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REGRET THEORY: A BOLD ALTERNATIVE TO THE
ALTERNATIVES*
Han Bleichrodt and Peter P. Wakker
In their famous 1982 paper in this JOURNAL, Loomes and Sugden introduced regret theory. Now,
more than 30 years later, the case for the historical importance of this contribution can be made.
Until the late 1970s, economists focused on the rational homo economicus, not only for
normative but also for descriptive purposes. It was well understood that there were
many empirical deviations from rationality, as signalled, for example, by preference
reversals (Lichtenstein and Slovic, 1971; Lindman, 1971; Grether and Plott, 1979). But
it was believed that irrational behaviour was too chaotic to be modelled and should just
be taken as noise. For example, Arrow (1951, p. 406) wrote:
In view of the general tradition of economics, which tends to regard rational
behavior as a first approximation to actual, I feel justified in lumping the two
classes of theory [normative and descriptive] together.
The early 1980s saw a big shift, first in decision under risk, the topic of this article,
and then in other fields including intertemporal choice, game theory and ambiguity
(unknown probabilities). Kahneman and Tversky (1979) provided the first model of
decision under risk that explicitly and deliberately deviated from the rational expected
utility of homo economicus, but that could still be sufficiently tractable to permit
economic modelling and predictions. Unfortunately, their model had some theoretical
problems. It led their student Chew Soo Hong to co-author the unpublished Chew and
MacCrimmon paper (1979), followed up by Chew (1983), with the first theoretically
sound and axiomatised non-expected utility model. It also led John Quiggin (1982),
then an unknown Australian student, to introduce his now famous rank-dependent
utility. Machina (1982) gave a further boost to non-expected utility by providing
constructive generalisations of optimality results. With the exception of Kahneman and
Tversky, the aforementioned authors did not restrict their model to descriptive
applications but also claimed a normative status of their models.
All the aforementioned generalisations maintained one of the most basic assump-
tions of economic optimisations: transitivity. Transitivity underlies the axioms of
revealed preference for choices between multiple options. As good things often come
*Corresponding author: Peter P. Wakker, Erasmus School of Economics, Erasmus University Rotterdam,
P.O. Box 1738, Rotterdam, 3000 DR, the Netherlands. Email: wakker@ese.eur.nl
This paper benefited from inputs from Mark Machina and comments from Graham Loomes and Robert
F. Sugden.
This is an open access article under the terms of the Creative Commons Attribution-NonCommercial
License, which permits use, distribution and reproduction in any medium, provided the original work is
properly cited and is not used for commercial purposes.
[ 493 ]
The Economic Journal, 125 (March), 493–532. Doi: 10.1111/ecoj.12200 ©2015 The Authors. The Economic Journal published by John W iley & Sons Ltd on behalf of
Royal Economic Society. Published by John Wiley & Sons, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main Street, Malden, MA 02148, USA.
in threes, so it happened in 1982 when three papers independently proposed a theory
that gave up transitivity: regret theory. One paper, Fishburn (1982), focused on
mathematical and axiomatic elaborations. The second paper, Bell (1982), focused on
decision analytic applications, taking regret as an extra attribute of consequences. The
third paper, Loomes and Sugden (1982; LS henceforth), the topic of this review,
focused on conceptual features and interpretations and most clearly described the
empirical and normative status of regret theory. The three papers reinforced each
other, with cross-references and mutual recognitions from the beginning.
Good ideas usually do not appear out of the blue but grow from seeds planted
before. While the linguistic and the psychological concept of regret have existed for
ages and have been studied in psychology for over a century (see Zeelenberg and
Pieters, 2007 and their references), formal roles in decision theory have appeared since
the 1950s. LS cite Savage’s (1951) minimax regret theory and Fishburn (1988, p. 274)
cites the bilinear mathematical functional of Kreweras (1961)
1
as a predecessor. Yet, it
was not until 1982 that a complete decision theory of regret became available.
1. Regret Theory and Expected Utility
The LS paper, reproduced in this issue, gives a careful exposition of regret theory and its
full details, with motivations and discussions added. The high quality and depth of their
presentation has made the paper a classic.Our presentation aims to be didactical,focusing
on the simplest and most popular special case of regret theory and on the simplest
implications. Although our notation and terminology is usually as close as possible to LS,
in a few instances we deviate and use conventions that are common in the field today.
S={s
1
,... ,s
n
}denotes a state space, assumed finite for simplicity. Exactly one state
s
j
is true but a decision-maker is uncertain which state that is. Throughout this article,
we use an example of an urn containing 100 balls numbered 1–100. One ball is drawn
randomly. The true state of nature is the number of the ball actually drawn and
S={1, ... , 100}, so that n=100. Subsets of Sare events, which are true if they contain
the true state of nature. Thus, the event odd is {1, 3, ... ,99}.Actions, with generic
notation A, specify for each state swhat the consequence A(s) (money amount) is if sis
true. In the example, a bet Aon event odd, yielding £2ifsis odd and nothing
otherwise but costing £1, would be the action Asuch that A(s)=1 whenever sis odd
and A(s)=1 whenever sis even. We assume that Sis endowed with a probability
measure P, and write p
j
=P(s
j
). In the example, every number has probability 1/100,
and every event with jstates has probability j/100.
By ¤we denote the preference relation of the decision-maker over actions, with
strict preference ≻, indifference ~and reversed preferences ^and ≺as usual. The
most used model of decision under uncertainty is expected utility (EU). We then have
A1¤A2,X
n
j¼1
pjC½A1ðsjÞ X
n
j¼1
pjC½A2ðsjÞ (1)
1
Fishburn learned about this work in French from personal communication with the French economist
Denis Bouyssou.
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494 THE ECONOMIC JOURNAL [MARCH
for all actions A
1
,A
2.
Here, Cdenotes the utility function, which is subjective. The
probabilities may be objective, as in the example, but in the absence of objective
information they are subjective. We can rewrite (1) as
A1¤A2,X
n
j¼1
pjfC½A1ðsjÞ C½A2ðsjÞg 0:(2)
Table 1 illustrates a pair of actions, with M denoting million,A
R
designating a risky
action and A
S
designating a safe action. In the Table, A
R
yields 5M for ball numbers 1–
10, 1M for numbers 11–99 and 0 for number 100.
Although A
R
has the higher expected value, most people prefer A
S
because of its
safety, with no risk of ending up with 0. The regret of having missed a sure £1M if ball
100 is drawn is unbearable to many people. The preference for A
S
can be
accommodated by Bernoulli’s (1738) EU. Scaling C(0) =0, substitution readily shows
that the preference A
S
≻A
R
then holds if and only if
Cð1MÞ=Cð5MÞ[10=11;(3)
reflecting diminishing marginal utility.
We now consider a general choice situation between two actions A
1
and A
2.
Regret
theory generalises expected utility by assuming that the utility C[A
1
(s
j
)] experienced
under A
1
is affected by what would have happened had A
2
been chosen instead of A
1
,
and vice versa. People feel regret about A
1
(s
j
) if the result of the alternative choice,
A
2
(s
j
), had been better. Because of this regret, under choice A
S
, in Table 1, people may
feel less happy if ball 1–10 is drawn than if ball 11–99 is drawn, even though the same
consequence, 1M, results in all these cases. If ball 1–10 is drawn, then winning £5M has
been forgone due to an own decision, which arouses regret and reduces happiness
relative to balls 11–99.
The other side of the coin of regret is rejoicing, felt if the most favourable
consequence under some state s
j
has resulted. After a choice A
S
, people will rejoice if
ball 100 is selected and, for the preference assumed in Table 2, this rejoicing is enough
to prefer A
S
, despite the regret felt for balls 1–10.
Regret theory holds if, for general actions A
1
and A
2
, we have
A1¤A2,X
n
j¼1
pjQ fC½A1ðsjÞ C½A2ðsjÞg 0:(4)
The strictly increasing function Qcaptures the utility difference, but also the regret
and rejoicing experienced at A
1
(s
j
) and A
2
(s
j
). Rejoicing being the other side of the
regret-coin is captured by setting Q(x)=Q(x). This equality ensures consistency of
Table 1
First Choice
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2015] REGRET THEORY 495
overweight omission relative to commission (Ritov and Baron, 1995)? Should one-
sided legal liability be imposed on doctors to induce such regret and overweighting
externally, at the cost of societal efficiency? Should tests for Down syndrome and
vaccinations that demonstrably reduce the mortality rate be provided to the general
public even though they may lead to lifelong emotions of regret that would not
have occurred otherwise (Ritov and Baron, 1990; Murray and Beattie, 2001)? Or in
another domain, should seeding hurricanes be forbidden if it leads to regret
with some parties affected, even though total damage is reduced (Howard et al.,
1972)?
The 2000s has seen the emergence of neuroeconomics which has led to new insights
into regret. Camille et al. (2004) find that the orbitofrontal cortex has a fundamental
role in mediating regret and that people with lesions in the orbitofrontal cortex
7
who
do not experience regret make worse decisions than normal subjects who do anticipate
regret. Giorgetta et al. (2013) found different neural localisations for regret and
disappointment.
Bleichrodt et al. (2010) developed methods to obtain precise quantitative measure-
ments of the parameters of regret theory. These measurements allow us to derive exact
predictions, for example, about how much more supply is needed next year if regret is
increased by advertisement campaigns. To illustrate another application, in Example 2
we showed that there are values of Cand Qfor which regret theory predicts preference
reversals. By measuring these values individually, we can predict exactly when
preference reversals will occur for each subject and we can then test whether they
actually do (Baillon et al., 2014).
Whereas regret theory accommodates intransitivities by allowing state-wise com-
parisons of consequences, it maintains the classical linear weighting of probabilities.
Two recent approaches relax the latter assumption. Loomes’s (2010) new model, the
perceived relative argument model, is a rich model defined for the probability
triangle and uses paired comparisons of consequences like regret theory, but it also
uses similar paired comparisons of probabilities. It can explain many empirical
regularities including the aforementioned similarity cycles that are inconsistent with
regret theory.
Bordalo et al.’s (2012) salience theory uses pairwise comparisons of consequences
to readjust the weights (salience), rather than utility differences, of states of nature.
As did LS, salience theory assumes that large differences are overweighted but it
does not use an extremity overweighting function Qfor differences of utilities to
model this. Instead, it assumes that the salience function overweights the states of
nature that have large utility differences for the actions under consideration.
Salience theory shares the implication of the sure-thing principle with LS, with the
salience of state snot affected by consequences outside s. As with regret theory, the
novelty of salience theory resides in where it violates transitivity. Unlike LS, Bordalo
et al. (2012) did not analyse or discuss intransitivities extensively, but left this to
future work.
7
People with lesions in the orbitofrontal cortex are not emotionally unresponsive as they did experience
disappointment.
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506 THE ECONOMIC JOURNAL [MARCH
and Tversky, 1979; see also LS p. 812).
2
Other scenarios, violating EU but not regret
theory, can be triggered in other setups, which we discuss in the next Section.
2. Regret Theory’s Deviations from Expected Utility
2.1. A Deviation Illustrating the Regret Functional
This subsection discusses a theoretical deviation of regret theory from EU to illustrate
the nature of the regret functional. The next subsections present empirical implica-
tions. Imagine that x
0
,... ,x
4
is an increasing sequence of outcomes that is equally
spaced in Cunits. That is,
Cðx4ÞCðx3Þ¼ ¼Cðx1ÞCðx0Þ[0:3(6)
We denote these utility differences by d. Consider the two actions in Table 3.
