Disappointment without Prior Expectation:
A Unifying Perspective on Decision under Risk
Philippe Delquié and Alessandra Cillo*
INSEAD
Decision Sciences area
Boulevard de Constance
Fontainebleau F-77300
France
Send correspondence to:
Philippe Delquié
INSEAD
Decision Sciences area
Boulevard de Constance
Fontainebleau F-77300
France
Tel.: +33 1 6072 4376
*
We are grateful to Peter Wakker for his comments on a previous version of this paper. We also
thank an anonymous referee for a detailed review.
Abstract
The central idea of Disappointment theory is that an individual forms an expectation about a
risky alternative, and may experience disappointment if the outcome eventually obtained falls
short of the expectation. We abandon the hypothesis of a well-defined prior expectation:
disappointment feelings may arise from comparing the outcome received with any of the
gamble’s outcomes that the individual failed to get. This leads to a new, general form of
Disappointment model.
It encompasses Rank Dependent Utility with an explicit one-
parameter probability transformation, and Risk-Value models with a generic risk measure
including Variance, providing a unifying behavioral foundation for these models.
Key words: Disappointment Theory; Rank-Dependent Utility; Risk-Value models; MeanVariance; EU violations.
JEL Classification: D80, D81
2
Models of decision under risk often attempt to fulfill two rather conflicting objectives: (1)
reflecting the values and concerns of the decision maker accurately, while (2) being grounded
on sound, logical principles that would provide a rationale for decision making. The first
objective is a descriptive one, whereas the second is normative in nature. For example,
Expected Utility (EU) rigorously fulfills the normative requirement, but does so at the
expense of the descriptive objective. Prospect theory (Kahneman and Tversky, 1979) may
have the opposite: although it can accommodate a wide range of choice behavior, it lacks
normative underpinnings.
Disappointment theory, proposed by Bell (1985) and Loomes and Sugden (1986), can be
viewed as an attempt to achieve a compromise between descriptive flexibility and normative
rigor. The basic proposition of Disappointment theory is that individuals will evaluate the
outcome of a risky situation by comparison to a prior expectation they entertain about the
situation.
If the outcome is inferior to their expectation, individuals will experience
disappointment; if it is superior, they will experience elation.
A key assumption of
Disappointment theory is that individuals will (and perhaps should) anticipate these possible
feelings beforehand, and take them into account in their ex-ante evaluation of risks.
Bell (1985) sketches a model of disappointment for two-outcome gambles, in which the prior
expectation is taken as the mathematical expected value of the payoffs.
Bell (1985)
recognizes the difficulties of articulating a prior expectation and goes on to provide several
principles or “axioms” for modeling disappointment while avoiding being explicit about the
formulation of prior expectation. Loomes and Sugden (1986) propose the following model.
Consider a gamble X that yields outcome xi with probability pi, i = 1,..., n. The subjective
value of outcomes is captured by a function v(·). The expected subjective value of the
gamble, E[v(X )] = ∑i =1 pi v( xi ) , is taken as a measure of the decision maker’s prior
n
expectation. Disappointment and elation arise from comparing the value obtained, v(xi), with
the expectation, E[v(X)]. Therefore, the utility experienced upon receiving xi is:
u ( xi ) = v( xi ) + D(v( xi ) − E[v(X )]) ,
(1)
3
where D(·) is an increasing function capturing the effects of disappointment and elation, by
taking negative values for v(xi)–E[v(X)] < 0 (disappointment) and positive values otherwise
(elation). The overall utility of the gamble is then the mathematical expectation of the
modified utility expressed in Eq. 1:
n
U ( X ) = E[u ( X )] = ∑ pi (v( xi ) + D(v( xi ) − E[v( X )])) .
(2)
i =1
Loomes and Sugden’s (1986) argument regarding the normative status of Disappointment
theory has some appeal: if individuals can’t help but be affected by disappointment and
elation, then it may be quite legitimate for them to take those feelings into account in making
choices. Bell (1985) also makes a case for such rationality: he argues that trading off some
expected financial gain to reduce the possible negative effects of disappointment may make as
much sense as trading off expected financial gain to reduce risk, as is typically allowed under
EU. In addition, these authors show how their models can account for a number of wellknown behavioral violations of EU. Therefore, Disappointment theory can be viewed as an
attractive alternative among competing theories of risky choice because (1) it offers standard
descriptive capabilities, and (2) it is based on a parsimonious, intuitively appealing
psychological hypothesis.
Here, we argue that the idea of a single prior expectation may be fuzzy, and we propose to
relax that hypothesis. This leads to a model structure different from previous models of
disappointment, which provides a clean integration with other, independently developed
theories of decision under risk. Let us clarify at the outset that our purpose here is not to
provide experimental testing of a new descriptive model, nor to propose an axiomatic
treatment of Disappointment theory. Rather, our goal is to expose a novel approach to
Disappointment theory, which avoids issues related to the formation of prior expectations,
while bearing some behavioral appeal. In Section 1 we develop the motivation for, and the
analytical form of, our model. Key properties of the model, and a comparison to previous
models, are examined in Section 2. Section 3 will focus on the particular case of linear
disappointment: this yields an explicit, cumulative probability transformation function. A
4
Risk-Value formulation of the model is proposed in Section 4: this yields a general measure
of risk, including Mean-Variance as a particular case.
1. Modeling Disappointment without a Prior Expectation
We can take issue with one of the key premises of Disappointment theory: the notion of a
single, well-defined, prior expectation as a basis for the experience of disappointment. This
postulate is at the core of the different models we are aware of: Bell (1985), Loomes and
Sugden (1986) as briefly reviewed above, but also Gul (1991), and Jia, Dyer and Butler
(2001). The argument can be made on both behavioral and theoretical considerations.
The first issue concerns what would constitute an appropriate prior expectation for a gamble.
Many reference points are candidates to serve the role of prior expectation: the Expected
Value (EV), a more general Certainty Equivalent (CE), the Median, the Mode, any other
summary measure of the probability distribution, or perhaps an exogenously specified
aspiration level. It is not clear how to choose among these possibilities. Second, one could
argue that an expectation benchmark should be a real, tangible outcome of the gamble. For
discrete gambles, the EV, CE, or other summary measures will generally not correspond to an
outcome that could actually be obtained from the gamble; thus they may be rather abstract,
artificial references, unlikely to hold the status of a clear expectation in the decision maker’s
mind.
Next, there is a possible issue of recursiveness in thinking about the effect of disappointment
on a gamble valuation. Since disappointment is anticipated ex-ante, a prior expectation about
a gamble could or should reflect some adjustment for disappointment potential, which would
lower expectation, which in turn modifies disappointment, and so on. This loop could spiral
down to converge on an extremely low prior expectation.1 At the extreme, it could be argued
that if an individual cares to protect him/herself from disappointment, then perhaps he/she’d
1
It should be noted, however, that Gul’s (1991) definition of prior expectation is immune to this kind of
reasoning.
5
better expect nothing but the worst. The more general point here is that different individuals
are liable to form their prior expectations in different ways. An optimist and a pessimist may
entertain different prior expectations about a gamble. This makes the pinpointing of a prior
expectation rather elusive.
Finally, the uniqueness of a prior expectation can be disputed in favor of multiple
expectations. Arguably, any outcome of a prospect is “expected” to some degree, in fact, to
the degree afforded by its probability.
Therefore, any outcome is liable to trigger
disappointment feelings when compared to better outcomes.
Ordóñez, Connolly and
Coughlan (2000) provide direct experimental evidence in that respect.
They show that
subjects’ satisfaction with their salary outcomes is influenced by multiple comparisons to
other outcomes (higher or lower) separately, rather than comparison to a single reference
point formed by prior integration. Mellers et al. (1997) also provide relevant experimental
evidence: they find that a subject’s emotional response to a given outcome depends on the
unobtained outcomes with which it is paired.
Our hypothesis for modeling Disappointment is that individuals are liable to compare the
outcomes they obtain, not to a prior expectation, but to the other outcomes that they failed to
get. Specifically, upon obtaining a particular outcome xi, an individual will be disappointed
with regard to outcomes better than xi, but also comforted with regard to outcomes worse than
xi. In preliminary work (Delquié and Cillo, 2006), we proposed an implementation of this
idea, but different from what is developed below. To go any further with modeling, it will
now be practical to rank and number the outcomes by order of decreasing preference, that is,
x1 ≥ x2 ≥ ... ≥ xn. Also, assume that the subjective value of outcomes is captured by a nondecreasing function v(·).
Now, let us imagine receiving xi. Intuitively, the intensity of disappointment experienced
from receiving xi as compared to a better xk, k < i, must increase with the difference
(v( xk ) − v( xi ) )
between the subjective values of the two outcomes.
6
This psychological
assumption is indeed built into Bell’s (1985) and Loomes and Sugden’s (1986) approaches.
Likewise, disappointment should increase with the probability of the missed outcome xk: the
more likely—thus, more expected—xk was, the more painful it is to have failed to get it. That
psychological assumption is clearly supported by Van Dijk and Zeelenberg (2002), as they
state (p. 790): “More probable (positive) outcomes give rise to more disappointment when
these outcomes are (unexpectedly) not obtained.”
Based on these considerations, we posit that the disappointment of receiving xi as opposed to
xk is driven by pk D(v( xk ) − v( xi ) ) , where D(·) is a function capturing sensitivity to
disappointment, non-decreasing, and such that D(0) = 0. If either pk = 0 or v( xk ) − v( xi ) = 0 ,
there is no disappointment; furthermore, as either of these terms increases, disappointment
will increase. Now, considering all the outcomes superior to xi, we can measure the total
disappointment accruing to xi with respect to the other outcomes of the gamble as:
∑
i −1
k =1
pk D (v( xk ) − v( xi ) ) . Note that this term is positive, and it is zero if and only if xi is the
best outcome, that is, x1. Also, the lower xi ranks among the other outcomes of the gamble,
the higher the disappointment associated with it. A parallel reasoning applied to the elation
generated
∑
n
k =i +1
by
receiving
xi
leads
to
the
following
measure
of
total
elation:
pk E (v( xi ) −v( xk ) ) , where E(·) is a non-decreasing function capturing sensitivity to
elation, such that E(0) = 0.
In sum, the utility from receiving xi in gamble X is composed of: the value of having xi itself;
the disappointment arising from comparing xi to all the outcomes that are better, that is, the xk,
k = 1, ..., i–1; and the relief from comparing xi to the outcomes that are worse, i.e., xk, k = i+1,
..., n. We denote this adjusted utility by uX(·), defined as:2
 i
  n

