Priors and Desires
A Model of Payoff-Dependent Beliefs∗
Guy Mayraz†
August 2010‡
Abstract
This paper introduces a model of Bayesian decision making where
a person’s beliefs about the likelihood of different outcomes depend
upon the anticipated payoff consequences of those outcomes. Based on
the twin assumptions of consequentialism and invariance to additive
shifts in payoffs, I characterize the unique representation of payoffdependent beliefs. For agents in the model the desirability of an event
is effectively part of the evidence about its likelihood. A parameter determines both the direction and weight of this ‘evidence’, with positive
values corresponding to optimism, and negative values to pessimism.
The resulting bias depends on the strength of the payoff ‘evidence’
relative to the true evidence, and is greatest in situations that combine high stakes with much uncertainty. Incentives for accuracy do
not affect the bias. A change in the expected payoff consequences of
an event amounts to new ‘evidence’, and can thus affect beliefs, even
in the absence of any real new evidence. Economic consequences are
explored in various settings, including the economics of crime, where
increased punishment may fail to deter crime.
JEL classification: D01,D03,D80,D81,D83,D84.
Keywords: Optimism, pessimism, cognitive-dissonance, wishful thinking, self-serving beliefs, optimal expectations.
∗
I am grateful to Erik Eyster, Wolfgang Pesendorfer and Matthew Rabin for stimulating
discussions and valuable suggestions at key points in the writing of the paper, and to
seminar participants at LSE, MIT, Berkeley, CalTech, UCSB, UBC, UCL, Oxford, Essex,
Royal Holloway, Warwick, Hebrew University, Tel-Aviv, Ben-Gurion, Bar-Ilan, Collegio
Carlo-Alberto, and Maastricht.
†
Department of Economics and Nuffield College, Oxford, and Centre for Economic
Performance, London School of Economics.
‡
First version: November 2008.
1
1
Introduction
Bayesian decision makers are frequently required to make subjective judgments of likelihood. A manager contemplating a merger deal has to assess
the likelihood that it would succeed. An individual considering the purchase
of an investment property has to take a view on the future of the housing
market. Parents who have an opportunity to move their child to a different
school have to judge the quality of the available schools. As these judgments
are subjective, it is impossible to tell what the right answer is.1 People do,
however, frequently know exactly what they want the right answer to be.
The manager wants the deal to succeed, as success would lead to a bonus
and a promotion. The individual wants the housing market to continue to
rally, as she is already exposed to the housing market as a property owner.
The parents want it to be the case that the school their child has been allocated to is the best one, as they would face costs if they were to move their
child to a different school.
This paper explores the hypothesis that what people want to be true can
and does sway their judgment. The manager may have viewed the merger
differently if its success were to benefit her opponent, the individual may have
formed a different view of the housing market if she did not already stand to
benefit from rising prices, and the parents may have come to view the other
school as better if the random allocation had placed their child differently.
The goal is to be able to answer the counter-factual question of how beliefs would differ if the payoff-consequences were different. With in mind, I
seek not so much to explain the bias, but to model its effects. Part of the motivation is the possibility of parsimoniously accounting for such psychological
phenomena as over-optimism, self-serving beliefs, and cognitive dissonance.
Another is applications to choice under uncertainty. One such application involves the modeling of belief change. For example, the purchase of a property
increases an investor’s exposure to the housing market. What effect would
this have on the investor’s beliefs about the housing market, and by implication, on future investment decisions? Another application is to strategic
environments, where different agents face different payoffs. Suppose some
1
This is exactly what makes these judgments subjective. This distinction between
objective and subjective judgments of likelihood parallels the one Knight (1921) drew
between (objective) risk and (subjective) uncertainty. According to Knight, “Business
decisions, for example, deal with situations which are far too unique, generally speaking,
for any sort of statistical tabulation to have any value for guidance. The conception of an
objectively measurable probability or chance is simply inapplicable.” (III.VII.47), but that
“Yet it is true, and the fact can hardly be overemphasized, that a judgment of probability
is actually made in such cases.” (III.VII.40)
2
people stand to gain from an increase in house prices, while others stand to
lose. What effect would this difference in payoffs have on their beliefs, and
therefore on their choices?2 Finally, if we accept a weak version of rational
expectations, according to which disinterested observers have rational expectations, we can assess how a given exposure to an economic risk results in an
overestimation or underestimation of the risk relative to its objective level.
I model subjective beliefs by a probability measure over a set of states S.
Rather than model explicitly the mapping between states and consequences,
I use the device of a payoff-function to represent the expected utility in each
subjective state. A payoff-function f is thus a mapping f : S → R where
f (s) represents the expected-utility in state s. The payoff f (s) can also
be thought of as representing the desire for s to be the true state. Using
this definition, the subjective judgment of likelihood can be modeled by a
distortion mapping π, which associates with each payoff-function f a payoffdependent probability measure πf .
Given this framework, the aim of the model is a parsimonious representation of π. The simplest case is if π is constant, which would be the case if
beliefs were independent of desires. At the opposite extreme π may be very
complicated, with arbitrary non-local dependencies and discontinuous transitions. Such arbitrary distortions have no simple representation, no grounding
in evidence, and little or no predictive power. I therefore introduce a number
of restrictions in an attempt to find a middle ground of realism and practicality. The restrictions take the form of exceptions—special situations in
which differences in the payoff-function do not result in differences in beliefs.
The two key assumptions are shift-invariance, and consequentialism.3
Shift-invariance is the assumption that the judgment bias is affected only
by utility differences—by how desirable states are relative to each other.4
For example, according to Shift-Invariance, the bias in the beliefs of a risk
neutral property investor over the future of the housing market is not affected
by her winning a lottery prize.5 Consequentialism is the assumption that if
two payoff-functions coincide over some event then the probability measures
2
In order to predict differences in beliefs on the basis of differences in payoffs it is
necessary to assume that there are no unmodeled sources for a systematic difference in
beliefs between the two groups.
3
I also assume continuity: small changes in payoffs lead only to small changes in beliefs,
and absolute continuity: the set of zero probability events is the same for all payofffunctions.
4
Formally, f 0 = f + a ⇒ πf 0 = πf .
5
If she is risk-averse then winning a big lottery prize would affect the payoff-difference
between different housing market outcomes, and could therefore result in a change in
beliefs.
3
conditional on this event also coincide.6 In other words, the payoff in a state
can affect the likelihood of that state relative to other states, but not the
likelihood of other states relative to each other.
In a model with no payoff-dependent belief distortion πf is constant, and
hence there exists a probability measure p, such that πf = p for all f . The
key result of the paper is that as long as the above assumptions are satisfied,
a payoff-dependent belief distortion can be represented with the aid of only
one additional real-valued parameter ψ. The distorted subjective probability
measure is related to p, f , and ψ via the following expression, where A is any
general event:
Z
πf (A) ∝
eψf (s) dp
(1)
A
If f is constant over A, we obtain the following simpler expression:
πf (A) ∝ p(A) · eψf (A)
(2)
Computing the probability of an event using Equations 1 or 2 requires a
normalization constant. The formula is simpler if we are interested in the
odds-ratio between two events, as the normalization term drops out. Suppose
A and B are two non-null events with a well-defined payoff. The log-odds
ratio between them can be written as follows:
log
p(A)
πf (A)
= log
+ ψ · [f (A) − f (B)]
πf (B)
p(B)
(3)
To understand Equation 3 suppose first that the agent is indifferent between
A and B, so that f (A) = f (B). The second term on the right then drops out,
and we obtain that πf = p. The probability measure p therefore represents
the agent’s undistorted beliefs—the beliefs she would have held if she were
indifferent whether A or B is the case. Suppose now that the agent is not
indifferent, for example that f (A) > f (B). The second term on the right
hand-side of Equation 3 then has the same sign as ψ. Thus, if ψ > 0 the
agent would assign a higher probability to A than she would if she were
indifferent, and if ψ < 0 her probability would be lower. A positive value of
ψ therefore corresponds to optimistic bias, and a negative value to pessimistic
bias. The case of ψ = 0 corresponds to a Spock like agent, whose judgment
is completely unaffected by desires. Moreover, the higher ψ is, the greater
is the subjective probability of the more desirable event. ψ can therefore be
thought of as the coefficient of relative optimism.