Under expected utility, the two actions are equivalent because the Cdifference for
balls 51–75 is twice as big as the Cdifference for balls 1–50 but it has half the
probability. However, many decision-makers may prefer the lower action A
ℓ
. They
regret the small utility loss (x
3
instead of x
4
) after choosing A
ℓ
(balls 1–50) much less
than the double and more salient utility loss (x
0
instead of x
2
) after choosing A
u
(balls
51–75). This is captured by 2Q[C(x
4
)C(x
3
)] <Q[C(x
2
)C(x
0
)]. That is:
2QðdÞ\Qð2dÞ:(7)
This condition is satisfied by functions Qthat are convex on Rþ(and, hence, concave
on R).
The reversed preference A
u
≻A
ℓ
can also be accommodated by regret theory. Some
decision-makers may prefer A
u
because the probability of regret is only small (0.25 for
balls 51–75), whereas the probability of regret is higher for A
ℓ
(0.5 for balls 1–50). Such
decision-makers do not discriminate much between utility losses dand 2dand for them
the inequality in (7) is reversed. The most common case, however, is (7). It was recently
confirmed empirically by Bleichrodt et al. (2010) and it is mostly assumed by LS (end
of their Section II). Then extreme utility differences are salient and are overweighted.
We now turn to some empirically important deviations from EU.
Table 3
Violation of Expected Utility Explained by Regret Theory
2
Loomes and Sugden (1998), in yet another critical test of their theory, still found violations here,
providing evidence against their theory. Birnbaum (2008, p. 481 ff.) also reports some violations.
3
Bleichrodt et al. (2010) demonstrated that these equalities can be revealed from preferences as follows.
Using obvious notation, we measure indifferences (odd: x
j+1
, even: g)~(odd: x
j
, even: G) for j=0, ... ,3,
and outcomes G>gconveniently chosen. Equation (4) then implies Q[C(G)C(g)] =Q[C(x
j
+1) C
(x
j
)] for all j. Because Qis strictly increasing, (6) follows.
©2015 The Authors.
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2015] REGRET THEORY 497
2.2. Violating the Equivalence Axiom
See Table 4 (LS 6), where the subscript din A
d
refers to decreasing outcomes, whereas
this is not the case for A
n
.
Under the common assumption that large utility differences are overweighted, the
superiority of A
n
for balls 76–100 decides and A
n
is preferred. Convexity of Qfor gains
implies this preference under regret theory:
0:25 Q½Cð30ÞCð20Þ þ 0:25 Q½Cð20ÞCð10Þ þ 0:25 Q½Cð10ÞCð0Þ\
0:25 Q½Cð30ÞCð0Þ:
However, A
i
and A
n
induce the same probability distribution over outcomes!
Apparently, such actions need not be equivalent under regret theory. Under EU, to the
contrary, they must be equivalent, which LS (p. 818) calls the equivalence axiom. This
requirement appears most clearly from (1). Then the correlation of the two actions,
and the particular matching of their outcomes, is immaterial. It does not matter if A
n
resulted from another independent drawing from the urn. By contrast, the matching
of outcomes is crucial for regret theory, as is shown in (4). As surprising as this
implication may be, it is a natural consequence if we experience regret. This point will
be further discussed in the next subsection.
2.3. Accommodating the Allais Paradox,and a Comparison with Other Non-expected Utility
Theories
Papers on non-expected utility of the 1980s usually started with a description of the
Allais paradox and then showed how a newly introduced model could accommodate it.
We now show how regret theory can accommodate this paradox. Consider a variation
in Table 2, called the independent variation, where the lower action A
S
is generated by a
second, independent, drawing from the urn. Under EU’s equivalence axiom, this
change should not affect preference. However, under regret theory it may matter,
because the matching of the outcomes changes and, hence, regret effects will change.
We use a simple Qfunction to illustrate the basic idea. Imagine that the decision-maker
feels no strong regret for utility losses up to C(£0)–C(£1M) and C(£1M)–C(£5M),
and Qis close to linear for such and smaller losses. However, larger losses such as C
(£0)–C(£5M) exceed a tolerance threshold and result in strong regret. Choosing A
S
risks experiencing such strong regret because, given the independence of the two
actions and unlike the original choice situation in Table 2, outcome 5M for A
r
and
outcome 0 for A
S
can occur simultaneously (with probability 0.10 90.89 =0.089). If
the regret Q[C(£0) C(£5M)] is strong enough, then A
r
will be preferred.
Table 4
Violation of the Equivalence Axiom by Regret Theory
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498 THE ECONOMIC JOURNAL [MARCH
i982] REGRET THEORY 807
However, in the next section we shall outline the framework
of an alternative
theory which not only explains the reflection effect and simultaneous
gambling
and insurance, but also predicts the behaviour described
in (a), (b) and (c). We
shall then argue that, besides
being predictable,
such behaviour
can be defended
as rational, and that our model therefore
provides the basis for an alternative
theory of rational choice under uncertainty.
II. THE FRAMEWORK OF AN ALTERNATIVE THEORY
We consider an individual in a situation where there is a finite number, n, of
alternative states
of the
world,
any one of which might occur. Each state
j has a
probability
pj where
o < pj < I and
p1 + ... +?p = I. These probabilities
may
be
interpreted either as objective probabilities known to the individual or, in the
absence of firm knowledge of this kind, as subjective probabilities
which repre-
sent the individual's degree of belief or confidence in the occurrence of the
corresponding states. The individual's problem is to choose between actions.
Each action is an n-tuple of consequences,
one consequence for each state of the
world. We shall write the consequence
of the ith action in the event that the
jth
state occurs as xij. Consequences
need not take the form of changes in wealth,
although in our applications
of our theory, we shall interpret
xij as an increment
or decrement of wealth, measured
relative to some arbitrary
level (which need
not be the individual's current wealth). Notice that actions, unlike prospects,
associate consequences with particular states of the world. Thus a number of
different actions might correspond
with the same prospect. We shall recognise
this difference
by using the symbol A for actions, reserving
X for prospects.
Thus
far, our theory has a close resemblance to Savage's, except in that we take
probabilities
as given, just as von Neumann and Morgenstern
do.
A choice problem may involve any number of available actions, but we shall
begin by analysing problems where there is only a pair of actions to choose
between. All of Kahneman and Tversky's evidence concerns the behaviour of
people choosing between pairs of prospects. Choices between three or more
actions raise some additional issues, which we shall discuss
in Section IV.
Our first assumption
is that for any given individual there is a choiceless
utility
Junction
C(.), unique up to an increasing
linear transformation,
which assigns
a
real-valued utility index to every conceivable consequence. The significance
of
the word 'choiceless' is that C(x) is the utility that the individual would derive
from the consequence
x if he experienced it without
havilg
chosen
it. For example,
he might have been compelled to have x by natural forces,
or x might have lbeen
imposed on him by a dictatorial government. Thus -in contrast to the von
Neumann-Morgenstern concept of utility - our concept of choiceless utility is
defined independently of choice. Our approach is utilitarian in the classical
sense. What we understand by 'choiceless utility' is essentially what Bernoulli
and Marshall understood
by 'utility' - the psychological experience
of pleasure
that is associated
with the satisfaction of desire. We believe that it is possible to
introspect about utility, so defined, and that it is therefore meaningful to talk
about utility bcing experienced
in choiceless
situations.
©2015 The Authors.
The Economic Journal published by John Wiley & Sons Ltd on behalf of Royal Economic Society.
2015] REGRET THEORY 515
insights (see their Table 1) while maintaining tractability. Cubitt and Sugden (1998,
p. 761) argue that giving up transitivity is like giving up separability but in another
direction than when giving up the sure-thing principle. Although accommodating the
Allais paradox was not LS’s main goal, they could still do so.
2.4. Violating Transitivity
We now turn to the main goal of LS, incorporating the boldest and most controversial
deviation from classical models available in the literature. Consider Table 5, an
extension of Table 4 (combining LS’s Tables 4 and 6), which is obtained by further
shifts of outcomes. Each preference in the Table follows from the same line of
reasoning as used in Table 4. The largest regret of 0 versus £30M each time overrules
the multiple smaller regrets in the other direction. A preference cycle results and
transitivity is violated. LS thus challenged one of the most standard assumptions
of economic optimisations. They provided detailed arguments against transitivity
(pp. 820–22), extended in later papers (Sugden, 1991, pp. 760–61).
Loomes and Sugden (1987b, beginning of Section 4) and Sugden (2004, Section
II.7) showed that regret theory deviates from expected utility and can bring new
phenomena only where it deviates from transitivity. Luce and Raiffa (1957, pp. 280–82)
explained a similar point for earlier forms of regret. Hence, the violations of transitivity
are central to regret theory. A generalisation is in Bikhchandani and Segal (2011,
Theorem 1).
3. Empirical Support for Regret Theory
Regret theory received much support during the first decade after its introduction.
Most empirical studies, several by Loomes and Sugden in collaboration with Chris
Starmer, confirmed the predictions of the theory.
EXAMPLE 1. Loomes (1988a) tested the juxtaposition effects described in the
preceding Section by asking subjects to state the money amount £a
0
for which they
were indifferent between the two actions in Table 6.
Table 5
A Preference Cycle Implied by Regret Theory
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500 THE ECONOMIC JOURNAL [MARCH
Next in a second problem, subjects were asked to state the money amount £a
1
for
which they were indifferent between the two actions in Table 7.
Any theory based on the equivalence axiom predicts that a
0
=a
1
. Regret theory
makes a different prediction. The proof of the following claim is in the Appendix.
CLAIM 1. Under regret theory with Q convex for gains, a
1
>a
0
.
Loomes (1988a) indeed found that the average value of a
1
was much larger than the
average value of a
0
(£22.58 versus £17.52), confirming the prediction of regret theory.
Other studies on juxtaposition effects that supported regret theory include Loomes
and Sugden (1987a), Loomes (1988b, 1989), Starmer and Sugden (1989) and Starmer
(1992). Moreover, Loomes et al. (1992) confirmed violations of stochastic dominance
predicted by regret theory.
A particularly desirable feature of regret theory is that it can explain preference
reversals (PR). PRs were first discovered by Lichtenstein and Slovic (1971) and
Lindman (1971), and were brought to the attention of economists by Grether and
Plott (1979). PRs occur when subjects are confronted with two prospects, a £-bet which
offers a relatively large sum of money, but a relatively small probability of winning, and
a P-bet, which offers a more modest sum of money, but a greater probability of
winning. Subjects are then asked to perform three tasks: to choose between the two
prospects, and to attach a certainty equivalent to each prospect. The typical finding is
that subjects prefer the P-bet, while paradoxically, the £-bet is given the higher
valuation. The opposite pattern, choosing the £-bet but valuing the P-bet higher, is
rarely observed.
Preference reversals challenge those who wish to explain economic behaviour in
terms of rational theories of choice. Psychologists often interpreted PRs as evidence
that individuals do not have a single system of preferences and respond differently to
choice and valuation tasks (Slovic and Lichtenstein, 1983; Tversky et al., 1988, 1990).
Regret theory provides a different interpretation based on intransitive preferences as
Table 7
Second Choice
Table 6
First Choice
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2015] REGRET THEORY 501
8i8 THE ECONOMIC JOURNAL [DECEMBER
assume that the individual has a preference ordering over prospects. Thus two
of the fundamental principles of expected utility theory are retained: that pair-
wise choices are transitive and that courses of action associated with identical
probability distributions of consequences are equivalent to one another. (We
shall call this latter principle the equivalence axiom.) Allais and Machina break
away from expected utility theory by dropping the independence axiom; given
that the equivalence axiom is retained, this amounts to abandoning the sure-
thing principle. Our strategy is radically different: we retain the sure-thing
principle while jettisoning both the equivalence axiom and the transitivity
axiom. As a result we are able to predict the isolation effect in two-stage gambles,
a form of observed behaviour that contravenes the equivalence axiom and
therefore cannot be explained by either Allais or Machina. We are also able to
predict the systematic occurrence of the reflection effect. Although Allais's and
Machina's theories are not contradicted by the reflection effect, they do not
predict it.
