Prospect Theory: An Analysis of Decision under Risk
Daniel Kahneman; Amos Tversky
Econometrica, Vol. 47, No. 2. (Mar., 1979), pp. 263-292.
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ECONOMETRICAPROSPECT THEORY: AN ANALYSIS O F DECISION UNDER RISKThis paper presents a critique of expected utility theory as a descriptive model ofdecision making under risk, and develops an alternative model, called prospect theory.
Choices among risky prospects exhibit several pervasive effects that are inconsistent withthe basic tenets of utility theory. In particular, people underweight outcomes that are
merely probable in comparison with outcomes that are obtained with certainty. Thistendency, called the certainty effect, contributes to risk aversion in choices involving suregains and to risk seeking in choices involving sure losses. In addition, people generally
discard components that are shared by all prospects under consideration. This tendency,
called the isolation effect, leads to inconsistent preferences when the same choice ispresented in different forms. An alternative theory of choice is developed, in which value
is assigned to gains and losses rather than to final assets and in which probabilities are
replaced by decision weights. The value function is normally concave for gains, commonly
convex for losses, and is generally steeper for losses than for gains. Decision weights are
generally lower than the corresponding probabilities, except in the range of low probabilities. Overweighting of low probabilities may contribute to the attractiveness of both
insurance and gambling.

1. INTRODUCTIONEXPECTEDUTILITY THEORY has dominated the analysis of decision making underrisk. It has been generally accepted as a normative model of rational choice [24],and widely applied as a descriptive model of economic behavior, e.g. [15,4].
Thus, it is assumed that all reasonable people would wish to obey the axioms of thetheory [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 thelight of these observations we argue that utility theory, as it is commonlyinterpreted and applied, is not an adequate descriptive model and we propose analternative account of choice under risk.
2. CRITIQUE
Decision making under risk can be viewed as a choice between prospects or
gambles. A prospect (xl, pl; . .. ;x,,, p,) is a contract that yields outcome xi withprobability pi, where pl + p z + . . . +p, = 1. To simplify notation, we omit nulloutcomes and use (x, p) to denote the prospect (x, p ; 0, 1 -p) that yields x withprobability p and 0 with probability 1-p. The (riskless) prospect that yields x
with 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 basedon the following three tenets.
(i) Expectation: U(x1, p ~. ;. . ;x,, p,) = p l u ( x l ) + . . . +p,u(x,).

'

This work was supported in part by grants from the Harry F. Guggenheim Foundation and fromthe Advanced Research Projects Agency of the Department of Defense and was monitored by Officeof Naval Research under Contract N00014-78-C-0100 (ARPA Order No. 3469) under Subcontract
78-072-0722 from Decisions and Designs, Inc. to Perceptronics, Inc. We also thank the Center for
Advanced Study in the Behavioral Sciences at Stanford for its support.
263

264

D. KAHNEMAN AND A. TVERSKY

That is, the overall utility of a prospect, denoted by U, is the expected utility of
its outcomes.
(ii) Asset Integration: (xl, pl; . . . ;x,, p,) is acceptable at asset position w iff
U(w + X I ,p1;. . . ; w +x", p,)> u(w).
That is, a prospect is acceptable if the utility resulting from integrating theprospect with one's assets exceeds the utility of those assets alone. Thus, thedomain of the utility function is final states (which include one's asset position)rather than gains or losses.
Although the domain of the utility function is not limited to any particular classof consequences, most applications of the theory have been concerned with
monetary outcomes. Furthermore, most economic applications introduce thefollowing additional assumption.
(iii) Risk Aversion: u is concave (u"< 0).
A person is risk averse if he prefers the certain prospect (x) to any risky prospectwith expected value x. In expected utility theory, risk aversion is equivalent to theconcavity of the utility function. The prevalence'of risk aversion is perhaps thebest known generalization regarding risky choices. It led the early decision
theorists of the eighteenth century to propose that utility is a concave function ofmoney, and this idea has been retained in modern treatments (Pratt [33], Arrow[41).
In the following sections we demonstrate several phenomena which violatethese tenets of expected utility theory. The demonstrations are based on the
responses of students and university faculty to hypothetical choice problems. The
respondents were presented with problems of the type illustrated below.
Which of the following would you prefer?
A:

50°/o 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 theamounts involved, note that the median net monthly income for a family is about
3,000 Israeli pounds. The respondents were asked to imagine that they wereactually faced with the choice described in the problem, and to indicate thedecision 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,
and 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 adozen problems per booklet. Several forms of each questionnaire were constructed 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 positionof the prospects was reversed.
The problems described in this paper are selected illustrations of a series ofeffects. Every effect has been observed in several problems with different
outcomes and probabilities. Some of the problems have also been presented togroups of students and faculty at the University of Stockholm and at the265

PROSPECT THEORY

University of Michigan. The pattern of results was essentially identical to theresults obtained from Israeli subjects.
The reliance on hypothetical choices raises obvious questions regarding thevalidity of the method and the generalizability of the results. We are keenly awareof these problems. However, all other methods that have been used to test utility
theory 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 insuch contexts. Laboratory experiments have been designed to obtain precise
measures of utility and probability from actual choices, but these experimental
studies typically involve contrived gambles for small stakes, and a large number ofrepetitions of very similar problems. These features of laboratory gambling
complicate the interpretation of the results and restrict their generality.
By default, the method of hypothetical choices emerges as the simplest procedure by which a large number of theoretical questions can be investigated. The
use of the method relies on the assumption that people often know how theywould 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 arereasonably accurate in predicting their choices, the presence of common andsystematic violations of expected utility theory in hypothetical problems provides
presumptive evidence against that theory.

Certainty, Probability, and Possibility
In expected utility theory, the utilities of outcomes are weighted by theirprobabilities. The present section describes a series of choice problems in whichpeople's preferences systematically violate this principle. We first show thatpeople overweight outcomes that are considered certain, relative to outcomes
which are merely probable-a phenomenon which we label the certainty effect.
The best known counter-example to expected utility theory which ekploits thecertainty effect was introduced by the French economist Maurice Allais in 1953[2]. Allais' example has been discussed from both normative and descriptive
standpoints by many authors [28,38].The following pair of choice problems is avariation of Allais' example, which differs from the original in that it refers tomoderate 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 betweenA:

2,500 with probability.33,

2,400 with probability.66,

0 with probability

.01;

N=72

[I81

B:

2,400 with certainty.

266

D. KAMNEMAN AND A. TVERSKYPROBLEM2: Choose between
C: 2,500 with probability.33,

0 with probability

.67;

D:

2,400 with probability.34,

0 with probability.66.

The data show that 82 per cent of the subjects chose B in Problem 1, and 83 percent of the subjects chose C in Problem 2. Each of these preferences is significant
at the .O1 level, as denoted by the asterisk. Moreover, the analysis of individual
patterns of choice indicates that a majority of respondents (61 per cent) made themodal choice in both problems. This pattern of preferences violates expectedutility theory in the manner originally described by Allais. According to thattheory, with u(0) = 0, the first preference implies

while the second preference implies the reverse inequality. Note that Problem 2 isobtained from Problem 1by eliminating a .66 chance of winning 2400 from bothprospects. under consideration. Evidently, this change produces a greater reduction in desirability when it alters the character of the prospect from a sure gain to aprobable one, than when both the original and the reduced prospects are
uncertain.
A simpler demonstration of the same phenomenon, involving only twooutcome gambles is given below. This example is also based on Allais [2].