6
If f = f 0 over an event E then πf 0 (·|E) = πf (·|E).
4
An important feature of Equation 3 is that it is formally identical to Bayes
Rule:
p(A|e)
p(A)
L(e|A)
log
= log
+ log
(4)
p(B|e)
p(B)
L(e|B)
where p(A)/p(B) is the prior odds ratio, p(A|e)/p(B|e) is the posterior odds
ratio, and L(e|A)/L(e|B) is the likelihood ratio of the evidence e. A comparison of Equation 3 with Equation 4 reveals a perfect correspondence, with p
standing for undistorted or prior beliefs, πf for distorted or posterior beliefs,
and ψf (A) for the log likelihood in A, with an analogous expression for B.
In this analogy, payoffs play the role of evidence. It is thus as if an optimist
observes that an event is desirable, and concludes that it is likely to obtain.
A pessimist would behave similarly, but in the same situation would draw
the opposite conclusions. For realist agents, of course, the desirability or
undesirability of an event is uninformative about its likelihood.
In Section 3 I prove the representation theorem, highlighting the role of
each the individual assumptions in the proof. Shift-Invariance and Continuity are only necessary for the linearity of the log-likelihood function. If
we assume instead only that any two constant payoff-functions result in the
same probability measure, we obtain a representation that is formally similar, but the log-likelihood function is then completely unrestricted. If we
drop even this assumption (and so assume only Absolute Continuity and
Consequentialism) we obtain a representation in which the likelihood function is state-dependent. In the present context, however, it is far from clear
that such more general representations are genuinely useful. In a practical
application one would want to take care in selecting the appropriate utility
function, but once payoffs are expressed in utility terms Equations 1–3 may
well be sufficiently general.
Holding the direction of the bias constant, its magnitude is increasing
in the following three factors: (1) the degree of subjective uncertainty (the
closer log p(A)/p(B) is to zero), (2) the degree of optimism or pessimism (the
absolute value of ψ), and (3) how strongly the person feels about the outcome (the absolute value of the payoff-difference f (A)−f (B)). The Bayesian
updating analogy (Equation 4) provides a perspective on these results. In
the Bayesian analogy p corresponds to prior beliefs, and ψ(f (A) − f (B))
is the evidence term. A higher level of optimism/pessimism and a larger
payoff-difference both correspond to stronger evidence, while a greater degree of subjective uncertainty corresponds to greater uncertainty in prior
beliefs. The comparative statics results then follow simply from the observation that the shift in beliefs following Bayesian updating is increasing with
5
the strength of the evidence and the degree of uncertainty in prior beliefs.7
Thus, the beliefs of the property investor about the housing market would
be particularly distorted if (i) she has a lot to gain or lose, and (ii) there
isn’t much genuine evidence to determine the issue with any certainty. A
particularly clear illustration of the comparative statics is provided by the
case of normally distributed random variables. For example, let X denote the
dollar profit from a merger deal, and suppose the manager’s personal payoff
in utility terms is linear in X: f = aX, and that her undistorted probability
distribution function for X is normal: PX (X) ∼ N (µ, σ 2 ). It can be readily
shown (Section 4.2) that her distorted probability distribution function for
X would also be normal, and that it has the same variance, with the mean
shifted in proportion to the variance σ 2 , the stakes a, and the coefficient of
relative optimism ψ: ΠX (X) ∼ N (µ + ψaσ 2 , σ 2 ).
While a person’s stakes in what the state of the world is have a central
role in determining the size of the bias, the stakes in knowing what the state
is are not part of the comparative statics. In particular, beliefs are not any
less distorted just because belief distortion results in costly decisions. Of
course, if the agent has important decisions to make, we would expect her
to make a big effort to acquire relevant information. Thus, if the state of
the world can be readily discovered, we would indeed observe a decreased
bias when the incentives for accurate beliefs go up. However, controlling
for information, the cost of holding distorted beliefs has no effect on the
bias. Consequently, in situations in which there is limited scope to reducing
uncertainty,8 the prediction is of a substantial bias, however great the loss
from holding distorted beliefs.
Suppose some new evidence e is observed. The effect e has on the agent’s
beliefs can be readily obtained by conditioning Equations 1–3 on the new
evidence. In the case of Equation 3 the following expression can be derived:
log
πf (A)
L(e|A)
eψ(f (A|e)−f (A))
πf (A|e)
= log
+ log
+ log ψ(f (B|e)−f (B))
πf (B|e)
πf (B)
L(e|B)
e
(5)
Posterior beliefs are thus related to prior beliefs via Bayesian updating,
but there are two updating terms, corresponding to two separate channels through which new information can affect beliefs. The first channel
7
A minor subtlety: the greatest bias is obtained not at the point of maximum uncertainty (f (A) = f (B)), but at the point such that distorted beliefs are a mirror image of
undistorted beliefs. For example, if eψ(f (A)−f (B)) = 2, the maximum change in probability
would occur if p(A) = 1/3, so that πf (A) = 2/3.
8
Perhaps the most highly regarded experts disagree, as is often the case in an asset
bubble.
6
is the standard Bayesian log likelihood term. The second channel is a quasiBayesian updating term representing change in the payoff-consequences. As
long as these are fixed, belief updating is purely Bayesian, but to outside
observers it may well seem to be non-Bayesian. For example, consider an
optimistic manager who needs one of two independent projects to succeed in
order to get her promotion. Being optimistic, she overestimates the probability that she gets promoted, which given the payoff-function results in an
overestimation of the ex-ante probability of success in both projects, but only
conditional on the other project failing. Consequently, if she were to learn
that the first project has succeeded, the result of Bayesian updating would be
to lower her subjective probability of success in the second project back to its
undistorted level. From the point of view of an outside observers, however,
this change in beliefs may well seem inconsistent with Bayesian updating.
The second channel for belief update involves a change to the payofffunction itself. For example, consider the beliefs of optimistic parents whose
child is to be randomly allocated to one of two schools with even odds. Since
the parents have no idea which school their child would end up in, they are
initially indifferent which school is better, and their beliefs are not distorted.
However, after they learn which school their child is allocated to, they want
that school to be the better school, and they become biased about that school.
In certain situation the second channel for belief update can go in the
opposite direction to Bayesian updating, and can even lead to a change in
beliefs opposite to what the Bayesian update would suggest. For example,
an entrepreneur may become an even stronger believer in her business falling
a setback, as the belief update following the increased need for the business
to succeed overcomes the negative Bayesian update.
The choice implications of the model are, in principle, straightforward:
preferences depend on beliefs via expected utility maximization, and beliefs
depend on payoff expectations via Equations 1–3. As an example of how
this could play out, consider an optimistic investor with a significant preexisting investment in some asset. The existing investment would result in an
overestimation of the probability that this asset would do well, and hence
in a ceteris paribus tendency to invest more in the same asset.9 There are,
however, situations in which it can make a significant difference what choice
the agent anticipates making when she makes up her mind: if she anticipates
choosing A then she wants A to be a good choice, and her beliefs are biased
accordingly. If, instead, she expected to choose B, she would have wanted B
to be a good choice, and would have had different beliefs guiding her decision.
9
If the investor is risk-averse she would benefit from diversifying her investments. Optimism would then result in less diversification than the optimal level.
7
In the opposite direction it is possible to define revealed-preferences axioms for the model. That is, preferences that respect these axioms can be
represented by a subjective expected utility functional, where the probability
measure in this functional is payoff-dependent in accordance with Equation 1.