Fishburn's model is more like regret theory (although he does not mention
any notion of regret) in that he also drops the transitivity axiom. However, his
model is presented in terms of prospects rather than actions, and therefore does
not accommodate the isolaticn effect. On the other hand, if we restrict ourselves
to statistically independent prospects (and Fishburn does so - see his p. 9), then
our theory and his basic axioms are compatible, and provide an interesting
example of how an axiomatic treatment and a more introspective psychologically-
based approach may complement each other.'
However, having indicated that our theory provides certain predictions and
explanations that the other theories mentioned do not, we should make it clear
that we are not claiming that regret theory can explain all of the behavioural
regularities revealed by experimental research into choice under uncertainty. So
far we have focused on a number of patterns of behaviour observed by Kahneman
and Tversky; but we have not dealt with every one of their observations, still less
with the vast amount of evidence accumulated by other researchers.
Some of the experimental findings do not appear to be completely consistent.
In relation to this paper, the most significant case concerns the reflection effect.
Hershey and Schoemaker (i 980 a) and Payne et al. (i 980) have published results
that show this effect to be not nearly as strong or as general as Kahneman and
Tversky's evidence suggests. However, this nmay
not present any great difficulties
for regret theory since, as we noted in Section III (d), the general prediction of the
reflection effect requires C(.,) to be linear. Instances in which the reflection effect
is weak or absent may well be explicable if C(.) is assumed to be concave.
There are nevertheless certain observations that simply cannot be accounted
for by regret theory in the form presented here. One example is the 'framing'
effect discussed by Tversky and Kahneinan (I 98 I) and the very similar 'context'
effect observed by Hershey and Schoemaker (i q8o b). In these cases exactly the
I At a late stage, we have received a copy of a Working Paper by David E. Bell (I98I) which is of
great interest. Quite independently he has developed a model which also explicitly incorporates a notion
of regret, using multi-attribute utility theory along the lines suggested by Keeney and Raiffa (1976).
We note that when both models are applied to the same phenomena - the original Allais paradox,
simultaneous insuring and gambling, and the reflection effect - the conclusions are strikingly similar.
©2015 The Authors.
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526 THE ECONOMIC JOURNAL [MARCH
Another challenge to regret theory came from other studies, starting with Tversky
(1969), that observed systematic cycles that could not be explained by regret theory.
Consider the three actions in Table 9, which is problem T’U’V’ in Day and Loomes
(2010). Regret theory is consistent with the cycle C≻B≻A≻Cbut not with the
opposite cycle. However, Day and Loomes (2010) observed that this opposite cycle
prevailed. Their finding can be explained by Rubinstein’s (1988) similarity theory.
When comparing Aand B, people may consider that the 5% extra probability that B
offers is so small that they pay little or no attention to the probability dimension and
instead concentrate on the dissimilar pay-off dimension and choose A. Likewise, they
consider the winning probabilities of Band Cto be similar and choose B. However,
they may also find that the 10% difference in winning probability between Aand Cis
large enough to make Aand Clook dissimilar on the probability dimension and this
may shift their attention back to the probability dimension and they then choose C.
Lindman and Lyons (1978), Budescu and Weiss (1987), Leland (1994, 1998), Mellers
and Biagini (1994), Bateman et al. (2007) and Day and Loomes (2010) reported
evidence for such similarity cycles, which cannot be explained by regret theory.
Starmer (1999) reported a comparable cycle although he did not explain it by
similarity but by original prospect theory (Kahneman and Tversky, 1979).
Yet another challenge came from mathematical psychologists. Starting with
Iverson and Falmagne (1985), several papers showed that asymmetric cycles need
not necessarily be inconsistent with transitive preferences if the stochastic nature
of human preferences is taken into account (overviewed by Regenwetter et al.,
2011). Even though they mainly concentrated on the similarity cycles observed by
Tversky (1969) and showed that these could be explained by transitive
preferences with error, their objections also applied to the regret cycles that
were observed.
However, a serious blow to regret theory came from Starmer and Sugden (1993).
They discovered that previously observed support for regret theory could, to a large
extent, be explained by event-splitting effects by which splitting an event with a given
consequence into two sub-events increases its weight.
EXAMPLE 3. Consider the four problems in Tables 10–13. According to regret
theory, Problems I and III are equivalent and so are Problems II and IV. Regret theory
predicts that choices AB0(Ain Problems I and III and B0in Problems II and IV) will
occur more often than choices BA0. However, according to event splitting, choices AB0
should be more likely than choices BA0in Problems I and II, suggesting regret effects,
Table 9
Opposite Cycle
©2015 The Authors.
The Economic Journal published by John Wiley & Sons Ltd on behalf of Royal Economic Society.
2015] REGRET THEORY 503
but not in Problems III and IV: in Problem I the £7 is not split, whereas in Problem II it
is, which may make B0appear more attractive, but both in Problems III and in Problem
IV the £7 is split.
The prediction of event splitting was, indeed, what Starmer and Sugden (1993)
observed: clear regret effects in Problems I and II but no effects in Problems III and IV.
Their study suggested strong event-splitting effects and weaker regret effects (see also
Humphrey, 1995). On the other hand, Starmer and Sugden (1998) found that not all
regret effects were due to event-splitting effects but that some were mainly due to
framing, as had been suggested before by Harless (1992).
Table 10
Problem I
Table 11
Problem II
Table 12
Problem III
Table 13
Problem IV
©2015 The Authors.
The Economic Journal published by John Wiley & Sons Ltd on behalf of Royal Economic Society.
504 THE ECONOMIC JOURNAL [MARCH
Starmer and Sugden subjected ‘their’ regret theory to rigorous testing
6
and thereby
discovered the remarkable fact that splitting states can make prospects substantially
more attractive. Camerer (1995, pp. 655–56) praised the authors’ work on regret
theory, and the resulting progress of our understanding, and wrote ‘this is a story of
successful detective work’.
5. Recent Applications
Even though event-splitting effects may provide an alternative explanation for some of
the phenomena that led to the introduction of regret theory, as Starmer (2000, p. 376)
notes ‘insights from [regret theory] have proved useful in understanding real
behaviour’. The authors of this article benefited from regret theory’s insight that
pairs of outcomes for different actions provide a natural basis for decision-making and
used this idea in trade-off techniques (Bleichrodt et al., 2010; Wakker, 2010). This
insight was also used by Bouyssou and Pirlot (2003, especially table 1) and Vind (2003).
The 2000s have witnessed many applications that either use regret theory or extensions
of the model, with LS cited as a source of inspiration. For example, Barberis et al.
(2006) use regret theory to explain the stock market participation puzzle: few people
invest in stocks even though rational economic theory predicts that they should. Other
applications of regret models to financial decisions include Muermann et al. (2006)
and Michenaud and Solnik (2008) who study asset allocation decisions.
Braun and Muermann (2004) apply regret theory to the demand for insurance and
show that regret theory can explain the frequently observed preference for low
deductibles. Smith (1996) applies regret theory to health and Perakis and Roels (2008)
use it in the newsvendor model. Filiz-Ozbay and Ozbay (2007) and Engelbrecht-
Wiggans and Katok (2008) explain how regret theory can explain overbidding in first
price auctions. Other regret models include Sarver (2008) and Hayashi (2008). These
models differ from LS in that they study preferences over menus, i.e. sets of prospects,
in which decision-makers experience regret if their choice turns out to be inferior ex
post.
A critical aspect of regret is the extent to which decision-makers, after their choices,
are informed about the outcomes that would have resulted had they chosen differently.
This issue has been explored in the experimental and theoretical literature on
feedback-conditional regret. It has been found that people prefer options which screen
them from discovering the outcome of forgone choices. The anticipated pain of regret
is reduced or eliminated if people do not know the outcome of the forgone choice.
Thus, the option of not entering a lottery is more attractive if, conditional on not
entering, one will never know whether one would have won or lost. This tendency is
exploited in postal code lotteries in which a postal code rather than an anonymous
number is drawn (Zeelenberg, 1999; Humphrey, 2004).
Regret theory has been widely applied in the health domain, raising fundamental
ethical questions. Should doctors be allowed to use excessive diagnostic testing just
to avoid the regret about missing the occasional serious case, just because they
6
As was done in many papers by Loomes and Sugden, including Loomes and Sugden (1998).
©2015 The Authors.
The Economic Journal published by John Wiley & Sons Ltd on behalf of Royal Economic Society.
2015] REGRET THEORY 505
overweight omission relative to commission (Ritov and Baron, 1995)? Should one-
sided legal liability be imposed on doctors to induce such regret and overweighting
externally, at the cost of societal efficiency? Should tests for Down syndrome and
vaccinations that demonstrably reduce the mortality rate be provided to the general
public even though they may lead to lifelong emotions of regret that would not
have occurred otherwise (Ritov and Baron, 1990; Murray and Beattie, 2001)? Or in
another domain, should seeding hurricanes be forbidden if it leads to regret
with some parties affected, even though total damage is reduced (Howard et al.,
1972)?
The 2000s has seen the emergence of neuroeconomics which has led to new insights
into regret. Camille et al. (2004) find that the orbitofrontal cortex has a fundamental
role in mediating regret and that people with lesions in the orbitofrontal cortex
7
who
do not experience regret make worse decisions than normal subjects who do anticipate
regret. Giorgetta et al. (2013) found different neural localisations for regret and
disappointment.
Bleichrodt et al. (2010) developed methods to obtain precise quantitative measure-
ments of the parameters of regret theory. These measurements allow us to derive exact
predictions, for example, about how much more supply is needed next year if regret is
increased by advertisement campaigns. To illustrate another application, in Example 2
we showed that there are values of Cand Qfor which regret theory predicts preference
reversals. By measuring these values individually, we can predict exactly when
preference reversals will occur for each subject and we can then test whether they
actually do (Baillon et al., 2014).
Whereas regret theory accommodates intransitivities by allowing state-wise com-
parisons of consequences, it maintains the classical linear weighting of probabilities.
Two recent approaches relax the latter assumption. Loomes’s (2010) new model, the
perceived relative argument model, is a rich model defined for the probability
triangle and uses paired comparisons of consequences like regret theory, but it also
uses similar paired comparisons of probabilities. It can explain many empirical
regularities including the aforementioned similarity cycles that are inconsistent with
regret theory.
Bordalo et al.’s (2012) salience theory uses pairwise comparisons of consequences
to readjust the weights (salience), rather than utility differences, of states of nature.
As did LS, salience theory assumes that large differences are overweighted but it
does not use an extremity overweighting function Qfor differences of utilities to
model this. Instead, it assumes that the salience function overweights the states of
nature that have large utility differences for the actions under consideration.
Salience theory shares the implication of the sure-thing principle with LS, with the
salience of state snot affected by consequences outside s. As with regret theory, the
novelty of salience theory resides in where it violates transitivity. Unlike LS, Bordalo
et al. (2012) did not analyse or discuss intransitivities extensively, but left this to
future work.
7
People with lesions in the orbitofrontal cortex are not emotionally unresponsive as they did experience
disappointment.
©2015 The Authors.
The Economic Journal published by John Wiley & Sons Ltd on behalf of Royal Economic Society.
