are almost no satisfactory formal theories concerning, for example, drive and incentive, and those that exist are best discussed as aspects of learning. So this chapter on mathematical theories of motivation is limited to a study of preference and to the closely related constructs of utility and subjective probability. 1. GENERAL REMARKS ON THE STUDY OF PREFERENCE 1.1 Origins of the Mathematical Theories of Preference Although we are correct in viewing formal theories of preference as a part of mathematical psychology, it would badly wrench history to suggest that the research was mostly carried out by people who classify themselves as psychologists. Much of the theory, and certainly the best of it, was worked out by economists and statisticians who needed psychological GENERAL REMARKS ON THE STUDY OF PREFERENCE 253 underpinnings for their decision theories. Only in the last half dozen years have psychologists begun to isolate for separate study these inherently psychological theories of preference. While being elaborated as distinct and testable psychological theories, the theories of preference have begun to acquire a richness and complexity-hopefully reflecting a true richness and complexity of beh~vior-that renders them largely useless as bases for economic and statistical theories. Perhaps we may ultimately find simple, yet reasonably accurate, approximations to the more exact descriptions of behavior that can serve as psychological foundations for other theoretical developments, but at the moment this is not the main trend. The psychological reader should be warned that statisticians and economists have long employed a technique of nonempirical, rationalistic argument which is totally foreign to many psychologists, who tend to the other extreme of automatically rejecting plausible hypotheses unless they have powerful experimental support. Psychologists sometimes seem slightly naive in their reverence for what are held to be pure empirical facts, when actually most experimental inferences depend on some more or less implicit theoretical position, often partially embodied in a statistical model. Be that as it may, many theories of preference were formulated and evaluated initially in terms of what a "rational" person having unlimited computational abilities and resources ought to do. Frequently these debates have a somewhat tenuous quality for psychologists, especially since there does not exist any really satisfactory comprehensive definition of rationality and only a few of its properties are generally agreed on. Some psychologists have simply rejected work of this genre as sterile, but those who have taken an active interest in it have found two research paths open. First, can these so-called normative theories also be adequate as descriptions of behavior? Attempts to answer this question have led to laboratory tests of the normative theories-with, however, mostly ambiguous results. Second, can theories be devised that are frankly descriptive in intent, that try to encompass explicitly some of the phenomena that seem to be found in the laboratory? Several attempts are discussed. Our attitude in this chapter is primarily psychological: we ask whether a theory seems to describe behavior, not whether it characterizes a rational man; we report experiments, although admittedly not enough experimental work has yet been done to provide us with either completely satisfactory designs or highly reliable phenomena; and we explore some of the relations between theories of preference and other psychological theories. At the same time, we try to recount the normative considerations that led originally to the theories and to cite the more important rationalistic criticisms that have been leveled against them. PREFERENCE, UTILITY, AND SUBJECTIVE PROBABILITY 1.2 Preference, Discrimination, and Bias Irwin (1958, p. 152) developed in detail the widely accepted thesis that, " . preference is exactly as fundamental as discrimination and that, indeed, the two are so intimately related in behavior that if the organism exhibits a discrimination, it must also exhibit a preference, and conversely." Without reproducing the supporting argument in detail, we may suggest the main point. Suppose that outcome x, results either when response r, is made to stimulus presentation s, or when r, is made to s,; whereas x, results when r, is made to s, or when r, is made to s,. Before we can expect a subject to respond differentially and thus to show his ability to discriminate between s, and s,, it must matter to him which outcome occurs-he must prefer one outcome to the other. Equally well, he can evidence a preference between x, and x, only if he is capable of discriminat- ing the discriminative stimuli, for only then can he know (or learn) which response is appropriate on each trial to achieve the preferred outcome. A careful examination of Irwin's discussion suggests that this view is correct: the notions of discrimination and preference are both fundamental to an understanding of behavior and they are profoundly inter- twined in parallel roles. If so, it appears perverse, if not worse, to divide psychology-in particular, that portion based upon choice experiments- into a part concerned primarily with the discrimination of stimuli and a part concerned primarily with preferences among outcomes. We are saved from total chaos, however, by the experiments that in principle we should perform, but rarely do because we are certain of the results on the basis either of the informal experimentation of experience or of studies deeply buried in the past of psychology. In a psychophysical or learning experiment we assume that we know what a subject's preferences are among the outcomes. We are confident that hungry animals prefer receiving food pellets to not receiving them, that a student prefers winning five cents to losing three, etc. Equally well, in (human) preference experiments, we always attempt to select discriminative stimuli, for example, labels to identify the outcomes, that we are sure from our knowledge of people are perfectly discriminable. We could perform the necessary auxiliary experiments to prove these assumptions, but usually we do not because we are so certain of the results. This means, for example, that when a subject exhibits some inconsistency in his choices in a preference experiment, we automatically attribute it to an ambivalence about the worth of the outcomes, not to his inability to tell which outcomes are associated with which stimulus-response pairs nor to his inability to discriminate among the relevant stimuli.

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