In this pair of problems as well as in all other problem-pairs in this section, overhalf the respondents violated expected utility theory. To show that the modal
pattern of preferences in Problems 3 and 4 is not compatible with the theory, setu(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 the prospect
C = (4,000, .20) can be expressed as (A, .25), while the prospect D = (3,000, .25)
can be rewritten as (B,.25). The substitution axiom of utility theory asserts that if
B is preferred to A, then any (probability) mixture (B, p) must be preferred to the
mixture (A, p). Our subjects did not obey this axiom. Apparently, reducing theprobability of winning from 1.0 to .25 has a greater effect than the reduction from

267

PROSPECT THEORY.8 to .2. The following pair of choice problems illustrates the certainty effect with
non-monetary outcomes.

A:

50% chance to win a threeweek tour of England,
France, and Italy;

C: 5% chance to win a threeweek tour of England,
France, and Italy;

N=72

[671*

B:

A one-week tour ofEngland, with certainty.

D:

10% chance to win a oneweek tour of England.
[331

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 followingproblems.

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 areminuscule (.002 and .001) in both prospects. In this situation where winning ispossible 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 thatcannot be captured by the expected utility model. The results suggest thefollowing empirical generalization concerning the manner in which the substitution axiom is violated. If ( y , pq) 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.

268

D. KAHNEMAN AND A. TVERSKYThe Reflection Effect
The previous section discussed preferences between positive prospects, i.e.,
prospects that involve no losses. What happens when the signs of the outcomes arereversed 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 section,and the right-hand column displays choice problems in which the signs of theoutcomes are reversed. We use -x to denote the loss of x, and > to denote theprevalent preference, i.e., the choice made by the majority of subjects.
TABLE I

Posltlve prospects

Problem 3:
N=95
Problem 4 :N = 95
Problem 7 :N=66
Problem 8:
N=66(4,000, .80) <(3,000).
[201(4,000, .20) > (3,000, .25).
[651*
[351
(3,000, .90) > (6,000, .45).
[86]*
[I41
(3,000, ,002) < (6,000, ,001).
~271
[731*

Negative prospects

Problem 3': (-4,000, .80) 1(-3,000).
N=95
[921*
[81
Problem 4': (-4,000, .20) < (-3,000, .25).
N=95
[421
[581
Problem 7 ' : (-3,000, .90) < (-6,000, .45).
N=66
[81
r921*
Problem 8': (-3,000, ,002) > (-6,000, ,001).
N=66
r701*
~301

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 0 reverses the preference order. We label thispattern 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 ofsubjects were willing to accept a risk of .80 to lose 4,000, in preference to a sureloss of 3,000, although the gamble has a lower expected value. The occurrence ofrisk seeking in choices between negative prospects was noted early by Markowitz[29]. Williams [48] reported data where a translation of outcomes produces adramatic shift from risk aversion to risk seeking. For example, his subjects wereindifferent between (100, .65; - 100, .35) and (O), indicating risk aversion. They
were also indifferent between (-200, .80) and (-loo), indicating risk seeking. A
recent review by Fishburn and Kochenberger [14] documents the prevalence ofrisk 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 corresponding negative prospects also violate the expectation principle in the samemanner. For example, Problems 3' and 4', like Problems 3 and 4, demonstrate thatoutcomes which are obtained with certainty are overweighted relative to
uncertain outcomes. In the positive domain, the certainty effect contributes to arisk 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 psychologicalprinciple-the overweighting of certainty-favors risk aversion in the domain ofgains 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 prevalentpreferences 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 expectedvalue. When the prospects are reduced, however, the difference in variance
between (3,000, .25) and (4,000, .20) may be insufficient to overcome thedifference 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 thenotion 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 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 topurchase insurance policies at a price that exceeds the expected actuarial cost?
However, an examination of the relative attractiveness of various forms of
insurance does not support the notion that the utility function for money isconcave everywhere. For example, people often prefer insurance programs thatoffer limited coverage with low or zero deductible over comparable policies thatoffer higher maximal coverage with higher deductibles-contrary to risk aversion(see, e.g., Fuchs [16]). Another type of insurance problem in which people's
responses are inconsistent with the concavity hypothesis may be called probabilistic insurance. To illustrate this concept, consider the following problem,
which was presented to 95 Stanford University students.

PROBLEM9: 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 newprogram called probabilistic insurance. In this program you pay half of the regularpremium. In case of damage, there is a 50 per cent chance that you pay the otherhalf 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 thelosses. For example, if an accident occurs on an odd day of the month, you pay theother 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 yourlosses 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:

N=95

Yes, No.
[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, andthe 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, reducing the probability of a loss from p to p/2 is less valuable than reducing theprobability of that loss from p/2 to 0.
In contrast to these data, expected utility theory (with a concave u) implies thatprobabilistic 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 theprobability 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

implies

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 thatwhich 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 regularinsurance, 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
vulnerable to many financial and other risks which his policies do not cover. Thereappears 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

27 1

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 areequated. Thus, two prospects that are equivalent in probabilities and outcomes
could have different values depending on their formulation. Several demonstrations of this general phenomenon are described in the next section.
The Isolation Effect
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 decompositions sometimes lead to different preferences. We refer to this phenomenon as
the isolation effect.
PROBLEM10: Consider the following two-stage game. In the first stage, there isa 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
between(4,000, .80)

and(3,000).

Your choice must be made before the game starts, i.e., before the outcome of thefirst stage is known.
Note that in this game, one has a choice between .25 x .80 = .20 chance to win
4,000, and a .25 x 1.0 = .25 chance to win 3,000. Thus, in terms of final outcomesand probabilities one faces a choice between (4,000, .20) and (3,000, .25), as inProblem 4 above. However, the dominant preferences are different in the two
problems. Of 141 subjects who answered Problem 10,78 per cent chose the latterprospect, contrary to the modal preference in Problem 4. Evidently, people
ignored 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 inProblem 3 above.
The standard and the sequential formulations of Problem 4 are represented asdecision trees in Figures 1 and 2, respectively. Following the usual convention,
squares 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 I), the decision maker faces a choice between two
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 adependency between the prospects without changing either probabilities or

D. KAHNEMAN A N D A. TVERSKYFIGURE 1.-The

FIGURE2.-The

representation of Problem 4 as a decision tree (standard formulation).

representation of Problem 10 as a decision tree (sequential formulation).

outcomes. Specifically, the event 'not winning 3,000' is included in the event 'notwinning 4,000' in the sequential formulation, while the two events are independent in the standard formulation. Thus, the outcome of winning 3,000 has acertainty advantage in the sequential formulation, which it does not have in thestandard formulation.
The reversal of preferences due to the dependency among events is particularly
significant because it violates the basic supposition of a decision-theoreticalanalysis, that choices between prospects are determined solely by the probabilities
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 theother hand, the following problem is most likely to be represented in thesequential form. One may invest money in a venture with some probability oflosing one's capital if the venture fails, and with a choice between a fixed agreed
return and a percentage of earnings if it succeeds. The isolation effect implies thatthe contingent certainty of the fixed return enhances the attractiveness of thisoption, relative to a risky venture with the same probabilities and outcomes.