506 THE ECONOMIC JOURNAL [MARCH
6. Discussion
LS were not only bold in taking issue with some of the most widely accepted
assumptions in decision theory, transitivity and the equivalence axiom but also in
their interpretations, showing insights ahead of their time. When their paper was
published in the early 1980s, a strict ordinal revealed preference view was dominant
in economics. Utility modelled decisions and nothing else. Introspective interpre-
tations were not made. The situation has changed today, with Kahneman (1994),
Loewenstein and Ubel (2008, Section 2) and others pleading for broader
interpretations of utility, and with introspective happiness studies, a popular and
influential field in economics (van Praag and Ferrer-i-Carbonell, 2004; Diener and
Biswas-Diener, 2008; Benjamin et al., 2014). Although LS could have avoided
introspection because, as they show in their Appendix, all components in their
model can be revealed from preferences, they chose a psychologically sound
interpretation of their model: They interpreted the function Cin Section 3 as the
inherent utility, resulting when the individual experiences a consequence ‘without
having chosen it’ (emphasis in original). Then regret or rejoicing plays no role. Ccan
be felt through introspection. Hence, LS used the term choiceless utility for C. Next,
in a second stage, regret comes in, captured through the function Q. LS (Section V,
2nd para.) explicitly distanced themselves from a narrow empirical approach to
preference theory.
In 1982, the prevailing hypothesis of prospect theory was a total reflection of
preference, with risk aversion for gains coupled with equally strong risk seeking for
losses. LS (Section III end) immediately predicted weaker, only partial, reflection with
risk seeking for losses weaker than risk aversion for gains. Their prediction has since
been confirmed empirically (surveyed by Wakker, 2010 Section 9.5). LS also carefully
presented evidence against regret theory (Section V, middle) and recommend
reference-dependent generalisations.
A limitation of regret theory, as of any intransitive theory of binary choice, is that it is
unclear how to extend the theory to choices among three or more actions. LS (Section
IV) provided the first ideas about such extensions, with defences against book making
and money pump criticisms in Section V; Loomes and Sugden (1987b) provided an
elaborated theory. A preference foundation is in Sugden (1993). Hayashi (2008)
suggested an alternative extension.
Although LS are firm on a normative status of regret theory and provide strong and
cogent arguments, the authors of this comment have different views. LS argue that
feelings of regret are a fact of life and that it is irrational to ignore them, a view
supported by Bourgeois-Gironde (2010) using neurodata. We are less tolerant and
more paternalistic about such feelings. In its everyday meaning, regret is a useful
emotion to signal possible improvements of future actions in situations of incomplete
information. The formal decision-theoretic meaning, however, is different. Consider
Table 2, with a choice of A
r
. A rational person should maximise happiness, given the
external constraints. The latter are the same if ball 11 is drawn as if ball 100 is drawn, in
both cases the consequence being £0. Having feelings of regret for ball 100 because of
the forgone £1M leads to harm for no good reason. We believe that such voluntary self-
harming is irrational.
©2015 The Authors.
The Economic Journal published by John Wiley & Sons Ltd on behalf of Royal Economic Society.
2015] REGRET THEORY 507
Note that, unlike in everyday life situations where regret can be a useful signal,
nothing can be learned from the ball drawn in Table 7, given that all probabilities and
consequences were known beforehand. We also assume complete modelling and,
hence, for instance, we assume that there are no outsiders blaming the decision-maker
after ball 100 was drawn. Taking any emotion as rational just because it exists is too
permissive and applies Hume’s adage ‘reason is, and ought only to be the slave of the
passions’ too leniently. Although we see no normative status for regret theory, it is
obvious that its descriptive value is huge, making it one of the most important
contributions to decision theory. LS’s careful arguments for the rationality of regret
theory, challenging something as basic as transitivity, are thought provoking and have
also produced many new insights.
7. Conclusion
In our perception, salient features of Sugden’s work during the last three decades have
been great originality and breadth, and salient features of Loomes’ work have been
great sharpness and depth. In retrospect, it is then no surprise that when these two
strong and complementary minds came together in 1982, something lasting resulted.
Appendix A. Proofs
Proof of claim 1. Informally, writing afor both a
0
and a
1
, in Table 7, 40 differences (£a versus
£0) and 40 differences (£0versus £12) of Table 6 have been replaced by 40 differences (£a versus
£12) and 40 differences (£0versus £0). By the extremity overweighting of Q, the removals of (£a
versus £0) count most, weakening the case for the upper prospect. Hence, a larger value a
1
is
needed in Table 7.
Formally, according to regret theory the first indifference implies
0:40 Q½Cða0ÞCð0Þ ¼ 0:60 Q½Cð12ÞCð0Þ:(A.1)
The second indifference implies
0:40 Q½Cða1ÞCð12Þ ¼ 0:20 Q½Cð12ÞCð0Þ:(A.2)
By Q’s extremity overweighting, Q[C(a
0
)C(0)] >Q[C(a
0
)C(12)] +Q[C(12) C(0)].
Hence 0.40 9Q[C(a
0
)C(12)] <0.40 9Q[C(a
0
)C(0)] 0.40 9Q[C(12) C(0)] =(by
(A.1)) 0.60 9Q[C(12) C(0)] 0.40 Q[C(12) C(0)] =0.20 9Q[C(12) C(0)]. Because Q
is strictly increasing, it follows that to obtain the equality in (A.2) we must have a
1
>a
0
.
Proof of claim 2. For contradiction, assume the opposite cycle P-bet ≺£-bet ≺c≺P-bet.
Then
0:30Q½Cð18ÞCð8Þ þ 0:30Q½Cð8Þ [0;(A.3)
0:30Q½Cð8ÞCðcÞ þ 0:30Q½Cð8ÞCðcÞ þ 0:40Q½CðcÞ [0;(A.4)
©2015 The Authors.
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508 THE ECONOMIC JOURNAL [MARCH
0:30Q½CðcÞCð18Þ þ 0:30Q½CðcÞ þ 0:40Q½CðcÞ [0:(A.5)
Adding the left-hand sides of (A.3)–(A.5) and using Q(x)=Q(x) for all x>0 gives
0:30fQ½Cð18ÞCð8Þ þ Q½Cð8ÞCðcÞ Q½Cð18ÞCðcÞgþ
0:30fQ½Cð8Þ þ Q½Cð8ÞCðcÞ þ Q½CðcÞg [0:(A.6)
Because Qoverweights extremes, the terms in square brackets are negative and we have a
contradiction.
Erasmus University Rotterdam
Submitted: 18 March 2014
Accepted: 18 June 2014
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©2015 The Authors.
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512 THE ECONOMIC JOURNAL [MARCH
The Economic
Journal, 92 (December
I982), 805-824
Printed in Great
Britain
REGRET THEORY: AN ALTERNATIVE THEORY
OF RATIONAL CHOICE UNDER UNCERTAINTY*
Graham
Loomes and Robert
Sugden
The main body of current economic analysis of choice under uncertainty is built
upon a small number of basic axioms, formulated in slightly different ways by
von Neumann and Morgenstern (I 947), Savage (1
954) and others. These
axioms are widely believed to represent the essence of rational behaviour under
uncertainty. However, it is well known that many people behave in ways that
systematically violate these axioms.'
We shall initially focus upon a paper by Kahneman and Tversky (I 979) which
presents extensive evidence of such behaviour. Kahneman and Tversky offer a
theory, which they call 'prospect theory ', to explain their observations. We shall
offer an alternative theory which is much simpler than prospect theory and
which, we believe, has greater appeal to intuition.
The following notation will be used throughout. The ith prospect is written as
Xi. If it offers increments or decrements of wealth xl, ..., x. with probabilities
Pi, .Pn (where p, + ... +pn = I) it may be denoted as (xi,pi; .. .; XwPn). Null
consequences are omitted so that the prospect (x,p; o, I -p) is written simply as
(x,p). Complex prospects, i.e. those which offer other prospects as consequences,
may be denoted as (Xi,pi; ...; Xn,pn). We shall use the conventional notation
>, > and to represent the relations of strict preference, weak preference and
indifference. We shall take it that for all prospects Xi and Xk, Xi > Xk or Xi < Xk;
but we shall not in general require that the relation > is transitive.
I. KAHNEMAN AND TVERSKY S EVIDENCE
Kahneman and Tversky's experiments offered hypothetical choices between
pairs of prospects to groups of university faculty and students. Table I lists a
selection of their results, which reveal three main types of violation of con-
ventional expected utility theory:
(a) The 'certainty effect' or 'common ratio effect', e.g. the conjunction of
X5 -<
X6 and Xg >-
Xlo and the conjunction X13 < AX14 and XA5
>_ X16.
There is also a 'reverse common ratio effect', e.g. the conjunction of
X7 >- X8 and
Xll -< A'12.
(b) The original 'Allais Paradox' or 'common consequences effect', e.g. the
conjunction of X1 -< X2 and X3 >- X4.
(c) The 'isolation effect' in two-stage gambles, e.g. the conjunction of
X9 >- XAO
and X17 < X18.
* W\e particularly wish to thank Michael Jones-Lee, Mark Machina and two anionymous referees
for many helpful suggestions and criticisms.
I For a survey and discussion of much of the evidence, see Allais and Hagen (I979) and Schoemaker
(1 980, I 982). [ 805 ]
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8o6 THE ECONOMIC JOURNAI [DECEMBER
Table I also reveals a 'reflection effect' where a change of sign on the con-
sequences is associated with a reversal of the modal preference and the risk
attitude that characterises it, e.g. X5 -< X6 and X7 >- X8. One instance of the
reflection effect, revealed in Problems I4 and 14', may be interpreted as an
Table I
Percentage
Kahneman of subjects Characterisation of
and Tversky with modal
problem no. Prospects offeredt Modal preference preference modal preference
I X1i- (2,500, o033; X1 -< X2 82* Risk averse
2,400, o-66)
X2 =(2,400, I OO)
2 X3= (2,500, 0.33) X3 >- X4 83* Not clear
X4= (2,400, 0-34)
3 X5= (4,000, o-8o) X5 .< X6 80* Risk averse
X6= (3,ooo, I 00)
3' X7 = (-4,000, o-80) X7 >- X8 92* Risk loving
X8 (- 3,000, I o00)
4 Xg= (4,000, 0.20) X9 >. X10 65* Not clear
Xo= (3,ooo, 0o25)
4 X = (-4,000, o020) Xll < X12 58 Not clear
X12= (-3,000, 0.25)
7 X13 = (6,ooo, 0-45) X13 < X14 86* Risk averse
X14= (3,000, 0.90)
8 X15 = (6,ooo, o-ooI) X15 >- X16 73* Risk loving
X16= (3,ooo, 0002)
IO X17 = (X5, 0.25) X17 .< X18 78* Risk averse
X18= (X6, 025)
14 X = (5,ooo, O.OOI) X19 >- X20 72* Risk loving
X2= (5, 1-ooo)
14 X21 = (-5,ooo, o.OOI) X21 e< X22 83* Risk averse
X2s = (- 5, I 000)
* Statistically significant at the o-oI level.
t Consequences are increments or decrements of wealth, measured in Israeli pounds.
example of simultaneous gambling and insurance, since Xlg >- X20 indicates a
willingness to enter an actuarially fair lottery offering a small probability of a
large prize, while X21
.< X22 signifies a willingness to take out actuarially fair
insurance against a small probability of a large loss. We also nlote an interesting
mixture of risk attitudes. Sometimes risk aversion is associated with problems
involving increments of wealth, e.g. X13 _< X14, and sometimes witb problems
involving decrements, e.g. X21
.< X22. Likewise, risk loving is sometimes associated
with problems involving increments, e.g. X15 >- X6, and sometimes witlh
problems
involving decrements, e.g. X7 >- Xg.
Simultaneous gambling and insurance, the reflection effect, and the mixture
of risk attitudes may all be accommodated by conventional expected utility
theory, though only at the cost of certain fairly arbitrary assumptions and some
rather unsatisfactory implications.' But no accommodation is possible for the
effects listed in (a), (b) and (c) above - the observations here simply violate one
or more of the conventional axioms.