u X ( xi ) = v( xi ) −  ∑ pk D(v( xk ) − v( xi )) +  ∑ pk E(v( xi ) − v( xk )) .
 k =1
  k =i

2
(3)
Note how the subscript k runs to i, instead of i−1, and from i, instead of i+1, in the sums in Eq. 3. The terms
corresponding to k = i are zero in any case, but this just makes Eq. 3 work cleanly for i = 1 and n.
7
It is clear in Eq. 3 that the contribution of xi to the gamble utility depends on each and every
other outcome individually and its probability, that is, on the entire context created by X. The
overall valuation of the gamble is then:
n
i
n


U ( X ) = E[u X ( X )] = ∑ pi  v( xi ) − ∑ pk D(v( xk ) − v( xi )) + ∑ pk E(v( xi ) − v( xk )) ,
i =1
k =1
k =i


(4)
where again the outcomes are numbered by order of decreasing preference.
The preference functional expressed in Eq. 4 is a Disappointment model without prior
expectation in the sense that it does not assume a single prior expectation for all outcomes.
Rather every outcome plays a dual role: it is at once an expectation relative to which other
outcomes will be judged, and it bears a mixture of disappointment and elation when judged in
the context of the other outcomes.
Our previous attempt to model multiple comparison of outcomes (Delquié and Cillo, 2006)
differed from the above by assuming v(·) linear and defining D(·) and E(·) to apply to the
sums
∑
i −1
k =1
pk ( xk − xi ) and
∑
n
k =i +1
pk ( xi − xk ) , respectively, rather than differences in
outcomes. That formulation had limitations because it did not allow us to derive the full set
of results that model (4) can accomplish, for example, on stochastic dominance (see Section
2), or the link to a Mean-Variance formulation.
2. Model Properties
Relationship to other models
Expected Utility. It is fairly easy to verify that model (4) includes EU as a special case,
obtained when D(y) = E(y) = cy with c a constant, for any y = v( xi ) − v( x j ) . If c = 0, then
neither disappointment nor elation are present and we clearly have EU.
If c > 0,
disappointment and elation are present, but they exactly cancel each other out to yield EU
8
preferences again. In Section 4, we will further be able to see that EU is obtained whenever
D(y) = E(y) for all y, regardless of whether these functions are linear or not.
Loomes and Sugden (1986). Next, it can be verified that Eq. 4 is different from Loomes and
Sugden’s (1986) Eq. 2. Both models coincide only if the functions D and E in our Eq. 4 are
linear and equal, and D in Eq. 2 is linear, in which case both our and Loomes and Sugden’s
models reduce to EU.
Barring that case, the two models have distinct mathematical
structures. Note that although Eq. 4 is specified with two functions D and E, it does not
necessarily imply more degrees of freedom than Eq. 2. Loomes and Sugden’s function D
covers both disappointment and elation domains: the negative and positive branches of this
function may be different and they correspond to D and E in Eq. 3, respectively. The
application of the disappointment/elation effect is just split differently in Eq. 3.
Bell (1985). Eq. 4 with linear (but not necessarily equal) D and E functions and linear v(·)
coincides with Bell’s (1985) model for two-outcome lotteries, although our model is not built
on the idea of single psychological expectation, as are the models by Bell and Loomes and
Sugden. However, unlike Bell’s (1985, p. 8), our model applies to multiple outcomes and
continuous probability distributions as well: the summations expressed in discrete form in Eq.
4 can be generalized to integrals over continuous variables. In this sense, our model can be
seen as a generalization of Bell (1985).
Jia, Dyer and Butler (2001). These authors reinterpret Bell’s (1985) formulation in a riskvalue framework (Jia and Dyer, 1996) to arrive at the following form:
n


U ( X ) = ∑ pi xi −  d ⋅ ∑ pi (E[ X ] − xi ) − e ⋅ ∑ pi ( xi − E[ X ])
i =1
xi >E[ X ]
 xi ≤E[ X ]

with d and e positive constants. Let us further rearrange Eq. 5 in the following form:
n
U ( X ) = ∑ pi xi − (d − e)
i =1
∑ p (E[X ] − x ) .
xi ≤E[ X ]
i
i
9
(5)
This
form
highlights
a
property
of
Jia,
Dyer
and
Butler’s
(2001)
model:
disappointment/elation depends only on the part of the distribution that lies below E[X].3 In
other words, if X was modified by just changing that part of its distribution above the mean
but without altering E[X], the valuation of the gamble would not change. By contrast, our
model predicts that disappointment, hence, the gamble valuation, is liable to change if
changes occur in any outcomes (and/or their probabilities) above the mean. Thus, given the
same degrees of freedom, our model may offer more flexibility than Jia, Dyer and Butler’s
(2001) in capturing preference patterns.
As another noteworthy difference, Jia, Dyer and Butler (2001) show (Fig. 2, p. 67) that there
can be kinks or breakpoints in the indifference curves generated by their model as an outcome
changes from being above or below the prior expectation. By contrast, Eq. 4 is smooth with
respect to variations in any of the gamble’s outcome or its probability. Finally, Jia, Dyer and
Butler (2001) need to assume specific forms of v(·) (either piecewise linear or power
functions) to obtain their general Disappointment model, while we have no such restriction.
Descriptive account of EU violations
It is straightforward to verify that the preference functional (4) is able to account for some
classic behavioral deviations from EU, such as the Common Ratio Effect. As an example, we
work out the Allais Paradox in Appendix A.
Risk aversion
A couple of immediate properties concerning risk aversion can readily be inferred from Eq. 4.
Any pair of outcomes xi, xj with xi ≥ xj, will give rise to two factors weighing with opposite
signs in the gamble evaluation: (1) the potential elation of receiving xi as opposed to xj,
3
Note that it could alternatively be formulated as depending on the part of the distribution lying above E[X].
The point is that disappointment/elation in Jia, Dyer and Butler’s (2001) model depends on just a part of the
distribution, not its entire structure.
10
pi p j E(v( xi ) − v( x j ) ) ; and (2) the potential disappointment of receiving xj as opposed to xi,
p j pi D(v( xi ) − v( x j ) ) . Clearly, if the disappointment resulting from any difference between
actual and potential outcomes is stronger than the elation resulting from the same, i.e., D(y) >
E(y) for any y = v( xi ) − v( x j ), there will be a net discount in the evaluation of gambles,
enhancing risk averse choices. Conversely, risk-seeking choices will result if elation is
systematically stronger than disappointment. Therefore, D(y) – E(y) drives risk attitude to a
large extent.
In another vein, consider adding a constant to all payoffs of a gamble, that is, a translation of
the distribution. Under the assumption of linear utility of outcomes, all differences (xi – xj)
entering the functions D(·) and E(·) will be unaffected by addition of a constant, and thus the
gamble’s certainty equivalent will change by the same constant. That is, if economic payoffs
are valued linearly, the preference functional in Eq. 4 exhibits constant risk aversion, for any
choice of D(·) and E(·) functions. Therefore, any decreasing risk aversion, often regarded as a
desirable feature, should have to be carried by a non-linear valuation of wealth. It seems right
to have wealth valuation, an economic consideration, be clearly separate from inclination to
disappointment/elation, a psychological reaction to the presence of risk. Bell (1983) made a
similar claim in the context of a Regret model.
After all, one’s being subject to
disappointment for receiving a relatively inferior outcome should be independent from how
one values absolute wealth. This separation between risk attitude and the utility of sure
economic payoffs is reminiscent of that occurring in the RDU model. The similitude between
the two models is even more deeply rooted as we will see in Section 3.
Stochastic dominance
Our focus here is on establishing the conditions under which the preference functional (4)
follows the Stochastic Dominance (SD) principle. For this, we use Machina’s (1982) result
concerning the “local” utility function. If the local utility defined as V ( xi ; X ) = ∂U ( X ) ∂pi is
everywhere non-decreasing in any outcome xi of any probability distribution X, then the
11
preference functional U(X) satisfies first order stochastic dominance. We show, in Appendix
B, that a necessary and sufficient condition for this is:
− 1 ≤ D ′( y ) − E ′( y ) ≤ 1
(6)
everywhere. This seems to match with intuition: as attributes of a gamble vary, the net
resulting variations in disappointment and elation should not have more leverage on overall
utility than the variations in the payoff valuations themselves (which have derivative equal to
1).
Condition (6) is parallel to what Loomes and Sugden (1986) obtained for their
Disappointment/Elation function ( 0 ≤ D′( y ) ≤ 1 ). However, Loomes and Sugden (1986) have
only a sufficient condition, while we have a necessary and sufficient condition.
We now turn to the special case of linear effects of disappointment and elation.
3. Special Case: Linear Disappointment Effects
In this section, we assume linear functions for disappointment and elation, that is, D(y) = d⋅y,
E(y) = e⋅y, with d , e ≥ 0 . Bell (1985) and Jia, Dyer and Butler (2001) make this assumption
throughout their analysis. With linear functions, Eq. 4 takes the form:
n
i
n


