Journal of Risk and Uncertainty, 5:297-323 (1992) �� 1992 Kluwer Academic Publishers Advances in Prospect Theory: Cumulative Representation of Uncertainty AMOS TVERSKY Stanford University, Department of Psychology, Stanford, CA 94305-2130 DANIEL KAHNEMAN* University of California at Berkeley, Department of Psychology, Berkeley, CA 94720 Key words: cumulative prospect theory Abstract We develop a new version of prospect theory that employs cumulative rather than separable decision weights and extends the theory in several respects. This version, called cumulative prospect theory, applies to uncertain as well as to risky prospects with any number of outcomes, and it allows different weighting functions for gains and for losses. Two principles, diminishing sensitivity and loss aversion, are invoked to explain the characteris- tic curvature of the value function and the weighting functions. A review of the experimental evidence and the results of a new experiment confirm a distinctive fourfold pattern of risk attitudes: risk aversion for gains and risk seeking for losses of high probability risk seeking for gains and risk aversion for losses of low probability. Expected utility theory reigned for several decades as the dominant normative and descriptive model of decision making under uncertainty, but it has come under serious question in recent years. There is now general agreement that the theory does not provide an adequate description of individual choice: a substantial body of evidence shows that decision makers systematically violate its basic tenets. Many alternative mod- els have been proposed in response to this empirical challenge (for reviews, see Camerer, 1989 Fishburn, 1988 Machina, 1987). Some time ago we presented a model of choice, called prospect theory, which explained the major violations of expected utility theory in choices between risky prospects with a small number of outcomes (Kahneman and Tver- sky, 1979 Tversky and Kahneman, 1986). The key elements of this theory are 1) a value function that is concave for gains, convex for losses, and steeper for losses than for gains, *An earlier version of this article was entitled "Cumulative Prospect Theory: An Analysis of Decision under Uncertainty." This article has benefited from discussions with Colin Camerer, Chew Soo-Hong, David Freedman, and David H. Krantz. We are especially grateful to Peter P. Wakker for his invaluable input and contribution to the axiomatic analysis. We are indebted to Richard Gonzalez and Amy Hayes for running the experiment and analyzing the data. This work was supported by Grants 89-0064 and 88-0206 from the Air Force Office of Scientific Research, by Grant SES-9109535 from the National Science Foundation, and by the Sloan Foundation.

298 AMOS TVERSKY/DANIEL KAHNEMAN and 2) a nonlinear transformation of the probability scale, which overweights small probabilities and underweights moderate and high probabilities. In an important later development, several authors (Quiggin, 1982 Schmeidler, 1989 Yaari, 1987 Weymark, 1981) have advanced a new representation, called the rank-dependent or the cumulative functional, that transforms cumulative rather than individual probabilities. This article presents a new version of prospect theory that incorporates the cumulative functional and extends the theory to uncertain as well to risky prospects with any number of out- comes. The resulting model, called cumulative prospect theory, combines some of the attractive features of both developments (see also Luce and Fishburn, 1991). It gives rise to different evaluations of gains and losses, which are not distinguished in the standard cumulative model, and it provides a unified treatment of both risk and uncertainty. To set the stage for the present development, we first list five major phenomena of choice, which violate the standard model and set a minimal challenge that must be met by any adequate descriptive theory of choice. All these findings have been confirmed in a number of experiments, with both real and hypothetical payoffs. Framing effects. The rational theory of choice assumes description invariance: equiva- lent formulations of a choice problem should give rise to the same preference order (Arrow, 1982). Contrary to this assumption, there is much evidence that variations in the framing of options (e.g., in terms of gains or losses) yield systematically different prefer- ences (Tversky and Kahneman, 1986). Nonlinear preferences. According to the expectation principle, the utility of a risky prospect is linear in outcome probabilities. Allais's (1953) famous example challenged this principle by showing that the difference between probabilities of .99 and 1.00 has more impact on preferences than the difference between 0.10 and 0.11. More recent studies observed nonlinear preferences in choices that do not involve sure things (Cam- erer and Ho, 1991). Source dependence. People's willingness to bet on an uncertain event depends not only on the degree of uncertainty but also on its source. Ellsberg (1961) observed that people prefer to bet on an urn containing equal numbers of red and green balls, rather than on an urn that contains red and green balls in unknown proportions. More recent evidence indicates that people often prefer a bet on an event in their area of competence over a bet on a matched chance event, although the former probability is vague and the latter is clear (Heath and Tversky, 1991). Risk seeking. Risk aversion is generally assumed in economic analyses of decision under uncertainty. However, risk-seeking choices are consistently observed in two classes of decision problems. First, people often prefer a small probability of winning a large prize over the expected value of that prospect. Second, risk seeking is prevalent when people must choose between a sure loss and a substantial probability of a larger loss. Loss' aversion. One of the basic phenomena of choice under both risk and uncertainty is that losses loom larger than gains (Kahneman and Tversky, 1984 Tversky and Kahne- man, 1991). The observed asymmetry between gains and losses is far too extreme to be explained by income effects or by decreasing risk aversion.

