American Economic Association
A Rationale for Preference Reversal
Author(s): Graham Loomes and Robert Sugden
Source: The American Economic Review, Vol. 73, No. 3 (Jun., 1983), pp. 428-432
Published by: American Economic Association
Stable URL: http://www.jstor.org/stable/1808127
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A Rationale for Preference Reversal
By GRAHAMLOOMESAND ROBERTSUGDEN*
Ever since the formulationby John von
Neumannand OskarMorgensternof a set of
axioms of rationalchoice underuncertainty,
a numberof situationshave been identified
in which there are significantand repeated
violations of one or more of those basic
axioms.A comprehensivesurveyof relevant
evidence has been providedby Paul Schoemaker (1982), together with an outline of
various attempts that have been made to
explainthe observedbehavior.
One such attempt was our 1982 article,
and was given the name regret theory.We
showed that several behavioral patterns
which contradict standard expected utility
theory (the "common consequenceseffect"
or Allais' paradox, the "common ratio
effect," and the "isolation effect" in twostage gambles) are predictedby regret theory; and other patterns,such as the "reflection effect" and simultaneousgamblingand
insurance,if not firmlypredicted,are at least
consistentwith regrettheory.
However,a numberof problematicalcases
were not explicitly treated in our earlier
paper. One such case, the phenomenonof
preferencereversal, was the subject of an
article by David Gretherand CharlesPlott
(1979). This phenomenon,first reportedby
Sarah Lichtensteinand Paul Slovic (1971)
and by Harold Lindman (1971), takes the
following general form: asked to make a
direct choice betweengambleA and gamble
B, an individualstates a preferencefor A;
but asked to considerthe two gambles separately,he places a highercertaintyequivalent value,or reservationprice, on B.
This phenomenon is particularly interesting because, as Gretherand Plott pointed out, with such reversalsit appearedthat
....
no optimization principles of any sort lie
behind even the simplestof humanchoices"
(p. 623). The purpose of this paper is to
*Department of economics, University of Newcastle
upon Tyne.
428
suggest that this is not so, and that regret
theory may provide the optimizationprinciples whichexplainthe asymmetricpatternof
choice observed by Grether and Plott, and
others.
L.AnOutlineof RegretTheory
The essentialnotion underlyingregrettheory is that people tend to compare their
actual situations with the ones they would
have been in, had they madedifferentchoices
in the past. If they realize that a different
choice would have led to a better outcome,
people may experiencethe painful sensation
of regret;if the alternativewould have led to
a worse outcome, they may experience a
pleasurable sensation we call "rejoicing."
When faced with new choice situations,people remembertheirpreviousexperiencesand
form expectations about the rejoicing and
regretthat the presentalternativesmightentail. They then take these expectationsinto
account when making their decisions. We
model all this as follows.
In any pairwisechoice, individualsare regarded as choosing between pairs of actions
denotedAi and Ak9 where each action is an
The
n-tupleof state-contingentconsequences.
consequenceof choosing Ai if the jth state
occursis denoted xij; and the probabilityof
thejth state occurringis given as pj.
It is proposed that, for any individual,
there exists a choicelessutility function, C(.),
unique up to an increasinglinear transformation, which assigns a real-valuedutility
index number to every conceivable consequence. The value C(xii) is written as cij,
and representsthe utilityan individualwould
derive if the consequencewere experienced
other than as a resultof choice, for example,
if the consequence were imposed in some
way.
The choice an individualfaces is either to
accept Ai and simultaneouslyreject Ak, or
else to accept Ak and simultaneouslyreject
Ai. Suppose he contemplates accepting Ai
VOL. 73 NO. 3
LOOMES AND SUGDEN: PREFERENCE REVERSAL
and rejectingAk. Should thejth state occur,
he will experiencex1Jbut simultaneouslymiss
out on Xkj. So in additionto c 1, he may also
derive some incrementor decrementof utility due to rejoicingor regret:an incrementif
cij > Ckj; a decrementif Cij < Ckj.
