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This is the most cited paper ever to appear in Econometrica, a top ...

Daniel Kahneman and Amos Tversky famously wrote Prospect Theory. Wi...

Expected utility is one of the first theories of decision making an...

The axioms of expected utility theory that the authors refer to her...

Kahneman and Tversky selected a highly educated group of university...

Utility and preferences are difficult to measure. In an ideal exper...

This is a violation of expected utility theory and introduces the c...

In Problem 2, the expected utility of Choice C (825) is greater tha...

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ECONOMETRICA

PROSPECT THEORY: AN ANALYSIS OF DECISION UNDER RISK

This paper presents a critique of expected utility theory as a descriptive model of

decision making under risk, and develops an alternative model, called prospect theory.

Choices among risky prospects exhibit several pervasive effects that are inconsistent with

the basic tenets of utility theory. In particular, people underweight outcomes that are

merely probable in comparison with outcomes that are obtained with certainty. This

tendency, called the certainty effect, contributes to risk aversion in choices involving sure

gains 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 is

presented 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 prob-

abilities. Overweighting of low probabilities may contribute to the attractiveness of both

insurance and gambling.

INTRODUCTION

EXPECTED

UTILITY

THEORY

has dominated the analysis of decision making under

risk. 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 the

theory

[47,36],

and that most people actually do, most of the time.

The present paper describes several classes of choice problems in which

preferences systematically violate the axioms of expected utility theory. In the

light of these observations we argue that utility theory, as it is commonly

interpreted and applied, is not an adequate descriptive model and we propose an

alternative account of choice under risk.

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 with

probability pi, where pl

+pz+

+p,

To simplify notation, we omit null

outcomes and use (x, p) to denote the prospect (x, p;

-p) that yields x with

probability p and

with probability

-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 based

on the following three tenets.

(i) Expectation: U(x1, p~;

. .

;

x,, p,) =plu(xl)+

. . +p,u(x,).

This work was supported in part by grants from the Harry

Guggenheim Foundation and from

the Advanced Research Projects Agency of the Department of Defense and was monitored by Office

of 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

KAHNEMAN

AND A.

TVERSKY

That is, the overall utility of a prospect, denoted by

is the expected utility of

its outcomes.

(ii) Asset Integration: (xl, pl;

. .

;

x,, p,) is acceptable at asset position w iff

U(w +XI, p1;.

. .

;

w +x", p,)> u(w).

That is, a prospect is acceptable if the utility resulting from integrating the

prospect with one's assets exceeds the utility of those assets alone. Thus, the

domain 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 class

of consequences, most applications of the theory have been concerned with

monetary outcomes. Furthermore, most economic applications introduce the

following additional assumption.

(iii) Risk Aversion: u is concave

(u"< 0).

A person is risk averse if he prefers the certain prospect (x) to any risky prospect

with expected value

In expected utility theory, risk aversion is equivalent to the

concavity of the utility function. The prevalence'of risk aversion is perhaps the

best known generalization regarding risky choices. It led the early decision

theorists of the eighteenth century to propose that utility is a concave function of

money, and this idea has been retained in modern treatments (Pratt

[33], Arrow

[41).

In the following sections we demonstrate several phenomena which violate

these 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?

50°/o chance to win 1,000,

450 for sure.

50% chance to win nothing;

The outcomes refer to Israeli currency. To appreciate the significance of the

amounts 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 were

actually faced with the choice described in the problem, and to indicate the

decision they would have made in such a case. The responses were anonymous,

and the instructions specified that there was no 'correct' answer to such problems,

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 a

dozen problems per booklet. Several forms of each questionnaire were con-

structed so that subjects were exposed to the problems in different orders. In

addition, two versions of each problem were used in which the left-right position

of the prospects was reversed.

The problems described in this paper are selected illustrations of a series of

effects. Every effect has been observed in several problems with different

outcomes and probabilities. Some of the problems have also been presented to

groups of students and faculty at the University of Stockholm and at the

265

PROSPECT

THEORY

University of Michigan. The pattern of results was essentially identical to the

results obtained from Israeli subjects.

The reliance on hypothetical choices raises obvious questions regarding the

validity of the method and the generalizability of the results. We are keenly aware

of these problems. However, all other methods that have been used to test utility

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 in

such 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 of

repetitions 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 pro-

cedure 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 they

would behave in actual situations of choice, and on the further assumption that the

subjects have no special reason to disguise their true preferences. If people are

reasonably accurate in predicting their choices, the presence of common and

systematic violations of expected utility theory in hypothetical problems provides

presumptive evidence against that theory.