See Friedman and Savage (1948), Markowitz (I952) and Hirschleifer (I966).
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514 THE ECONOMIC JOURNAL [MARCH
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However, in the next section we shall outline the framework
of an alternative
theory which not only explains the reflection effect and simultaneous
gambling
and insurance, but also predicts the behaviour described
in (a), (b) and (c). We
shall then argue that, besides
being predictable,
such behaviour
can be defended
as rational, and that our model therefore
provides the basis for an alternative
theory of rational choice under uncertainty.
II. THE FRAMEWORK OF AN ALTERNATIVE THEORY
We consider an individual in a situation where there is a finite number, n, of
alternative states
of the
world,
any one of which might occur. Each state
j has a
probability
pj where
o < pj < I and
p1 + ... +?p = I. These probabilities
may
be
interpreted either as objective probabilities known to the individual or, in the
absence of firm knowledge of this kind, as subjective probabilities
which repre-
sent the individual's degree of belief or confidence in the occurrence of the
corresponding states. The individual's problem is to choose between actions.
Each action is an n-tuple of consequences,
one consequence for each state of the
world. We shall write the consequence
of the ith action in the event that the
jth
state occurs as xij. Consequences
need not take the form of changes in wealth,
although in our applications
of our theory, we shall interpret
xij as an increment
or decrement of wealth, measured
relative to some arbitrary
level (which need
not be the individual's current wealth). Notice that actions, unlike prospects,
associate consequences with particular states of the world. Thus a number of
different actions might correspond
with the same prospect. We shall recognise
this difference
by using the symbol A for actions, reserving
X for prospects.
Thus
far, our theory has a close resemblance to Savage's, except in that we take
probabilities
as given, just as von Neumann and Morgenstern
do.
A choice problem may involve any number of available actions, but we shall
begin by analysing problems where there is only a pair of actions to choose
between. All of Kahneman and Tversky's evidence concerns the behaviour of
people choosing between pairs of prospects. Choices between three or more
actions raise some additional issues, which we shall discuss
in Section IV.
Our first assumption
is that for any given individual there is a choiceless
utility
Junction
C(.), unique up to an increasing
linear transformation,
which assigns
a
real-valued utility index to every conceivable consequence. The significance
of
the word 'choiceless' is that C(x) is the utility that the individual would derive
from the consequence
x if he experienced it without
havilg
chosen
it. For example,
he might have been compelled to have x by natural forces,
or x might have lbeen
imposed on him by a dictatorial government. Thus -in contrast to the von
Neumann-Morgenstern concept of utility - our concept of choiceless utility is
defined independently of choice. Our approach is utilitarian in the classical
sense. What we understand by 'choiceless utility' is essentially what Bernoulli
and Marshall understood
by 'utility' - the psychological experience
of pleasure
that is associated
with the satisfaction of desire. We believe that it is possible to
introspect about utility, so defined, and that it is therefore meaningful to talk
about utility bcing experienced
in choiceless
situations.
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Now suppose that an individual experiences a particular consequence as the
result of an act of choice. Suppose that he has to choose between actions Al and A2
in a situation of uncertainty. He chooses Al and then the jth state of the world
occurs. He therefore experiences the consequence xjj. He now knows that, had he
chosen A2 instead, he would be experiencing x2j. Our introspection suggests to us
that the psychological experience of pleasure associated with having the con-
sequence xjj in these circumstances wili depend not only on the nature of xlj
but also on the nature of x2j. If x2j is a more desirable consequence than xlj, the
individual may experience regret: he may reflect on how much better his position
would have been, had he chosen differently, and this reflection may reduce the
pleasure that he derives from xlj. Conversely, if xlj is the more desirable conse-
quence, he may experience what we shall call rejoicing.
the extra pleasure associated
with knowing that, as matters have turned out, he has taken the best decision.
We guess that many readers will recognise these experiences. For example,
compare the sensation of losing /ioo as the result of an increase in income tax
rates, which you could have done nothing to prevent, with the sensation of
losing Cioo on a bet on a horse race. Our guess is that most people would find
the latter experience more painful, because it would inspire regret. Conversely,
compare the experience of gaining C
I
oo from an income tax reduction with that
of winning C
I oo on a bet. Now we should guess that most people would find the
latter experience more pleasurable. This concept of regret resembles Savage's
(I95i) notion in some ways, but it will emerge that our theory is very different
from his minimax regret criterion.
We shall incorporate the concepts of regret and rejoicing into our theory by
means of a modified
utilityfunction. Suppose that an individual chooses action Ai in
preference to action Ak, and that the jth state of the world occurs. The actual
consequence is xij while, had he chosen differently, Xk would have occurred.
We shall write C(xij) as cij and we shall then say that the individual experiences
the modified
utility
mkj where:
M = (Cij, ckj). (I)
The function M(.) assigns a real-valued index to every ordered pair of choiceless
utility indices. The difference between mz?j
and cij may be interpreted as an
increment or decrement of utility corresponding with the sensations of rejoicing
or regret. To formulate regret and rejoicing in this way is to assume that the
degree to which a person experiences these sensations depeilds only on the
choiceless utility associated with the two consequences in question - 'what is'
and 'what might lhave been' - and is independent of any other characteristics
of these consequences. Given this assumption, it is natural to assume in addition
that if cij = c1j then mj = c^: if what occurs is exactly as pleasurable as what
might have occurred, there is neither regret nor rejoicing. It is equally natural
to assume that am"Ij /Dcki < o: the more pleasurable the consequence that might
have been, the more regret - or less rejoicing - is experienced. (We include as a
limiting case the possibility that a person might not experience regret or rejoicing
at all.) We also make the uncontroversial assumption that amkO/acij > o: that,
other things being equal, modified utility increases with clioiceless utility.
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516 THE ECONOMIC JOURNAL [MARCH
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Our theory is that the individual chooses between actions so as to maximise
the mathematical expectation of modified utility. We may define the expected
modified
utility Ek of action Al, evaluated with respect to action Ak, by:
n
Ei pmij. (2)
Faced with a choice between Ai and Ak, the individual will prefer Ai, prefer Ak
or be indifferent between them according to whether Eih is greater than, less
than or equal to Ei.
Why, it may be asked, do we assume that people maximise the mathematical
expectation of modified utility? Principally because this is a simple assumption
which yields implications consistent with empirical evidence. We do not claim
that maximising expected modified utility is the only objective that is con-
sistent with a person being rational. However - and we shall say more about this
in Section V - we believe that this is not irrational, and that, given the utili-
tarian premises of our approach, there is at least a presumption that people who
experience regret and rejoicing will seek to maximise expected modified utility.
Notice that, in our theory, someone who does not feel regret or rejoicing at all
will simply maximise expected choicelcss utility. This special case of our theory
corresponds with expected utility theory in its traditional or Bernoullian formn,
in which utility is interpreted as a psychological experience. To assume that
people maximise expected modified utility is to gencralise Bernoulli's theory in a
very natural way, since the individual who does
experience rejoicing and regret
can be expected to try to anticipate those feelings and take them into account
when making a decision under uncertainty.
We shall now show that all of the experimental evidence described in Section I
is consistent witlh regret theory. We shall do this by taking a restricted form of our
general theory and by showing that the experimental evidence is consistent with
this restricted form.
The particular restriction involves a simplifying assumption about the
function M(.). We shall assume that the degree of regret or rejoicing that a
person experiences depends only on the difference between the choiceless
utility of 'what is' and the choiceless utility of 'what might have been'. This
allows us to define a regret-rejoice
function R(.) which assigns a real-valued index
to every possible incrcment or decrement of choiceless utility, and then to write:
m1= cij + R(cij - c.1)* (3)
It follows from the assumptions we have inade about M(.) that R(o) = o and
that R(.) is non-decreasing. In the limiting case in which R(_) = o for all 6,
regret thcory would yield exactly the saine prcdictions as expected utility thcory.
Since we wish to einphasise the differences bctwccn the two theories we shall
assume that R(.) is strictly increasing and thiee times diffcrentiable.
Now suppose, as before, that an individual has to choosc between the actions
Ai and A,. The individual will have the weak prefeirence Ai > Ak if and only if:
'fl
E ij - c1.j
+ R(cj - c.j) - R(c,.j - oii) - (4)
j I
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2015] REGRET THEORY 517
8io THE ECONOMIC JOURNAL [DECEMBER
It is convenient to define a function Q(.) such that for all 6,
Q(6) = 6+R(6) -R( -6). (5)
Thus Ai > Ak if and onily if: n
I iIQ (ci - ii)] o (6)
j=1
Q(.) is an increasing function which has the following property of symmetry:
for all 6, Q((6) - Q( -). Thus to know the value of Q(6) for all 6 > o is to know
the value of Q(f) for all 6.
Three altern'ative simplifying assumptions about Q(.) can be distinguished:
Assumption i. Q(.) is linear or equivalently, for all 6, R"(6) = R"(-6). It
follows immediately from (6) that in this case the individual will behave exactly
as if he were maximising expected choiceless utility. Thus regret theory would
yield the same predictions as expected utility theory and choiceless utility
indices would be operationally indistinguishable from von Neumann-Morgen-
stern utility indices.
Assumption 2. Q
(.) is concave for all positive values of 6 or equivalently, for
all > o, R"(g) <R'"(-6).
Assumption 3. Q(.) is convex for all positive values of 6 or equivalently, for all
> o, R"(6) > R"(
-g).
On the face of it, there seems to be no a priori reason for preferring any one of
these assumptions to the others. They are simply alternative assumptions about
human psychology and a choice between them should be made mainly on the
basis of empirical evidence.' We shall therefore show that all the evidence listed
in Table I is consistent with the restricted form of our theory under Assumption 3.
In contrast, Assumption I would predict no violations of expected utility theory,
while Assumption X would predict violations, but in the opposite direction to
those generally observed.
III. SOME IMPLICATIONS OF REGRET THEORY
We shall now derive some implications of our theory concerning choices between
pairs of statistically independent
prospects. In our theory, a choice problem cannot
be analysed unless a matrix of state-contingent consequences can be specified,
and a given pair of prospects (i.e. probability distributions of consequences) may
I We say 'mainly' because there may be some theoretical reasons for expecting Assumption 3 to be
true more often than either of the other two assumptions. Notice that it is a sufficient (but not a necessary)
condition for Assumption i to hold that, for all g, R1"'(g)
= o. Similarly it is sufficient for Assumption 2
to hold that, for all 6, R'..(6) < o; and it is sufficient for Assumption 3 to hold that, for all g, R".(6) > o.
Consider the following three alternative cases: that R(.) is linear, that it is everywhere con-vex, and that
it is everywhere concave. Linearity entails that for all g, R"'(6) = o and so entails Assumption i.
Convexity entails that for all 6, R'(g) > o and R'(g) > o. Since, given these two conditions, R"'(6) ? o
cannot hold for all 6, the simplest assumption to make about R"'(.) is that for all 6, R"'(g) > o. This in
turn entails Assumption 3. Concavity entails that for all g, R'(g) > o and R'(g) < o. Since R".(g) < o
cannot hold for all g, the simplest assumption to make is again that for all g, R"'(6) > o. So Assump-
tion 3 fits with both convexity and concavity, while Assumption I is appropriate only for linearity -
which is only one point on a continuous spectrum which ranges from extreme convexity to extreme
concavity.