U ( X ) = ∑ pi  v( xi ) − d ⋅∑ pk (v( xk ) − v( xi ) ) + e ⋅∑ pk (v( xi ) − v( xk ) ) .
i =1
k =1
k =i


(7)
Probabilities and outcomes seem intertwined in a rather intricate fashion in Eq. 7. Yet, this
expression can be rearranged as follows, as we show in Appendix C:
n
n

 i

U ( X ) = ∑ pi 1 + (d − e ) ∑ pk − ∑ pk   v( xi ) .
i =1
k =i
 k =1


(8)
Probabilities and payoffs values are now clearly separated in a multiplicative fashion in Eq. 8,
that is, the model can be written as:
n
n

 i

U ( X ) = ∑ πi v( xi ) with π i = pi 1 + (d − e ) ∑ p k − ∑ p k   .
k =i
i =1
 k =1


The above formulation looks like a general subjectively weighted utility model, where πi
depends only on the probabilities, not the outcomes. Can the πis, i = 1, …, n, be regarded as
12
subjectively weighted probabilities, though? First, it can be verified that
∑
n
i =1
πi = 1 for any
d, e. In addition, it is easy to show that πi ≥ 0 if and only if − 1 ≤ d − e ≤ 1 . Remember that
this is exactly the necessary and sufficient condition for satisfying SD. Therefore, we see that
if linear disappointment in Eq. 7 is to satisfy SD, then it is equivalent to what may be properly
called a probability transformation defined by:
n