273

PROSPECT THEORY

The preceding problem illustrated how preferences may be altered ky different
representations of probabilities. We now show how choices may be altered byvarying the representation of outcomes.
Consider the following problems, which were presented to two different groupsof subjects.
PROBLEM11: In addition to whatever you own, you have been given 1,000.
You are now asked to choose between

A: (1,000, .50),
N=70

and

B : (500).

[16]

[841*

PROBLEM12: In addition to whatever you own, you have been given 2,000.
You are now asked to choose betweenC:(-1,000, .50),

N=68

and

D : (-500).

[69*]

[311

The majority of subjects chose B in the first problem and C in the second. These
preferences conform to the reflection effect observed in Table I, which exhibits
risk aversion for positive prospects and risk seeking for negative ones. Note,however, that when viewed in terms of final states, the two choice problems are
identical. Specifically,A = (2,000, .50; 1,000, .50) = C,andB = (1,500) = D.

In fact, Problem 12 is obtained from Problem 11 by adding 1,000 to the initial
bonus, and subtracting 1,000 from all outcomes. Evidently, the subjects did not
integrate the bonus with the prospects. The bonus did not enter into thecomparison of prospects because it was common to both options in each problem.
The pattern of results observed in Problems 11 and 12 is clearly inconsistent withutility theory. In that theory, for example, the same utility is assigned to a wealthof $100,000, regardless of whether it was reached from a prior wealth of $95,000
or $105,000. Consequently, the choice between a total wealth of $100,000 andeven chances to own $95,000 or $105,000 should be independent of whether one
currently owns the smaller or the larger of these two amounts. With the added
assumption of risk aversion, the theory entails that the certainty of owning$100,000 should always be preferred to the gamble. However, the responses to
Problem 12 and to several of the previous questions suggest that this pattern will
be obtained if the individual owns the smaller amount, but not if he owns thelarger amount.
The apparent neglect of a bonus that was common to both options in Problems
11 and 12 implies that the carriers of value or utility are changes of wealth, rather
than final asset positions that include current wealth. This conclusion is thecornerstone of an alternative theory of risky cho'ice, whkh is described in thefollowing sections.

274

D. KAHNEMAN AND A. TVERSKY

3. THEORY
The preceding discussion reviewed several empirical effects which appear toinvalidate expected utility theory as a descriptive model. The remainder of thepaper presents an alternative account of individual decision making under risk,
called prospect theory. The theory is developed for simple prospects with
monetary outcomes and stated probabilities, but it can be extended to more
involved choices. Prospect theory distinguishes two phases in the choice process:an early phase of editing and a subsequent phase of evaluation. The editing phaseconsists of a preliminary analysis of the offered prospects, which often yields a
simpler representation of these prospects. In the second phase, the edited
prospects are evaluated and the prospect of highest value is chosen. We next
outline the editing phase, and develop a formal model of the evaluation phase.
The function of the editing phase is to organize and reformulate the options so
as to simplify subsequent evaluation and choice. Editing consists of the application of several operations that transform the outcomes and probabilities
associated with the offered prospects. The major operations of the editing phaseare described below.
Coding. The evidence discussed in the previous section shows that people
normally perceive outcomes as gains and losses, rather than as final states of
wealth or welfare. Gains and losses, of course, are defined relative to some neutral
reference point. The reference point usually corresponds to the current asset
position, in which case gains and losses coincide with the actual amounts that arereceived or paid. However, the location of the reference point, and theconsequent coding of outcomes as gains or losses, can be affected by theformulation of the offered prospects, and by the expectations of the decision
maker.
Combination. Prospects can sometimes be simplified by combining the probabilities associated with identical outcomes. For example, the prospect(200, .25; 200, .25) will be reduced to (200, .SO). and evaluated in this form.
Segregation. Some prospects contain a riskless component that is segregatedfrom the risky component in the editing phase. For example, the prospect(300, .80; 200, .20) is naturally decomposed into a sure gain of 200 and the risky
prospect (100, .80). Similarly, the prospect (-400, .40; -100, .60) is readily seen
to consist of a sure loss of 100 and of the prospect (-300, .40).
The preceding operations are applied to each prospect separately. The following operation is applied to a set of two or more prospects.
Cancellation. The essence of the isolation effects described earlier is thediscarding of components that are shared by the offered prospects. Thus, our
respondents apparently ignored the first stage of the sequential game presented in
Problem 10, because this stage was common to both options, and they evaluatedthe prospects with respect to the results of the second stage (see Figure 2).
Similarly, they neglected the common bonus that was added to the prospects in
Problems 11 and 12. Another type of cancellation involves the discarding ofcommon constituents, i.e., outcome-probability pairs. For example, the choicePROSPECT THEOKY

275

between (200, .20; 100, .50; -50, .30) and (200, .20; 150, .50; -100, .30) can be
reduced by cancellation to a choice between (100, .50;-50, .30) and(150, S O ; -100, .30).
Two additional operations that should be mentioned are simplification and thedetection of dominance. The first refers to the simplification of prospects byrounding probabilities or outcomes. For example, the prospect (101, .49) is likely
to be recoded as an even chance to win 100. A particularly important form ofsimplification involves the discarding of extremely unlikely outcomes. The second
operation involves the scanning of offered prospects to detect dominated alternatives, which are rejected without further evaluation.
Because the editing operations facilitate the task of decision, it is assumed thatthey are performed whenever possible. However, some editing operations either
permit or prevent the application of others. For example, (500, .20; 101, .49) willappear to dominate (500, .15; 99, .5 1) if the second constituents of both prospects
are simplified to (100, .50). The final edited prospects could, therefore, depend on
the sequence of editing operations, which is likely to vary with the structure of theoffered set and with the format of the display. A detailed study of this problem isbeyond the scope of the present treatment. In this paper we discuss choiceproblems where it is reasonable to assume either that the original formulation of
the prospects leaves no room for further editing, or that the edited prospects canbe specified without ambiguity.
Many anomalies of preference result from the editing of prospects. For example, the inconsistencies associated with the isolation effect result from the cancellation of common components. Some intransitivities of choice are explained by asimplification that eliminates small differences between prospects (see Tversky[43]). More generally, the preference order between prospects need not be
invariant across contexts, because the same offered prospect could be edited indifferent ways depending on the context in which it appears.
Following the editing phase, the decision maker is assumed to evaluate each of
the edited prospects, and to choose the prospect of highest value. The overallvalue of an edited prospect, denoted V, is expressed in terms of two scales, .rr
and v.
The first scale, .rr, associates with each probability p a decision weight .rr(p),which reflects the impact of p on the over-all value of the prospect. However, .rr is
not a probability measure, and it will be shown later that .rr(p)+.rr(l- p ) is
typically less than unity. The second scale, v, assigns to each outcome x a numberv ( x ) ,which reflects the subjective value of that outcome. Recall that outcomes aredefined relative to a reference point, which serves as the zero point of the value
scale. Hence, v measures the value of deviations from that reference point, i.e.,
gains and losses.
The present formulation is concerned with simple prospects of the form
(x,p; y, q ) , which have at most two non-zero outcomes. In such a prospect, one
receives x with probability p, y with probability q, and nothing with probability
1 - p - q, where p + q s 1. An offered prospect is strictly positive if its outcomesare all positive, i.e., if x, y > 0 and p + q = 1; it is strictly negative if its outcomes

