PROSPECT THEORY: AN ANALYSIS OF DECISION UNDER RISK ECoNome mea (pre-/9 6i: Ma 1979: 47, 2: ABVINFORM Global ECONOMETRICA VOLUME 47 MARCH 1979 NUMBER 2 PROSPECT THEORY: AN ANALYSIS OF DECISION UNDER RISK BY DANIEL KAHNEMAN AND AMOS TVERSKY ndency, called the certainty effect, contr losses. In addition, people general nsurance and gambling 1. INTRODUCTION EXPECTED UTILITY THEORY has dominated the analysis of nmaking under risk. It has been generally accepted ormative model onal choice [24 and widely applied as a descriptive model of economic or,eg.[15,4 Thus, it is assumed that all reasonable people would wish to obey the axioms of the theory [47, 36], and that most people actually do, most of the time The present paper describes several classes of choice problems in which preferences systematically violate the axioms of expected utility theory. In the light of these observations we argue that utility theory, as it is common interpreted and applied is not an adequate descriptive model and we propose an alternative account of choice under risk 2. CRITIQUE Decision making under risk can be viewed as a choice between prospects or gambles. a prospect(x1, Pi;...; xm, Pa)is a contract that yields outcome x with probability pi, where p+p2+,,,+Pa =1. To simplify notation, we omit null outcomes and use(x, p) to denote the prospect(x, p: 0, 1-p) that yields x with probability p and o with probability 1-p. The(riskless) prospect that yields x th certainty is denoted by(x). The present discussion is restricted to prospects with so-called objective or standard probabilities The application of expected utility theory to choices between prospects is based n the following three tenets (i)Expectation: U(x1, Pi;...;tm, Pn)=Piu(x1)+..+pnu(a,n) his work was supported in part by grants from the Harry F Guggenheim Foundation and from 8-072-0722 from Decisions and Designs, Inc. to Perceptronies, Ine. We also thank the Center Advanced Study in the Behavioral Sciences at Stanford for its support Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. PROSPECT THEORY: AN ANALYSIS OF DECISION UNDER RISK DANIEL KAHNEMAN; AMOS TVERSKY Econometrica (pre-1986); Mar 1979; 47, 2; ABI/INFORM Global pg. 263
264 D. KAHNEMAN AND A. TVERSKY That is, the overall utility of a prospect, denoted by U, is the expected utility of ii)Asset Integration: (x1, Pi;...; tm, pn)is acceptable at asset position w iff U(w +x1, Pi That is, a prospect is acceptable if the utility resulting from integrating the prospect with one's assets exceeds the utility of those assets alone. Thus, the main of the utility function is final states (which include one's asset position) ather than gains or losses. Although the domain of the utility function is not limited to any particular class of consequences, most applications of the theory have been concerned with monetary outcomes. Furthermore, most economic applications introduce the following additional assumption iii)Risk Aversion: u is concave(u"<O). <s person is risk averse if he prefers the certain prospect(x)to any risky prospect xpected value x. In expected utility theory, risk aversion is equivalent to the concavity of the utility function. The prevalence of risk aversion is perhaps the best known generalization regarding risky choices. It led the early decisi theorists of the eighteenth century to propose that utility is a concave function of money, and this idea has been retained in modern treatments(Pratt [33, Arrow In the following sections we demonstrate several phenomena these tenets of expected utility theory. The demonstrations are niversity faculty to hypothetical choice espondents were presented with problems of the type illustrate thich of the following would you prefer? A: 50% chance to win 1.000 B: 450 for sure 50% chance to win nothing The outcomes refer to Israeli currency. To appreciate the significance of the ote that the median net monthly income for a family is about 3,000Isr The respondents were asked to imagine that they were actually faced with the choice described in the problem, and to indicate the decision they would have made in such a case. The responses were anonymous and the instructions specified that there was no'correct'answer to such problems, nd that the aim of the study was to find out how people choose among risky prospects. The problems were presented in questionnaire form, with at most ozen problems per booklet. Several forms of each questionnaire were con- ructed so that subjects were exposed to the problems in different orders. In addition, two versions of each problem were used in which the left-right position s reversed he problems described in this paper are selected illustrations of a series of effects. Every effect has been observed in several problems with differen outcomes and probabilities. Some of the problems have also been presented to groups of students and faculty at the University of Stockholm and at the