Certainty, Probability, and Possibility

In expected utility theory, the utilities of outcomes are weighted by their

probabilities. The present section describes a series of choice problems in which

people's preferences systematically violate this principle. We first show that

people overweight outcomes that are considered certain, relative to outcomes

which are merely probable-a phenomenon which we label the

certainty effect.

The best known counter-example to expected utility theory which ekploits the

certainty 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 a

variation of Allais' example, which differs from the original in that it refers to

moderate rather than to extremely large gains. The number of respondents who

answered each problem is denoted by

and the percentage who choose each

option is given in brackets.

PROBLEM

Choose between

2,500

with probability

2,400

with probability

N=72 [I81

.33,

.66,

.01;

B: 2,400

with certainty.

266

KAMNEMAN

AND

TVERSKY

PROBLEM2: Choose between

C: 2,500 with probability .33, D: 2,400 with probability .34,

0 with probability .67; 0 with probability .66.

The data show that 82 per cent of the subjects chose

in Problem 1, and 83 per

cent 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 the

modal choice in both problems. This pattern of preferences violates expected

utility theory in the manner originally described by Allais. According to that

theory, with

u(0)

0, the first preference implies

while the second preference implies the reverse inequality. Note that Problem 2 is

obtained from Problem

by eliminating a .66 chance of winning 2400 from both

prospects. under consideration. Evidently, this change produces a greater reduc-

tion in desirability when it alters the character of the prospect from a sure gain to a

probable one, than when both the original and the reduced prospects are

uncertain.

A simpler demonstration of the same phenomenon, involving only two-

outcome 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, over

half 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, set

u(0)

0, and recall that the choice of B implies u(3,000)/u(4,000)>4/5,

whereas the choice of

implies the reverse inequality. Note that the prospect

(4,000, .20) can be expressed as (A, .25), while the prospect

(3,000, .25)

can be rewritten as (B,.25). The substitution axiom of utility theory asserts that

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 the

probability of winning from 1.0 to .25 has a greater effect than the reduction from

267

PROSPECT THEORY

.2.

The following pair of choice problems illustrates the certainty effect with

non-monetary outcomes.

50%

chance to win a three-

A one-week tour of

week tour of England, England, with certainty.

France, and Italy;

chance to win a three-

10%

chance to win a one-

week tour of England, week tour of England.

France, and Italy;

N=72 [671* [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

the probabilities of winning are substantial

(.90

and

.45),

and most people choose the prospect where winning is more probable. In Problem

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 substitu-

tion axiom is violated. If

(y,

pq)

is equivalent to

(x,

p),

then

(y,

pqr)

is preferred to

(x,

pr), 0 <p, q, r

This property is incorporated into an alternative theory,

developed in the second part of the paper.

268

KAHNEMAN

AND

TVERSKY

The 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 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 section,

and the right-hand column displays choice problems in which the signs of the

outcomes are reversed. We use

-x

to denote the loss of

and

to denote the

prevalent preference, i.e., the choice made by the majority of subjects.

TABLE

Posltlve prospects Negative prospects

Problem 3: (4,000, .80)

(3,000). Problem 3': (-4,000, .80)

(-3,000).

N=95

[201

N=95

[921* [81

Problem 4: (4,000, .20)

(3,000, .25). Problem

4':

(-4,000, .20)

(-3,000, .25).

[651* [351

N=95

[421 [581

Problem 7:

(3,000, .90)

(6,000, .45). Problem 7': (-3,000, .90)

(-6,000, .45).

N=66 [86]* [I41 N=66

[81 r921*

Problem 8: (3,000, ,002)

(6,000, ,001). Problem 8': (-3,000, ,002)

(-6,000, ,001).

N=66

~271 [731*

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 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

risk 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 were

indifferent between (100,

.65;

100, .35) and (O), indicating risk aversion. They

were also indifferent between (-200, .80) and (-loo), indicating risk seeking.

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

269

PROSPECT

THEORY

that is merely probable over a smaller loss that is certain. The same psychological

principle-the overweighting of certainty-favors risk aversion in the domain of

gains 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 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 losses

has been regarded by many as strong evidence for the concavity of the utility

function for money. Why otherwise would people spend so much money to

purchase 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 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's

responses 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

KAHNEMAN

AND

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:

Yes, No.

