A note on optimism and pessimism when the reference point
corresponds to the individual’s expectations
Fabrice Le Lec*, Serge Macé†
31 January 2017 – Preliminary Draft
__________________
Abstract: When the reference point of the individual is endogenously determined by expectations that
she has about her possible future outcomes as in the Kőszegi-Rabin 2007’s model, optimism and
pessimism generate preferences that differ from those predicted in a standard expected utility framework
in which decisions weights are equal to subjective probabilities. In particular, being optimistic about the
occurrence of a good outcome can reduce the perceived expected utility, whereas being pessimistic can
increase it. Furthermore, in some cases, it pays to be pessimistic even if it leads the individual to make
a choice that differ from the choice the individual would have made having correctly predicted
probabilities.
Keywords: pessimism, optimism, multiple reference point, loss aversion, expectations
JEL Classification : D03, D81
__________________
*
Université Paris 1, Centre d’Economie de la Sorbonne, umr CNRS 8174, 116-112 Boulevard de l’Hôpital, 75013
Paris, France, [email protected]
†
Corresponding author. Edhec Business School, 24 avenue Gustave Delory, Roubaix. [email protected]
1 Introduction
When the objective probability of an event that matters for an individual is unknown, this
individual must assign it a subjective probability. And it often occurs that people are excessively
optimistic or pessimistic in their predictions, which influences virtually all possible major economic
decisions under risk, from health prevention and portfolio diversification to business creation or
insurance. For this reason, optimism and pessimism have been extensively studied in the psychological
literature (Chang, 2001) and through the main models of decision under risk, notably rank-dependent
utility (Quiggin, 1982; Gonzales and Wu, 1999), ambiguity aversion (Schmeidler, 1989) or cumulative
prospect theory (Kahneman and Tversky, 1992).
More recently, Kőszegi and Rabin (2006, 2007) have proposed a model that generalizes the
standard expected utility model, while integrating prospect theory assumptions and an endogenous
stochastic reference point corresponding to expectations. In this model, an individual facing a future
uncertain prospect updates her reference point to her (rational) expectations: her new stochastic
reference point mimics the future lottery. She then evaluates this future prospect according to its
expected utility, with the utility of each outcome being the weighted average of how it feels relative to
each possible realization of this stochastic reference point. This model has become a reference model
for reference-dependent preferences (Masatlioglu and Raymond, 2016). It is particularly relevant to
study economic preferences in a context in which expectations form natural reference points like the
expectation of a wage increase for instance. And it has received partial empirical support (Abeler, Falk,
Goette, and Huffman, 2011; Crawford and Meng, 2011; Card and Dahl, 2011; Gill and Prowse, 2012;
Bartling, Brandes, and Schunk, 2015).
The objective of this note is to show that the Kőszegi and Rabin’s model (hereafter, KR model)
makes predictions about the effect of optimism and pessimism that contradict several results derived
from the standard expected utility framework (with subjective probabilities). In particular, being
optimistic about the occurrence of a good outcome can reduce the perceived expected utility, whereas
being pessimistic can increase it. Furthermore, in some cases, it pays to be pessimistic even if it leads
the individual to make a choice that differ from the choice the individual would have made, having
correctly predicted probabilities.
The remainder of this note is organized as follows: In section 2, we briefly expose the
consequences of optimism and pessimism in the standard expected utility model in the simplified case
of an individual facing a binary lottery. In section 3, we contrast these intuitive results with those derived
from the KR model. Section 4 concludes.
2 Optimism and Pessimism in the standard expected utility model
Consider an individual facing a simple binary lottery L  ( y, x ; p,1  p) with y  x , p being the
probability of the good outcome. Denote L  ( y, x ; p ,1  p ) with p  p the perception of the
lottery by an optimist individual and L  ( y, x ; p ,1  p ) with p  p its perception by a pessimist
individual. Denote also m(.) with m  0 and m  0 the standard utility function (this notation helps
2
future comparisons with the KR model). In the standard model without reference point or anticipatory
emotions, the expected utility given by any lottery L is equal to
U e ( L)  pe m( y)  (1  pe )m( x)
(1)
with p e the subjective estimated probability of the good outcome.