 i

pi a hi ( p1 ,K, pn ) = πi = pi 1 + (d − e ) ⋅  ∑ pk − ∑ pk   .
k =i
 k =1


(9)
The probability transformation (9) suggests that the decision weight πi of an outcome is
obtained by multiplying the outcome’s probability pi by a factor. This factor may be greater
or less than 1, implying overweighting or underweighting of the outcome’s probability,
respectively. For example, assuming d > e, that is to say, disappointment feelings loom larger
than elation, the weighting factor is greater than 1 for
∑
k ≤i
pk ≥ ∑k ≥i p k , less than 1
otherwise. In other words, if d > e, the outcomes below the median of the distribution are
overweighted, while outcomes above the median are underweighted.
It is clear from (9) that the probability weight πi of an individual outcome depends on the
whole distribution, and on the rank of the outcome in the distribution. Therefore, we clearly
have here a probability transformation in the spirit of Quiggin (1982). However, for a true
RDU
representation,
we
should
exhibit
a
weighting
function
w(·)
such
that:
π i = w(∑k =1 pk ) − w(∑k =1 pk ) . In Appendix D, we show that the RDU weighting function
i
i −1
underlying our disappointment model, that is, implicit in the transformation described by (9),
must be of the following form:
w(p) = p(1 – (d–e)) + p2(d–e).
(10)
Remember that this function operates on cumulative probabilities, that is, the argument p of
w(p) is not the probability of an outcome, but rather the probability of obtaining a given
outcome x or better.
It is easy to verify that w(p) is convex whenever d – e > 0,
disappointment weighs more than elation; concave whenever d – e < 0, elation is stronger
than disappointment; and linear when d – e = 0, in which case there is, of course, no
13
probability distortion. Figure 1 provides graphs of the probability weighting w(p) expressed
in Eq. 10 for different values of d – e.
[insert Figure 1 about here]
Jia, Dyer and Butler (2001) also link their model to the probability weighting of Cumulative
Prospect theory (Tversky and Kahneman, 1992). For this, however, they require piecewise
linear utility over gains and losses, and their probability transformation is like a step function,
depending only on whether an outcome is above or below the mean: all outcomes below the
mean have their probabilities augmented by the same multiplicative constant, and those above
the mean reduced by another constant. By contrast, the probability transformation in Eq. 9
smoothly affects each outcome in a different manner.
The family of functions (10) produces cumulative probability transformations that are either
convex or concave. Experimental evidence typically suggests an inverse-S shape (Wu and
Gonzalez, 1996; Abdellaoui, 2000; Bleichrodt and Pinto, 2000)–although all convex shapes
are also commonly encountered. Therefore, concerns may be raised about the descriptive
capabilities of our model. We should underline, however, that we had to assume D, E linear
to obtain the separation of probabilities and outcomes that led to RDU. Introducing nonlinearity in the disappointment and elation functions would significantly augment the variety
of preference patterns that can be captured by our model.
4. A Risk-Value Representation
Bell (1985) was prompt to observe the parallel between a discount for disappointment and a
risk premium. Indeed, any Disappointment model based on a prior expectation can be readily
reframed as a trade-off between risk and value. Such decomposition is not readily obvious in
our model, because all the payoff values are intertwined in each term of the mathematical
expectation in Eq. 4. The key to it is as follows. Consider any two outcomes, xi, xj, of gamble
X; suppose that xi ≥ xj. Then, the ordered difference v( xi ) − v( x j ) ≥ 0 will appear exactly
14
twice in U(X): once in the ith term of the sum, entering function E, and once in the jth term,
entering function D. This is true for all possible ordered pairs of outcomes in the gamble.
Recognizing this, it is possible to mathematically rearrange Eq. 4 to collect the terms
containing v( xi ) − v( x j ) together, and arrive at the following form:
U ( X ) = ∑ pi v( xi ) − ∑ ∑ pi p j (D (v( xi ) −v( x j ) ) − E (v( xi ) −v( x j ) )) ,
(11)
U ( X ) = ∑ p i v( xi ) − ∑∑ p i p j H (v( xi ) −v( x j ) ) ,
(12)
n
i =1
n
or
i =1
n
n
i =1 j =i +1
n
i =1 j >i
with H(y) = D(y) – E(y). It now becomes apparent that only the difference between the
disappointment and elation functions is ultimately relevant to determining the overall utility
of a gamble. Eq. 12 is a Risk-Value model, stating that U(X) is composed of the economic
value of the gamble in the absence of risk considerations, minus a term, let us call it
R ( X ) = ∑∑ pi p j H (v( xi ) −v( x j ) ) , measuring a subjective discount for risk in X.
n
i =1 j >i
H(·) captures the individuals’ preference for dispersion. By definition of D and E, H is
defined on the domain y ≥ 0 and such that H(0) = 0. Also, even though D and E are
monotonic increasing, H may be non-monotonic. Three broad cases may be considered for H.
•
If the disappointment of receiving the inferior of two outcomes is always stronger than the
elation from receiving the opposite: then we will have H(y) ≥ 0 for all y, making R(X) a
positive discount for risk, leading to conservative choices.
•
If, on the contrary, elation feelings consistently dominate disappointment for any given
comparison of outcomes, then H(y) ≤ 0 for all y, leading to risk seeking choice behavior.
•
If the comparison of xi to xj is weighed exactly as much as the reverse comparison of xj to
xi, for all xi, xj, then D(y) = E(y) for all y, thus H(y) = 0. In that case, R(X) = 0 for any
gamble, and Eq. 12 reduces to EU.
Mixed cases are possible, of course, whereby the risk adjustment could be positive for some
gambles, negative for others, and even zero for still other gambles, all depending on the shape
15
of H. For the sake of focusing on H, the above discussion has ignored possible risk aversion
effects distinctly induced by the non-linearity of v. The risk attitudes resulting from v would
combine with those due to H.
Further, R(X) can be viewed as a general measure of the dispersion of a probability
distribution. Indeed, R(X) is equal to 0 whenever all outcomes are equal, and it is invariant to