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PROSPECT THEORY 265 University of Michigan. The pattern of results was essentially identical to the The reliance on hypothetical choices raises obvious questions regarding the alidity of the method and the generalizability of the results. We are keenly aware of these problems. However, all other methods that have been used to test utility heory also suffer from severe drawbacks, Real choices can be investigated either in the field, by naturalistic or statistical observations of economic behavior, or in the laboratory. Field studies can only provide for rather crude tests of qualitative predictions, because probabilities and utilities cannot be adequately measured in such contexts. Laboratory experiments have been designed to obtain precise measures of utility and probability from actual choices, but these experimental udies typically involve contrived gambles for small stakes, and a large number of complicate the interpretation of the results and restrict their generality By default, the method of hypothetical choices emerges as the simplest pro cedure by which a large number of theoretical questions can be investigated. The of the method relies on the assumption that people often know how they would behave in actual situations of choice, and on the further assumption that the subjects have no special reason to disguise their true preferences. If people casonably accurate in predicting their choices, the presence of common and systematic violations of expected utility theory in hypothetical problems provides presumptive evidence against that theory. In expected utility theory, the utilities of outcomes are weighted by their robabilities. The present section describes a series of choice problems in which people's preferences systematically violate this principle. We first show that people overweight outcomes that are considered certain, relative to outcomes which are merely probable-a phenomenon which we label the certainty effect pected utility theory which exploits th certainty effect was introduced by the French economist Maurice Allais in 1953 ] Allais' example has been discussed from both normative and descriptive standpoints by many authors [28, 38]. The following pair of choice problems is a variation of Allais example, which differs from the original in that it refers to moderate rather than to extremely large gains. The number of respondents who answered each problem is denoted by N, and the percentage who choose each option is given in brackets PRObLEM 1: Choose between A: 2, 500 with probability 33, B: 2, 400 with certainty 0 with probability [82] produoed with perm is sion of the copyright owner. Further reproduction prohibited without permission
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266 PROBLEM 2: Choose between The data show that 82 per cent of the subjects ent of the subjects chose C in Problem 2. Each of these preferences is significant at the 01 level, as denoted by the asterisk. Moreover, the analysis of individual hoice indicates that a majority of respondents(61 per cent)made the Dice in both problems. This pattern of preferences violates expected y in the manner originally described by Allais. According to that th u(0)=0, the first preference implies (2,400)>33(2,500)+66(2,400)or34(2,400)>33(2,500) while the second preference implies the reverse inequality. Note that Problem 2 is obtained from Problem 1 by eliminating a. 66 chance of winning 2400 from both prospects under consideration. Evidently, this chang tion in desirability when it alters the character of the prospect from a sure gain to a probable one, than when both the original and the reduced prospects are A simpler demonstration of the same phenomenon, involving only two- outcome gambles is given below. This example is also based on Allais [2] A:(4,00,80),orB:(3,000) [80] C:(4,000,20),orD:(3,00025), 95[65] [35 In this pair of problems as well as in all other problem-pairs in this section, over half the respondents violated expected utility theory. To show that the modal attern of preferences in Problems 3 and 4 is not compatible with the theory, set (0)=0, and recall that the choice of B implies u(3, 000)/u(4,000)>4/5 whereas the choice of C implies the reverse inequality. Note that C=(4,000,. 20)can be expressed as (A, 25), while the prospect D rewritten as(B,. 25). The substitution axiom of utility theory asserts that ferred to A, then any(probability)mixture(B, p)must be preferred to the (A, p). Our subjects did not obey this axiom. Apparently, reducing the bability of winning from 1.0 to 25 has a greater effect than the reduction from Reproduced with permission of the copyright owner. Further reproduction prohibited without pemission
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PROSPECT THEORY 8 to. 2. The following pair of choice problems illustrates the certainty effect with LEM 5 A: 50% chance to win a three- B: A one- week tour of ur of England England, with certainty. [78] PROBLEM 6: lance to win a three- D: 10% chance to win a one week tour of England, week tour of England N=72[67] [33] s not the only type of violation of the substitution axiom hich this axiom fails is illustrated by the following problems. PROBLEM 7 A:(6000,45),B:(3,000,,90) PROBLEM 8 N=66[73] Note that in Problem 7 the probabilities of winning are substantial (.90 and. 45), d most people choose the prospect where winning is more probable. In Problem 8, there is a possibility of winning, although the probabilities of winning are minuscule (.002 and. 001)in both prospects. In this situation where winning is possible but not probable, most people choose the prospect that offers the large gain. Similar results have been reported by MacCrimmon and Larsson[28] The above problems illustrate common attitudes toward risk or chance that nnot be captured by the expected utility model. The results suggest the following empirical generalization concerning the manner in which the substitu- tion axiom is violated If (y, pq) is equivalent to(x, p), then(y, pqr) is preferred to (x, pr),0<p, a, r<1. This property is incorporated into an alternative theory, developed in the second part of the paper
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