Assume first exogenous probabilities. We have the two following propositions:
Proposition 1: The future utility will be equal to either m( y ) or m( x) and does not depend on the
pessimism or optimism of the individual.
Corollary 1: There is no interest for the individual to be strategically pessimist or optimist
Corollary 2: The true expected utility does not depend on optimism or pessimism and is equal to
U ( L)  pm( y)  (1  p)m( x)
Proposition 2: Perceived expected utility increases monotonically with the perceived probability of the
good outcome.
This second proposition is the direct consequence of first-order stochastic dominance. It implies that
U ( L )  U ( L)  U ( L ) : Optimism increases perceived expected utility, pessimism lowers it.
Assume now that the probability p depends positively on an effort c  0 made by the individual
p  p(c) . Optimism and pessimism correspond to monotonic transformation of p(c) . Optimism
implies p (c)  p(c) for all c and p (c)  p (c) for all c. To keep things simple, also assume first that
efforts enter as an additional argument in the utility function1. The estimated expected utility is now
given by:
U e ( L, c)  pe (c)m( y)  (1  pe (c))m( x)  c
(2)
with c  0 if the individual does an effort and c  0 if he does not. This estimated expected utility
corresponds to the true expected utility when p e (.)  p . The individual chooses to make the effort c if
U e ( L, c)  U e ( L,0)  0 , that is after rearrangement if  pe (c)  pe (0)   m( y)  m(c)   c . As a result,
the true expected utility now depends on the optimism or pessimism of the individual, but only through
their influence on the level of effort made by the individual, which may not correspond to the one chosen
by the individual when she correctly predicts probabilities.
The consequences of optimism and pessimism on the choices of the individual depend on the
exact form of this deformation of the true probability function p(c) and on how the effort enters the
utility function. Many cases are indeed possible. Depending on the exact specification, optimism for
instance can lead the individual to make more effort and so increase the probability of the good outcome
(self-fulfilling prophecies), push her on the contrary to make less effort (self-defeating prophecies) or
1
An
alternative
assumption
is
that
U ( L, c)  p(c)m( y  c)  (1  p(c))m( x  c) .
efforts
are
made
of
the
same
argument
3
leave her efforts unchanged. Given that our objective in this note is to contrast meaningfully the
consequences of optimism and pessimism predicted in the expected utility model and the KR (2007)’s
model, it is helpful to restrict our attention to a simplified well-defined case. Let us so assume that the
individual can only exert two possible levels of efforts: c  0 and 0 . Denote p  p(c)  p(0) ,
p  p (c)  p (0) and p  p (c)  p (0) . For our purpose, we adopt the following definition:
Definition: An individual is optimistic regarding her efforts if p  p and pessimistic if p  p
.
Hence, an optimistic individual as previously defined overestimates the influence of the effort c on the
probability to get the good outcome, whereas the pessimistic individual underestimates it. Both
optimism and pessimism can generate self-fulfilling prophecies but self-defeating prophecies are no
longer possible. For instance, suppose that the pessimist individual chooses c  0 , whereas she would
have chosen c  0 , had she correctly predicted probabilities (self-fulfilling prophecy). Given the
previous reasoning, it means that U  ( L, c)  U  ( L,0)  U ( L,0)  U ( L, c) or after rearrangement,
that:
p m
 c 
estimated extra utility if c >0
pm
(3)
true extra expected utility if c > 0
with m  m( y)  m( x) the increase in utility of having y instead of x.