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518 THE ECONOMIC JOURNAL [MARCH
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be capable of being represented by many different matrices. However, the
assumption of statistical independence ensures that there is a unique matrix for
each pair of prospects. In most of Kahneman and Tversky's experiments,
subjects were simply asked to choose between pairs of prospects. In such cases,
we suggest, the most natural assumption for subjects to make is that the prospects
are independent. Given this assumption, we can show that the evidence of
Table i is entirely consistent with regret theory. As before, we shall use x1 and x2
to represent consequences. We shall use c1 and c2 to represent the choiceless
utility indices C(x1) and C(x2). For simplicity, we choose a transformation of
C(.) such that C(o) = o; and we assume that C(.) is an increasing function.
(a) The 'common ratio effect', and its reverse
Our theory yields the following prediction, which violates expected utility
theory:
Let Xi = (x1,
Ap) and Xk = (x2,p) be independent prospects, where i > p > O
and I > A > o. If there exists some probability fi such that Xi Xk when
p =l, then (i) (the common
ratio effect) if xl > x2 > o, then p < f =>
Xi >- Xk
and p > fi =>
Xi -< Xk and (ii) (the reverse common ratio effect)
if o > x2 > x1, then
p < fi =>Xi
-< Xk and p > fi =>
Xi X-
In proving this result, it is convenient to begin by stating a general property of
our theory. Let X' = (x1,pl) and X" = (x2,p2) be any two independent prospects.
The choice between these prospects may be represented by the matrix given in
Table 2
Action
corresponding
with prospect P1P2 1(I -P2) (I -P2)P2 (I -P1) (I -P2)
X' xl X10 0
X" X2 0 X2 0
Table 2, where each column represents a different state of the world, and the
probability that each state will occur is given at the top of its column. Applying
Expression (6) to Table 2, we find: that
X' X" iffpl Q(cO) -P2 Q(C2) -p1p2[Q(Cl) - Q(Cl-C2) - Q(2)] o. (7)
Thus in the case where X, =(xl, Ap) and XI =(X2,p),
Xi XlXk
iffp{AQ(c1) - Q(c2) -Ap[Q(cC) - Q(cl-c2) - Q(c2)]} o. (8)
By assumption, Q(c) is convex for all c > o so that when cl > c2 > o, [Q(cl) -
Q
(cl - c2) - Q (c2)] > o. Given this inequality, the common ratio effect follows
straightforwardly from Expression (8). Conversely, when o > c2 > cl, [Q(C1)
-Q(cl - c2) - Q(c2)] < o; and this implies the reverse common ratio effect.
The evidence of Problems 3 and 4 is consistent with the existence of the
common ratio effect. Let x1 = 4,000, X2 = 3,ooo and A = o-8. Then if p = I *0, X5
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8I2 THE ECONOMIC JOURNAL [DECEMBER
= (xl, Ap) and X6= (X2,P). IfP = -25, X9
= (x1,
Ap) and X10 = (x2,p). The con-
junction of preferences X5 -< X6 and Xg >- X10 violates expected utility theory
but is consistent with regret theory (corresponding with the case i o > 1b
> 0-25').
Over half of Kahneman and Tversky's subjects had this conjunction of pre-
ferences. Further evidence of the common ratio effect is provided by Problems
7 and 8, while Problems 3' and 4' reveal the reverse common ratio effect.
(b) The 'common
consequences
effect'
or Allais paradox
Our theory yields a further prediction, which also violates expected utility
theory:
Let Xi = (xl,pl; x2,
a) and Xk = (x2,p2 + o) be independent prospects where
I > P2 > P1 > o and (I-P2) > a >, o. If there exists some probability it such
that X, Xk when ac =
o, then (i) (the common
consequences
effect)
if xl > x2 > o,
then a < =>
Xi >- Xk and as > o=> Xi -< Xk and (ii) (the reverse
common
con-
sequences
effect)
if o > x2 > xl, then a < c
=:>
Xi -< Xkand a > =Xi > Xk.
According to regret theory,
Xi CXk iff Pl Q(Cl)-P2 Q (C - Pl (P2+ a) [ Q (CO - Q (Cl- C2) Q
Q(C2) ;< o.- (9)
Because Q(c) is assumed to be convex for all c > o, [Q(cC) - Q(cl - c2) - Q(c2) is
positive if xl > x2 > o and negative if o > x2 > xl. Given these two propositions,
Expression (g) entails both the common consequences effect and the reverse
common consequences effect.
The evidence of Problems I and 2 is consistent with the existence of the
common consequences effect. Letxl = 2,500, X2
= 2,400,P1 = 033 andp2 = 0o34.
Then if a' = (I -P2), X1 = (xl,pl; x2,x ) and X2 = (x2,p2+xa). If c =-OX3 =
(x1,p1; x2, o) and X4 (X2,P2
+ tx). The conjunction of preferences XA1
< AT2
and
X3 >- X4 violates expected utility theory but is consistent with regret theory
(corresponding with the case o'66 > oc
> o). At least 65 % of Kahneman and
Tversky's subjects had this conjunction of preferences. Kahneman and Tversky
did not publish any results relevant to our prediction of a reverse common con-
sequences effect.
(c) The 'Isolation
effect'
in the two-stage
gambles
In Kahneman and Tversky's Problem Io, their respondents were offered a
two-stage gamble. In the first stage there was a 0o75 probability of the gamble
ending with a null consequence and a 0o25 probability of going through to the
second stage. Before embarking on the first stage, respondents were asked to
choose which of X5 or X6 they would prefer if they got through to the second stage.
According to the compound probability axiom of expected utility theory,
X17 = (X5,
,025) is equivalent to (4,000, 0 20) which is simply prospect X9; and
X18 = (X6,0 25) is equivalent to (3,000, 0o25) which is prospect X10. Thus
expected utility theory makes no distinction between Problem I
o and Problem 4.
However, regret theory does make a distinction. The simple prospects X9
and
X10 are regarded as statistically independent, and Problem 4 is therefore repre-
sented by the matrix of state-contingent consequences shown in Table 3 a. By
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520 THE ECONOMIC JOURNAL [MARCH
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contrast, prospects
X17
and X8 are not statistically independent: the first stage
of the gamble is common to both, and if the state occurs
under which the gamble
comes to an end, the individual receives the same null consequence whichever
prospect was chosen. Hence Problem io is represented
by the matrix of state-
contingent consequences shown in Table 3b. Since Tables 3a and 3b are
Table 3 a
Action
corresponding
with prospect o-6o 0120 015 0-05
Xg 0 0 4,000 4,000
X1o 0 3,000 0 3,000
Table 3 b
Action
corresponding
with prospect o075 0120 0-05
X17 0 4,000 0
X18 0 3,000 3,000
different, our theory provides no reason to suppose that an individual will have
the same preferences
between X17
and X18
as between X9 and XIO.
Before analysing this example further, we present a result which holds for
regret theory in its most general form, and which we shall call the separability
principle.
Let S1, . .., Sn be mutually exclusive events (i.e. non-intersecting
sets of states
of the world) with the non-zero probabilities
P1,
. ..,Pn where
p1
+ ... +Pn =
Let S', ..., Sn4+
be mutually exclusive
events
with the probabilities
jtp,, . . .,
I -It, wlhere o < # < i. Let Ai = (xII, ...,x1n) and Ak = (x21,
...x2n) be any
two actions defined in relation to the events S1,..., S,. Let Let Aa and Abbe
actions defined in relation to the events S, ..., Sn+, such that Aa = (x11,.
X1n, y) and Ab= ..., x2,y) y being any consequence common to both
actions. Then Aa > Ab if and only if Ai > Ak.
The proof is straightforward. If E+k and E1t
are the expected modified utilities
of Ai and Ak,
evaluated in relation to one another, then E. = tEi + (i -t) C(y)
and Eb = jtE,k + (i -,a) C(y). Hence Eik > E, ab > Eba, which entailsAi > Ak
Aa > Ab. The separability principle entails Savage's sure-thing principle as a
special case. Let 4a
remain constant, and let us construct two new actions, A.
and Ad, which are the same as Aa and Ab except that the common consequence y
is replaced by the common consequence
z. It is clear thatAi > AkA, > Ad, and
hence it follows that Aa > Ab A, > Ad, which is Savage's sure-thing principle.
Returning to Kahneman and Tversky's
evidence, let A5 and A6 be the actions
corresponding
to the independent prospects
X5 and X6, and let A17
and A18
be
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the actions corresponding to X17
and 718
in Table 3 b. Since E1 =aE6 + ( -a) C(o)
and Et8 = gE65+ (i- ) C(o), it follows that X5 -< X6 X7- <X18 We have
already seen in (a) above that the conjunction X5 -< X6 and Xg >- XAo
is con-
sistent with our theory. Thus it follows that the conjunction X9 >- Xlo and
X17 -< X18, which violates conventional expected utility, is also consistent with
regret theory.
(d) The
'reflection
efect'
The results in (a), (b) and (c) above were derived without making any assump-
tion about C(.) other than that it is monotonically increasing. We shall derive
our results in (d) and (e) by making the additional assumption that U(.) is
linear; and, for convenience, we shall choose a transformation of that linear
function such that for all x, C(x) = x.
Consider two independent prospects, Xi = (xl,pl) and A = (x2,p2). Their
'reflections' are denoted Xi' = (-xl,pl) and XAk
= (-x2,p2) . From Expression
(7) we know that Xi > Xk if and only if:
P1
Q
(X1)
-P2 Q
(X2) -P1 P2[Q (X1)
-Q (X1-X2) -Q (X2)] > O. (Io)
Now exactly the same inequality is necessary and sufficientfor XT -< XA'.
Hence
Xi > XYG
XiT
< X'. Thus if C(.) is linear, the reflection effect is always observed.
Our intuition is that C(.) is not linear but concave. If this is correct, the reflec-
tion effect will not always be observed, and in particular, individuals will reject
actuarially fair 50-50 gambles, rather than being indifferent towards them. This
point is discussed further in Section V.
(e) Mixed
risk
attitudes;
simultaneous
gambling
and
insurance
Consider two independent prospects which offer an actuarially fair gamble:
Xi = (o, I) and XIG
= (x,p; -px/(i-p), I -p), where o <p < I and x > o.
Maintaining our previous assumption about C(.) we can apply Expression (7)
and rearrange to give:
Xi
_<Xl
if I_)I_ Q(
): O. (I I)
From the assumption that Q
(x) 'is convex for all x > o, it follows that
Xi _ AX,,
as p -o5. So the individual will accept small-stake large-prize fair
gambles (p < o.5) but reject large-stake small-prize fair gambles (p > os).
Insurance typically involves paying a small premium to avoid a smali prob-
ability of a large loss; thus in terms of our theory - which does not use the concept
of a 'reference point' - to buy actuarially fair insurance is to reject a large-stake
small-prize fair gamble, and thus it is consistent with our theory that an indi-
vidual may simultaneously insure and accept small-stake large-prize gambles.
Moreover, we can construct both small-stake large-prize fair gambles, and
large-stake small-prize fair gambles either with all consequences positive or
with all consequences negative. Thus a mixture of risk attitudes in both the
positive and the negative domain is also consistent with our theory.
These conclusions would require some modification if C(.) were assumed to be
concave rather than linear. In this case it can be shown that Xi >- Xk if p o-5,
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522 THE ECONOMIC JOURNAL [MARCH
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but it is no longer possible to make a firm prediction when p < o5. However, if
an individual is more strongly influenced by the shape of Q ( ) than by the non-
linearity of C(.), simultaneous gambling and insurance is still consistent with
our tbeory.
IV. TRANSITIVITY OF PREFERENCES AND
MULTI-ACTION PROBLEMS
One controversial property of our theory is that >, the relation of weak
preference, is not necessarily transitive. Consider the three actions shown in
Table 4 in relation to an individual for whom C(.) is linear. Relative to A1, A2
is a large-stake small-prize fair gamble, so that the individual would have the
preference A1 >- A2 if he had to choose between these two actions. If, as our
theory entails, the individual acts according to the separability principle outlined
in Section III (c), state S, can be ignored in a comparison between A2 and A3.