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D. KAHNEMAN AND A. TVERSKY

are all negative. A prospect is regular if it is neither strictly positive nor strictly
negative.
The basic equation of the theory describes the manner in which 7~ and v are
combined to determine the over-all value of regular prospects.
If (x,p; y , q ) i s a r e g u l a r p r o s p e c t ( i . e . , e i t h e r p + q < 1 , o r x > O > y , o r x ~ O ~Y 1, then(1)

V(x, p ; Y, q) = .rr(p)u(x)+ .rr(q)u(y)where v(O)= O; ~ ( 0=)0, and ~ ( 1=) 1. As in utility theory, V is defined onprospects, while v is defined on outcomes. The two scales coincide for sureprospects, where V(x, 1.O) = V(x) = v (x).
Equation (1) generalizes expected utility theory by relaxing the expectation
principle. An axiomatic analysis of this representation is sketched in the Appendix, which describes conditions that ensure the existence of a unique .rr and aratio-scale v satisfying equation (1).
The evaluation of strictly positive and strictly negative prospects follows adifferent rule. In the editing phase such prospects are segregated into two
components: (i) the riskless component, i.e., the minimum gain or loss which iscertain to be obtained or paid; (ii) the risky component, i.e., the additional gain orloss which is actually at stake. The evaluation of such prospects is described in thenext equation.
I f p + q = l a n d e i t h e r x > y > O o r x < y < O , then(2)

V(x,e; Y, q ) = v ( ~ ) + . r r ( e ) [ v ( x ) - v ( ~ ) l .

That is, the value of a strictly positive or strictly negative prospect equals the valueof the riskless component plus the value-difference between the outcomes,
multiplied by the weight associated with the more extreme outcome. For example,
V(400, .25; 100, .75) = u(100)+.rr(.25)[~(400)-v(100)I. The essential featureof equation (2) is that a decision weight is applied to the value-differencev(x) - v (y), which represents the risky component of the prospect, but not to v (y),which represents the riskless component. Note that the right-hand side ofequation (2) equals .rr(p)v(x)+[l- .rr(p)]v(y). Hence, equation (2) reduces toequation (1) if ~ ( p+)~ (- p1) = 1. AS will be shown later, this condition is not
generally satisfied.
Many elements of the evaluation model have appeared in previous attempts tomodify expected utility theory. Markowitz [29] was the first to propose that utility
be defined on gains and losses rather than on final asset positions, an assumption
which has been implicitly accepted in most experimental measurements of utility(see, e.g., [7,32]). Markowitz also noted the presence of risk seeking in preferences among positive as well as among negative prospects, and he proposed autility function which has convex and concave regions in both the positive and thenegative domains. His treatment, however, retains the expectation principle;
hence it cannot account for the many violations of this principle; see, e.g., Table I.
The replacement of probabilities by more general weights was proposed by
Edwards [9], and this model was investigated in several empirical studies (e.g.,

PROSPECT THEORY

277[3,42]). Similar models were developed by Fellner [12], who introduced theconcept of decision weight to explain aversion for ambiguity, and by van Dam [46]
who attempted to scale decision weights. For other critical analyses of expectedutility theory and alternative choice models, see Allais [2], Coombs [6], Fishburn[13], and Hansson [22].
The equations of prospect theory retain the general bilinear form that underlies
expected utility theory. However, in order to accomodate the effects described in
the first part of the paper, we are compelled to assume that values are attached tochanges rather than to final states, and that decision weights do not coincide withstated probabilities. These departures from expected utility theory must lead tonormatively unacceptable consequences, such as inconsistencies, intransitivities,
and violations of dominance. Such anomalies of preference are normally corrected by the decision maker when he realizes that his preferences are inconsistent, intransitive, or inadmissible. In many situations, however, the decision
maker does not have the opportunity to discover that his preferences could violate
decision rules that he wishes to obey. In these circumstances the anomalies
implied by prospect theory are expected to occur.

The Value Function
An essential feature of the present theory is that the carriers of value arechanges in wealth or welfare, rather than final states. This assumption is compatible with basic principles of perception and judgment. Our perceptual apparatus is
attuned to the evaluation of changes or differences rather than to the evaluation ofabsolute magnitudes. When we respond to attributes such as brightness, loudness,or temperature, the past and present context of experience defines an adaptationlevel, or reference point, and stimuli are perceived in relation to this reference
point [23]. Thus, an object at a given temperature may be experienced as hot orcold to the touch depending on the temperature to which one has adapted. The
same principle applies to non-sensory attributes such as health, prestige, andwealth. The same level of wealth, for example, may imply abject poverty for one
person and great riches for another--depending on their current assets.
The emphasis on changes as the carriers of value should not be taken to imply
that the value of a particular change is independent of initial position. Strictly
speaking, value should be treated as a function in two arguments: the asset
position that serves as reference point, and the magnitude of the change (positiveor negative) from that reference point. An individual's attitude to money, say,could be described by a book, where each page presents the value function forchanges at a particular asset position. Clearly, the value functions described on
different pages are not identical: they are likely to become more linear withincreases in assets. However, the preference order of prospects is not greatly
altered by small or even moderate variations in asset position. The certainty
equivalent of the prospect (1,000, .50), for example, lies between 300 and 400 for
most people, in a wide range of asset positions. Consequently, the representation

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D. KAHNEMAN AND A. TVERSKY

of value as a function in one argument generally provides a satisfactory approximation.
Many sensory and perceptual dimensions share the property that the psychological response is a concave function of the magnitude of physical change. For
example, it is easier to discriminate between a change of 3" and a change of 6" inroom temperature, than it is to discriminate between a change of 13" and a changeof 16". We propose that this principle applies in particular to the evaluation ofmonetary changes. Thus, the difference in value between a gain of 100 and a gainof 200 appears to be greater than the difference between a gain of 1,100 and a gainof 1,200. Similarly, the difference between a loss of 100 and a loss of 200 appears
greater than the difference between a loss of 1,100 and a loss of 1,200, unless thelarger loss is intolerable. Thus, we hypothesize that the value function for changes
of wealth is normally concave above the reference point ( v U ( x<) 0 , for x > 0 ) andoften convex below it ( v V ( x>
) 0 , for x <O). That is, the marginal value of bothgains and losses generally decreases with their magnitude. Some support for thishypothesis has been reported by Galanter and Pliner [17],who scaled theperceived magnitude of monetary and non-monetary gains and losses.
The above hypothesis regarding the shape of the value function was based on
responses to gains and losses in a riskless context. We propose that the valuefunction which is derived from risky choices shares the same characteristics, as
illustrated in the following problems.