N=95

[20] [80]*

Although Problem

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

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/2

is less valuable than reducing the

probability of that loss from

p/2

In contrast to these data, expected utility theory (with a concave

implies that

probabilistic insurance is superior to regular insurance. That is, if at asset position

one is just willing to pay a premium

to insure against a probability

of losing

then one should definitely be willing to pay a smaller premium

to reduce the

probability of losing

from

r)p, 0

Formally, if one is indifferent

between

(w -x, p; w, 1 -p)

and

y),

then one should prefer probabilistic

insurance

x, (1

r)p; w

y, rp; w

ry,

-p)

over regular insurance

y).

To prove this proposition, we show that

implies

Without loss of generality, we can set

u(w -x)

and

u(w)

Hence,

u(w

1 -p,

and we wish to show that

which holds

and only

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

vulnerable 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

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 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 decom-

positions 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 is

a probability of

.75

to end the game without winning anything, and a probability of

.25 to move into the second stage. If you reach the second stage you have a choice

between

(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 .25

.80

.20 chance to win

4,000, and a .25

1.0

.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

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 in

Problem 3 above.

The standard and the sequential formulations of Problem 4 are represented as

decision trees in Figures

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 a

dependency between the prospects without changing either probabilities or

KAHNEMAN AND A. TVERSKY

FIGURE 1.-The representation of Problem

as a decision tree (standard formulation).

FIGURE 2.-The representation of Problem

decision tree (sequential formulation).

outcomes. Specifically, the event 'not winning

3,000'

is included in the event 'not

winning

4,000'

in the sequential formulation, while the two events are indepen-

dent in the standard formulation. Thus, the outcome of winning

3,000

has a

certainty advantage in the sequential formulation, which it does not have in the

standard formulation.

The reversal of preferences due to the dependency among events is particularly

significant because it violates the basic supposition of a decision-theoretical

analysis, 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 the

other hand, the following problem is most likely to be represented in the

sequential form. One may invest money in a venture with some probability of

losing one's capital if the venture fails, and with a choice between a fixed agreed

return and a percentage of earnings if it succeeds. The isolation effect implies that

the contingent certainty of the fixed return enhances the attractiveness of this

option, 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 by

varying the representation of outcomes.

Consider the following problems, which were presented to two different groups

of subjects.

PROBLEM

11:

In addition to whatever you own, you have been given

1,000.

You are now asked to choose between

(1,000, .50),

and

(500).

N=70 [16] [841*

PROBLEM

12:

In addition to whatever you own, you have been given

2,000.

You are now asked to choose between

(-1,000, .50),

and

(-500).

N=68 [69*]

[311

The majority of subjects chose

in the first problem and

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,

(2,000, .50; 1,000, .50)

and

(1,500)

In fact, Problem

is obtained from Problem

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 the

comparison of prospects because it was common to both options in each problem.

The pattern of results observed in Problems

and

is clearly inconsistent with

utility theory. In that theory, for example, the same utility is assigned to a wealth

$100,000,

regardless of whether it was reached from a prior wealth of

$95,000

$105,000.

Consequently, the choice between a total wealth of

$100,000

and

even chances to own

$95,000

$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

and to several of the previous questions suggest that this pattern will

be obtained

the individual owns the smaller amount, but not

he owns the

larger amount.

The apparent neglect of a bonus that was common to both options in Problems

and

implies that the carriers of value or utility are changes of wealth, rather

than final asset positions that include current wealth. This conclusion is the

cornerstone of an alternative theory of risky cho'ice, whkh is described in the

following sections.

274

KAHNEMAN

AND

TVERSKY

THEORY

The preceding discussion reviewed several empirical effects which appear to

invalidate expected utility theory as a descriptive model. The remainder of the

paper 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 phase

consists 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 appli-

cation of several operations that transform the outcomes and probabilities

associated with the offered prospects. The major operations of the editing phase

are 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 are

received or paid. However, the location of the reference point, and the

consequent coding of outcomes as gains or losses, can be affected by the

formulation of the offered prospects, and by the expectations of the decision

maker.

Combination.

Prospects can sometimes be simplified by combining the prob-

abilities 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 segregated

from 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 follow-

ing operation is applied to a set of two or more prospects.

Cancellation.

The essence of the isolation effects described earlier is the

discarding 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 evaluated

the 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

and 12. Another type of cancellation involves the discarding of

common constituents, i.e., outcome-probability pairs. For example, the choice

275

PROSPECT

THEOKY

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,

SO;

-100, .30).