ADVANCES IN PROSPECT THEORY 299 The present development explains loss aversion, risk seeking, and nonlinear prefer- ences in terms of the value and the weighting functions. It incorporates a framing pro- cess, and it can accommodate source preferences. Additional phenomena that lie be- yond the scope of the theory--and of its alternatives--are discussed later. The present article is organized as follows. Section 1.1 introduces the (two-part) cu- mulative functional section 1.2 discusses relations to previous work and section 1.3 describes the qualitative properties of the value and the weighting functions. These properties are tested in an extensive study of individual choice, described in section 2, which also addresses the question of monetary incentives. Implications and limitations of the theory are discussed in section 3. An axiomatic analysis of cumulative prospect theory is presented in the appendix. 1. Theory Prospect theory distinguishes two phases in the choice process: framing and valuation. In the framing phase, the decision maker constructs a representation of the acts, contingen- cies, and outcomes that are relevant to the decision. In the valuation phase, the decision maker assesses the value of each prospect and chooses accordingly. Although no formal theory of framing is available, we have learned a fair amount about the rules that govern the representation of acts, outcomes, and contingencies (Tversky and Kahneman, 1986). The valuation process discussed in subsequent sections is applied to framed prospects. 1.1. Cumulative prospect theory In the classical theory, the utility of an uncertain prospect is the sum of the utilities of the outcomes, each weighted by its probability. The empirical evidence reviewed above suggests two major modifications of this theory: 1) the carriers of value are gains and losses, not final assets and 2) the value of each outcome is multiplied by a decision weight, not by an additive probability. The weighting scheme used in the original version of prospect theory and in other models is a monotonic transformation of outcome prob- abilities. This scheme encounters two problems. First, it does not always satisfy stochastic dominance, an assumption that many theorists are reluctant to give up. Second, it is not readily extended to prospects with a large number of outcomes. These problems can be handled by assuming that transparently dominated prospects are eliminated in the edit- ing phase, and by normalizing the weights so that they add to unity. Alternatively, both problems can be solved by the rank-dependent or cumulative functional, first proposed by Quiggin (1982) for decision under risk and by Schmeidler (1989) for decision under uncertainty. Instead of transforming each probability separately, this model transforms the entire cumulative distribution function. The present theory applies the cumulative functional separately to gains and to losses. This development extends prospect theory to

300 A M O S T V E R S K Y / D A N I E L K A H N E M A N uncertain as well as to risky prospects with any number of outcomes while preserving most of its essential features. The differences between the cumulative and the original versions of the theory are discussed in section 1.2. Let S be a finite set of states of nature subsets of S are called events. It is assumed that exactly one state obtains, which is unknown to the decision maker. Let X be a set of consequences, also called outcomes. For simplicity, we confine the present discussion to monetary outcomes. We assume that X includes a neutral outcome, denoted 0, and we interpret all other elements of X as gains or losses, denoted by positive or negative numbers, respectively. An uncertain prospect f is a function from S into X that assigns to each state s e S a consequencefls) -- x inX. To define the cumulative functional, we arrange the outcomes of each prospect in increasing order. A prospectf is then represented as a sequence of pairs (xi,Ai), which yieldsxi ifAi occurs, wherexi xj iffi j, and (Ai) is a partition of S. We use positive subscripts to denote positive outcomes, negative subscripts to denote negative outcomes, and the zero subscript to index the neutral outcome. A prospect is called strictly positive or positive, respectively, if its