a translation of the distribution of v(xi).
Interestingly, R(X) includes the variance as a
particular case, obtained for v(·) linear and H(y) = y2 (shown in Appendix E). Thus, with v(·)
linear and H(y) = βy2, β ≥ 0, Eq. 12 just becomes:
U ( X ) = E[ X ] − βσ 2 [X ] ,
(13)
the Mean-Variance model widely used in finance. From Condition (6), we know that Eq. 13
will satisfy stochastic dominance if and only if − 1 ≤ H ′( y ) = 2β y ≤ 1 , i.e. y ≤ 1/(2β) for all y.
This yields a general result: the Mean-Variance model will satisfy stochastic dominance for
all gambles having a maximum spread ≤ 1/(2β), regardless of the shape of their distributions.
As β reflects the weight put on risk, i.e., risk aversion, this shows that a Mean-Variance model
will satisfy SD for all gambles whose total spread is small enough relative to risk tolerance.
Synthesizing the findings of Sections 3 and 4, we can now summarize the connections
between Disappointment without Prior Expectation and other standard models as in Figure 2.
[Insert Figure 2 about here]
It may be surprising that there exists a linkage between RDU and Mean-Variance, two models
that have evolved from quite distinct intellectual paths. The connection all stems out of a
simple behavioral proposition: that individuals are sensitive to the differences between the
outcomes of a gamble, not just their absolute values. Thus, beneath the mathematical links
described in Figure 2, Disappointment without Prior Expectation offers a unifying
psychological rationale for the different models, which may have been lacking in some cases.
For RDU, the cumulative probability transformation was originally engineered to produce
desired behavioral results without violating stochastic dominance. It has sometimes been
16
observed that it lacks clear psychological underpinnings (Diecidue and Wakker, 2001). On
the Risk-Value side, our model provides a clear behavioral foundation to the Mean-Variance
model, conceptually attractive and widely used because it features two well understood
summary measures of a gamble, which are cornerstones of the analysis of random variables.
The gamble outcomes are net payoff levels in our model, not changes in payoffs; thus, we do
not distinguish between positive and negative outcomes. Differences in outcomes, which may
be felt as gains or losses depending on the direction in which the difference is experienced, do
matter however, as they enter the disappointment and elation functions. Thus, D and E are
reminiscent of the negative and positive branches of the Prospect theory (Kahneman and
Tversky, 1979) value function, respectively.
5. Conclusion
To model disappointment, we assumed that each and every outcome may act as an
expectation. This behavioral hypothesis can be deemed a priori as compelling as the standard
assumption of a single prior expectation. Our model leads naturally to other classic models of
choice under risk.
On the practical side, those interested in studying the economic
implications of disappointment in various areas of decision under risk, e.g., finance or
insurance, may find our result helpful: using RDU with a weighting function as in Eq. 10, or
using the Risk-Value formulation of Eq. 12 would be ways of implementing disappointment
aversion.
From among the different Disappointment models reviewed here, which one provides a better
fit to choice data remains an open question. We intend no claim here about the descriptive
performance of our model compared to alternatives. Experimental tests should be carried out
to shed light on this.
Also, our approach was inspired by plausible psychological
considerations regarding expectations and reference points. Future research efforts may help
identify a set of behavioral axioms underlying Disappointment without Prior Expectation.
17
6. Appendix A
Allais Paradox. Loomes and Sugden (1986) and Jia, Dyer and Butler (2001) show that they
can explain the Allais paradox. Bell (1985) does not address it since his model deals with
two-outcome lotteries only. Let us consider the following choice problems:4
Problem 1: Choose between A and B
A:
$2 million for sure
B:
0.10 chance of $3 million
0.89 chance of $2 million
0.01 chance of $0
D:
0.10 chance of $3 million
0.90 chance of $0
Problem 2: Choose between C and D
C:
0.11 chance of $2 million
0.89 chance of $0
A prevalent pattern of choices is A f B and D f C , which is unallowable under EU. Take
D(y) = d⋅y, E(y) = e⋅y, and v(x) = x. With d – e = 0.9, we obtain: U(A) = 2 > U(B) = 1.98 and
U(C) = 0.044 < U(D) = 0.057.
7. Appendix B
Theorem 1. The preference functional (4) satisfies first order stochastic dominance if and
only if − 1 ≤ D ′( y ) − E ′( y ) ≤ 1 for any y ≥ 0.
Proof. Based on Machina (1982), the local utility function V ( xi ; X ) = ∂U ( X ) ∂pi for an
outcome xi can be interpreted as the rate at which valuation changes as the probability
associated with xi changes.
Machina (1982) shows that the preference functional U(X)
satisfies first order stochastic dominance if and only if V ( xi ; X ) is non-decreasing in xi.
4
We use payoffs that are slightly different from the traditional Allais choice questions. In our numerous
classroom demonstrations, we have observed that the payoffs used here produce more Allais-type violations of
EU nowadays than Allais’ original values.
18
Let us derive V ( xi ; X ) = ∂U ( X ) ∂pi . The main thing to be careful about in taking the partial
derivative of U(X) with respect to a given pi is to realize that pi is present not just in the ith
term but in every term of the sum in (4), due to its appearance in either the disappointment or
the elation part of every outcome. We obtain the following:
V ( xi ; X ) =
i
n
∂U ( X )
= v( xi ) − ∑ p k D(v( xk ) − v ( xi )) + ∑ p k E(v ( xi )−v( xk )) +
∂pi
k =1
k =i
n
i
k =i
k =1
− ∑ p k D(v( xi )− v( xk )) + ∑ p k E (v( xk ) − v( xi ))
i −1
V ( xi ; X ) = v( xi ) − ∑ p k (D(v( xk ) − v( xi )) − E(v( xk ) − v( xi )) ) +
k =1
−
(14)
n
∑ p (D(v( x )− v( x )) − E(v( x )−v( x )) )
k =i +1
k
i
k
i
k
Sufficiency. Assume − 1 ≤ D ′( y ) − E ′( y ) ≤ 1 for any y ≥ 0. Let us show that V ( xi ; X ) is nondecreasing in xi, that is, ∂V ( xi ; X ) ∂xi ≥ 0 . From (14), we readily obtain:
∂V ( xi ; X )
 i −1
= v′( xi )1 + ∑ p k (D ′(v( xk ) −v( xi )) − E ′(v( xk ) − v( xi )) )
∂xi
 k =1
−