The undistorted probability measure is uniquely determined, and the utility
function is determined up to a positive affine transformation, just as in standard subjective expected utility models. The coefficient of relative optimism
is determined jointly with the utility function, so that the right hand side of
Equation 1 remains invariant.10 Operationalizing these axioms is, however,
made more difficult by the potential for choices to be affected by unobserved
choice expectations. This problem can be solved by offering the agent instead of a choice between g and h, a choice between f with probability 1 − and g with probability , and an analogous mix of f and h. Because of the
properties of expected utility maximization, the preference between the first
pair of alternatives can be revealed by the choice between the second pair.
Moreover, with a sufficiently small , the potential for the unobserved choice
expectations to affect the choice disappears. Using this device it is possible
to reveal the preference relation that is induced by different payoff-functions,
and except in degenerate cases11 this is sufficient to recover all the parameters
of the model.
The psychology evidence for payoff-dependent beliefs is extensive. Perhaps best known are over-optimism studies, such as the one showing that
most people believe themselves to be better drivers than most other people (Svenson, 1981). Optimistic bias is the most natural explanation for this
and many other findings, but the fact that people know more about themselves than about other people opens the door to other explanations. Stronger
evidence,therefore, is provided by experiments in which payoffs are randomly
assigned. One example of such a study is an experiment reported in Klein
and Kunda (1992), in which subjects had to assess the ability of a player
in a history trivia game. Subjects who were told the person would play on
their team (and so wanted him to be a good player) had considerably higher
ratings of the player’s ability than subjects who expected the person to play
on the opposing team (and so wanted him to be a weak player). Cognitive
dissonance experiments provide evidence that a change in payoffs can lead to
a change in beliefs absent any normative reason for belief change. While such
evidence has been originally interpreted as to do with dissonance between an
action and beliefs Festinger and Carlsmith (1959); Cooper and Fazio (1984)
10
If payoffs are scaled by some number a ∈ R+ then the coefficient of relative optimism
is scaled by 1/a.
11
If two payoff-functions are shifted versions of one another
8
the evidence can also be explained by payoff-dependent beliefs.12 Finally,
experiments on people suffering from depression make clear that while optimism may be the prevalent bias in the population, many people are instead
pessimistically biased, and in particular accept unrealistically pessimistic interpretation of events that are important to them (Seligman, 1998).13
Payoff-dependent beliefs have been used to analyze a range of economic
phenomena, such as bargaining (Babcock and Loewenstein, 1997; Yildiz,
2004), monopolistic contract design (Spiegler and Eliaz, 2008), failure rates
and credit rationing in small-business borrowing (De Meza and Southey,
1996), and capital structure in corporate finance (Heaton, 2002). These
and other papers have produced many interesting results, demonstrating
the range of economic phenomena in which payoff-dependent belief distortion plays an important role. Empirical studies have linked over-optimism in
CEOs with poor over-investments (Malmendier and Tate, 2005) and the making of too many acquisitions (Malmendier and Tate, 2008). Mullainathan and
Washington (2009) have found that casting a vote affects beliefs, consistent
with payoff-dependent belief updating. Puri and Robinson (2005) use survey
data to relate over-optimism to a range of important economic choices.
A key feature of the model of this paper is that it is non-strategic. Agents
in the model strive to make optimal instrumental choices, just as in any
standard economic model—it just so happens that their judgment of probabilities is biased. An analogy may be useful: we observe that a person asked
to choose the longer of two lines makes the wrong choice. Rather than interpret this as a non-instrumental preference for the shorter line, we interpret
it as a mistake: the person did want to choose the longer line, but was the
victim of a visual illusion.
In taking this approach I depart from the strategic belief distortion model
introduced by Akerlof and Dickens (1982), and perhaps best represented by
Brunnermeier and Parker (2005). In the strategic belief distortion model
belief bias is assumed to be the result of deliberate choice. The assumptions
are: (i) that people have preferences not only over what is true, but also over
what they believe to be true,14 (ii) that beliefs are can be chosen at will,
12
In the psychology literature Kunda (1990) suggested that the manipulations in cognitive dissonance experiments lead to a change in motivation, and that this change in
motivation leads to a change in beliefs.
13
Many people have heard of ‘depressive realism’—the idea that normal people are
biased, and that depressed people hold accurate beliefs. While it is true that the depressed
do not suffer from the illusion of control in situation in which people have no control (Alloy
and Abramson, 1979), the findings reported in Seligman (1998) and more generally in the
psychiatric literature, make it clear that depressed people can be very far indeed from
being realistic.
14
Note these are direct preferences over beliefs. Standard agents sometimes have in-
9
and (iii) that once chosen, beliefs are subject to standard Bayesian updating
(in particular, beliefs cannot be updated at will). Agents therefore trade the
desire to consume positively biased beliefs, against the cost such beliefs may
have in distorting future choices.
An important advantage of strategic belief distortion models is that they
offer an explanation of optimistic belief bias, rather than just a model. On
the other hand, while static optimistic bias can be modeled as a strategic
bias, non-normative belief change cannot, and static pessimistic bias is also
a non-starter.15 The prediction of extreme bias in situations in which belief
bias is costless is another disadvantage.16
Whether payoff-dependent belief bias is strategic or not is a highly important question for applications. In many economically important situations,
holding biased beliefs can result in large financial and other losses, and so
strategic models of belief distortion would predict only a relatively small
bias. On the other hand, the non-strategic model offered in this paper would
predict a significant bias whenever agents have much at stake, regardless of
the potential cost to their future choices. Thus, much of the importance of
payoff-dependent belief distortion depends on whether the bias is or is not
strategic.
The reminder of this paper is organized as follows. In Section 2 I review the psychology and economics evidence for payoff-dependent beliefs,
and discuss briefly how the model suggested in this paper can account for
the available evidence. In Section 3 I formally develop the model. In Section 4 is devoted to the implication of the model for beliefs in static situations,
and Section 5 to belief updating. Section 5 also includes an application to
the economics of crime, showing that an increase in the severity of punishment can end up reducing deterrence. Section 6 discusses the implications
of the model for choice, and offers a revealed preferences axiomatisation of
the model. Section 7 concludes. The proofs that are not in the text are in
Appendix A.
strumental preferences over beliefs if, for example, certain beliefs may help or hinder
performance in some real-world task.
15
Note that it is not possible to have agents choose their beliefs freshly in each period,
because they would then choose to be always completely biased except at the precise time
when they have to make choices.
16
For example, while there is evidence that voters overestimate the probability that their
candidate would win (Hayes Jr, 1936; Dolan and Holbrook, 2001; Granberg and Brent,
1983) the bias in their belief does not seem unlimited, despite the fact that there are no
obvious costs to holding such biased beliefs.
10
2
Evidence
In this section I describe some of the key psychology and economics empirical
findings that relate to payoff-dependent beliefs. The following assumptions
are useful in relating these findings to the model of this paper:
O1 (optimism). The large majority of people are optimistic.
O2 (rational expectations). Undistorted beliefs (the beliefs a person would
hold if she were indifferent what state obtains) are rational in the sense
of Muth (1961).
2.1
Psychology evidence
The psychology evidence can be usefully separated into the following five
groups:
1. Over-optimism (sometimes also wishful-thinking, or over-confidence.17 )
Studies in this group demonstrate an overestimation of the probability
of desirable events, and a corresponding underestimation of the probability of undesirable events. Probably the best known studies in this
group are those that show that a majority of people view themselves
as better in some valuable skills than most others. For example, in
driving (Svenson, 1981), and in teaching ability (Cross, 1977). In a
broader study Weinstein (1980) had students rate the relative likelihood of various events happening to them compared to the likelihood
of the same event happening to other students (e.g. getting a good job,
falling seriously ill). The finding is that on average students see desirable (undesirable) events as more (less) likely to occur to them than to
other students. Sjöberg (2000) and Weinstein (1989) report comparable results in risk perception, and Weinstein et al. (2005) shows that
smokers underestimate the risks of smoking.