outcomes are all positive or nonneg- ative. Strictly negative and negative prospects are defined similarly all other prospects are called mixed. The positive part off, denotedf + , is obtained by lettingf + (s) = f(s) if f(s) 0, and f + (s) = 0 if f(s) O. The negative part of f, denoted f - , is defined similarly. As in expected utility theory, we assign to each prospectf a number V ( f ) such thatfis preferred to or indifferent tog iff V(f) _ V(g). The following representation is defined in terms of the concept of capacity (Choquet, 1955), a nonadditive set function that gener- alizes the standard notion of probability. A capacity Wis a function that assigns to eachA C S a number W(A) satisfying W((b) = 0, W(S) = 1, and W(A) _ W(B) wheneverA D B. Cumulative prospect theory asserts that there exist a strictly increasing value function v:X--+ Re, satisfying v(x0) = v(0) = 0, and capacities W + and W - , such that f o r f = (xi, Ai), - m - i n, V(f) = V ( f +) + V ( f - ) , n 0 V ( f +) = ~'Tr/+v(x,), V ( f - ) = 2 "rr,-v(xi), (1) i - O i = m where the decision weights "rr + (f+) = (nv~-, ... , v +) and ~ r - ( f - ) = ('rr_-m, "" , Wo) are defined by: + = W + = W - ( A - m ) , nvi + = W + ( A i U ... U A n ) - W+(Ai+I U ... U A n ) , O _ i _ n - 1, "rr i- = W - ( A - m U ... U Ai) - W - ( A - m O ... U A i - 1 ) , l - m - i - O. Letting qr i = "rr? if/ -- 0 and Tri = q'r/- if/ O, equation (1) reduces to V(f) = 2 "rriP(xi) ��� i = - - m (2)

ADVANCES IN PROSPECT THEORY 301 The decision weight 7ri +, associated with a positive outcome, is the difference between the capacities of the events "the outcome is at least as good asxi" and "the outcome is strictly better than xi." The decision weight vi-, associated with a negative outcome, is the difference between the capacities of the events "the outcome is at least as bad asxi" and :'the outcome is strictly worse than xi." Thus, the decision weight associated with an outcome can be interpreted as the marginal contribution of the respective event, 1 de- fined in terms of the capacities W + and W - . If each W is additive, and hence a proba- bility measure, then Wi is simply the probability of Ai. It follows readily from the defini- tions of'rr and Wthat for both positive and negative prospects, the decision weights add to 1. For mixed prospects, however, the sum can be either smaller or greater than 1, because the decision weights for gains and for losses are defined by separate capacities. If the prospectf = (xi,Ai) is given by a probability distributionp(Ai) = Pi, it can be viewed as a probabilistic or risky prospect (xi, Pi). In this case, decision weights are defined by: 7 + = w + ( p . ) , ~ _ - = w-(p_m), "rr+ = w + ( p i + .-. + Pn) - w + ( P i + l + ... + pn),O i - n - 1, vr i = w - ( p m + ... + P i ) - w - ( p - m + ... + p i - l ) , l - m _i_ O. where w + and w - are strictly increasing functions from the unit interval into itself satisfyingw+(0) = w-(0) = 0, andw+(1) = w-(1) = 1. To illustrate the model, consider the following game of chance. You roll a die once and observe the result x = 1, ... , 6. Ifx is even, you receive Sx ifx is odd, you pay Sx. Viewed as a probabilistic prospect with equiprobable outcomes, f yields the conse- quences ( - 5, - 3, - 1, 2, 4, 6), each with probability 1/6. Thus,f + = (0, 1/2 2, 1/6 4, 1/6 6, 1/6), and f - = ( - 5, 1/6 - 3, 1/6 - 1, 1/6 0, 1/2). By equation (1), therefore, V ( f ) = V ( f +) + V ( f - ) = v(2)[w+(1/2) - w+(1/3)] + v(4)[w+(1/3) - w+(1/6)] + v(6)[w + (1/6) - w + (0)] + v ( - 5 ) [ w (1/6) - w (0)] + v ( - 3 ) [ w - (1/3) - w - ( 1 / 6 ) ] + v ( - 1 ) [ w (1/2) - w - ( 1 / 3 ) ] . 1.2. Relation to previous work Luce and Fishburn (1991) derived essentially the same representation from a more elaborate theory involving an operation O of joint receipt or multiple play. Thus,f O g is the composite prospect obtained by playing b o t h f and g, separately. The key feature of their theory is that the utility function U is additive with respect to O, that is, U ( f O g) = U ( f ) + U(g) provided one prospect is acceptable (i.e., preferred to the status quo) and the other is not. This condition seems too restrictive both normatively and descriptively. As noted by the authors, it implies that the utility of money is a linear function of money