Applying equation 1 to the modal preference in these problems yields~ ( . 2 5 ) v ( 6 , 0 0<0 )~ ( . 2 5 ) [ v ( 4 , 0 0+0v(2,000)])and~ ( . 2 5 ) v ( - 6 , 0 0 0>) ~ ( . 2 5 ) [(-4,000)v+ v (-2,OOO)l.
Hence, ~ ( 6 , 0 0 0<) ~ ( 4 , 0 0 0+)~ ( 2 , 0 0 0and) v(-6,000) > v(-4,000) + v(-2,000).
These preferences are in accord with the hypothesis that the value function isconcave for gains and convex for losses.
Any discussion of the utility function for money must leave room for the effectof special circumstances on preferences. For example, the utility function of anindividual who needs $60,000 to purchase a house may reveal an exceptionallysteep rise near the critical value. Similarly, an individual's aversion to losses may
increase sharply near the loss that would compel him to sell his house and move toPROSPECT THEORY

279

a less desirable neighborhood. Hence, the derived value (utility) function of anindividual does not always reflect "pure" attitudes to money, since it could be
affected by additional consequences associated with specific amounts. Suchperturbations can readily produce convex regions in the value function for gains
and concave regions in the value function for losses. The latter case may be
more common since large losses often necessitate changes in life style.
A salient characteristic of attitudes to changes in welfare is that losses loom
larger than gains. The aggravation that one experiences in losing a sum of money
appears to be greater than the pleasure associated with gaining the same amount[17].Indeed, most people find symmetric bets of the form (x, SO; -x, .50)
distinctly unattractive. Moreover, the aversiveness of symmetric fair bets
generally increases with the size of the stake. That is, if x > y 2 0 , then(y, SO; -y, SO) is preferred to (x, SO; -x, .50). According to equation (I),therefore,v(y)+v(-y)>v(x)+v(-x)and

v(-y)-u(-x)>v(x)-v(~).

Setting y = 0 yields v(x) < -v(-x), and letting y approach x yields vl(x) <v'(-x), provided v', the derivative of v, exists. Thus, the value function for losses is
steeper than the value function for gains.
In summary, we have proposed that the value function is (i) defined on
deviations from the reference point; (ii) generally concave for gains and commonly convex for losses; (iii) steeper for losses than for gains. A value function
which satisfies these properties is displayed in Figure 3. Note that the proposed
S-shaped value function is steepest at the reference point, in marked contrast to
the utility function postulated by Markowitz [29] which is relatively shallow in that
region.

VALUE

FIGURE3.-A hypothetical value function.

280

D. KAHNEMAN AND A. TVERSKY

Although the present theory can be applied to derive the value function frompreferences between prospects, the actual scaling is considerably more complicated than in utility theory, because of the introduction of decision weights. For
example, decision weights could produce risk aversion and risk seeking even witha linear value function. Nevertheless, it is of interest that the main properties
ascribed to the value function have been observed in a detailed analysis of von
Neumann-Morgenstern utility functions for changes of wealth (Fishburn andKochenberger [14]). The functions had been obtained from thirty decision makers
in various fields of business, in five independent studies [5,18,19,21,40].Most
utility functions for gains were concave, most functions for losses were convex,
and only three individuals exhibited risk aversion for both gains and losses. With asingle exception, utility functions were considerably steeper for losses than forgains.
The Weighting Function
In prospect theory, the value of each outcome is multiplied by a decision weight.
Decision weights are inferred from choices between prospects much as subjective
probabilities are inferred from preferences in the Ramsey-Savage approach.
However, decision weights are not probabilities: they do not obey the probability
axioms and they should not be interpreted as measures of degree or belief.
Consider a gamble in which one can win 1,000 or nothing, depending on the tossof a fair coin. For any reasonable person, the probability of winning is .50 in thissituation. This can be verified in a variety of ways, e.g., by showing that the subject
is indifferent between betting on heads or tails, or by his verbal report that heconsiders the two events equiprobable. As will be shown below, however, thedecision weight ~ ( . 5 0which)
is derived from choices is likely to be smaller than.50. Decision weights measure the impact of events on the desirability of prospects, and not merely the perceived likelihood of these events. The two scales
coincide (i.e., ~ ( p=)p) if the expectation principle holds, but not otherwise.
The choice problems discussed in the present paper were formulated in terms of
explicit numerical probabilities, and our analysis assumes that the respondents
adopted the stated values of p. Furthermore, since the events were identified onlyby their stated probabilities, it is possible in this context to express decision
weights as a function of stated probability. In general, however, the decision
weight attached to an event could be influenced by other factors, e.g., ambiguity[lo, 111.
We turn now to discuss the salient properties of the weighting function T , which
relates decision weights to stated probabilities. Naturally, T is an increasing
function of p, with ~ ( 0=)0 and ~ ( 1=) 1. That is, outcomes contingent on an
impossible event are ignored, and the scale is normalized so that ~ ( pis) the ratioof the weight associated with the probability p to the weight associated with thecertain event.
We first discuss some properties of the weighting function for small probabilities. The preferences in Problems 8 and 8' suggest that for small values of p, T

PROSPECT THEORY

281

is a subadditive function of p, i.e., .rr(rp)>r.rr(p) for 0 < r < I . Recall that in
Problem 8, (6,000, .001) is preferred to (3,000,.002). Hence.rr(.001)> ~ ( 3 , 0 0 0 ) 1
a(.002) v (6,000)>2

by the concavity of v.

The reflected preferences in Problem 8' yield the same conclusion. The pattern ofpreferences in Problems 7 and 7', however, suggests that subadditivity need not
hold for large values of p.
Furthermore, we propose that very low probabilities are generally overweighted, that is, ~ ( p> p) for small p. Consider the following choice problems.