Two additional operations that should be mentioned are simplification and the

detection of dominance. The first refers to the simplification of prospects by

rounding 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 of

simplification involves the discarding of extremely unlikely outcomes. The second

operation involves the scanning of offered prospects to detect dominated alter-

natives, which are rejected without further evaluation.

Because the editing operations facilitate the task of decision, it is assumed that

they are performed whenever possible. However, some editing operations either

permit or prevent the application of others. For example,

(500, .20; 101, .49)

will

appear 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 the

offered set and with the format of the display. A detailed study of this problem is

beyond the scope of the present treatment. In this paper we discuss choice

problems 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 can

be specified without ambiguity.

Many anomalies of preference result from the editing of prospects. For exam-

ple, the inconsistencies associated with the isolation effect result from the cancel-

lation of common components. Some intransitivities of choice are explained by a

simplification 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 in

different 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 overall

value of an edited prospect, denoted

is expressed in terms of two scales,

.rr

and

The first scale,

.rr,

associates with each probability

a decision weight

.rr(p),

which reflects the impact of

on the over-all value of the prospect. However,

.rr

not a probability measure, and it will be shown later that

.rr(p)+.rr(l -p)

typically less than unity. The second scale,

assigns to each outcome

a number

v(x),

which reflects the subjective value of that outcome. Recall that outcomes are

defined relative to a reference point, which serves as the zero point of the value

scale. Hence,

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;

q),

which have at most two non-zero outcomes. In such a prospect, one

receives

with probability

and nothing with probability

1 -p

where

An offered prospect is strictly positive if its outcomes

are all positive, i.e., if

and

it is strictly negative if its outcomes

276

KAHNEMAN

AND

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

.rr

and v are

combined to determine the over-all value of regular prospects.

(x,p;

y,q)isaregularprospect(i.e.,eitherp+q<1,orx>O>y,orx~O~

then

(1) V(x, p; Y, q)

.rr(p)u(x)+ .rr(q)u(y)

where v(O)=

~(0)

0, and ~(1)

1. As in utility theory, V is defined on

prospects, while

is defined on outcomes. The two scales coincide for sure

prospects, where V(x,

.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 Appen-

dix, which describes conditions that ensure the existence of a unique

.rr

and a

ratio-scale v satisfying equation (1).

The evaluation of strictly positive and strictly negative prospects follows a

different rule. In the editing phase such prospects are segregated into two

components: (i) the

riskless component, i.e., the minimum gain or loss which is

certain to be obtained or paid; (ii) the risky component, i.e., the additional gain or

loss which is actually at stake. The evaluation of such prospects is described in the

next equation.

Ifp+q=l andeitherx>y>Oorx<y<O, then

(2) V(x,e; Y,

q)=v(~)+.rr(e)[v(x)-v(~)l.

That is, the value of a strictly positive or strictly negative prospect equals the value

of 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 feature

of equation (2) is that a decision weight is applied to the value-difference

v(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 of

equation (2) equals ~(p)v(x)+[l- .rr(p)]v(y). Hence, equation (2) reduces to

equation (1) if ~(p) -p) 1. AS will be shown later, this condition is not

~(1

generally satisfied.

Many elements of the evaluation model have appeared in previous attempts to

modify 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 pref-

erences among positive as well as among negative prospects, and he proposed a

utility function which has convex and concave regions in both the positive and the

negative 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.,

277

PROSPECT THEORY

[3,42]). Similar models were developed by Fellner [12], who introduced the

concept of decision weight to explain aversion for ambiguity, and by van Dam [46]

who attempted to scale decision weights. For other critical analyses of expected

utility 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 to

changes rather than to final states, and that decision weights do not coincide with

stated probabilities. These departures from expected utility theory must lead to

normatively unacceptable consequences, such as inconsistencies, intransitivities,

and violations of dominance. Such anomalies of preference are normally cor-

rected by the decision maker when he realizes that his preferences are inconsis-

tent, 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 are

changes in wealth or welfare, rather than final states. This assumption is compati-

ble with basic principles of perception and judgment. Our perceptual apparatus is

attuned to the evaluation of changes or differences rather than to the evaluation of

absolute magnitudes. When we respond to attributes such as brightness, loudness,

or temperature, the past and present context of experience defines an adaptation

level, 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 or

cold to the touch depending on the temperature to which one has adapted. The

same principle applies to non-sensory attributes such as health, prestige, and

wealth. 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 (positive

or 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 for

changes at a particular asset position. Clearly, the value functions described on

different pages are not identical: they are likely to become more linear with

increases 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

278

KAHNEMAN

AND

TVERSKY

of value as a function in one argument generally provides a satisfactory approxi-

mation.