∑ p (D ′(v( x ) −v( x )) − E ′(v( x )− v( x )))
n
k =i +1
k
i
k
i
k
Since D ′(v( xk )−v( xi )) − E ′(v( xk ) − v( xi )) ≥ −1 for each xk, we have:
i −1
i −1
k =1
k =1
A = ∑ p k (D ′(v( xk ) −v( xi ) ) − E ′(v ( xk ) − v ( xi ) )) ≥ −∑ p k .
Since D ′(v( xi ) −v( xk )) − E ′(v( xi ) − v ( xk )) ≤ 1 for each xk, we have:
B=
n
∑ pk (D ′(v( xi ) −v( xk )) − E ′(v( xi )− v( xk )) ) ≤
k =i +1
n
n
k =i +1
k =i +1
∑ pk that is, − B ≥ − ∑ pk .
Adding the two above inequalities we get:
i −1
A − B ≥ −∑ pk −
k =1
n
∑p
k =i +1
k
= −(1 − pi ), that is : 1 + A − B ≥ pi ≥ 0 .
We have shown that, if − 1 ≤ D ′( y ) − E ′( y ) ≤ 1 for any y, then
is, V ( xi ; X ) is non-decreasing in xi.
19
∂V ( xi ; X )
≥ v′( xi )( pi ) ≥ 0 , that
∂xi
Necessity. Assume that V ( xi ; X ) is non-decreasing in any xi of any probability distribution X.
In particular, for x1 we have:
n
∂V ( x1 ; X )


= v′( x1 )1 − ∑ p k (D ′(v( x1 ) −v( xk )) − E ′(v( x1 )− v( xk )) ) ≥ 0 for any pk and xk ≤ x1.
∂x1
 k =2

Suppose there existed y ≥ 0 such that D ′( y ) − E ′( y ) > 1 , that is, D ′( y ) − E ′( y ) = 1 + ε , with ε >
0. Then, take the gamble with: p2 = 1/(1+ε/2), v( x2 ) = v( x1 )− y , and pk = 0, for k = 3, …, n
(therefore p1 = 1−p2 = (ε/2)/(1+ε/2)). For this gamble, we would have:
∂V ( x1 ; X )
1
= v′( x1 )(1 − p 2 (D ′(v( x1 ) −v ( x2 )) − E ′(v( x1 )− v( x2 )) )) = v′( x1 )(1 −
(1 + e)) < 0, a
∂x1
1 + e/2
contradiction.
In a parallel fashion, for xn we have:
∂V ( xn ; X )
 n−1