Similarly, an optimist individual chooses c  0 , whereas he would have chosen c  0 , had she correctly
predicted probabilities if U ( L,0)  U  ( L, c)  U  ( L,0) or after rearrangement if
p m
estimated extra utility if c > 0
 c 
pm
(4)
true extra utility if c > 0
Note however that even if optimism can create a self-fulfilling prophecy pushing the individual to make
the effort, whereas he would not have made it, this effort is still ex ante suboptimal. More precisely, we
have the following proposition:
Proposition 3: Ex ante, the true expected utility is maximized when the individual chooses a level of
effort that corresponds to the level chosen by the individual when she correctly assesses the
probabilities.
Put another way, the optimal effort corresponds to the one that maximizes true expected utility. There
is no strategic advantage to be either optimistic or pessimistic. It is often claimed that optimism and
pessimism can be sometimes self-serving biases (Chang, 2001) but if they are, it is for reasons outside
the standard expected model and that are related to the counteraction of other bias (lack of confidence),
other phenomena like positive anticipatory emotions or as we see now, through their influence on the
reference points.
3. Optimism and pessimism when the reference point corresponds to past expectations
4
3.1 The model
Following KR (2007), suppose now that the utility derived by an individual from a deterministic
outcome x is given by:
u( x r )  m( x)  µ(m(x)  m(r))
(5)
The utility is the sum of two components: the intrinsic utility that corresponds to the traditional
utility function. It is increasing with x and concave in the general case. The second component µ
corresponds to gain-loss utility, that is the additional sensation of loss or gain created from the departure
from the reference point r. µ is continuous for all x, twice differentiable for x  0 and strictly increasing
with µ(0)=0. Loss aversion for large deviations from the reference point is captured by assuming that,
if y  x  0 , then µ( y)  µ( x)  µ( x)  µ( y) and near the reference point by stating that
u (0) u (0)    1 with u the left derivative and u the right derivative. Though it does not matter
directly for most of our results, we also assume, in coherence with prospect theory, some diminishing
sensitivity in the domains of losses and gains, that is u( x)  0 for x  0 and u( x)  0 for x  0 .
If the individual fully adjusts her anticipation to the future lottery and forms rational
expectations (that is, she has an accurate view of the distribution), the expected utility of an individual
facing the lottery L is given by
U ( L | L)  pm( y)  (1  p)m( x)  p(1  p) µ(m( y)  m( x))  (1  p) pµ(m( y)  m( x))
exp ected intrinsic utility
(6)
exp ected gain loss utility
Thus, the KR model adds an expected gain-loss utility to the expected intrinsic utility of the standard
EU model. It is also useful to rewrite slightly differently this utility function. Denote, µ  µ(m) and
µ  µ(m) with m  m( y)  m( x) . After rearrangement, we have:
U ( L | L)  m(x)  pm+ p(1  p)µ  (1  p) pµ
(7)
Equation (7) shows that when an individual has a probability p  0 to get y instead of x for sure that
would give him the utility level m( x) , three effects occur. It gives to the individual : i) a probability
p  0 to get an increase in intrinsic utility equal to m(h) ii) a probability p  0 to benefit from an
additional sensation of gain equal to (1  p) µ corresponding to the fact that the individual can get y
whereas he had a probability (1  p) to have x, iii) a probability (1  p) to get a loss sensation equal
to pµ if she has x whereas she was expecting y with a probability p. The first two effects play
positively while the last one enters the total utility negatively. In the absence of loss aversion, ii) and iii)
cancel out. But if loss aversion is strong enough, it may more than offset effects i) and ii). For our
purpose, this utility function has two important properties. Denote ( x,1) the certainty of having x . We
have:
5
Property 1: ( x,1)
Lp iff U ( Lp | Lp )  m(x)  pm+ p(1  p)(µ  µ )  0 , that is after
rearrangement if (µ  µ )  m 1  p 2
Property 2: Denote Lp  ( y, x ; p,1  p) the lottery with the probability p variable to get the good
outcome y . Then, if conditions of proposition 1 hold, U ( Lp | Lp ) decreases on [0, p 2] with
p  (m  µ) / µ > 0 .3
Hence, in the presence of loss aversion, property 1 indicates that the individual may prefer the certainty
of having x for sure to the lottery Lp  ( y,x ; p,1  p) when the latter inflates too much the expected
loss sensation if y does not occur. Property 2 indicates that not only the individual may prefer the
certainty of having x for sure, but that for small values of p , an increase in the probability of the bad
outcome may even reinforce the interest of the individual to reject this small probability of gain.