In the simplest restricted form of the
model, it is proposedthat regret-rejoicing
be
representedby the function R(.), where the
degree of regret-rejoicingdepends only on
the difference (cij - ckj). We assume that
R(.) is strictly increasing and three times
differentiable, and that R(O)= O. On this
basis, the modified utility that results if Ai is
chosen and Ak is rejectedand the jth state
occurs can be written as m k where
(1)
m
k
=
cij + R(Ci
k).
We can then define the expected modified
utilityof acceptingAi and rejectingAk as Eik
where
n
(2)
Ei
E
'
pmj.
j=l
The proposedchoice rule that the individual
will prefer Ai, prefer Ak, or be indifferent
between them according to whether E/k is
greaterthan, less than, or equalto Ek, generates the propositionthat
Q(t) is assumed to be convex for all ( > 0,
regret theory predicts a numberof patterns
of behaviorwhich,althoughinconsistentwith
expected utility theory, have been observed
in repeatedexperiments,includingthose referred to in the introductionto this paper.
We shall now show that under the same
assumptions,regret theory can explain the
main kind of preferencereversalobservedby
Gretherand Plott.
II. PreferenceReversalandRegretTheory
Grether and Plott asked individuals to
consider pairs of bets where the "P bet"
offered a high probability of a relatively
small money gain, while the "$ bet" offered
a lower probability of a larger gain. Both
bets had the same mean value,' but the variance was greaterfor the $ bet. All pairs of
bets can be described with the following
notation.
The P bet offered the dollarconsequences
yp and xp with probabilitiesHJpand 1- Hp;
yp > 0 > xp. The $ bet offered the consequencesy$ and x$ with probabilities1II$and
1- fl$; y$ > 0 > x$. In all Gretherand Plott's
experiments, FIp > 0.5 >1$ and y$ > yp. In
some pairs of bets xp > X$, while in others
xp < X$, but in all cases the differencebetween the two negative consequences was
relativelysmall; in particular:
lxp - xs$ < IyP - ysl
n
(3) Ai : Ak iff E pj[cj -ckj
+ R(c
-CkI)-R(Ckj
429
and
-
Cij)]
>
0.
For compactness, we define a function
lxp - x5l < lyp- x$1.
Each individual was asked to state the
lowest price at which he would sell each of
the bets; we shall denote these two reservation prices by rp and r$. He was also at a
+ R(()Q(.) such that for all (, Q(()=
R(- ). (Note that for all (, Q(()= -
differenttime asked to make a direct choice
betweenthe P bet and the $ bet. In 561 cases
Q(-c).) Thus
(74.4 percent) out of 754,2 individuals placed
n
(4) Ai > Ak iff E pJQ(cij -ckj) < 0
j=
If Q(.) is linear,regrettheorypredictschoices
that are consistentwith expectedutility theory, but if the function is nonlinear, the
predictionsof the two theories diverge. If
'In Grether and Plott's experiments, the two bets did
not have exactly the same expected money values, but
only because consequences were rounded to the nearest
dollar and probabilities to the nearest one-thirty-sixth.
2These figures are based on Grether and Plott, Tables 5, 6, 8 and 9 (pp. 632-33), but omitting 14 cases
where P ~ $, since Grether and Plott gave no information about reservation prices in these cases.
a higherreservationprice on the $ bet, despite the greater variance of those bets.
MoreoverGretherand Plott reported(p. 632)
that the reservationprices for $ bets were
frequentlygreaterthan theirexpectedvalues.
These observationscan be squaredwith expected utility theory, but only by dropping
the usualassumptionof risk aversion.
Much more serious for standard theory,
however,was the incidenceof preferencereversal. In 76 cases (10.1 percent)P -< $ was
combinedwith rp > r, and in 205 cases (27.2
percent)P >- $ was combinedwith rp < r,.
These observationsare completelyinconsistent with conventional theory. But now
consider the implications of regret theory.