Many sensory and perceptual dimensions share the property that the psy-

chological response is a concave function of the magnitude of physical change. For

example, it is easier to discriminate between a change of

and a change of

room temperature, than it is to discriminate between a change of

13"

and a change

16".

We propose that this principle applies in particular to the evaluation of

monetary changes. Thus, the difference in value between a gain of

100

and a gain

200

appears to be greater than the difference between a gain of

1,100

and a gain

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 the

larger loss is intolerable. Thus, we hypothesize that the value function for changes

of wealth is normally concave above the reference point

(vU(x)

for

and

often convex below it

(vV(x)

for

<O).

That is, the marginal value of both

gains and losses generally decreases with their magnitude. Some support for this

hypothesis has been reported by Galanter and Pliner

[17],

who scaled the

perceived 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 value

function which is derived from risky choices shares the same characteristics, as

illustrated in the following problems.

Applying equation

to the modal preference in these problems yields

~(.25)v(6,000)

v(2,000)]

~(.25)[v(4,000)

and

~(.25)v(-6,000)

~(.25)[v(-4,000)

v (-2,OOO)l.

Hence,

~(6,000)

~(4,000)

~(2,000)

and

v(-6,000)

v(-4,000)

v(-2,000).

These preferences are in accord with the hypothesis that the value function is

concave for gains and convex for losses.

Any discussion of the utility function for money must leave room for the effect

of special circumstances on preferences. For example, the utility function of an

individual who needs

$60,000

to purchase a house may reveal an exceptionally

steep 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

279

PROSPECT

THEORY

a less desirable neighborhood. Hence, the derived value (utility) function of an

individual does not always reflect "pure" attitudes to money, since it could be

affected by additional consequences associated with specific amounts. Such

perturbations 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 20,

then

(y, SO;

y, SO) is preferred to (x, SO; -x, .50). According to equation

(I),

there-

fore,

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 com-

monly 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

FIGURE

3.-A

hypothetical value function.

280

KAHNEMAN

AND

TVERSKY

Although the present theory can be applied to derive the value function from

preferences between prospects, the actual scaling is considerably more compli-

cated than in utility theory, because of the introduction of decision weights. For

example, decision weights could produce risk aversion and risk seeking even with

a 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 and

Kochenberger

[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 a

single exception, utility functions were considerably steeper for losses than for

gains.

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 toss

of a fair coin. For any reasonable person, the probability of winning is

.50

in this

situation. 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 he

considers the two events equiprobable. As will be shown below, however, the

decision weight

~(.50)

which is derived from choices is likely to be smaller than

.50.

Decision weights measure the impact of events on the desirability of pros-

pects, 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

Furthermore, since the events were identified only

by 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

which

relates decision weights to stated probabilities. Naturally,

is an increasing

function of

with

~(0)0

and

~(1)

That is, outcomes contingent on an

= =

impossible event are ignored, and the scale is normalized so that

~(p)

is the ratio

of the weight associated with the probability

to the weight associated with the

certain event.

We first discuss some properties of the weighting function for small prob-

abilities. The preferences in Problems

and

suggest that for small values of

281

PROSPECT

THEORY

is a subadditive function of

i.e.,

.rr(rp)> r.rr(p)

for

0< r

Recall that in

Problem

8, (6,000, .001)

is preferred to

(3,000, .002).

Hence

.rr(.001)

~(3,000) 1

a(.002) v (6,000)

by the concavity of

The reflected preferences in Problem

yield the same conclusion. The pattern of

preferences in Problems

and

7',

however, suggests that subadditivity need not

hold for large values of

Furthermore, we propose that very low probabilities are generally over-

weighted, that is,

~(p)

for small

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 a

small loss, which can be viewed as the payment of an insurance premium, over a

small probability of a large loss. Similar observations have been reported by

Markowitz

[29].

In the present theory, the preference for the lottery in Problem

implies

v(5),

hence

.rr(.001)~(5,000)> .rr(.001)

v(5)/v(5,000)> .001,

assuming the value function for gains is concave. The readiness to pay for

insurance in Problem

14'

implies the same conclusion, assuming the value

function for losses is convex.