Note that in Problem 14, people prefer what is in effect a lottery ticket over the
expected value of that ticket. In Problem 14', on the other hand, they prefer asmall loss, which can be viewed as the payment of an insurance premium, over asmall probability of a large loss. Similar observations have been reported byMarkowitz [29]. In the present theory, the preference for the lottery in Problem
14 implies .rr(.001)~(5,000)>v(5), hence .rr(.001)> v(5)/v(5,000)>.001,
assuming the value function for gains is concave. The readiness to pay forinsurance in Problem 14' implies the same conclusion, assuming the valuefunction for losses is convex.
It is important to distinguish overweighting, which refers to a property ofdecision weights, from the overestimation that is commonly found in the assessment of the probability of rare events. Note that the issue of overestimation does
not arise in the present context, where the subject is assumed to adopt the statedvalue of p. In many real-life situations, overestimation and overweighting may
both operate to increase the impact of rare events.
Although .rr(p)>p for low probabilities, there is evidence to suggest that, for all
0 <p < 1,~ ( p+ ~) (- p1) < 1. We label this property subcertainty. It is readily
seen that the typical preferences in any version of Allias' example (see, e.g.,
Problems 1 and 2) imply subcertainty for the relevant value of p. Applying

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D. KAHNEMAN AND A. TVERSKY

equation ( 1 )to the prevalent preferences in Problems 1 and 2 yields, respectively,[ I - r ( . 6 6 ] ~ ( 2 , 4 0 0 )r>( . 3 3 ) ~ ( 2 , 5 0 0 ) and7 4 . 3 3 )(2,500)~> r ( . 3 4 ) ~ ( 2 , 4 0 0 ) ; hence,
Applying the same analysis to Allais' original example yields r ( . 8 9 )+ r ( . l l )< 1,
and some data reported by MacCrimmon and Larsson [28]imply subcertainty for
additional values of p.
The slope of r in the interval ( 0 , 1) can be viewed as a measure of the sensitivityof preferences to changes in probability. Subcertainty entails that r is regressivewith respect to p, i.e., that preferences are generally less sensitive to variations ofprobability than the expectation principle would dictate. Thus, subcertainty
captures an essential element of people's attitudes to uncertain events, namelythat the sum of the weights associated with complementary events is typically lessthan the weight associated with the certain event.
Recall that the violations of the substitution axiom discussed earlier in thispaper conform to the following rule: If (x,p) is equivalent to ( y ,pq) then ( x ,pr) is
not preferred to ( y ,pqr), 0 <p, q, r s 1. By equation ( I ) ,r ( p ) v( x ) = r ( p q ) v (y )impliesr ( p r ) v ( x ) r ( p q r ) v( Y 1;

hence,

Thus, for a fixed ratio of probabilities, the ratio of the corresponding decision
weights is closer to unity when the probabilities are low than when they are high.
This property of r,called subproportionality, imposes considerable constraints on
the shape of r :it holds if and only if log r is a convex function of log p.
It is of interest to note that subproportionality together with the overweightingof small probabilities imply that r is subadditive over that range. Formally, it canbe shown that if r ( p )> p and subproportionality holds, then r ( r p )> r r ( p ) ,0 <r < 1, provided r is monotone and continuous over ( 0 , l ) .
Figure 4 presents a hypothetical weighting function which satisfies overweighting and subadditivity for small values of p, as well as subcertainty and subproportionality. These properties entail that r is relatively shallow in the open
interval and changes abruptly near the end-points where r ( 0 )= 0 and r ( 1 )= 1.
The sharp drops or apparent discontinuities of r at the endpoints are consistent
with the notion that there is a limit to how small a decision weight can be attached
to an event, if it is given any weight at all. A similar quantum of doubt could
impose an upper limit on any decision weight that is less than unity. This quanta1
effect may reflect the categorical distinction between certainty and uncertainty.
On the other hand, the simplification of prospects in the editing phase can lead theindividual to discard events of extremely low probability and to treat events ofextremely high probability as if they were certain. Because people are limited in

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PROSPECT THEORY

their ability to comprehend and evaluate extreme probabilities, highly unlikely
events are either ignored or overweighted, and the difference between highprobability and certainty is either neglected or exaggerated. Consequently, T is
not well-behaved near the end-points.

0

.5

1.0

STATED PROBABILITY: p

FIGURE4.-A hypothetical weighting function.

The following example, due to Zeckhauser, illustrates the hypothesized
nonlinearity of T.Suppose you are compelled to play Russian roulette, but are
given the opportunity to purchase the removal of one bullet from the loaded gun.
Would you pay as much to reduce the number of bullets from four to three as you
would to reduce the number of bullets from one to zero? Most people feel thatthey would be willing to pay much more for a reduction of the probability of deathfrom 1/6 to zero than for a reduction from 416 to 3/6. Economic considerations
would lead one to pay more in the latter case, where the value of money
is presumably reduced by the considerable probability that one will not live toenjoy it.
An obvious objection to the assumption that ~ ( pf jp involves comparisons
between prospects of the form ( x , p ; x, q ) and ( x , p'; x, q ' ) , where p + q = p' + q ' <1 . Since any individual will surely be indifferent between the two prospects, it
could be argued that this observation entails ~ ( p+)~ ( q=)~ ( p '+)~ ( q ' )which,inturn implies that T is the identity function. This argument is invalid in the present
theory, which assumes that the probabilities of identical outcomes are combinedin the editing of prospects. A more serious objection to the nonlinearity of T
involves potential violations of dominance. Suppose x > y > 0, p >p', and p + q =p' + q' < 1; hence, ( x , p ; y, q ) dominates ( x , p ' ; y, q ' ) . If preference obeys

284

D. KAHNEMAN AND A. TVERSKY

dominance, then.rr(p)v(x)+.rr(q)v(y)> .rr(pf)v(x)+.rr(q')4y),

Hence, as y approaches x , ~ ( p-).rr(pf)approaches .rr(qt)- ~ ( q )Since.
p -p' =
q'- q, .rr must be essentially linear, or else dominance must be violated.
Direct violations of dominance are prevented, in the present theory, by theassumption that dominated alternatives are detected and eliminated prior to theevaluation of prospects. However, the theory permits indirect violations ofdominance, e.g., triples of prospects so that A is preferred to B, B is preferred toC, and C dominates A. For an example, see Raiffa [34, p. 751.
Finally, it should be noted that the present treatment concerns the simplest
decision task in which a person chooses between two available prospects. We havenot treated in detail the more complicated production task (e.g., bidding) where
the decision maker generates an alternative that is equal in value to a given
prospect. The asymmetry between the two options in this situation could introduce systematic biases. Indeed, Lichtenstein and Slovic [27] have constructedpairs of prospects A and B, such that people generally prefer A over B, but bid
more for B than for A. This phenomenon has been confirmed in several studies,with both hypothetical and real gambles, e.g., Grether and Plott [20]. Thus, itcannot be generally assumed that the preference order of prospects can berecovered by a bidding procedure.
Because prospect theory has been proposed as a model of choice, the inconsistency of bids and choices implies that the measurement of values and decision
weights should be based on choices between specified prospects rather than onbids or other production tasks. This restriction makes the assessment of v and .rr
more difficult because production tasks are more convenient for scaling than paircomparisons.
4 . DISCUSSION

In the final section we show how prospect theory accounts for observed
attitudes toward risk, discuss alternative representations of choice problems
induced by shifts of reference point, and sketch several extensions of the present
treatment.
Risk Attitudes
The dominant pattern of preferences observed in Allais' example (Problems 1
and 2) follows from the present theory iff

PROSPECT THEORY

285

Hence, the violation of the independence axiom is attributed in this case to
subcertainty, and more specifically to the inequality ~ ( . 3 4<) 1- ~ ( . 6 6 )This.
analysis shows that an Allais-type violation will occur whenever the v-ratio of thetwo non-zero outcomes is bounded by the corresponding T - ratios.
Problems 3 through 8 share the same structure, hence it suffices to consider one
pair, say Problems 7 and 8. The observed choices in these problems are implied by
the theory iff

The violation of the substitution axiom is attributed in this case to the subproportionality of T . Expected utility theory is violated in the above manner,
therefore, whenever the v-ratio of the two outcomes is bounded by the respective
T-ratios. The same analysis applies to other violations of the substitution axiom,
both in the positive and in the negative domain.
We next prove that the preference for regular insurance over probabilistic
insurance, observed in Problem 9, follows from prospect theory-provided theprobability of loss is overweighted. That is, if ( - x , p ) is indifferent to ( - y ) , then
( - y ) is preferred to ( - x , p / 2 ; - y , p / 2 ; - y / 2 , l - p ) . For simplicity, we define forx z 0, f ( x ) = - v ( - x ) . Since the value function for losses is convex, f is a concave
function of x. Applying prospect theory, with the natural extension of equation 2 ,we wish to show that

Substituting for f ( x ) and using the concavity off, it suffices to show that

7 r ( p ) / 26 7 r ( p / 2 ) ,which follows from the subadditivity of T .