3.2 Optimism and pessimism with a stochastic reference point
In the standard EU framework without reference point, we stated the following propositions:
Proposition 1: With exogenous probabilities, the true future utility is either m( y ) or m( x) and
does not depend on the pessimism or optimism of the individual. As a result, there is no interest
for the individual to be strategically pessimist or optimist
Proposition 2: With exogenous probabilities, perceived expected utility increases monotonically
with the perceived probability of the good outcome
Proposition 3: With endogenous probability, the true expected utility is maximized when the
individual chooses a level of effort that corresponds to the level chosen by the individual when
she correctly assesses the probabilities.
In a KR model described above, these three propositions are no longer true and can be replaced by the
following ones:
Proposition 1’: Assume exogenous probabilities. Whatever the outcome, optimism reduces future utility
by reducing the gain sensation if the favorable outcome occurs and increases the loss sensation if it
does not. Pessimism increases future utility for the opposite reasons.
Corollary 1’: It is of interest for the individual for the individual to be pessimistic.
Corollary 2’: The true expected utility increases when people are pessimistic and decreases when they
are optimistic.
2
This possibility of violating first-order stochatic dominance is underlined by Kőszegi and Rabin in their
proposition 7.
3
Proof :
U ( Lp | Lp )
p
 m  (1  2 p) µ  (1  2 p) µ . It is positive if p 
m  µ
p
that is if p  . Property
2
µ
2 ensues.
6
In the standard expected utility framework, with exogenous probabilities, pessimism and optimism
influence the perceived expected utility but does not influence the possible true future utilities. This is
no longer true when we add a gain-loss utility component in which the multiple reference point
corresponds to the expected outcomes. The reason is that even if being pessimistic or optimistic does
not change the true probability of getting either y or x when this probability is exogenous, it changes
the reference point and with it, the gain/loss sensations.
With p e the estimated probability of getting y (that can be equal or not to p ), the true future utility is
given by m( y)  (1  p e )µ if the favourable outcome y occurs and by m( y)  p e µ if it is the
unfavourable outcome. Both depend negatively on p e . As a result, if the individual is pessimistic for
example, the extra sensation of loss is lower in the unfavorable state and equal to p µ  pµ .
Simultaneously, the extra sensation of gain is higher and equal to (1  p ) µ  (1  p) µ in the
favorable state. Corrolary 1’ is another way to express the common idea that with exogenous
probabilities, “having low expectations is the secret of happiness”. Optimism has opposite
consequences.
In the presence of gain/loss utility, the true expected utility is given :
U  m(x)  pe m+ p(1  pe )µ  (1  p) p e µ
(8)
Given that it corresponds to the weighted sum by true probabilities of the possible utility levels that
depends negatively on p e , the true expected utility also increases when people are pessimistic and
decreases when they are optimistic.
Now, optimism and pessimism affect not only the true expected utility but also the perceived expected
utility. The latter is given by U e  m(x)  pe m+ pe (1  pe )µ  (1  pe ) pe µ . It depends on the
perceived probability p e . But as indicated by the next proposition, in this model, the relation may not
to be monotonically positive.
Proposition 2’: Assume exogenous probabilities. A higher perceived probability to get the good
outcome can reduce the perceived expected utility.