Tables 1-3 portraythree matricesof statecontingent consequences correspondingto
the threechoice problemsfacing an individual for any pair of bets. (The columns in
each table representstates of the world,with
the probabilityof each state given above.)
Table 1 representsthe choice between retainingthe P bet and sellingit for the sum rp
(an action we denote by P*). If rp is the
reservationprice, then P- P*. Similarlyin
Table 2, $* is the action of selling the $ bet,
and if r, is the reservationprice, then $- $*.
Table 3 representsthe choice between the
two bets. This matrixis constructedto correspond with the random device used in
Gretherand Plott'sexperiments(see p. 629).
To producefirm predictions,and to highlight the role of regret and rejoicingin our
theory, we shall make the additional simplifyingassumptionthat the choicelessutility
functionC(.) is linearoverthe relevantrange.
On that basis, we obtain the following general result.
Let Ai be an action that gives the consequence w with certainty while Ak is an
actuariallyfair gamble offering w + u with
probability 11, and w - flu/(I - 1) with
probability1- f; 1> H > 0 and u > 0. To
simplify notation, we choose a transformation of the linear function C(.) such that
C(4) =
JUNE 1983
THE A MERICAN ECONOMIC REVIEW
430
(
for all (. Applying expression (4)
and rearranging,
we have
(5) Ai :Ak
iff
(u-O
ib-A-
1
TABLE I
nip
I - rip
P
yp
Xp
P*
rp
rp
TABLE 2
$*
1-$
1- n$
r$
r$
TABLE 3
s
Hp -
I - rnp
P
yp
yp
Xp
$
y$
Xs
Xs
all ( > 0, it follows
With Q(4) convex for
>
that Ai >j:Ak iff II 0.5. Thus, under the
special assumptionthat C(.) is linear, regret
theory predicts that individualswill accept
small-stakelarge-prizefair gambles(HI< 0.5)
but reject large-stakesmall-prizefair gambles (I > 0.5).
In the light of this result,considerTable 1.
If rp were equal to the expecteddollar value
of the P bet (which we shall write as V(P))
then P (seen in relation to P*) would be a
large-stakesmall-prizefairgamble(recallthat
FIp > 0.5). Thus P
-< P*.
So to achieve P -
whichmust be the case if rp is a reservation price, it is necessary that rp < V(P).
Now considerTable 2. In this case rI$ < 0.5,
so to achieve$ $* it is necessarythat r, >.
P*,
V($). But V(P) = V($). Combining these re-
sults, rp< r,. Thus,underthe specialassumption of linearity,regrettheory explainsboth
the tendencyfor individualsto place a higher
reservationprice on the $ bet, and the tendency for that price to exceed the expected
value of the bet.
Now considerthe choice betweenP and $
representedin Table 3. Applying expression
LOOMES AND SUGDEN: PREFERENCE REVERSAL
VOL. 73 NO. 3
(4) again,and retainingour assumptionabout
C(.), we have
(6)
iff
P<$
U$Q(yp - y$)
431
preferencereversal remains consistent with
regret theory and with its presumptionthat
individualsare rational,optimizingagents.
APPENDIX
+ (1Ip - I$) Q(yp -Xs)
To simplifynotation,let:
+ 0 -p) Q(xp -X$) :;i
HFJ $
Because both bets have the same expected
dollarvalue,we know that
(7)
-IP
-X$)
= 0.
(i)
- Hp > 0;
lI> -$9PtI-tI$,
(iii)
yp - X$ > 0;
(iv) jxp - x$1 < Iyp - y$j, IyP- x$1.
Given all this, the assumptionthat Q(J) is
convex for all > 0 entails the following
results(see the Appendixfor proof):
CaseI:
P-<$ifxp-x5>0
andy
CaseII:
CaseIII:
-
yp> yp -x
P-<$ifxp-x5<0
and Illp- rI$ > 0.5
P>-$ifxp -x5<0
X$
P
a33-x
$.