Findings in this group can all be parsimoniously explained by payoffdependent belief bias together with assumptions O1 and O2. For example, let Ax denote the event that person x is a better driver than most
other people, and let Bx denote the event that this is not the case. It is
17
The term over-confidence is, however, also used to describe the finding that people
select too small confidence intervals to represent the accuracy of their predictions (Alpert
and Raiffa, 1982). While over-optimism is very likely a factor, this misuse of confidence intervals may well have to do with the broader difficulty people seem to have in working with
probabilities (manifested in such unrelated biases as the ‘Law of Small Numbers’ (Tversky
and Kahneman, 1971)).
11
clear that x wants Ax to be true, whether because of direct ego-utility,
or for instrumental reasons (lower risk of accidents). Thus, if f denotes
x’s payoff-function then f (Ax ) > f (Bx ). Therefore, according to equation 3 and assumption O1, x would overestimate the probability of Ax
relative to its undistorted probability (i.e. πf (Ax ) > p(Ax )). Moreover,
according to O2, p(Ax ) coincides with rational expectations. Combining these two observations we obtain that the agent overestimates the
probability of Ax as compared to rational expectations.18
2. Self-serving beliefs. This is a closely related group of studies, but instead of demonstrating a bias compared to rational expectations, studies in this group compare the beliefs of two groups of subjects that
differ in their payoff-function. The underlying assumption is that there
is no other reason for systematic bias in beliefs between the two groups,
so that the difference in beliefs can be attributed to the difference in
the payoff-function. Some of the studies in this group compare the
beliefs of supporters of political candidates or sport fans. For example, Hayes Jr (1936) found that in the 1932 presidential elections 93%
of Roosevelt supporters and 73% of Hoover supporters predicted that
their own candidate would win. Similar findings have been found since
in other elections, making this a very robust finding (Dolan and Holbrook, 2001; Granberg and Brent, 1983). One weakness of such studies
is that they are not incentive compatible. Babad and Katz (1991),
however, found a similar bias for sports fans in incentive compatible
bets. A more serious weakness is that there may be other reasons for a
difference in beliefs between the two groups. For example, it may be the
case that people judge the level of support for their favorite candidate
partly on the basis of the views of their friends, and that supporters
of a given party are more likely to have friends who supports the same
party. In some cases, however, there is relatively limited scope for
such confounds. For example, Sherman and Kunda (1989) presented
to a group of women a study claiming to link caffeine consumption to
breast cancer. The finding was that women who were significant consumers of coffee tended to find the study less trust-worthy than women
who were not significant consumers of caffeine. Since the women first
saw the study as part of the experiment, it is difficult to imagine other
18
The finding that relative risk is underestimated appears to be highly robust (Sjöberg,
2000; Weinstein, 1989; Weinstein et al., 2005). By contrast, there are cases Sjöberg (2000)
in which the absolute risk level is overestimated. Such findings can be explained by consistent optimism (O1) combined with selective failure of rational expectations (O2).
12
reasons for a systematic difference in beliefs between the two groups.19
Studies in this group can also be readily explained by the model. For
example, in the case of Sherman and Kunda (1989) let T (F ) denote the event that finding of a link between caffeine consumption
and breast cancer is true (false). The relevant outcome for a woman
participating in the experiment is that she develops breast cancer, or
that she does not. For a coffee-drinking woman the probability of cancer conditional on T is higher than its probability conditional on F .
Hence, in her payoff-function fcoffee (T ) < fcoffee (F ). For non coffeedrinking women, however, the two lotteries are the same, and hence
fno coffee (T ) = fno coffee (F ). Thus, by assumption O1, πcoffee (T ) < pcoffee ,
but πno coffee (T ) = pno coffee . If we assume that there is no other reason
for systematic belief bias between the two groups of women then on
average pcoffee = pno coffee , and therefore πcoffee < πno coffee .
3. Motivated beliefs. Studies in this group randomly assigns subjects into
two groups with different payoff-functions, and test for differences in
beliefs. For example, Klein and Kunda (1992) had subjects assess the
ability of a player in a history trivia game. Subjects who were told
the person would play on their team (and so wanted him to be a good
player) had considerably higher ratings of the player’s ability than subjects who expected the person to play on the opposing team (and so
wanted him to be a weak player).
Studies in this group provide the most direct causal evidence for payoffdependent beliefs, as the random allocation eliminates other potential
sources for a systematic difference in beliefs between the two groups.
The results of such experiments can be explained by the model together
with assumption O1.
4. Cognitive dissonance. The term ‘cognitive-dissonance’ is used differently by different people, and in some uses is more or less synonymous
with any effect of what a person wants to be true on her beliefs. I will
use the term here to refer to a more specific type of finding, namely
a change in beliefs, which occurs not in response to relevant evidence,
but in response to some action by the subject which affects her payoffs. For example, in a classic experiment by Festinger and Carlsmith
(1959) students took part in a boring task, following which some of the
students (the treatment group) were put in a position where they had
19
By contrast, evidence showing that smokers underestimate the risks of smoking (Weinstein et al., 2005) can also be potentially explained as the result of selection bias.
13
to tell another student (unknown to them, a confederate of the experimenters) that the task is interesting. The finding was that subjects in
the treatment group perceived the task to have been more interesting
than subjects in the control group.
The explanation offered in Festinger and Carlsmith (1959) is that
recommending the task to another student created a dissonance in the
mind of the students in the treatment group, and that they resolved
this dissonance by adopting a more favorable view of the task. From
the point of view of the model of this paper we can view the act of
recommending the task to another student as effecting a change in
the payoff-difference between the state in which the task is interesting
and the state in which it is boring (compared to the control group,
subjects in the treatment group would also be guilty of misleading a
fellow student if the task is boring). Hence, the change in beliefs can
readily explained as the effect of the change in the payoff-function.20
Festinger and Carlsmith (1959) also included a treatment in which subjects were paid a large sum of money to recommend the task to another
student, and subjects in this group showed only a small change in beliefs. The explanation offered by the authors is that subjects in this
group explained to themselves the act of recommending the task to another student by the amount of the money they received. Consequently,
there was no dissonance for them between the belief that the task was
boring and the act of recommending it to another student. Translating
this idea into the model of this paper, we can say that the amount
of money the students received and the degree to which the task was
interesting are substitutes in their utility function: subjects are happy
either if the task was interesting, or if they get paid a lot of money to
recommend the task, and are only unhappy if they recommend a boring task without getting much for it. In particular, the more subjects
are paid for recommending the task, the less they care whether it was
interesting. If this assumption is accepted then the result follow from
the model of this paper.
5. Unrealistic pessimism. Studies in this group are quite different from
the other studies. Whereas those studies are performed by social psychologists looking to understand belief biases, studies in this group are
typically run by researchers interested in depression. The focus, therefore, is not on the average bias in the population, but on differences
between individuals. Seligman (1998) views a number of studies which
20
Section 5 discusses such changes in belief in more detail.
14
find that there is a group of people who seem to be biased in an opposite direction to the average bias in the population. In particular,
they tend to interpret failures as broad, long-lasting, and personal, and
successes as narrow, fleeting, and likely due to favorable external circumstances. Such individuals are also much more likely to be diagnosed
with depression, or become depressed in the future.
In all these cases the pessimism exhibited by the subjects is consistent
with pessimism as defined in this paper. For example, consider the
interpretation of a failure in some task. In one state the failure is the
result of a permanent problem that would also affect the future, and
in the second state the failure is the result of transient circumstances.
For both ego and instrumental reasons people would prefer the second
state to be the true state, and hence pessimists would overestimate the
probability that the first is true, i.e. that their failure is indeed the
result of a permanent problem.