It is important to distinguish overweighting, which refers to a property of

decision weights, from the overestimation that is commonly found in the assess-

ment 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 stated

value of

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

~(p) -p)

We label this property subcertainty. It is readily

~(1

seen that the typical preferences in any version of Allias' example (see, e.g.,

Problems

and

imply subcertainty for the relevant value of

Applying

282

KAHNEMAN

AND

TVERSKY

equation

(1)

to the prevalent preferences in Problems

and

yields, respectively,

[I-

r(.66]~(2,400)>

and

r(.33)~(2,500)

74.33)~(2,500)

r(.34)~(2,400);

hence,

Applying the same analysis to Allais' original example yields

r(.89)

r(.ll)

and some data reported by MacCrimmon and Larsson

[28]

imply subcertainty for

additional values of

The slope of

in the interval

(0, 1)

can be viewed as a measure of the sensitivity

of preferences to changes in probability. Subcertainty entails that

is regressive

with respect to

i.e., that preferences are generally less sensitive to variations of

probability than the expectation principle would dictate. Thus, subcertainty

captures an essential element of people's attitudes to uncertain events, namely

that the sum of the weights associated with complementary events is typically less

than the weight associated with the certain event.

Recall that the violations of the substitution axiom discussed earlier in this

paper conform to the following rule: If

(x, p)

is equivalent to

(y, pq)

then

(x, pr)

not preferred to

(y, pqr), 0 <p, q, r

By equation

(I),

r(p)v

(x)

r(pq)v( y)

implies

r(pr)v(x) r(pqr)v

(

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

called subproportionality, imposes considerable constraints on

the shape of

it holds if and only if log

is a convex function of log

It is of interest to note that subproportionality together with the overweighting

of small probabilities imply that

is subadditive over that range. Formally, it can

be shown that if

r(p)

and subproportionality holds, then

r(rp)

rr(p), 0

provided

is monotone and continuous over

(0,l).

Figure

presents a hypothetical weighting function which satisfies overweight-

ing and subadditivity for small values of

as well as subcertainty and sub-

proportionality. These properties entail that

is relatively shallow in the open

interval and changes abruptly near the end-points where

r(0)

and

r(1)

The sharp drops or apparent discontinuities of

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 the

individual to discard events of extremely low probability and to treat events of

extremely high probability as if they were certain. Because people are limited in

283

PROSPECT

THEORY

their ability to comprehend and evaluate extreme probabilities, highly unlikely

events are either ignored or overweighted, and the difference between high

probability and certainty is either neglected or exaggerated. Consequently,

not well-behaved near the end-points.

1.0

STATED PROBABILITY:

FIGURE

4.-A

hypothetical weighting function.

The following example, due to Zeckhauser, illustrates the hypothesized

nonlinearity of

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 that

they would be willing to pay much more for a reduction of the probability of death

from 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 to

enjoy it.

An obvious objection to the assumption that

~(pj

involves comparisons

between prospects of the form

(x, p

;

x, q)

and

(x, p'; x, q'),

where

p' +q'

Since any individual will surely be indifferent between the two prospects, it

could be argued that this observation entails

~(p)

~(q)

~(q'),~(p')

which in

turn implies that

is the identity function. This argument is invalid in the present

theory, which assumes that the probabilities of identical outcomes are combined

in the editing of prospects. A more serious objection to the nonlinearity of

involves potential violations of dominance. Suppose

p',

and

;

hence,

(x, p;

dominates

(x, p';

q').

If preference obeys

284

KAHNEMAN

AND

TVERSKY

dominance, then

.rr(p)v(x) +.rr(q)v(y)

.rr(pf)v(x)+.rr(q')4y),

Hence, as y approaches

~(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 the

assumption that dominated alternatives are detected and eliminated prior to the

evaluation of prospects. However, the theory permits indirect violations of

dominance, e.g., triples of prospects so that

is preferred to

and

dominates

For an example, see Raiffa

[34,

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 have

not 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 intro-

duce systematic biases. Indeed, Lichtenstein and Slovic

[27]

have constructed

pairs of prospects

and

such that people generally prefer

over

but bid

more for

than for

This phenomenon has been confirmed in several studies,

with both hypothetical and real gambles, e.g., Grether and Plott

[20].

Thus, it

cannot be generally assumed that the preference order of prospects can be

recovered by a bidding procedure.

Because prospect theory has been proposed as

model of choice, the inconsis-

tency of bids and choices implies that the measurement of values and decision

weights should be based on choices between specified prospects rather than on

bids or other production tasks. This restriction makes the assessment of

and

.rr

more difficult because production tasks are more convenient for scaling than pair

comparisons.

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

and

follows from the present theory iff

285

PROSPECT

THEORY

Hence, the violation of the independence axiom is attributed in this case to

subcertainty, and more specifically to the inequality

~(.34)

~(.66).