Thus, relative to A2, A3 is also a large-stake small-prize fair gamble, and so
A2 >- A3. However, relative to A1, A3 is a small-stake large-prize fair gamble, so
that A3 >- A1. This is not to say that our theory specifically predicts non-transitive
pairwise choices (since the C(.) function need not be linear); but such choices
can be consistent with the theory.
Table 4
S1 S2 S3
Action 0-4 0-2 0?4
A1 6 6 6
A2 0 10 10
A3 0 0 15
The example shows that an individual will necessarily make non-transitive
choices if (i) he acts according to the separability principle (or according to the
sure-thing principle), (ii) he always accepts small-stake large-prize fair gambles
and (iii) he always rejects large-stake small-prize fair gambles. In the light of the
evidence that many people simultaneously gamble and insure one might well
argue that a satisfactory theory of choice under uncertainty should encompass
the case of the individual who acts according to (ii) and (iii). To say this is to say
that either the sure-thing principle or the axiom of transitivity must be dropped.
Our theory differs from many of its rivals by dropping transitivity rather than
the sure-thing principle.
This raises two questions. One is whether a theory that allows non-transitive
pairwise choices can be regarded as a theory of rational behaviour; this issue is
discussed in Section V. The other question is how to extend our theory to deal
with multi-action choice problems: since in our theory the relation > is not
necessarily transitive, we cannot deal with choices from sets of three or more
actions simply by invoking the idea of a preference ordering. We shall argue that
the logic of regret and rejoicing points towards a different way of generalising a
theory of pairwise choice.
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Consider the problem of choosing one action from a set S. The logic of our
approach requires that the individual should evaluate each action in turn by
asking himself what sensations of regret or rejoicing he would experience in each
state of the world, were he to choose that action. Since to choose one action is to
reject all of the others, the individual could experience regret or rejoicing in
contemplating any of the rejected actions. This idea might be formulated in the
following way. As before, we use Ek to represent the expected modified utility
of choosing action Ai in a situation where the only, alternative is action Ak. Now
let Es represent the expected modified utility of choosing Ai from the set of
actions S. It seems natural to make ES a weighted average of the values of Ek
for each of the actions Ak
in S (other than Ai itself). One way of building this idea
into our theory would be to assign action
weights
ax to each action A in S, normal-
ised so that these weights sum to unity. Then ES could be defined as:
EsV
= E s (k * i) . ( I 2)
keS I -a
The individual's decision rule, as in the case of pairwise choice, would be to
maximise expected modified utility. We hope in the future to formulate a theory
of action weights, but in the example which follows we shall just make the
simplest assumption - that each action has the same weight.
Table 5
Action 1/3 1/3 1/3
A1 I I I
A2 ? ? 3
A3 0 3 0
This illustrative example refers to the choice problem shown in Table 5. As
before, we shall assume that C(x) = x, and we shall make a particular assumption
about the regret-rejoice function, that over the relevant range, R(6) = i - o.86.
In this case, and for these three actions, the relation > happens to be transitive;
A2 > Al, A3 > Al) A2 A3 .
It is tempting (but, we suggest, wrong) to conclude
from this that A, will not be chosen from the set {A,, A2, A3}. If the action weights
are equal to one another then Es = o0946, Es = o899 and Es = o-899, so that,
according to the decision rule, A1 will be chosen. Whether or not such behaviour
can be defended as rational will be discussed in Section V.
V. THE POSITIVE AND NORMATIVE STATUS
OF REGRET THEORY
The experimental results published by Kahneman and Tversky, wide-ranging
though they are, form only a small fraction of the evidence accumulated in the
past 30 years to show consistent and repeated violations of certain axioms of
expected utility theory. Regret theory is one of a number of alternative theories
that have been proposed in the light of this evidence; other theories have been
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524 THE ECONOMIC JOURNAL [MARCH
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presented by, for example, Allais (I953), Kahneman and Tversky (I979),
Fishburn ( I98 I) and Machina (i 982). We shall shortly compare our theory with
these others, but first let us discuss a possible argument against regret theory.
It might be objected that regret theory is limited to cases where probabilities
are known, and that it rests on assumptions about non-observable functions,
whereas expected utility theory is built on clear behaviourai axioms which make
it possible, in principle, to construct a series of choice problems which will reveal
the individual's von Neumann-Morgenstern utility function.
While we do not share the methodological position that the only satisfactory
theories are those formulated entirely in terms of empirical propositions, we
would point out that if an individual behaves according to our model, it is
possible in principle to infer from observations of his choices: his subjective
probabilities; his C(.) function (unique up to a positive linear transformation);
and his Q(.) function (which, for any given transformaticn of C(.), will be
unique up to a positive linear transformation with a fixed point at the origin).
Thus each of the assumptions about C(.) and Q(.) required to generate our
predictions is in principle capable of empirical refutation. (For an outline of the
procedures involved, see the Appendix.)
The other criteria that are commonly used to evaluate positive theories are
predictive power, simplicity and generality. Regret theory yields a wide range of
firm predictions that are supported by experimental evidence, and it does so on
the basis of a remarkably simple structure. Only the two functions C(.) and Q(.)
are required. As far as C(.) is concerned, some of the most important predictions
of our model - the common ratio effect, the common consequences effect, their
reverses, and the isolation effect - require only that this function is monotonically
increasing; the additional assumption of linearity yields clear predictions con-
cerning the reflection effect and simultaneous gambling and insurance. In
generating all these predictions, the other crucial assumption is simply that Q
(6)
is convex for all 6 > o.
Thus in comparison with Kahneman and Tversky's 'prospect theory' - which
is also consistent with all the evidence in Table I - regret theory is very simple
indeed. Kahneman and Tversky's theory superimposes on expected utility theory
a theory of systematic violations. Among their many assumptions are: (i) the
rounding of probabilities up or down, and the complete editing out of 'small'
probabilities; (ii) a 'decision weight function' which overweights small prob-
abilities, underweights large probabilities, involves 'subcertainty', 'sub-
proportionality' and 'subadditivity', and which is discontinuous at both ends,
thus implying certain 'quantal effects'; and (iii) a 'value function' (essentially
a utility function) which must have at least one point of inflection (at the indi-
vidual's 'reference point' - which may or may not move around) but which can,
if required, have no less than five points of inflection. We believe that against the
complex and somewhiat ad hoc array of assumptions required by prospect
theory the principle of Occam's Razor strongly favours the straightforwardness
of regret theory.
Allais's and Machina's theories are considerably simpler thani prospect theory,
but they cannot explain all of the evidence in Table i. Both of these theories
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assume that the individual has a preference ordering over prospects. Thus two
of the fundamental principles of expected utility theory are retained: that pair-
wise choices are transitive and that courses of action associated with identical
probability distributions of consequences are equivalent to one another. (We
shall call this latter principle the equivalence axiom.) Allais and Machina break
away from expected utility theory by dropping the independence axiom; given
that the equivalence axiom is retained, this amounts to abandoning the sure-
thing principle. Our strategy is radically different: we retain the sure-thing
principle while jettisoning both the equivalence axiom and the transitivity
axiom. As a result we are able to predict the isolation effect in two-stage gambles,
a form of observed behaviour that contravenes the equivalence axiom and
therefore cannot be explained by either Allais or Machina. We are also able to
predict the systematic occurrence of the reflection effect. Although Allais's and
Machina's theories are not contradicted by the reflection effect, they do not
predict it.
Fishburn's model is more like regret theory (although he does not mention
any notion of regret) in that he also drops the transitivity axiom. However, his
model is presented in terms of prospects rather than actions, and therefore does
not accommodate the isolaticn effect. On the other hand, if we restrict ourselves
to statistically independent prospects (and Fishburn does so - see his p. 9), then
our theory and his basic axioms are compatible, and provide an interesting
example of how an axiomatic treatment and a more introspective psychologically-
based approach may complement each other.'
However, having indicated that our theory provides certain predictions and
explanations that the other theories mentioned do not, we should make it clear
that we are not claiming that regret theory can explain all of the behavioural
regularities revealed by experimental research into choice under uncertainty. So
far we have focused on a number of patterns of behaviour observed by Kahneman
and Tversky; but we have not dealt with every one of their observations, still less
with the vast amount of evidence accumulated by other researchers.
Some of the experimental findings do not appear to be completely consistent.
In relation to this paper, the most significant case concerns the reflection effect.
Hershey and Schoemaker (i 980 a) and Payne et al. (i 980) have published results
that show this effect to be not nearly as strong or as general as Kahneman and
Tversky's evidence suggests. However, this nmay
not present any great difficulties
for regret theory since, as we noted in Section III (d), the general prediction of the
reflection effect requires C(.,) to be linear. Instances in which the reflection effect
is weak or absent may well be explicable if C(.) is assumed to be concave.
There are nevertheless certain observations that simply cannot be accounted
for by regret theory in the form presented here. One example is the 'framing'
effect discussed by Tversky and Kahneinan (I 98 I) and the very similar 'context'
effect observed by Hershey and Schoemaker (i q8o b). In these cases exactly the
I At a late stage, we have received a copy of a Working Paper by David E. Bell (I98I) which is of
great interest. Quite independently he has developed a model which also explicitly incorporates a notion
of regret, using multi-attribute utility theory along the lines suggested by Keeney and Raiffa (1976).
We note that when both models are applied to the same phenomena - the original Allais paradox,
simultaneous insuring and gambling, and the reflection effect - the conclusions are strikingly similar.
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526 THE ECONOMIC JOURNAL [MARCH
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same choice problem - that is, exactly the same when formulated in terms of a
matrix of state-contingent consequences - receives markedly different responses,
depending on the way the choice is presented. Another example is the 'trans-
lation' effect observed by Payne et al. (i980). This effect occurs when an
individual prefers one prospect to another, but reverses his preference when the
same sum of money is deducted from every consequence of both prospects. The
observed pattern of reversal is not predicted by regret theory. Finally, systematic
violations of the sure-thing principle have been observed (cf. Moskowitz (I 974);
Slovic and Tversky (I974)). And although there is some evidence that individuals
violate the sure-thing principle much less often than they violate some other
axioms (Tversky and Kahneman (i 98I, footnote I 5)), as it stands our theory
does not explain that behaviour.
On the other hand, there is some additional evidence that gives further
support to regret theory. A particular instance is the form of' preference reversal'
observed by Lindman (i97i) and Lichtenstein and Slovic (i97i, I973) and
subsequently confirmed, after rigorous testing, by Grether and Plott (I979).
This preference reversal occurs when an individual, faced with a pairwise choice
between gambles A and B, chooses A; but when asked to consider the two
gambles separately, places a higher certainty equivalent value on B. We have
shown elsewhere (Loomes and Sugden (' 982)) that the most commonly observed
reversal pattern is predicted by regret theory even in its restricted form.
Of course, we acknowledge that there is no simple theory that gives a unified
explanation of all the experimental evidence, and regret theory is no exception
in this respect. But we have tried to construct a theory that explains as much of
the evidence as possible on the basis of very few assumptions. We do not believe
that choiceless utility and regret are the only factors that influence behaviour
under uncertainty, but just that these two factors seem to be particularly signifi-
cant. Indeed, we have become increasingly convinced by evidence of framing,
context and translation effects that the notion of reference points deserves further
consideration, although we have not tried to deal with that issue in this paper.
In constructing our theory we have avoided any assumptions of mnisperceptions
or miscalculations by individuals. We do not doubt that in reality misperceptions
and miscalculations occur, and sometimes in systematic rather than random
ways. Nonetheless, our inclination as economists is to explain as much human
behaviour as we can in terms of assumptions about rational and undeceived
individuals. Thus we believe that regret theory does more than predict certain
systematic violations of conventional expected utility theory: it indicates that
such behaviour is not, in any meaningful sense of the word, irrational.
