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, 0<r<l Formally, if one is indifferent between(w-x, P; w, 1-p)and(w-y, then one should prefer probabilistic insurance(w-x, (l-r)p; w-y, rp; w-ry, 1-p)over regular insurance(w -y) To prove this proposition, we show that p(w-x)+(1-p)(w)=(w-y) implies (1-r)p(w-x)+pa(w-y)+(1-p)(w-ry)>a(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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