= v′( xn )1 + ∑ p k (D ′(v( xk )−v ( xn )) − E ′(v( xk )− v( xn )) ) ≥ 0 for any pk and xk ≥ xn.
∂xn
 k =1

Suppose there existed y ≥ 0 such that D ′( y ) − E ′( y ) < −1 , that is, D ′( y ) − E ′( y ) = −1 − ε , with
ε > 0. Then, take the gamble with: p1 = 1/(1+ε/2), v( x1 ) = v( x2 )+ y and pk = 0, for k = 3, …, n
(therefore p2 = 1−p1 = (ε/2)/(1+ε/2)). For this gamble, we would have:
∂V ( x2 ; X )
= v′( x2 )(1 + p1 (D ′(v( x1 ) − v( x2 )) − E ′(v( x1 ) − v( x2 )) ))
∂x2

1
(− 1 − ε ) = 1 − 1 + e < 0,
= v′( x2 )1 +
1 + ε/2
 1 + ε/2

a contradiction.
8. Appendix C
Developing the summation in Eq. 7, we get:
20
n
i
i
n
n


U ( X ) = ∑ pi  v( xi ) − d ⋅∑ pk v(xk ) + d ⋅∑ pk v(xi ) + e ⋅∑ pk v(xi ) − e ⋅∑ pk v(xk ) 
i =1
k =1
k =1
k =i
k =i


n
i
n
n
i
n
n


= ∑ pi v(xi )1 + d ∑ pk + e∑ pk  − d ∑ pi ∑ pk v(xk ) − e∑ pi ∑ pk v(xk ).
i =1
k =1
k =i
i =1
k =1
i =1
k =i


n
i
i =1
k =1
(15)
The term d ⋅∑ pi ∑ p k v(xk ) can be rewritten as follows:
n
i
i =1
k =1
d ⋅∑ pi ∑ p k v(xk )
= d ( p1 ( p1v(x1)) + p 2 ( p1v(x1) + p 2 v(x2 ) ) + L + p n ( p1v(x1) + L + p n v(xn ) ))
= d ( p1v(x1)( p1 + p 2 + L + p n ) + p 2 v(x2 )( p2 + L + p n ) + L + p n v(xn )( p n ) )
n
n
i =1
k =i
= d ⋅∑ pi v( xi )∑ p k .
(16)
n
n
i =1
k =i
Likewise, we can rewrite e ⋅∑ pi ∑ p k v(xk ) in the following way:
n
n
i =1
k =i
e ⋅∑ pi ∑ p k v(xk )
= e( p1 ( p1v(x1) + L + p n v(xn ) ) + p2 ( p2 v(x2 ) + L + pn v(xn ) ) + L + pn ( pn v(xn )) )
= e( p1v(x1)( p1 ) + p2 v(x2 )( p1 + p2 ) + L + pn v(xn )( p1 + p2 + L + pn ) )
n
i
i =1
k =1
= e ⋅∑ pi v( xi )∑ p k .
(17)
We can substitute Eq. 16 and Eq. 17 into the Eq. 15 to get:
n
i
n
n
n
n
i


U ( X ) = ∑ pi v( xi )1 + d ⋅ ∑ p k + e ⋅∑ p k  − d ⋅∑ pi v( xi )∑ p k − e ⋅∑ pi v( xi )∑ p k
i =1
k =1
k =i
i =1
k =i
i =1
k =1


n
i
n
n
i


= ∑ pi v( xi )1 + d ⋅∑ p k + e ⋅∑ p k − d ⋅∑ p k − e ⋅∑ p k 
i =1
k =1
k =i
k =i
k =1


n
n

 i

= ∑ pi v( xi )1 + (d − e ) ∑ pk −∑ p k  ,
i =1
k =i
 k =1


which is nothing but Eq. 8.
9. Appendix D
Under RDU, the utility VRDU of a gamble X is given by:
21
n
i
i −1
i =1
k =1
k =1
VRDU ( X ) = ∑ π i u ( xi ), where π i = w(∑ pk ) − w(∑ pk ) for i = 1,K, n ,
with w strictly increasing from [0,1] to [0,1] and w(0) = 0, w(1) = 1.
Let us show that the probability weighting function w(p) = p(1 – (d–e)) + p2(d–e) of Eq. 10
produces the transformation of Eq. 9, that is:
n
i
i −1

 i

pi 1 + (d − e ) ⋅  ∑ pk − ∑ pk   = w(∑ pk ) − w(∑ pk ) for i = 1,..., n.
k =i
k =1
k =1
 k =1


Denote d − e = δ. We can rewrite πi as expressed in Eq. 9 as follows:
n

 i

π i = pi 1 + δ ∑ p k − ∑ p k  
k =i
 k =1


i −1
 i

= pi + δpi  ∑ p k − 1 + ∑ p k 
k =1
 k =1

(18)
i −1


= pi + δpi  pi + 2∑ p k − 1.
k =1


Now, let us apply w(p) = p(1 – (d–e)) + p2(d–e) of Eq. 10 to calculate the decision weight of
outcome i, for i = 1,..., n. We get:
i
i −1
k =1
k =1
w(∑ pk ) − w(∑ pk )
= ( p1 + ... + pi )(1 − δ) + δ( p1 + ... + pi ) 2 − ( p1 + ... + pi −1 )(1 − δ) − δ( p1 + ... + pi −1 ) 2
= (1 − δ) pi + δ( pi + 2 p1 pi + ... + 2 pi −1 pi )
2
= pi + δpi ( pi + 2 p + .... + 2 pi −1 − 1)
i −1
= pi + δpi ( pi + 2∑ pk − 1),
k =1
which is nothing but πi in Eq. 18.
10. Appendix E
Let us show that R ( X ) = ∑∑ pi p j H (v( xi ) −v( x j ) ) is equal to σ 2 [ X ] for v(·) linear and H(y)
n
i =1 j >i
= y2. With v(·) linear and H(y) = y2, we have:
22
n
n
R ( X ) = ∑∑ pi p j ( xi − x j ) 2 = ∑∑ p i p j ( xi2 − 2 xi x j + x 2j )
i =1 j > i
i =1 j > i
n
n
n
= ∑∑ pi p j xi2 + ∑∑ p i p j x 2j − 2∑∑ pi p j xi x j
i =1 j > i
i =1 j > i
i =1 j >i
n
n

 n


= ∑ p i xi2  ∑ p j  + ∑ p i  ∑ p j x 2j  − 2∑∑ p i p j xi x j .
i =1
i =1 j > i
 j >i  i =1  j >i