2.2
Economics evidence
Empirical studies by economists are a more recent phenomenon. Babcock
and Loewenstein (1997) divide subjects into pairs, each of which consisting
of a plaintiff and a defendant. The two are presented with a real legal case,
in which the plaintiff demanded $100,000 from the defendant. The defendant
in the experiment is initially given a notional $100,000, and the final payoff is
proportional to what each side ends up with. Subjects have the opportunity
to reach an agreement, failing which the outcome is determined by what
the judge had decided in the actual case. In the latter case both sides also
have to pay ‘court costs’. Prior to bargaining subjects are asked to make
a side bet as to what the judge awarded in the real case. The finding is
that subjects in the role of plaintiff tend to overestimate the award decided
by the judge compared to subjects in the role of defendant (the average
difference was $14,500). Moreover, pairs in which this difference was large
tended to be those which failed to reach an agreement, and so had to pay
court costs. The authors thus conclude that self-serving beliefs are present,
and can furthermore explain bargaining impasse. As the payoff-function is
experimentally assigned, this study can be compared with the motivatedbeliefs psychology studies. In both cases results can be readily explained by
the model in combination with assumption O1.
Malmendier and Tate (2005) argue that overconfidence would lead CEOs
to overestimate the return to investments, and so overinvest when they have
abundant internal funds. They identify over-confident CEOs as those CEOs
15
who do not take advantage of opportunities to reduce their personal exposure
to company specific risk, and find that their investment decisions are indeed
significantly more responsive to cash flow constraint. In Malmendier and
Tate (2008) they extend the analysis to show that overconfident CEOs also
make more acquisitions, and that their deals are judged by the market to be
worse than those of other CEOs. These results are consistent with optimism
in the sense used in this paper.
Mullainathan and Washington (2009) test the cognitive dissonance prediction (consistent also with payoff-dependent belief updating) that casting
a vote has an effect on beliefs. Using age vote restrictions as an instrument
they find that voting does indeed increase political polarization.
Puri and Robinson (2005) use survey data to relate optimism to economic choice. They identify optimists as those who overestimate their life
expectancy compared to actuarial estimates, and find, for example, that optimists are more likely to believe that future economic conditions would improve, and that they invest more in stocks as percentage of their portfolio
than non-optimists.
3
Model
In this section I describe the representation theorem for payoff-dependent
belief distortion and outline its proof. The result itself is discussed in Section 4. The framework, the properties we want distortions to satisfy, and
the statement of the theorem are in Section 3.1. The proof for a finite statespace is outlined in Section 3.2, including the role played by each of the
different assumptions, and partial results that can be established with fewer
assumptions. The extension to a general state-space is in Section 3.3.
3.1
Framework
Subjective uncertainty is defined over a measurable-space (S, Σ), where S
is the set of states, and Σ is a σ-algebra of subsets of S, or events. In the
presence of objective uncertainty each state is associated with an objective
lottery over final consequences (if there is no objective uncertainty this lottery
is degenerate). A payoff-function is a Σ-measurable mapping f : S → R,
where the payoff f (s) in state s should be interpreted as the utility of the
appropriate objective lottery. The payoff f (s) can also be identified loosely
with the desirability of s. Let F denote the set of all Σ-measurable payofffunctions, and let ∆ denote the set of all σ-additive probability measures over
(S, Σ). The key ingredient in the model is a distortion mapping π : F → ∆,
16
associating with each payoff-function a probability measure over (S, Σ). The
interpretation is that the agent’s subjective probabilities are swayed by her
desires: if her desires can be represented by f then her subjective probabilities
can be represented by πf .
In the following definitions f and f 0 stand for any payoff-functions, a for
an any constant payoff-function yielding a payoff of a in all states, and E for
an any event. The first definition states the properties we want the distortion
mapping to satisfy. The second describes the linear distortion formula. The
theorem says that the two definitions are equivalent.
Definition 1. π : F → ∆ is a well-behaved distortion if the following conditions are satisfied:
A1 (absolute continuity) πf 0 (E) = 0 ⇐⇒ πf (E) = 0.
A2 (consequentialism) If f = f 0 over a non-null21 event E then πf 0 (·|E) =
πf (·|E).
A3 (shift-invariance) If f 0 = f + a then πf 0 = πf .
A4 (prize-continuity) If fn → f uniformly then πfn (E) → πf (E).
Absolute Continuity limits belief distortion to situations of uncertainty.
If the agent is certain that some event obtains (and therefore assigns its complement zero probability) then she would make the same judgment whatever
her payoff-function. Absolute Continuity rules such belief distortions as complete denial, where a person confronted with conclusive evidence that some
undesirable event has occurred nevertheless persists in disbelieving it.
Consequentialism requires that if two payoff-functions coincide over some
event E then the corresponding probability measures conditional on E also
coincide. That is, the payoff in states outside E should not affect the relative
probabilities of events in E. The intuition for Consequentialism is that the
payoff in a state makes that state subjectively more or less likely. It therefore
necessarily affects the overall probability of other states, but should not affect
their relative probability.22 Consequentialism has something of the flavor of
21
That is, both πf (E) > 0 and πf 0 (E) > 0. Absolute Continuity ensures that these two
requirements coincide.
22
One way Consequentialism could fail is if subjective judgments were swayed not by
the payoff in a state, but by a global property of payoffs, such as the rank of a given
payoff-value compared to the payoff-values that may be obtained in other states.
17
the Sure-Thing Principle (Savage, 1954),23 and of Luce’s Choice Axiom (Luce,
1959).24
Shift-invariance is the assumption that the judgment bias depends only
on how desirable different states are relative to each other. Thus, beliefs may
be distorted because one state is better than another, and may also depend
on exactly how much better the first state is (which with additive utility
translates into a given willingness to pay for insurance), but a uniform shift
in expected utility is irrelevant.25
Finally, prize-continuity states that small differences in payoffs can have
only a small effect on beliefs.
Definition 2 (Linear distortion). π : F → ∆ is a linear distortion if there
exists a probability measure p (the undistorted measure), and a real-number
ψ (the coefficient of relative optimism), such that for any payoff-function f
and any event A,
Z
eψf (s) dp
(6)
πf (A) ∝
A
Theorem 1 (Representation theorem). Suppose that there are three disjoint
events with positive πf probability for at least one f ∈ F , then π is a linear
distortion if and only if it is a well-behaved distortion.
It is interesting to analyze non-strategic belief distortion models in terms
of the definition of a well-behaved distortion 1. Shift-invariance is respected,
as beliefs result from expected utility maximization, and the choice maximizing expected utility is invariant to uniform shifts in utility. The other
assumptions are all violated, however. Consider a choice between three alternatives, yielding 1 in states A, B, and C respectively, and 0 in other states.
Beliefs are chosen at t = 0 and the decision is made at t = 1. No new information is revealed between t = 0 and t = 1. Clearly, if, say, the probability
of A is highest then the agent would choose to assign A a probability of 1, as
this would maximize her t = 0 subjective expected utility, and she would still
23
If we denote a preference for g over h given the payoff-function f by g f h then
Consequentialism is equivalent to the following assumption, reminiscent of the Sure-Thing
Principle: Suppose f and f 0 agree on E, and g and h differ only on E, then g f h ⇐⇒
g f 0 h.
24
Luce’s Choice Axiom is that the relative probability of choosing two alternatives in
some set is independent of alternatives outside the set.
25
Analogy: the length of a line is independent of its position in space, and hence any
property of the length is preserved under translations. Shift-invariance would fail if judgment were affected by mood, so that a high level of utility can result in a given difference
in payoff having a different effect on beliefs than the same difference would have had if all
utility values were lower.
18
be making the optimal t = 1 choice. However, this choice results in violation
of Absolute Continuity, Prize Continuity, and Consequentialism.
3.2
Proof outline
In this section I outline the proof of Theorem 1 for a finite state-space (the
formal proof is in the appendix). In this section, therefore, S consists of
finitely many states, and the set of events Σ is simply the power set of S.
