PROSPECT THEORY 267 8 to 2. The following pair of choice problems illustrates the certainty effect with PROBLEM A: 50% chance to win a three- B: a one-week tour of C: 5% chance to win a three- D 10% chance to win a one k tour of england france, and Italy N=72[67] [33] The certainty effect is not the only type of violation of the substitution axiom Another situation in which this axiom fails is illustrated by the following problems PRO A:(6,000,45),B:(3,000,,90) [86] P C:(6,000,,001),D:(3,000,.002) 66[73] [27] Note that in Problem 7 the probabilities of winning are substantial (.90 and. 45), and 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 larger gain. Similar results have been reported by MacCrimmon and Larsson [28] The above problems illustrate common attitudes toward risk or chance that cannot be captured by the expected utility model. The results suggest the following empirical generalization concerning the manner in which the substit tion axiom is violated If (y, pa) is equivalent to(x, p), then(y, pqr)is preferred to (x, pr),0<p, q, r<1. This property is incorporated into an alternative theory, developed in the second part of the paper
D. KAHNEMAN AND A. TVERSKY The Reflection Efect The previous section discussed preferences between positive prospects, i. e prospects 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 previous sect and the right-hand column displays choice problems in which the signs of outcomes are reversed we use -x to denote the loss of x and to denote prevalent preference, 1. e, the choice made by the majority of subjects TABLE I PREFERENCES BETWEEN POSITIVE AND NEGATIVE PROSPECTS Positive prospects oblem3:(4,000,80)<(3,000) Problem4:(4,000,20)>(3,000,25) Problem 4: ( (3,000,90)>(6,000,,45) 6,000,45) Problem8:(3,000,002)<(6,000,001) Problen8':(-3,000,002)>(-6,000,001) [27] [73 N=66 [70 In each of the four problems in Table I the preference between negative prospects is the mirror image of the preference between positive prospects. Thus the reflection of prospects around O reverses 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 effect implies that risk aversion in the positive domain is accompanied by risk seeking in the negative domain, In Problem 3, for example, the majority of subjects were willing to accept a risk of 80 to lose 4,000, in preference to a sure loss of 3, 000, although the gamble has a lower expected value The occurrence of isk seeking in choices between negative prospects was noted early by Markowitz [29]. Williams [48] reported data where a translation of outcomes produces a dramatic shift from risk aversion to risk seeking. For example, his subjects wer indifferent between(100, 65;-100, 35)and(0), indicating risk aversion. They were 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 prospects Second, recall that the preferences between the positive prospects in Table I are 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.In the negative domain, the same effect leads to a risk seeking preference for a loss
PROSPECT THEORY 269 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 gains and risk seeking in the domain of losses Third the refection effect eliminates aversion for uncertainty or variability as in explanation of the certainty effect. Consider, for example, the prevalent 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 prefer 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 the difference 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 notion that certainty is generally desirable. Rather, it appears that certainty increases the aversiveness of losses as well as the desirability of gains Probabilistic Insurance The prevalence of the purchase of insurance against both large and small losse as 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 purchase 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 for money is concave everywhere. For example, people often prefer insurance programs that offer limited coverage with low or zero deductible over comparable policies that offer higher maximal coverage with higher deductibles-contrary to risk aversion (see, e. g, Fuchs [16]. Another type of insurance problem in which people esponses are inconsistent with the concavity hypothesis may be called prob- abilistic 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 some property against damage, e. g, fire or theft. After examining the risks and the premium you find that you have no clear preference between the options of purchasing 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 regular premium. In case of damage, there is a 50 per cent chance that you pay the other 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 on an odd day of the month, you pay the other half of the regular premium and your losses are covered; but if the accident
270 D, KAHNEMAN AND A. TVERSKY occurs on an even day of the month, your insurance payment is refunded and your losses are not covered Recall that the premium for full coverage is such that you find this insurance barely worth its cost. Under these circumstances, would you purchase probabilistic insurance Y N=9520][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 probabilistic insurance is superior to regular insurance. That is, if at asset position w one is just willing to pay a premium y to insure against a probability p of losing x, then one should definitely be willing to pay a smaller premium ry to reduce the probability of losing x from p to(1-r)p, 0<r<1. Formally, if one is indifferent between (w-x, P;w, 1-p)and (w-y, then one should prefer probabilistic insurance(w-x, (1-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)u(w)=u(w-y) implies (1-r)pu(w-x)+rpu(w-y)+(1-p)u(w-ry) Without loss of generality, we can set u(w-x)=0 and u(w)=1. Hence u(w y)=1-p, and we wish to show that r(1-p)+(1-p)u(w-ry)>1-por(w-ry)>1- which holds if and only if u is concave This is a rather puzzling consequence of the risk aversion hypothesis of utility theory, because probabilistic insurance appears intuitively riskier than regular insurance, which entirely eliminates the element of risk. Evidently, the intuitive notion 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 insurance is, in a sense, probabilistic. The most avid buyer of insurance remains ulnerable to many financial and other risks which his policies do not cover. There appears to be a significant difference between probabilistic insurance and what may be called contingent insurance, which provides the certainty of coverage for a
PROSPECT THEORY 271 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 that 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 attractive than probabilistic insurance when the probabilities of unprotected loss are equated. Thus, two prospects that are equivalent in probabilities and outcomes could have different values depending on their formulation. Several demon strations of this general phenomenon are described in the next section The Isolation effecr In order to simplify the choice between alternatives, people often disregard components that the alternatives share and focus on the components that distinguish them(Tversky [44]). This approach to choice problems may produce inconsistent preferences, because a pair of prospects can be decomposed into common and distinctive components in more than one way, and different decom positions sometimes lead to different preferences. We refer to this phenomenon the isolation effect PROBLEM 10: Consider the following two-stage game. In the first stage there 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 tween (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 Note that in this game, one has a choice between 25x 80=20 chance to win 000, and a. 25 x1.0=.25 chance to win 3, 000. Thus in terms of final outcomes 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 prospect, contrary to the modal preference in Problem 4. Evidently, people gnored the first stage of the game, whose outcomes are shared by both prospects, and considered Problem 10 as a choice between(3, 000)and (4, 000, .80, as in Problem 3 above The standard and the sequential formulations of Problem 4 are represented decision trees in Figures 1 and 2, respectively. Following the usual convention quare denote decision nodes and circles denote chance nodes. The essential difference between the two representations is in the location of the decision node In the standard form(Figure 1), the decision maker fac risky prospects, whereas in the sequential form(Figure 2)he faces a choice between a risky and a riskless prospect. This is accomplished by introducing a dependency between the prospects without changing either probabilities or