This

analysis shows that an Allais-type violation will occur whenever the v-ratio of the

two non-zero outcomes is bounded by the corresponding

T-

ratios.

Problems

through

share the same structure, hence it suffices to consider one

pair, say Problems

and

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

sub-

proportionality of

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

follows from prospect theory-provided the

probability 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 for

f (x)

v (-x).

Since the value function for losses is convex,

is a concave

function of

Applying prospect theory, with the natural extension of equation

we wish to show that

Substituting for

f(x)

and using the concavity off, it suffices to show that

7r(p)/2

7r(p/2),

which follows from the subadditivity of

According to the present theory, attitudes toward risk are determined jointly by

and

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

(px).

risk seeking is implied whenever

~(p)

v(px)/v(x),

which is greater than

if the value function for gains is concave. Hence, overweighting

(7r(p)>p)

necessary but not sufficient for risk seeking in the domain of gains. Precisely the

same condition is necessary but not sufficient for risk aversion when

This

analysis restricts risk seeking in the domain of gains and risk aversion in the

domain of losses to small probabilities, where overweighting is expected to hold.

286

KAHNEMAN

AND

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 to

inhibit 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 both

experimental 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 of

disaster are sometimes entirely ignored. Furthermore, the evidence suggests that

minor 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 and

money, such factors as the value of security, social norms of prudence, the

aversiveness of a large number of small payments spread over time, information

and 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 the

introduction of variables or concepts which have not been considered in this

treatment.

Shifts of Reference

So far in this paper, gains and losses were defined by the amounts of money that

are 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 for

most 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 experien-

ced as a loss, not as a reduced gain. Similarly, an

entrepi-eneur who is weathering a

slump with greater success than his competitors may interpret a small loss as a

gain, 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 in

a business venture, has already lost 2,000 and is now facing a choice between a

sure 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

287

PROSPECT

THEORY

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 just

acceptable, then

z, p

;

is preferred over

(

for

with

To prove this proposition, note that

V(x, p;

-p)=

iff

T(~)v(x)

-T(I

-P)v(-Y).

Furthermore,

~(1-p)v

(

by the properties of

+~(l

by substitution,

-p)v(-z)

v(-z)[~(p)

~(1

since

v(-z)

-v(z),

-p)]

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 to

losses 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 the

decision to pay a premium

to protect against a loss

as a choice between

(-x

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 the

former 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 this

case, 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, this

formulation 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 domain

of losses.

288

KAHNEMAN

AND

TVERSKY

Many economic decisions involve transactions in which one pays money in

exchange for a desirable prospect. Current decision theories analyze such prob-

lems 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

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 the

isolation effect, suggests that people are unlikely to perform the operation of

subtracting 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

.rr

(.01)v(1,000)+v(-10)>0.

If this hypothesis is correct, the decision to pay

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),

-px, 1

-p)

and

(0)

then one will not pay

to purchase the prospect

(x, p).

Thus, people are expected to exhibit more risk seeking in deciding whether to

accept a fair gamble than in deciding whether to purchase

gamble for a fair price.

The location of the reference point, and the manner in which'choice problems are

coded 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)

prospects with any number of outcomes is straightforward. When the number of

outcomes is large, however, additional editing operations may be invoked to

simplify evaluation. The manner in which complex options, e.g., compound

prospects, 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

consequence of a policy decision. The main properties of the proposed value

function 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 the

probabilities 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 the

weighting function. For example, if

and

are complementary events and

neither is certain,

.rr(A)

v(B)

should be less than unity-a natural analogue to

subcertainty.

289

PROSPECT

THEORY

The decision weight associated with an event will depend primarily on the

perceived likelihood of that event, which could be subject to major biases

[45].

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 two

themes. 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

and

Stanford 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

1. The extension to

regular prospects with p +q

is straightforward. Let

denote the relation of preference between

prospects that is assumed to be connected, symmetric and transitive, and let

denote the associated

relation of indifference. Naturally, (x, p; y,

(y, q; x, p). We also assume, as is implicit in our

notation, that (x, p; 0, q)

(x, p; 0, r), and (x, p; y, 0)

(x, p;

0). That is, the null outcome and the

impossible 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')

(x', p'; y, q) and (x', p'; y", q") ~(x", p"; y', q'), then (x, p; yo, q")

(x", PI';

q).

Solvabil~ty:If (x, p; y, q)?