The finite state-space definition of a linear distortion and of Theorem 1 are
as follows:
Definition 3 (Linear distortion, finite state-space). π : F → ∆ is a linear
distortion if there exists a probability distribution p (the undistorted distribution), and a real-number ψ (the coefficient of relative optimism), such that
for any payoff-function f and any state s,
πf (s) ∝ p(s) · eψf (s)
(7)
Theorem 2 (Representation theorem, finite-state space). Suppose that for
at least one f ∈ F the support of πf includes three or more states, then π is
a linear distortion if and only if it is a well-behaved distortion.
The proof is structured so as to demonstrate the role played by each
of the different assumptions, and make it possible to prove partial results
of independent interest along the way. Absolute Continuity is sufficient on
its own to obtain a formula of the same form as Equation 7. Choose some
constant payoff-function a and define p = πa . By Absolute Continuity for all
f and s, πf (s) = 0 if and only if p(s) = 0. Thus, we can define a mapping
h : F × S → R+ , such that26
πf (s) ∝ p(s) · hf (s)
(8)
Three steps separate Equation 8 from Equation 7:
1. hf (s) is a function of f in all states, whereas eψf (s)) is a function only of
the value of f in s itself. A state-dependent Bayesian distortion is one
for which there exists a mapping µ : S × Z → R+ , such that Equation 8
holds with hf (s) = µs (f (s)). That is, for all f and s,
πf (s) ∝ p(s) · µs (f (s))
(9)
26
hf (s) = πf (s)/p(s) on states in which p(s) > 0, and hf (s) = 1 (or any other value)
on the states in which p(s) = 0.
19
2. µs may be different from µs0 for s 6= s0 , whereas eψf (s) is the same
function in all states. A Bayesian distortion is a distortion for which
there exists a mapping ν : Z → R+ , such that for all f and s,
πf (s) ∝ p(s) · ν(f (s))
(10)
3. The function ν in Equation 10 need not be exponential (log ν may not
be linear). A linear distortion is a Bayesian distortion for which there
exists a real number ψ such that ν(z) ∝ eψz . In other words, Equation 8
holds with hf (s) = eψf (s) .
In the Introduction I noted that a linear distortion can be interpreted as
a Bayesian update, with the payoff in a state playing the role of evidence
about that state. In a state-dependent Bayesian distortion the ‘evidence’
is the payoff in the state, but the same payoff-value can have a different
‘meaning’ in each state. In a (state-independent) Bayesian distortion a given
payoff-value has the same meaning in all states, but the meaning of a given
payoff-value is completely unrestricted.27 A linear distortion is a Bayesian
distortion in which payoffs have a cardinal meaning.28
In order to be able to state the necessary and sufficient conditions for each
of the above three steps it is necessary to define a new property, Indifference,
which follows from Shift-Invariance, but is considerably weaker:
A3’ (Indifference). If both f and f 0 are constant payoff-functions then πf =
πf 0 .
Note that Indifference does not require the set of payoffs to have cardinal (or
even ordinal) meaning. With Indifference defined, the theorem can now be
stated:
Theorem 3. Suppose that for at least one f ∈ F the support of πf includes
three or more states, then:
1. Absolute Continuity and Consequentialism are necessary and sufficient
conditions for π to be a state-dependent Bayesian distortion.
27
Moreover, no structure whatsoever is assumed on the set of payoffs (not even an ordinal
structure).
28
It is not possible to define a non-trivial distortion that distorts the odds-ratio between
two states by a given factor whenever one is ordinally better than the other. To see this,
suppose p(s) = p(t) = p(v) = 1/3, f (s) > f (t) > f (v), and that this factor is α. By
assumption πf (s)/πf (t) = α, πf (t)/πf (v) = α and πf (s)/πf (v) = α, and so α2 = α,
implying α = 1.
20
2. Indifference is a necessary and sufficient additional condition for π to
be a Bayesian distortion.
3. Shift-Invariance and Prize-Continuity are necessary and sufficient additional conditions for π to be a linear distortion.
Proving in each case that the assumptions are necessary conditions is
trivial. I therefore focus on outlining the proof that the assumptions are
sufficient. Define p = πa for some constant-payoff function a, and let S ∗
denote the support of p (and hence, by Absolute Continuity, also of πf for
any f ). The first step is based on two observations: (i) by Consequentialism
with E = s ∨ t, the odds-ratio between any two states s, t ∈ S ∗ depends only
on the payoff-value in those two states, and (ii) the odds-ratio between s and
t in S can be expressed as the ratio of the odds-ratio between each of s and
t and any third state v. Let now f (s, x) denote the payoff-function mapping
s to x and all other states to a, then
π(s,f (s)) (s)/π(s,f (s)) (v)
p(s) · (π(s,f (s)) (s)/(p(s)π(s,f (s)) (v)))
πf (s)
=
=
(11)
πf (t)
π(t,f (t)) (s)/π(t,f (t)) (v)
p(t) · (π(t,f (t)) (t)/(p(t)π(t,f (t)) (v)))
Using Equation 11 it is possible to define µs (x) so that Equation 9 holds for
any s other than v, but the case of v itself presents a special problem. However, since the choice of v is arbitrary, it is possible to replace v with all the
other states together, and define µs (x) = [πf (s,x) (s)/(1−πf (s,x) (s))]/[p(s)/(1−
p(s))]. Finally, the case of s ∈
/ S ∗ is trivial, since πf (s) = p(s) = 0 for s ∈
/ S ∗,
so that any value for µs would do.
The following counter-example demonstrates that the requirement that
S ∗ contains at least three states is a necessary condition:
Example 1.
Let S = {s, t}. Define a distortion π by
πf (s) =
(f (s) − f (t))2 + 1
(f (s) − f (t))2 + 2
and
πf (t) =
1
(f (s) − f (t))2 + 2
(12)
This distortion satisfies Absolute Continuity and Consequentialism. Nevertheless, the odds ratio between s and t cannot be expressed as the ratio of
two functions, one dependent only on f (s) and the other only on f (t).
For the second part of Theorem 3, let z ∈ Z be some payoff-value. Setting
f = πz in Equation 9 we obtain that µs (z) ∝ πz (s)/p(s) for all s ∈ S ∗ . By
Indifference, πz = p. Hence µs (z) is the same for all s ∈ S ∗ . We can thus
choose any s∗ ∈ S ∗ , define ν(z) = µs∗ (z), and Equation 10 would hold for all
s ∈ S.
21
The key to the third part is that Shift-invariance establishes linear constraints on log ν. Using induction it is straightforward to show that there
exists a parameter ψ, such that ν(q) = ν(0)eψq for all q ∈ Q. Equation 10
is invariant to scaling, and hence we can select ν(0) = 1, and Equation 10
would hold with ν(q) = eψq . The extension of this result to all real-values
payoffs follows from Prize-Continuity.
3.3
General state-space
In this section I generalize the results of the previous section, and prove
Theorem 1 for the general states-space of Section 3.1. The first step is to
embed the result of Theorem 2 in the general state-space. By assumption
there exist three disjoint events D1 , D2 , and D3 with positive πf probability
for at least one f ∈ F . Define E1 = D1 , E2 = D2 and E3 = S \ (E1 ∪ E2 ),
let Σ1 denote the minimal algebra that includes these three events, and let
F1 denote the set of Σ1 -measurable payoff-functions. The events E1 , E2 and
E3 can be considered as states for the purpose of Theorem 2, and all the
required assumptions are implied by the assumptions for Theorem 1. Hence,
we obtain that π restricted to payoff-functions in F1 and events in Σ1 is a
linear distortion.
For the next step, it is convenient to define the distortion δ(f, A, B) introduced by a payoff-function f to the odds-ratio between two constant-payoff
non-null events A and B:
δ(f, A, B) = log
πf (A) p(A)
/
πf (B) p(B)
(13)
Note that for f ∈ F1 and A, B ∈ {E1 , E2 , E3 } it follows from Equation 7 that
δ(f, A, B) = ψ(f (A) − f (B)), where ψ is the coefficient of relative optimism
for F1 . The following Lemma generalizes this result to all payoff-functions:
Lemma 1. Suppose that for at least one f ∈ F the support of πf includes
three or more states, and that π is a well-behaved distortion, and let ψ denote
the coefficient of relative optimism for F1 . Then for any f ∈ F , and non-null
constant-payoff events A and B, δ(f, A, B) = ψ(f (A) − f (B)).
