Prospect Theory: An Analysis of Decision under Risk
Author(s): Daniel Kahneman and Amos Tversky
Source:
Econometrica,
Vol. 47, No. 2 (Mar., 1979), pp. 263-292
Published by: The Econometric Society
Stable URL: http://www.jstor.org/stable/1914185
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E
C
O
N
OMETRICA
I
C
I
VOLUME 47
MARCH,
1979
NUMBER 2
PROSPECT
THEORY:
AN
ANALYSIS
OF
DECISION
UNDER
RISK
BY
DANIEL KAHNEMAN
AND
AMOS
TVERSKY'
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
(x1,
Pi;
...
;
xn,
pn)
is
a
contract that
yields
outcome
xi
with
probability
Pi,
where
Pl
+
P2
+
...
+
pn
=
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,
Pi;
...
;
Xn,
Pn)
=
pi
u
(x1)
+...
+PnU
(Xn)
1
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:
(xi, Pi;
...
;
Xn,
P)
is
acceptable
at
asset
position
w
iff
U(w +x1,
pl;
...
;
w
+Xn,
Pn)
>
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
[4]).
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% 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
PROSPECT THEORY 265
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 e*ploits 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 .33, B: 2,400
with
certainty.
2,400
with
probability .66,
0 with
probability .01;
N=72
[18]
[82]*
266
D.
KAHNEMAN
AND
A.
TVERSKY
PROBLEM 2: Choose between
C: 2,500 with probability
.33, D: 2,400 with probability .34,
0
with probability
.67; 0 with probability .66.
N
=72
[83]*
[17]
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 .01 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
u(2,400)> .33u(2,500)
+
.66u(2,400)
or
.34u(2,400)> .33u(2,500)
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].
PROBLEM
3:
A:
(4,000,.80),
or
B:
(3,000).
N
=
95
[20]
[80]*
PROBLEM 4:
C:
(4,000,.20),
or
D:
(3,000,.25).
N=
95
[65]*
[35]
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