Fermat's Library | Prospect Theory annotated/explained version.

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...

The reflection effect is a startling and important discovery. While...

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

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