Proof. Note first that Indifference implies that δ(a, A, B) = 0 for any constant payoff-function a. Using Consequentialism with E = A ∪ B we obtain
that δ(f, A, B) = δ(f 0 , A, B) whenever f and f 0 agree on A ∪ B. Combining these observations, we obtain that if f is constant over A ∪ B then
δ(f, A, B) = 0. Let now f be any payoff-function and A and B be any two
non-null constant payoff-events. Without limiting generality suppose A ∩ E1
22
is not-null. Define f 0 , f 00 ∈ F2 by f 0 = f (A) on A ∩ E1 and f 0 = f (B)
elsewhere, and f 00 = f (A) on E1 and f 00 (E2 ∪ E3 ) = f (B). Using the above
observations, and the fact that f 00 ∈ F1 ,
δ(f, A, B) = δ(f, A ∩ E1 , B) = δ(f 0 , A ∩ E1 , B) = δ(f 0 , A ∩ E1 , E2 )
= δ(f 00 , A ∩ E1 , E2 ) = δ(f 00 , E1 , E2 ) = ψ(f (A) − f (B))
(14)
Consider now the class of simple payoff-functions (a payoff-function f is
simple if f (S) is finite). For a simple payoff-function f any event can be
expressed as the sum of the constant-payoff events, and so the case of simple
payoff-functions is a straightforward corollary of Lemma 1:
Corollary 1. Theorem 1 holds when restricted to simple payoff-functions.
Finally, the proof for a general payoff-function f uses a limit argument in
which f is approached by simple-functions. The first step in the proof is to
prove that eψf (s) is effectively bounded for all f :
Lemma 2. Suppose π is a linear distortion over the set F2 of simple payofffunctions, let ψ denote its coefficient of relative optimism, then for any (not
necessarily simple) payoff-function f ∈ F there exist positive real numbers m
and M , such that m < eψf (s) < M almost everywhere.
The proof sketch is that if eψf (s) is not bounded, then given a sequence of
disjoint non-null events {An } we can construct for any n ∈ N a simple payofffunction such that πf (An ) ≥ πf (A1 ). Combining
P this with the observation
that since πf is a probability measure then n πf (An ) ≤ 1, it follows that
πf (A1 ) = 0, which is a contradiction to the assumption that A1 is not null.
Using this result, f can be approached by a sequence of simple payofffunctions fn . By Prize-Continuity
fn → πf , and by the monotone-convergence
R πψf
R
R ψf
n
theorem,
limn→∞ A e dp = A e dp, and similarly limn→∞ S eψfn dp =
R ψf
e dp. By Lemma
are bounded, and hence we obtain that
S
R 2 these integrals
R
limn→∞ πfn (A)R = A eψfRdp/ S eψf dp. Combining these resultsR we obtain
that πf (A) = A eψf dp/ S eψf dp, and hence for all A, πf (A) ∝ A eψf dp.
4
Belief distortion
Equation 6 describes how subjective beliefs are distorted by what a person
wants to be true. The eψf (s) term drops out if the payoff consequences are
such that the person obtains exactly the same payoff in all states. The probability measure p therefore represents a person’s undistorted beliefs, namely
23
the beliefs she would have held if she were indifferent about the state of the
world, and payoff-dependent belief distortion had no effect on her beliefs.
In certain applications it is useful to assume that p coincides with rational expectations (Muth, 1961), but this would be an additional assumption.
Equation 6 itself only describes the relationship between πf and p.
In order to see what Equation 6 has to say in situations in which the
agent is not indifferent, it is useful to focus on events for which the agent has
a well-defined desire, namely events in which the payoff-function is constant.
If f is constant over an event A, Equation 6 can be rewritten as follows:
πf (A) ∝ p(A) · eψf (A)
(15)
The factor eψf (A) is increasing in the payoff if ψ is positive, decreasing in
payoff if ψ is negative, and independent of payoff if ψ = 0. Payoff-dependent
belief distortion therefore makes desirable events subjectively more probable
if ψ > 0, and subjective less probable if ψ < 0. If ψ = 0 the desirability of
an event has no effect on its subjective likelihood. Agents with ψ > 0 are
optimists, and agents with ψ < 0 are pessimists. Moreover, the larger ψ is
in absolute terms the stronger is the effect of payoffs on beliefs. In analogy
with the coefficient of relative risk aversion, ψ can therefore be thought of as
the coefficient of relative optimism.
If an event is neither the most desirable event, nor the least desirable
one, it is not clear from Equation 15 whether its subjective probability is
higher or lower than it would be if the agent were indifferent between all
events. However, if we look at the odds-ratio between pairs of events, the
normalization term drops out, and we obtain a simple expression.29 Suppose
that f is constant over two events A and B, and that B is not-null.30 The
log-odds ratio between the two events can be written as follows:
log
p(A)
πf (A)
= log
+ ψ · f (A) − f (B)
πf (B)
p(B)
(16)
Thus, the effect of payoff-dependent belief distortion on the relative probability of two events depends only on the payoff-difference between them,
or the degree to which one is more desirable than the other. The same
odds-ratio can result from different combinations of evidence (underlying the
29
If we wanted to replace the proportionality symbol in Equations 6 and 15 with an
equality, we would have needed to include a normalization term to ensure that probabilities
sum to 1. This normalization term would depend on the payoff in states outside A, as well
as the payoff in A itself.
30
Absolute Continuity guarantees that the term “non-null” is well-defined without having
to specify the payoff-function.
24
f (A) − f (B)
π (A)
log πff (B)
optimist believes A
p(A)
log p(B)
optimist believes B
−2
−1
0
1
2
Figure 1: Iso-belief lines for an optimist as a function of undistorted beliefs
on the x-axis and payoff-difference between the two events the y-axis. Isobelief lines are straight lines sloping down and to the right with slope 1/ψ.
The Iso-belief lines for pessimists slope upwards and to the right. Those of
a neutral agent are vertical.
undistorted probability measure p), and desire (represented by the payoffdifference). Since Equation 16 is linear, the resulting iso-belief lines are also
linear (Figure 1).
4.1
Payoffs as information
The belief distortion equations have a close analogue in Bayes Rule. For
Equation 15 the analogous equation is the following:
p(A|e) ∝ p(A) · L(e|A)
(17)
where e represents new evidence, p represents beliefs prior to observing the
new information, p(A|e) represents posterior beliefs, and L(e|A) the likelihood of the new evidence. Similarly, the analogue of Equation 16 is
log
p(A)
L(e|A)
p(A|e)
= log
+ log
p(B|e)
p(B)
L(e|B)
(18)
where p(A)/p(B) is the prior odds ratio, p(A|e)/p(B|e) is the posterior odds
ratio, and L(e|A)/L(e|B) is the likelihood ratio. A comparison of these
equations reveals a perfect correspondence, with p standing for undistorted
25
or prior beliefs, πf for distorted or posterior beliefs, and ψf (A) as the loglikelihood in A, with an analogous expression for B.
It is thus as if the payoff in an event were relevant evidence about its
likelihood. An optimist takes high payoff to be evidence that an event is
more likely, while a pessimist takes the same payoff-value as evidence that
the event is less likely. Realists see payoffs as completely irrelevant in judging
the likelihood of events. Of course, it is not necessary to assume that people
are consciously aware of the effect payoff-consequences have on their beliefs.