(19)
Now, the second summation in the preceding line can be expanded as follows:

n
∑ p  ∑ p
i =1
i

j >i
j

x 2j  = p1 p 2 x 22 + p1 p 3 x32 + L + p1 p i xi2 + L + p1 p n x n2

+ p 2 p 3 x32 + L + p 2 p i xi2 + L + p 2 p n x n2
(20)
M
+ p i −1 p i xi2 + L + p i −1 p n x n2
M
+ p n −1 p n x n2
The above sum may be arranged by gathering and factorizing all the terms containing xi first,
n
that is, by columns in Eq. 20. Doing so permits to rewrite Eq. 20 as:

∑ p x  ∑ p
i =1
i
2
i

j <i
j

 . (If the


subscript j should go outside of range (i.e., j = 0 or j = n+1) in the above summations, the
corresponding terms do not exist, i.e., they are 0, of course.) Eq. 19 can now be further
pursued as follows:
n
n

 n


R ( X ) = ∑ pi xi2  ∑ p j  + ∑ pi xi2  ∑ p j  − 2∑∑ pi p j xi x j
i =1
i =1 j >i
 j <i 
 j >i  i =1
n
n
= ∑ pi xi2 (1 − pi ) − 2∑∑ pi p j xi x j
i =1
i =1 j >i
n
n
i =1
i =1
n
= ∑ pi xi2 − ∑ pi2 xi2 − 2∑∑ pi xi p j x j
i =1 j >i
n
 n

= E[ X 2 ] −  ∑ pi2 xi2 + 2∑∑ pi xi p j x j 
i =1 j >i
 i =1

2
 n

2
= E[ X ] −  ∑ pi xi  = E[ X 2 ] − (E[ X ]) = σ 2 [ X ].
 i =1

2
23
References
Abdellaoui, Mohammed. (2000). “Parameter-Free Elicitation of Utility and Probability
Weighting Functions,” Management Science 46, 1497-1512.
Bell, David. (1983). “Risk Premiums for Decision Regret,” Management Science 29 (10),
1156-1166.
Bell, David. (1985). “Disappointment in Decision Making under Uncertainty,” Operations
Research 33, 1-27.
Bleichrodt, Han, and José Luis Pinto. (2000). “A Parameter-Free Elicitation of the Probability
Weighting Function in Medical Decision Analysis,” Management Science 46, 14851496.
Delquié, Philippe, and Alessandra Cillo. (2006). “Expectations, Disappointment, and RankDependent Probability Weighting,” Theory and Decision 60 (2-3), 193-206.
Diecidue, Enrico, and Peter Wakker. (2001). “On the Intuition of Rank-Dependent Utility,”
Journal of Risk and Uncertainty 23, 281-298.
Gul, Faruk. (1991). “A Theory of Disappointment Aversion,” Econometrica 59, 667-686.
Jia, Jianmin, and James S. Dyer. (1996). “A Standard Measure of Risk and Risk-Value
Models,” Management Science 45, 519-532.
Jia, Jianmin, Dyer, James S., and John C. Butler. (2001). “Generalized Disappointment
Models,” Journal of Risk and Uncertainty 22 (1), 59-78.
24
Kahneman, Daniel, and Amos Tversky. (1979). “Prospect Theory: An Analysis of Decision
under Risk,” Econometrica 47, 263-291.
Loomes, Graham, and Robert Sugden. (1986). “Disappointment and Dynamic Consistency in
Choice under Uncertainty,” Review of Economic Studies 53, 271-282.
Machina, Mark J. (1982). “Expected Utility Analysis Without the Independence Axiom,”
Econometrica 50, 277-323.
Mellers, Barbara A., Alan Schwartz, Katty Ho, and Ilana Ritov. (1997). “Decision Affect
Theory: Emotional Reactions to the Outcomes of Risky Options,” Psychological
Science 8 (6), 423-429.
Ordóñez, Lisa D., Connolly, Terry, and Richard Coughlan. (2000). “Multiple Reference
Points in Satisfaction and Fairness Assessment,” Journal of Behavioral Decision
Making 13 (3), 329-344.
Quiggin, John. (1982). “A Theory of Anticipated Utility,” Journal of Economic Behavior and
Organization 3, 323-343.
Tversky, Amos, and Daniel Kahneman. (1992). “Advances in Prospect Theory: Cumulative
Representation of Uncertainty,” Journal of Risk and Uncertainty 5, 297-323.
van Dijk, Wilco W., and Marcel Zeelenberg. (2002). “What Do We Talk about When We
Talk about Disappointment? Distinguishing Outcome-Related Disappointment from
Person-Related Disappointment,” Cognition and Emotion 16 (6), 787-807.
Wu, George, and Richard Gonzalez. (1996). “Curvature of the Probability Weighting
Function,” Management Science 42, 1677-1690.
25
1
0.9
0.8
0.7
d - e = -1
0.6
d - e = -0.5
0.5
d-e=0
d - e = 0.5
0.4
d-e=1
0.3
0.2
0.1
0
0
0.1 0.2
0.3
0.4
0.5 0.6
0.7
0.8 0.9
1
Figure 1. Probability weighting functions w(p) = p(1 – (d–e)) + p2(d–e).
26
Expected
Utility
H(y) = 0
Disappointment without
Prior Expectation
H(y) = dy
−1 ≤ d ≤ 1
Risk-Value
models
Rank-Dependent Utility
with quadratic probability
weighting
v(x) = x
H(y) = βy2
The Mean-Variance
Model
Figure 2. The relationship between Disappointment Without Prior Expectation
and other classic models of decision under risk. Dashed boxes
indicate the assumptions, if any, associated with each link.
27