According to the present theory, attitudes toward risk are determined jointly by
v and T , and not solely by the utility function. It is therefore instructive to examine
the conditions under which risk aversion or risk seeking are expected to occur.
Consider the choice between the gamble ( x , p ) and its expected value ( p x ) . If
x > 0, risk seeking is implied whenever ~ ( p>)v ( p x ) / v ( x ) ,which is greater than p
if the value function for gains is concave. Hence, overweighting ( 7 r ( p ) > p ) is
necessary but not sufficient for risk seeking in the domain of gains. Precisely thesame condition is necessary but not sufficient for risk aversion when x < 0. This
analysis restricts risk seeking in the domain of gains and risk aversion in thedomain of losses to small probabilities, where overweighting is expected to hold.

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D. KAHNEMAN AND A. TVERSKY

Indeed these are the typical conditions under which lottery tickets and insurance
policies are sold. In prospect theory, the overweighting of small probabilities
favors both gambling and insurance, while the S-shaped value function tends toinhibit both behaviors.
Although prospect theory predicts both insurance and gambling for small
probabilities, we feel that the present analysis falls far short of a fully adequate
account of these complex phenomena. Indeed, there is evidence from bothexperimental studies [37],survey research [26],and observations of economic
behavior, e.g., service and medical insurance, that the purchase of insurance often
extends to the medium range of probabilities, and that small probabilities ofdisaster are sometimes entirely ignored. Furthermore, the evidence suggests thatminor changes in the formulation of the decision problem can have marked effects
on the attractiveness of insurance (371.A comprehensive theory of insurance
behavior should consider, in addition to pure attitudes toward uncertainty andmoney, such factors as the value of security, social norms of prudence, theaversiveness of a large number of small payments spread over time, informationand misinformation regarding probabilities and outcomes, and many others.
Some effects of these variables could be described within the present framework,e.g., as changes of reference point, transformations of the value function, or
manipulations of probabilities or decision weights. Other effects may require theintroduction of variables or concepts which have not been considered in thistreatment.
Shifts of Reference
So far in this paper, gains and losses were defined by the amounts of money thatare obtained or paid when a prospect is played, and the reference point was taken
to be the status quo, or one's current assets. Although this is probably true formost choice problems, there are situations in which gains and losses are coded
relative to an expectation or aspiration level that differs from the status quo. For
example, an unexpected tax withdrawal from a monthly pay check is experienced as a loss, not as a reduced gain. Similarly, an entrepi-eneur who is weathering aslump with greater success than his competitors may interpret a small loss as again, relative to the larger loss he had reason to expect.
The reference point in the preceding examples corresponded to an asset
position that one had expected to attain. A discrepancy between the reference
point and the current asset position may also arise because of recent changes in
wealth to which one has not yet adapted [29].Imagine a person who is involved ina business venture, has already lost 2,000 and is now facing a choice between asure gain of 1,000 and an even chance to win 2,000 or nothing. If he has not yet
adapted to his losses, he is likely to code the problem as a choice between(-2,000, .50) and (-1,000) rather than as a choice between (2,000, .50) and(1,000). As we have seen, the former representation induces more adventurous
choices than the latter.
A change of reference point alters the preference order for prospects. In
particular, the present theory implies that a negative translation of a choicePROSPECT T H E O R Y

287

problem, such as arises from incomplete adaptation to recent losses, increases risk
seeking in some situations. Specifically, if a risky prospect ( x , p ; - y, 1 - p ) is justacceptable, then ( x - z , p ; - y - z, 1- p ) is preferred over ( - z ) for x, y, z >0 , with x > z .
To prove this proposition, note thatV ( x ,p ; y, 1- p ) = 0

iff

T ( ~ ) v ( x=) - T ( I - P ) v ( - Y ) .

Furthermore,+ ~ ( 1 p-) v ( - Z )by the properties of v ,+ ~ (- pl) v ( - z )by substitution,> v ( - z ) [ ~ ( p+) ~ ( - p1) ]>v ( - z )since v ( - z ) < - v ( z ) ,by subcertainty.

This analysis suggests that a person who has not made peace with his losses is likely
to accept gambles that would be unacceptable to him otherwise. The well known
observation [31] that the tendency to bet on long shots increases in the course of
the betting day provides some support for the hypothesis that a failure to adapt tolosses or to attain an expected gain induces risk seeking. For another example,
consider an individual who expects to purchase insurance, perhaps because he has
owned it in the past or because his friends do. This individual may code thedecision to pay a premium y to protect against a loss x as a choice between( - x + y, p ; y, 1 - p ) and (0)rather than as a choice between ( - x , p ) and ( - y ) . The
preceding argument entails that insurance is likely to be more attractive in theformer representation than in the latter.
Another important case of a shift of reference point arises when a person
formulates his decision problem in terms of final assets, as advocated in decision
analysis, rather than in terms of gains and losses, as people usually do. In thiscase, the reference point is set to zero on the scale of wealth and the value function
is likely to be concave everywhere [39]. According to the present analysis, thisformulation essentially eliminates risk seeking, except for gambling with low
probabilities. The explicit formulation of decision problems in terms of final assets
is perhaps the most effective procedure for eliminating risk seeking in the domainof losses.

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D. KAHNEMAN AND A. TVERSKY

Many economic decisions involve transactions in which one pays money inexchange for a desirable prospect. Current decision theories analyze such problems as comparisons between the status quo and an alternative state which
includes the acquired prospect minus its cost. For example, the decision whether
to pay 10 for the gamble (1,000, .01) is treated as a choice between(990, .01; -10, .99) and (0). In this analysis, readiness to purchase the positive
prospect is equated to willingness to accept the corresponding mixed prospect.
The prevalent failure to integrate riskless and risky prospects, dramatized in theisolation effect, suggests that people are unlikely to perform the operation ofsubtracting the cost from the outcomes in deciding whether to buy a gamble.
Instead, we suggest that people usually evaluate the gamble and its cost
separately, and decide to purchase the gamble if the combined value is positive.
Thus, the gamble (1,000, .01) will be purchased for a price of 10 if .rr(.01)v(1,000)+v(-10)>0.
If this hypothesis is correct, the decision to pay 10 for (1,000, .01), for example,
is no longer equivalent to the decision to accept the gamble (990, .01; -10, .99).
Furthermore, prospect theory implies that if one is indifferent between ( x ( 1 p), p ; -px, 1 - p ) and ( 0 )then one will not pay px to purchase the prospect ( x ,p).
Thus, people are expected to exhibit more risk seeking in deciding whether toaccept a fair gamble than in deciding whether to purchase a gamble for a fair price.
The location of the reference point, and the manner in which'choice problems arecoded and edited emerge as critical factors in the analysis of decisions.