Then the problemis to determinethe sign of
(Al)
We also know that
(ii) yp- y < 0;
a2-ypY
$
U3-lHP
HI$(Yp-Y$)+(n_Ip-I$)(Yp-X$)
+ (I 1-1p)(Xp
a,-yp-Y$
-
I11Q(al)+1+2Q(a2)+
I3Q(a3),
given that
f11a1+ f12a2 + fI3a3 = 0
(A2)
1 > 1kl
(A3)
(A4)
112 f13 > 0
a2>0>al
la,1,1a21 > 1a31
(A5)
NO
(A6)
=-Q(-
0,
and Q(() is convex for all ( > 0.
andy$ - yp < yp - X$
CaseIV:
P>-$ifxp -x5>0
and l $ > 0.5
Thesefour cases are not exhaustive-there
are other cases in which no firm prediction
can be made-but they cover all six pairs of
gamblesin Gretherand Plott's experiments;
four of those pairs were cases for which
P -< $ is predicted,and two were cases where
the prediction is P
>- $3
But the most signifi-
cant feature is that in cases III and IV,
preferencereversalis predicted,and the form
of reversal- P
>- $
Case I:
-a,
> a2 > a3 > 0
In this case, a, is the only ai to be negative,
and is also the ai with the largest absolute
value. Thus, because of (A2) and (A6), (Al)
must be negative,that is, P -< $.
Case III:
-a2
< a, < a3 < 0
In this case, a2 is the only positiveai, and is
the ai with the largest absolute value. Thus
(Al) must be positive,that is, P >- $.
and rp < r5 - is the one
most frequentlyobserved. Of course, if we
relax the assumptionthat C(.) is linear, we
can no longer make such clear-cut predictions. Nevertheless, the common form of
Casell:
since Q(() is concavefor all ( < 0:
(A7)
3See their Table 2. Case I covers pairs of gambles
numbers 1, 4 and 5; Case II covers pair number 2; Case
III covers pairs numbers 3 and 6.
a1,a3<0<a2;12>?0.5
-171Q(a1)?+113Q(a3)
trr
1
A{11
a,
+
1I3a38
432
THE AMERICAN ECONOMIC REVIEW
Considera choice between (i) the certainty
of a consequenceof zero and (ii) the consequencesa2, (Iia, + J?3a3)/(11I + 113) with
probabilities -I2r 1' + 113. Since (ii) is an
actuariallyfair gamble with the probability
of winning, H2I > 0.5, we know from the
result concerningfair gambles given in the
paper that
(A8)
fJ2Q(a2)
+(uI rl+3)QQ
(H pA +
:i3a3)
<o.
Combining(A7) and (A8):
(A9) IIIQ(al)+ LL2Q(a2)+r13Q(a3) <0,
that is, P
<
CaseIV:
$.
a2, a3>0>a,; L >~O.5
A similarargumentto that given for Case II
above establishesthat (Al) is positive, that
is, P
>- $.
JUNE 1983
REFERENCES
Grether,David and Plott, Charles,"Economic
Theory of Choice and the Preference Reversal Phenomenon," American Economic
Review, September 1979, 69, 623-38.
Lichtenstein,Sarah and Slovic, Paul, "Reversal
of Preferences Between Bids and Choices
in Gambling Decisions," Journal of Experimental Psychology, July .1971, 89, 4655.
Lindman, Harold, "Inconsistent Preferences
Among Gambles," Journal of Experimental Psychology, August 1971, 89, 390-97.
Loomes, Grahamand Sugden, Robert, "Regret
Theory: An Alternative Theory of Rational Choice Under Uncertainty," EconomicJournal, December 1982, 92, 805-24.
Schoemaker, Paul, "The Expected Utility
Model: Its Variants, Purposes, Evidence
and Limitations," Journal of Economic
Literature, June 1982, 20, 529-63.
Von Neumann,John and Morgenstern,Oskar,
Theory of Games and Economic Behavior,
Princeton: Princeton University Press,
1947.