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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268 D KAHNEMAN AND A. TVERSKY that involve no losses. what happens when the signs of the outcomes are reversed so that gains are replaced by losses? The left-hand column of Table I displays four of the choice problems that were discussed in the pret nd the right-hand column displays choice problems in which the signs of the to denote the loss of x, and to denote the prevalent preference, i.e., the choice made by the majority of subjects. REFERENCES BETWEEN POSTTIVE AND NEGATIVE PROSPECTS Problem3:(4,000,80)<(3,00 Problem 3 Proble8:(3,000,.002)<(6,000,001). Problem8 02 In each of the four proble le I the preferen rospects is the mirror image of the cts Thu ses the preference order We label this pattern the reflection effect Let us turn now to the implications of these data. First, note that the reflection fect implies that risk aversion in the positive domain is accompanied by risk subjects in the negative domain. In Problem 3, for example the majority of ere willing to accept a risk of 80 to lose 4,000, in preference to a sure risk seeking in choices between negative prospects was noted early by Markowitz [48] reported data where dramatic shift from risk aversion to risk seeking. For example, his subjects were indifferent between(100, 65;-100,35)and(O), indicating risk aversion They rere also indifferent between(200, 80)and (100), indicating risk seeking. a recent review by Fishburn and Kochenberger [14] documents the prevalence of risk seeking in choices between negative prospe Second, recall that the preferences between the positive prospects in Table I inconsistent with expected utility theory, The preferences between the cor- responding negative prospects also violate the expectation principle in the same manner. For example, Problems 3 and 4, like Problems 3 and 4, demonstrate that outcomes which are obtained with certainty are overweighted relative to uncertain outcomes. In the positive domain, the certainty effect contributes to a risk averse preference for a sure gain over a larger gain that is merely probable. It the negative domain the same effect leads to a risk seeking preference for a loss Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
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PROSPECT THEORY that is merely probable over a smaller loss that is certain. The same psychological principle-the overweighting of certainty-favors risk aversion in the domain of ains and risk seeking in the domain of losses. Third, the reflection effect eliminates aversion for uncertainty or variability as an explanation of the certainty effect. Consider, for example, the prevalen preferences for(3, 000)over (4, 000,. 80)and for(4,000, 20)over(3, 000,. 25).To resolve this apparent inconsistency one could invoke the assumption that people refer prospects that have high expected value and small variance(see, e. g, Allais [2]; Markowitz [30]; Tobin [41]). Since(3, 000)has no variance while(4,000, 80) has large variance, the former prospect could be chosen despite its lower expected value. When the prospects are reduced, however, the difference in variance between (3,000,. 25)and (4,000, 20) may be insufficient to overcome tl ference in expected value. Because (-3,000)has both higher expected value and lower variance than (4, 000, 80), this account entails that the sure loss should be preferred, contrary to the data. Thus, our data are incompatible with the otion that certainty is generally desirable. Rather, it appears that certainty eases the aversiveness of losses as well as the desirability of gains The prevalence of the purchase of insurance against both large and small losses has been regarded by many as strong evidence for the concavity of the utility function for money. Why otherwise would people spend so much money to urchase insurance policies at a price that exceeds the expected actuarial cost However, an examination of the relative attractiveness of various forms of nsurance does not support the notion that the utility function fo: money is ncave everywhere For example, people often prefer insurance programs that offer limited coverage with low or zero deductible over comparable policies that ffer higher maximal coverage with higher deductibles-contrary to risk aversion (see, e. g, Fuchs [16]. Another type of insurance problem in which people's sponses are inconsistent with the concavity hypothesis may be called prob bilistic insurance. To illustrate this concept, consider the following problem which was presented to 95 Stanford University students PROBLEM 9: Suppose you consider the possibility of insuring against damage, e. g,, fire or theft. After examining the risks and the prem find that you have no clear preference between the options of pure insurance or leaving the property uninsured It is then called to your attention that the insurance company offers a new program called probabilistic insurance. In this program you pay half of the regula remium. In case of damage, there is a 50 per cent chance that you pay the othe half of the premium and the insurance company covers all the losses; and there is a 50 per cent chance that you get back your insurance payment and suffer all the losses. For example, if an accident occurs or an odd day of the month, you pay the other half of the regular premium and your losses are covered; but if the accident Reproduced with permission of the copyright owner. Further reproduction prohibited without permission