In claiming this we are breaking the terms of a truce that many theorists (with
the notable exception of Allais) have tacitly accepted. Proponents of expected
utility theory often concede that their theory has serious limitations as a pre-
dictive device but insist that its axioms have strong norrnative appeal as principles
of rational choice. Thus Morgenstern (I979, p. i8o) argues for expected utility
theory on the grounds that 'if people deviate from the theory, an explanation of
the theory and of their deviation will cause them to re-adjust their behaviour'.
Similarly, Savage (I954, pp. I02-3) admits that when confronted with a pair of
28 ECS 92
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choice problems rather like Problems I and 2, he behaved in accordance with
the common consequences effect and in violation of his own axioms. But, he
says, he was able to convince himself that this behaviour was mistaken (though
even after realising
his 'mistake' he continued to feel an 'intuitive attraction' to
that behaviour)
. At the other side of the truce, proponents
of alternative
theories
have often been willing to accept these claims. Kahneman and Tversky (I979,
p. 277) maintain that the departures
from expected utility theory that prospect
theory describes 'must lead to normatively unacceptable consequences' which
a decision-maker
would, if he realised the error of his ways, wish to correct.
Similarly, Machina (i 982, p. 277) notes the 'normative appeal' of the axioms
of expected utility theory before going on to propose a positive theory that
dispenses
with one of these axioms.
However, we shall challenge the idea that the conventional axioms constitute
the only acceptable basis for rational choice under uncertainty. We shall argue
that it is no less rational to act in accordance with regret theory, and that con-
ventional expected utility theory therefore
represents
an unnecessarily
restrictive
notion of rationality.
Regret theory rests
on two fundamental assumptions:
first, that many people
experience
the sensations
we call regret
and rejoicing;
and second, that in making
decisions under uncertainty, they try to anticipate and take account of those
sensations.
In relation
to the first
assumption,
it seems
to us that psychological
experiences
of regret and rejoicing cannot properly be described in terms of the concept of
rationality: a choice may be rational or irrational, but an experience is just an
experience. As far as the second assumption
is concerned, if an individual does
experience such feelings, we cannot see how he can be deemed irrational for
consistently
taking those feelings into account.
We do not claim that acting according
to our theory
is the only
rational
way to
behave. Nor do we suggest that all individuals who act according to our theory
must violate the conventional axioms. Some individuals may experience no
regret or rejoicing at all, while some others may have linear Q(.) functions: in
these special
cases of our theory, we would predict
that the individual's
behaviour
would conform
with all the conventional axioms.
On the other hand, individuals with non-linear Q(.) functions of the kind
described in this paper may consistently and knowingly violate the axioms of
transitivity
and equivalence without ever accepting, even after the most careful
reflection, that they have made a mistake. So these axioms do not necessarily
have the self-evident or overwhelming normative appeal that many theorists
suppose.
We shall now try to show why we do not accept the idea that the transi-
tivity and equivalence axioms are necessary
conditions for rational choice under
uncertainty.
Underlying those two axioms is a common idea: that the value placed on any
action Ai depends only on the interaction between, on the one hand, the prob-
ability-weighted consequences
offered by Ai and, on the other hand, the indi-
vidual's pattern of tastes, including his attitude to risk.
That is what is symbolised when, for any individual, an expected utility
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number is assigned to an action, that expected utility number being quite
independent of the range and nature of the available alternative actions. From
this idea, that there is some value in 'having Ai' which is quite independent of the
value of 'having Ak', and that if 'having Ai' gives more value than 'having Ak'
then Ai >- Ak, it follows that there must exist a complete and transitive preference
ordering over all actions.
It also follows that the particular state pattern of consequences is of no special
significance: if each action is evaluated independently, it does not matter how
the consequence of that action under any state of the world compares with the
consequence(s) of any other action(s) under the same state. Thus only the
probability distribution of consequences matters, and all actions, simple or
complex, which share the same probability distribution will be assigned the
same expected utility number and must be regarded as equivalent for the
purposes of choice decisions.
But if people experience regret and rejoicing, these arguments are illegitimate.
In regret theory the proposition Ai > Ak cannot be read as 'having Ai is at least
as preferred as having Ak'; it should rather be read as 'choosing Ai and simul-
taneously rejecting Ak is at least as preferred as choosing A,. and simultaneously
rejecting Ai'. Thus the transitivity of the relation 'is at least as preferred as'
(which we do not dispute) does not entail the transitivity of our relation >; and
so non-transitive choices do not indicate any logical inconsistency on the part
of the decision-maker.
The idea that non-transitive choices are irrational is sometimes argued as
follows. Suppose (as in the example discussed in connection with Table 4 in
Section IV) that there are three actions A1, A2, A3, such that A1 >- A2, A2 >- A3,
and A3 >- A1. Then, it is said, no choice can be made from the set {A1, A2,A3}
without there being an inconsistency with one of the original preference state-
ments: whichever action is chosen, another is preferred to it (cf. MacKay (i 980,
p. go)). The principle that is being invoked here is Chernoff's axiom: if Ai is
chosen from some set S, and if S' is a subset of S that contains Ai, then Ai must be
chosen from S'. But we suggest the appeal of this axiom derives from the supposi-
tion that the value of choosing an action is independent of the nature and
combination of the actions simultaneously rejected; and regret theory does not
accept this supposition. Since A1 >- A2 means only that choosing A1 from the
set {A1,
A2}
is preferred to choosing A2 from the set {A1, A2}
there is no implication
that choosing A1 from the set {A1,
A2, A3}
is preferred to choosing A2 from the set
{A1,
A2,
A3}. A similar argument applies to the example discussed in connection
with Table 5 in Section IV, where (despite the fact that the relation > happens
to be transitive) there is another violation of Chernoff's axiom.
A second common objection to non-transitivity runs like this. If someone
prefers A1 to A2, A2 to A3, and A3 to A1, every one of the actions is less preferred
than another; so might he not get locked into an endless chain of choice in which
he can never settle on any one action? Worse, might not a skilful bookmaker
capture all his wealth by confronting him with a suitably constructed sequence of
pairwise choices? But these objections rest on a fallacy. To suppose that the
individual can get locked into a cycle of choices, it is necessary to suppose that all
28-2
©2015 The Authors.
The Economic Journal published by John Wiley & Sons Ltd on behalf of Royal Economic Society.
2015] REGRET THEORY 529
822 THE ECONOMIC JOURNAL [DECEMBER
three actions are feasible. But if this is indeed the case, then propositions about
pairwise choices - about how choices are made when there are only two feasible
actions - are not relevant. The bookmaker can bankrupt his client only if he can
successively persuade him to believe in each of a long chain of mutually in-
consistent propositions about the feasible set.
Finally, there is no reason why the equivalence axiom should be regarded as a
necessary condition for rational choice, even when the choice is between two
simple actions with identical probability distributions of consequences. Consider
Table 6
Action 0o25 0?25 0-25 0-25
Ai 3 2 I 0
Ak 0 3 2 I
Ai and Ak in Table 6. If each action were evaluated independently, there would
be no grounds for preferring 'having Ai' to 'having Ak', or vice versa. But in our
model the decision is between 'choosing Ai and simultaneously rejecting Ak' and
'choosing Ak and simultaneously rejecting Ai'. These two alternatives are
associated with different probability mixes of regret and rejoicing. (In terms of
our theory, to choose Ai and reject Ak is to incur a o 25 probability of R( +3) and
a o075 probability of R(- i), while to choose Ak and reject Ai is to incur a O'25
probability of R( -3) and a o 75 probability of R( + i).) So for an individual
who experiences regret and rejoicing, the two courses of action cannot be
regarded as identical. It would therefore not be unreasonable for such an
individual to prefer one to the other.
VI. CONCLUSION
The evidence presented by Kahneman and Tversky and many others points to a
number of cases where commonly observed patterns of choice violate conven-
tional expected utility axioms. The fact that these violations are neither small-
scale nor randomly distributed may indicate that there are some important
factors affecting many people's choices which have been overlooked or mis-
specified by conventional theory.
We suggest that one significant factor is an individual's capacity to anticipate
feelings of regret and rejoicing. We therefore offer an alternative model which
takes those feelings into consideration. This model yields a range of predictions
consistent with the behaviour listed in Table I and provides an account of these
and other choice phenomena which conventional theory has so far failed to
explain.
That is the positive side of regret theory. But we believe that our approach also
has strong normative implications. We have argued that our theory describes
a form of behaviour which, although contravening the axioms of expected utility
theory, is rational. Thus, while we do not suggest that behaving according to those
©2015 The Authors.
The Economic Journal published by John Wiley & Sons Ltd on behalf of Royal Economic Society.
530 THE ECONOMIC JOURNAL [MARCH
i982] REGRET THEORY 823
conventional axioms is irrational, we do suggest that those axioms constitute an
excessively restrictive definition of rational behaviour.
University
of Newcastle
Date of receipt
offinal typescript: April 1982
Appendix:
Inferring
subjective
probabilities
and C(. ) and Q(.)
finctionisjrom choices
The following procedure will reveal, for any individual, which of two events has
the higher subjective probability. Let S, and S2 be any two non-intersecting and
non-empty events (i.e. sets of states of the world). Let S3 be the event that
comprises all those states of the world not in S, or S2. Let x, y, z be any three
consequences such that the person in question prefers x to y (under certainty).
Consider the two actions Ai = (x, y, z) and Ak = (y,x,z), which are defined in
relation to the events Sl, S2, S3. It then follows from the separability principle (see
Section III) that Ai is preferred to, indifferent to, or less preferred than Ak as the
subjective probability of S, is greater than, equal to, or less than that of S2. This
procedure is broadly similar to the one proposed by Savage (I 954) for inferring
subjective probabilities for individuals who behave according to his postulates.
The restricted form of our theory (see Section III) uses two functions for the
analysis of modified utility: C(.) and Q(.). C(.) can be identified, up to a
positive linear transformation, by confronting the individual with choices
involving 50-50 gambles. Consider any two prospects of the form Xi = (xl, i),
Xk = (x2,
c)5; X3, o0) where X3 > X1 > X2, so that the corresponding choiceless
utility indices are C3
> C1 > C2. Then:
Xi <>-XI iff o 5Q
Q(Cl-C2)-O5Q Q(C3 -CO -O.
But since Q(.) is increasing, it follows that:
Xi Xi, i
ff o 5 (cl- C2)-o 05 (C3 -CO :'- O.
Thus in this case, the individual chooses as thouigh
maximising expected choiceless
utility. So C(.) can be identified from experiments in much the same way as
von Neumann-Morgenstern utility functions are identified.
If C(.) is known, and if a particular transformation has been chosen, it is
possible to define consequences in terms of their choiceless utilities. Let xl and x2
be consequences such that cl = o and C2
=- i. Let X3 be any consequence such
that c3 = 6 where 6 > o and 6 * i. Consider the two prospects
Xi = (xl, i) and
= (X2,P; X3, I-p). Then:
Xi Xh iff Q(I) I -p.
Thus if one can find a value of p such that the individual is indifferent between
Xi and XI,
it is possible to infer the value of Q
(6) IQ
(I). So if Q(
i) is set equal to any
arbitrary positive value, the value of Q
(6) can then be determined by experiment
for all 6 > o; hence the concavity, convexity or linearity of Q
(.) over any interval
can be established.
©2015 The Authors.
The Economic Journal published by John Wiley & Sons Ltd on behalf of Royal Economic Society.
2015] REGRET THEORY 531
824 THE ECONOMIC JOURNAL [DECEMBER I982]
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