Extensions
In order to encompass a wider range of decision problems, prospect theory
should be extended in several directions. Some generalizations are immediate;
others require further development. The extension of equations ( 1 ) and ( 2 ) toprospects with any number of outcomes is straightforward. When the number ofoutcomes is large, however, additional editing operations may be invoked tosimplify evaluation. The manner in which complex options, e.g., compoundprospects, are reduced to simpler ones is yet to be investigated.
Although the present paper has been concerned mainly with monetary
outcomes, the theory is readily applicable to choices involving other attributes,
e.g., quality of life or the number of lives that could be lost or saved as aconsequence of a policy decision. The main properties of the proposed valuefunction for money should apply to other attributes as well. In particular, we
expect outcomes to be coded as gains or losses relative to a neutral reference
point, and losses to loom larger than gains.
The theory can also be extended to the typical situation of choice, where theprobabilities of outcomes are not explicitly given. In such situations, decision
weights must be attached to particular events rather than to stated probabilities,
but they are expected to exhibit the essential properties that were ascribed to theweighting function. For example, if A and B are complementary events andneither is certain, .rr(A)+ v ( B ) should be less than unity-a natural analogue tosubcertainty.

PROSPECT THEORY

289

The decision weight associated with an event will depend primarily on theperceived likelihood of that event, which could be subject to major biases [45]. In
addition, decision weights may be affected by other considerations, such as
ambiguity or vagueness. Indeed, the work of Ellsberg [lo] and Fellner [12] implies
that vagueness reduces decision weights. Consequently, subcertainty should be
more pronounced for vague than for clear probabilities.
The present analysis of preference between risky options has developed twothemes. The first theme concerns editing operations that determine how prospects
are perceived. The second theme involves the judgmental principles that govern
the evaluation of gains and losses and the weighting of uncertain outcomes.
Although both themes should be developed further, they appear to provide a
useful framework for the descriptive analysis of choice under risk.

The University of British Columbia
andStanford University
Manuscript received November, 1977;final revision received March, 1978.

APPENDIX'
In this appendix we sketch an axiomatic analysis of prospect theory. Since a complete self-contained
treatment is long and tedious, we merely outline the essential steps and exhibit the key ordinal
properties needed to establish the bilinear representation of equation (1). Similar methods could be
extended to axiomatize equation (2).
Consider the set of all regular prospects of the form (x, p; y, q) with p + q < 1. The extension toregular prospects with p + q = 1 is straightforward. Let 2 denote the relation of preference between
prospects that is assumed to be connected, symmetric and transitive, and let = denote the associatedrelation of indifference. Naturally, (x, p; y, q ) = (y, q ; x, p). We also assume, as is implicit in ournotation, that (x, p ; 0, q) = (x, p; 0, r), and (x, p; y, 0) = (x, p; z, 0). That is, the null outcome and theimpossible event have the property of a multiplicative zero.
Note that the desired representation (equation (1))is additive in the probability-outcome pairs.
Hence, the theory of additive conjoint measurement can be applied to obtain a scale V which
preserves the preference order, and interval scales f and g in two arguments such that
V(x, p; y, q ) = f ( x , p)+g(y, q).
The key axioms used to derive this representation are:Independence: (x, p; y, q ) 2 (x, p; y'q') iff (x', p'; y, q ) > (x', p'; y', q').
Cancellation: If (x, p; y'q') e (x', p'; y, q ) and (x', p'; y", q") ~ ( x "p";, y', q'), then (x, p; yo, q") ?
(x", PI'; Y , q).
Solvabil~ty:If (x, p; y, q)? ( z , r ) Z (x, p; y' q') for some outcome z and probability r, then there existy", q" such that(x. P; y"qs) = ( z , r).
It has been shown that these conditions are sufficient to construct the desired additive representation, provided the preference order is Archimedean [8,25]. Furthermore, since (x, p; y, q ) =(Y,4; x, PI, f(x, p)+g(y, q) =f(y, q)+g(x, PI, and letting q = 0 yields f = g.
Next, consider the set of all prospects of the form (x, p) with a single non-zero outcome. In this case,the bilinear model reduces to V(x, p) = v(p)v(x).This is the multiplicative model, investigated in [35jand [25]. To construct the multiplicative representation we assume that the ordering of the probability-outcome pairs satisfies independence, cancellation, solvability, and the Archimedean axiom. In
addition, we assume sign dependence [25] to ensure the proper multiplication of signs. It should be
noted that the solvability axiom used in [35] and [25] must be weakened because the probability factor
permits only bounded solvability.
We are indebted to David H. Krantz for his help in the formulation of this section.

'

290

D. KAHNEMAN AND A. TVERSKY

Combining the additive and the multiplicative representations yields
V i x , p; Y, q ) = f l ~ ( p ) v ( x ) l + f 1 m ( q ) ~ ( ~ ) l .
Finally, we impose a new distributivity axiom:( x ~ P ; Y , P ) = ( z , P ) iff(x,q;y,q)=(z,q).

Applying this axiom to the above representation, we obtainf [ . r ( p ) v ( x ) l + f [ . r r ( ~ ) v (=f[.rr(p)v(z)l~)l

implies

f[dq)v(x)l+f[.rr(s)v(y)l= f l d q ) v ( z ) l .
Assuming, with no loss of generality, that ~ ( q<)~ ( p )and, letting a = .rr(p)v(x),p = ~ ( p ) v ( y ) ,y = a ( p ) v ( z ) . and 6 = r ( q ) / r r ( p ) ,yields f ( a ) +f ( @ )= f ( y ) implies f ( B a ) +f ( 6 p ) = f ( 6 y ) for all0<6<1.
Because f is strictlv monotonic we can set y = f ' [ f ( a ) +f ( p ) ] . Hence, By = 6 f 1 [ f ( a ) + f ( p ) 1 =rl[fcea)+'f(?p)l. The solutlon to this functional equation is f ( a ) = k a c [ I ] . Hence, V ( x ,p; y, q ) =k [ n ( p ) v ( x ) y+ k [ ~ ( q ) vy)y,( for some k , c > 0 .The desired bilinearform is obtained by redefining thescales T , v, and V so as to absorb the constants k and c.
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Prospect Theory: An Analysis of Decision under Risk
Daniel Kahneman; Amos Tversky
Econometrica, Vol. 47, No. 2. (Mar., 1979), pp. 263-292.
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