(z,

r)Z (x, p; y' q') for some outcome

and probability r, then there exist

y",

q" such that

(x.

P; y"qs)

(z,

r).

It has been shown that these conditions are sufficient to construct the desired additive represen-

tation, 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

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

[35j

and

[25].

To construct the multiplicative representation we assume that the ordering of the prob-

ability-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

Krantz for his help in the formulation of this section.

290

KAHNEMAN

AND

TVERSKY

Combining the additive and the multiplicative representations yields

Vix, p; Y, q)

=fl~(p)v(x)l+f1m(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 obtain

f[.r(p)v(x)l+f[.rr(~)v(~)l

=f[.rr(p)v(z)l

implies

f[dq)v(x)l+f[.rr(s)v(y)l=

fldq)v(z)l.

Assuming, with no loss of generality, that ~(q) and letting

a .rr(p)v(x),

~(p)v(y),

~(p),

a(p)v(z).

and

r(q)/rr(p),

yields

f(a)+ f(@)

f(y)

implies

f(Ba)+ f(6p)

f(6y)

for all

0<6<1.

Because

is strictlv monotonic we can set

'[f(a)+ f(p)].

Hence,

6f1[f(a)+f(p)1=

rl[fcea)+'f(?p)l.

The solutlon to this functional equation is

f(a)

kac [I].

Hence,

V(x, p; y, q)

k[n(p)v(x)y

k[~(q)v(

y)y,

for some

k, c

The desired bilinearform is obtained by redefining the

scales

and

so as to absorb the constants

and

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Discussion

Utility and preferences are difficult to measure. In an ideal experiment, the choices wouldn't be hypothetical and the experiment would involve the actual, large monetary payouts that are hypothetically offered here. Laboratory experiments involving small stakes could potentially have noisy/faulty results. Unfortunately, behavioral economics research budgets aren't as large as las vegas casino budgets! Daniel Kahneman and Amos Tversky famously wrote Prospect Theory. With it, they popularized the field of behavioral economics and brought psychology into the heart of economics analysis. Kahneman won the nobel prize in Economics in 2002 for his work in decision making theory and authored the popular book, Thinking Fast and Slow. The axioms of expected utility theory that the authors refer to here come from the revolutionary and normative decision making model created by von Neumann and Morgenstern. These economists flipped Bernoulli's model. In Bernoulli's model, utility defined preference (i.e. people prefer the option that has the highest utility). While in their model, utility actually describes preferences and observing utility will show an observer the individuals' preferences. The three major axioms in the subjective expected utility model are transitivity, dominance and invariance. In their model, people try to maximize their subjective expected utility with their decision making. This is the most cited paper ever to appear in Econometrica, a top economics academic journal. It was written by two psychologists. Expected utility is one of the first theories of decision making and before this seminal paper was the established model for decision making under risk. Expected utility was first proposed by Daniel Bernoulli in 1738. In particular, Bernoulli proposed a modification on one of the oldest theories of decision making under risk: expected value. The expected value of an outcome is the sum of each individual outcomes payoff adjusted for its probability or risk, Bernoulli noticed a systematic bias in expected value. In particular, Bernoulli noticed that the value of payoffs is subjective and that the normative decision rule of expected value does not account for the value that individuals attach to payoffs. Bernoulli proposed the utility function and built a model where individuals attempt to maximize utility in their decision making. The decision making function was concave (not a linear function of wealth) and also introduced the concept of diminishing marginal utility. Kahneman and Tversky selected a highly educated group of university students and professors for this study, including statisticians and economists. Systematic bias is present even in the most educated and theory-aware decision makers. In Problem 2, the expected utility of Choice C (825) is greater than the expected utility of Choice D (816) and the difference between the expected utilities is the same as before (9), but now, Choice D also has an uncertainty to its result-> 0 with probability 66%, so 83% of individuals chose Choice C. This is a violation of the substitution axiom of expected utility theory (respondents flipped their selections from problem 1 to problem 2 ). The reflection effect is a startling and important discovery. While people are risk averse and prefer the "for sure" option when maximizing their subjective utility, the prospect of a guaranteed loss is not as appealing. Instead, people are risk seeking and prefer to gamble when it's possible to eliminate a guaranteed loss. This is a violation of expected utility theory and introduces the certainty effect, a key part of prospect theory. It is a violation because the expected value of choice A, 2409 (66*2400+.33*2500), is greater than the expect value of choice B, 2400. Choice A includes a 1% chance of a 0 result, making the "certainty" of 2400 more attractive to 82% of respondents.

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