This paradox is related to the above mentioned paradox of the KR model (properties 1 and 2) indicating
that in the presence of loss aversion, the individual may in some situations prefer the certainty of having
x for sure to the lottery L  ( y, x ; p,1  p) . As we state, ( x,1)
Lp iff (µ  µ )  m 1  p ,
which is satisfied for some values of the parameters. Furthermore, assume that this condition holds,
property 2 indicates that U ( Lp | Lp ) decreases on [0, p 2] with p  (m  µ) / µ > 0 . As a
consequence, starting from a situation in which the individual prefers ( x,1) to L p with p  p 2 , if
an individual moderately overestimates the probability of the good outcome, anticipating p  such that
p  p   p 2 , this will reduce her perceived expected utility. On the contrary, moderately
underestimating the probability will increase perceived expected utility.
7
Proposition 3’: Assume endogenous probabilities. With gain-loss utility, it can still pay for the
individual to be pessimistic even if it leads the individual to choose a level of effort different from the
level chosen by the individual when she correctly assesses the probabilities.
To understand the idea, suppose that the pessimist individual chooses c  0 , whereas she would have
chosen c  0 , had she correctly predicted probabilities (self-fulfilling prophecy). It implies that
U  ( L, c | L, c)  U  ( L,0 | L,0)  U ( L, c | L, c) . Replacing each term of this inequality by its
expression from equation (7), we have after rearrangement:
p m  µp (1  p (0)  p (c))  c  pm  µp(1  p(0)  p(c))
(9)
If she is pessimist and she chooses c  0 , her true expected utility is then equal to:
U pessimist  m(x)  p(0)m+ p(0)(1  p (0))µ  (1  p(0)) p (0)µ
(10)
If she had predicted correctly probabilities, her true expected utility would have been equal to:
U realistic  m(x)  p(c)m+ p(c)(1  p(c))µ  c
(11)
U pessimistic  U realistic  0 if
And
p(0)(1  p (0))µ  (1  p(0)) p (0)µ  p(c)(1  p(c))µ  pm  c
(12)
0
This inequality can be satisfied for some values of the parameters and it is more likely the case when
p (0) is low. When p (0)  0 for instance, ( the individual is so pessimist that she thinks that without
effort,
she
will
have
no
chance
to
get
y)
the
previous
inequality
becomes:

p(0)µ  p(c)(1  p(c))µ  pm  c .
Equation 12 shows that when the individual is pessimistic, two effects appear:
i)
She makes less efforts than what it would be optimal for a given gain-loss utility (negative effect
on her true future expected utility) and she looses pm  c
ii) But for some values of the parameters, being pessimistic also increases future utility for a given
effort (positive effect) by increasing the magnitude of the gain sensation while reducing the
magnitude of the loss sensation
And it is possible for ii) to be greater than i). This paradox occurs because though the individual is able
to anticipate the influence of her effort on her future reference point, she does not control her level of
pessimism or optimism. When the probability is exogenous, it always pays to be pessimistic. When it is
endogenous, it can still pay despite the fact that it can lead the individual to choose a level of effort
different from the one she would have chosen with correctly predicted probabilities.
4 Conclusion
8
When an individual faces a lottery with two possible outcomes, as KR argue, these two possible
outcomes constitute two natural reference points that the individual can use to evaluate the outcome that
will be observed, generating simultaneous sensations of gains and losses. KR’s model of a stochastic
reference point is based on the assumption that the weights given to each possible outcome correspond
to their probabilities, to capture the intuitive idea that the additional sensations of gains and losses of
having one outcome should be more intense if the individual had a high probability of obtaining the
other.
Clearly, in many situations, people can use other possible reference points corresponding to the
status quo, past values or the individual’s objectives. But in those situations in which possible outcomes
reasonably represent the reference points that the individual has in mind, the previous reasoning shows
that by lowering the reference point, pessimism could in some cases increase both perceived and real
expected utility, and even if it leads the individual to reduce her efforts. In all these situations,
exaggeratedly low expectations continue to be the secret of happiness.
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