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270 occurs on an even day of the month, your insurance payment is refunded and your that the premium for full coverage is such that you find this insurance orth its cost Under these circumstances, would you purchase probabilistic insurance: Yes. No N=95[20][80] Although Problem 9 may appear contrived, it is worth noting that probabilistic insurance represents many forms of protective action where one pays a certain cost to reduce the probability of an undesirable event-without eliminating it altogether. The installation of a burglar alarm, the replacement of old tires, and the decision to stop smoking can all be viewed as probabilistic insurance The responses to Problem 9 and to several other variants of the same question indicate that probabilistic insurance is generally unattractive. Apparently, reduc ing the probability of a loss from p to p/2 is less valuable than reducing the probability of that loss from p/2 to 0. In contrast to these data, expected utility theory (with a concave u) implies that w one is just willing to pay a premium y to insure against a probability p of losin x, then one should definitely be willing to pay a smaller premium ry to reduce th probability of losing x from p to (l-r)p, 0a(w-y) without loss of generality, we can set u(w-x)=0 and u(w)=l. Hence, u(w y)=1-p, and we wish to show that (1-p)+(1-p)(w-ry)>1-port(w-ry)>1- theory, because probabilistic insurance appears intuitively riskier than regu insurance, which entirely eliminates the element of risk. Evidently, the intuitive otion of risk is not adequately captured by the assumed concavity of the utility function for wealth The aversion for probabilistic insurance is particularly intriguing because all nsurance is, in a sense, probabilistic. The most avid buyer of insurance remain rulnerable to many financial and other risks which his policies do not cover. There ppears to be a significant difference between robabilistic insurance and what may be called contingent insurance, which provides the certainty of coverage for a Reproduced with permission of the copyright owmer. Further reproduction prohibited without permissio
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PROSPECT THEORY specified type of risk. Compare, for example, probabilistic insurance against all forms of loss or damage to the contents of your home and contingent insurance nat eliminates all risk of loss from theft, say, but does not cover other risks, e. g fire. We conjecture that contingent insurance will be generally more attracti than probabilistic insurance when the pre es of unprotected loss are quated. Thus, two prospects that are equi in probabilities and outcomes uld have different values depending on their formulation. Several demon strations of this general phenomenon are described in the next section The Isolation Effect components nts that the alternatives share and focus on the components tha distinguish them(Tversky [44]). This approach to choice problems may produce inconsistent preferences, because a pair of prospects can be decomposed into ommon and distinctive components in more than one way, and different deco ositions sometimes lead to different preferences, We refer to this phenomenon as PROBLEM 10: Consider the following two-stage game. In the first stage there is a probability of. 75 to end the game without winning anything, and a probability of 25 to move into the second stage. If you reach the second stage you have a choice (4,000,80)and(3,000) Your choice must be made before the game starts, i. e before the outcome of the first stage is known 25x1.0=, 25 chance to win 3, 000. Thus, in and probabilities one faces a choice between(4,000, 20)and(3, 000, 25), as in Problem 4 above. However, the dominant preferences are different in the two problems Of 141 subjects who answered Problem 10, 78 per cent chose the latter rospect,contrary to the modal preference in Problem 4. Evidently, peop and considered Problem 10 as a choice between(3, 000)and (4, 000, 80), as in Problem 3 above. he standard and the sequential formulations of Problem 4 are represented as decision trees in Figures 1 and 2, respectively. Following the usual convention d circles denote chance nodes, The essential ifference between the two representations is in the location of the decision node In the standard form( Figure 1), the decision maker hoice between tw aspects, whereas in the sequential form(Fi he faces a choice a risky and a riskless prospect. This is accor ng a ncy between the prospects without changing either probabilities or produced with perm ission of the copyright owner. Further reproduction prohibited without permission
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272 3000 FIGURE I - The representation of Problem 4 as a decision tree (standard formulation). FIGURE 2. -The representation of Problem 10 as a decision tree(sequential formulation). outcomes. Specifically, the event 'not winning 3, 000is included in the event ' not winning 4,000in the sequential formulation, while the two events are indepen ent in the standard formulation. Thus, the outcome of winning 3, 000 has a rtainty advantage in the sequential formulation, which it does not have in the standard formulation significant because it violates the basic supposition of a decision-theoretical of final states. It is easy to think of decision problems that are most naturally represented in one of the forms above rather than in the other. For example, the choice between two different risky ventures is likely to be viewed in the standard form. On the ther hand, the following problem is most likely to be represented in the sequential form. One may invest money in a venture with some probability of losing one's capital if the venture fails, and with a choice between a fixed agreed eturn and a percentage of earnings if it succeeds. The isolation effect implies that he contingent certainty of the fixed return enhances the attractiveness of this option, relative to a risky venture with the same probabilities and outcomes Reproduced with perm ission of the copyright owner. Further reproduction prohibited without permission
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