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.
1.
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.
2.
CRITIQUE
Decision making under risk can be viewed as a choice between prospects or
gambles. A prospect (xl, pl;
.
.
.
;
x,,, p,) is a contract that yields outcome xi with
probability pi, where pl
+pz+
.
.
.
+p,
=
1.
To simplify notation, we omit null
outcomes and use (x, p) to denote the prospect (x, p;
0,
1
-p) that yields x with
probability p and
0
with probability
1
-p. The (riskless) prospect that yields x
with certainty is denoted by (x). The present discussion is restricted to prospects
with so-called objective or standard probabilities.
The application of expected utility theory to choices between prospects is 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
F.
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
D.
KAHNEMAN
AND A.
TVERSKY
That is, the overall utility of a prospect, denoted by
U,
is the expected utility of
its outcomes.
(ii) Asset Integration: (xl, pl;
. .
.
;
x,, p,) is acceptable at asset position w iff
U(w +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
x.
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?
A:
50°/o chance to win 1,000,
B:
450 for sure.
50% chance to win nothing;
The outcomes refer to Israeli currency. To appreciate the significance of 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
N,
and the percentage who choose each
option is given in brackets.
PROBLEM
1:
Choose between
A:
2,500
with probability
2,400
with probability
0
with probability
N=72 [I81
.33,
.66,
.01;
B: 2,400
with certainty.
266
D.
KAMNEMAN
AND
A.
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
B
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
1
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
C
implies the reverse inequality. Note that the prospect
C
=
(4,000, .20) can be expressed as (A, .25), while the prospect
D
=
(3,000, .25)
can be rewritten as (B,.25). The substitution axiom of utility theory asserts that
if
B is preferred to A, then any (probability) mixture (B, p) must be preferred to the
mixture (A, p). Our subjects did not obey this axiom. Apparently, reducing the
probability of winning from 1.0 to .25 has a greater effect than the reduction from
267
PROSPECT THEORY
.8
to
.2.
The following pair of choice problems illustrates the certainty effect with
non-monetary outcomes.
A:
50%
chance to win a three-
B:
A one-week tour of
week tour of England, England, with certainty.
France, and Italy;
C:
5%
chance to win a three-
D:
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
7
the probabilities of winning are substantial
(.90
and
.45),
and most people choose the prospect where winning is more probable. In Problem
8,
there is a
possibility
of winning, although the probabilities of winning are
minuscule
(.002
and
.001)
in both prospects. In this situation where winning is
possible but not probable, most people choose the prospect that offers the larger
gain. Similar results have been reported by
MacCrimmon and Larsson
[28].
The above problems illustrate common attitudes toward risk or chance that
cannot be captured by the expected utility model. The results suggest the
following empirical generalization concerning the manner in which the 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
<
1.
This property is incorporated into an alternative theory,
developed in the second part of the paper.
268
D.
KAHNEMAN
AND
A.
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
x,
and
>
to denote the
prevalent preference, i.e., the choice made by the majority of subjects.
TABLE
I
Posltlve prospects Negative prospects
Problem 3: (4,000, .80)
<
(3,000). Problem 3': (-4,000, .80)
1
(-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).
N
=
95
[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.
A
recent review by Fishburn and Kochenberger
[14]
documents the prevalence of
risk seeking in choices between negative prospects.
Second, recall that the preferences between the positive prospects in Table I are
inconsistent with expected utility theory. The preferences between the cor-
responding negative prospects also violate the expectation principle in the same
manner. For example, Problems 3' and
4',
like Problems 3 and 4, demonstrate that
outcomes which are obtained with certainty are overweighted relative to
uncertain outcomes. In the positive domain, the certainty effect contributes to a
risk averse preference for a sure gain over a larger gain that is merely probable. In
the negative domain, the same effect leads to a risk seeking preference for a loss
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
D.
KAHNEMAN
AND
A.
TVERSKY
occurs on an even day of the month, your insurance payment is refunded and your
losses are not covered.
Recall that the premium for full coverage is such that you find this insurance
barely worth its cost.
Under these circumstances, would you purchase probabilistic insurance:
Yes, No.
N=95
[20] [80]*
Although Problem
9
may appear contrived, it is worth noting that probabilistic
insurance represents many forms of protective action where one pays a certain
cost to reduce the probability of an undesirable event-without eliminating it
altogether. The installation of a burglar alarm, the replacement of old tires, and
the decision to stop smoking can all be viewed as probabilistic insurance.
The responses to Problem
9
and to several other variants of the same question
indicate that probabilistic insurance is generally unattractive. Apparently, reduc-
ing the probability of a loss from
p
to
p/2
is less valuable than reducing the
probability of that loss from
p/2
to
0.
In contrast to these data, expected utility theory (with a concave
u)
implies that
probabilistic insurance is superior to regular insurance. That is, if at asset position
w
one is just willing to pay a premium
y
to insure against a probability
p
of losing
x,
then one should definitely be willing to pay a smaller premium
ry
to reduce the
probability of losing
x
from
p
to
(1
-
r)p, 0
<
r
<
1.
Formally, if one is indifferent
between
(w -x, p; w, 1 -p)
and
(w
-
y),
then one should prefer probabilistic
insurance
(w
-
x, (1
-
r)p; w
-
y, rp; w
-
ry,
1
-p)
over regular insurance
(w
-
y).
To prove this proposition, we show that
implies
Without loss of generality, we can set
u(w -x)
=
0
and
u(w)
=
1.
Hence,
u(w
-
y)
=
1 -p,
and we wish to show that
which holds
if
and only
if
u
is concave.
This is a rather puzzling consequence of the risk aversion hypothesis of utility
theory, because probabilistic insurance appears intuitively riskier than regular
insurance, which entirely eliminates the element of risk. Evidently, the intuitive
notion of risk is not adequately captured by the assumed concavity of the utility
function for wealth.
The aversion for probabilistic insurance is particularly intriguing because all
insurance is, in a sense, probabilistic. The most avid buyer of insurance remains
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