In fact, much of the interest in the model is precisely that a person’s beliefs
may be much affected by optimistic or pessimistic bias, while she consciously
strives to minimize such bias.31
4.2
Distorted distributions
Suppose payoff is a non-decreasing function of some random variable X. To
fix ideas, suppose that X(s) is the profit in dollars that a given investment
yields in state s, and that f (s) = f (X(s)) is the utility of this amount of
money. In this section I look at how the distorted cumulative distribution
function (CDF) of X compares with the undistorted CDF, and what the
distorted CDF would have been if ψ were different.
The simplest case is if X is binary. Let a > b denote the payoff values.
The distorted distribution is then also binary, and according to Equation 15
the distorted odds-ratio is related to the undistorted odds-ratio by a factor of
eψ(a−b) . In particular, the probability of the better event is greater the more
positive ψ is. In order to generalize this observation beyond the binary case
it is useful to define stochastic dominance in the likelihood ratio:32
Definition 4. Let F (x) and G(x) denote two cumulative distribution functions of some real-valued random variable. F stochastically dominates G in
the likelihood ratio, written F LRR G, if there exists a non-decreasing funcx
tion h : R → R, such that F (x) ∝ −∞ h(x)dG(x). The relationship is strict,
written as F LR G, if F LR G, but not G LR F .
The generalization of the relative optimism result is that the CDFs corresponding to two different values of the coefficient of relative optimism are
related via stochastic dominance in the likelihood ratio:
31
It is possible to draw an analogy to racist bias, where one has to differentiate between
(i) having one’s judgment about a person’s ability (perhaps unconsciously) affected by that
person’s race, as if race was evidence about ability, (ii) anticipating that one’s judgment
would be so affected, and (iii) consciously believing that a person’s race is informative
about that person, in ways consistent with the bias.
32
This definition is closely related to the monotone likelihood ratio property.
26
Proposition 1 (relative optimism). Suppose that the payoff-function f is
a non-decreasing function of a random variable X : S → R. Let ΠψX (x) =
ψ
πX
(X ≤ x) denote the distorted cumulative distribution of X if the coefficient
of relative optimism is ψ, and let ψH and ψL be two possible values for ψ,
such that ψH ≥ ψL . Then Πψf H LR Πψf L . Moreover, if f is strictly increasing
in X, ψH > ψL and X is not almost always constant then the inequality is
strict.
The immediate corollary is that the distorted beliefs of an optimistic
stochastically dominate in the likelihood ratio her undistorted beliefs, with
the opposite being true for a pessimist:
Corollary 2. Suppose the payoff f is a non-decreasing function of a random variable X : S → R. Let PX and ΠX denote the undistorted and
distorted payoff-distributions respectively, then ΠX LR PX for an optimist,
and PX LR ΠX for a pessimist. Moreover, if f is strictly increasing in X
and X is not a.e. constant then the inequalities are strict.
The higher (lower) ψ is, the more probability shifts towards the highest
(lowest) possible payoff. This limit is not always well-defined. If, for example,
the undistorted payoff distribution has its support over some open interval
then the limit of distorted beliefs for extreme optimism or pessimism does not
exist. Suppose, however, that there only finitely many possible payoff values.
In this case the limit is always well-defined, and takes a particularly simple
shape: an extreme optimist is certain she would obtain the best possible
payoff, while an extreme pessimist is certain she would obtain the worst
possible payoff:33
Proposition 2 (extreme optimism/pessimism). Let f be a simple payofffunction, and let Amin and Amax denote respectively the event that the minimal
(maximal) payoff is obtained, then
lim πf (Amin ) = lim πf (Amax ) = 1
ψ→−∞
ψ→∞
(19)
Optimistic and pessimistic bias typically results in a distortion of the
shape of the distribution. For example, if the undistorted distribution is uniform over some interval then the distorted distribution would be exponential
over the same interval. However, since the distortion likelihood term is exponential, if the undistorted payoff-distribution is a member of the exponential
33
Extreme optimism coincides with Leibniz’s concept of the “best of all possibleworlds” (Leibniz, 1710) if ‘best’ is understood to mean best for the optimist. Yildiz (2007)
explores a model in which agents behave like extreme optimists in this sense.
27
family of distributions, then so is the distorted payoff-distribution. The case
of a normal distribution is particularly interesting. If payoff is linear is some
normally distributed random variable, then optimism and pessimism result
simply in a shift of the mean of the distribution, the shift being proportional
to the variance and to ψ:
Proposition 3 (normally distributed payoffs). Suppose X : S → R is a
random variable with undistorted distribution PX ∼ N (µ, σ 2 ), and that there
exist a ∈ R+ and b ∈ R, such that the payoff-function is f = aX + b, then
ΠX ∼ N (µ + a2 σ 2 ψ, σ 2 ).
4.3
Preservation of stochastic dominance order
Suppose a change is made that from a disinterested perspective is unambiguously good (bad). Would it also necessarily be perceived as such by the person
whose payoff is thereby affected? The general question can be modeled as follows. Suppose payoff is an increasing function of a random variable X, which
in one situation has an undistorted cumulative distribution function F and
in another G, and let F 0 and G0 denote the corresponding distorted CDFs.
Suppose further that F is better in the sense that F stochastically dominates
G. Does it follow that F 0 is better than G0 ? Consider first two events in
which X obtains a value of x and x0 respectively. Following belief distortion,
0
the odds ratio between the two events is multiplied by eψ(f (x)−f (x )) . This
factor depends on the payoff-difference between the two events, but it does
not depend on their undistorted probabilities. Thus, if in some distribution
F the odds ratio between the two events is greater by some factor than in
another distribution G, then exactly the same ratio would be retained in the
distorted distribution. Generalizing this observation it is straightforward to
show that the relationship of stochastic dominance in the likelihood ratio is
preserved under both optimism and pessimism:
Proposition 4. Let X : S → R be a random variable, and suppose the payoff
f is a non-decreasing function of X. Let F and G (F 0 and G0 ) denote the
undistorted (distorted) cumulative distribution function of X in two different
situations, then if F LR G then also F 0 LR G0 .
By contrast, first-order stochastic dominance is in general not invariant
to optimistic or pessimistic belief distortion. Intuitively, if a bad state is
made worse then the payoff-difference between good states and the bad state
increases, so that an optimist would be more biased about the good states
than before the change. Consequently, the change in distorted beliefs is
ambiguous. Consider the following example:
28
Example 2. violation of first order stochastic dominance
Let A, B and C denote events, with payoff 2, 1 and 0 respectively. Let F
be a payoff-distribution defined by pF (A) = pF (B) = 0.5 and pF (C) = 0,
and G a payoff-distribution defined by pG (A) = pG (C) = 0.5 and pG (B) = 0.
Consider an optimist with a coefficient of relative optimism ψ = log 2, so
that eψ = 2. Given F , the distorted payoff distribution would be πfF (A) =
2/3, πfF (B) = 1/3 and πfF (C) = 0. Given G the distorted payoff-distribution
would be πfG (A) = 4/5, πfG (B) = 0 and πfG (C) = 1/5. Thus, F first-order
stochastically dominates G, but this is not the case for the distorted distributions.
5
Belief updating
Suppose some new evidence e is observed. Posterior beliefs are obtained by
conditioning the elements of Equations 6 on the new information:
Z
Z
ψf (s|e)
eψf (s|e) L(e|s) dp
e
dp(s|e) ∝
πf (A|e) ∝
A
ZA
Z
(20)
ψ(f (s|e)−f (s))
ψf (s)
ψ(f (s|e)−f (s))
∝
e
L(e|s)e
dp ∝
e
L(e|s) dπ
A
A
Equation 20 describes the effect of new evidence on the agent’s beliefs. The
equation includes not one, but two update terms, corresponding to two separate channels by which new information can affect beliefs. The first channel
is standard Bayesian updating, represented by the likelihood term L(e|s).
The second channel is a quasi-Bayesian updating due to the change in the
anticipated payoff-consequences, represented by the quasi-likelihood term
eψ(f (s|e)−f (s)) . The corresponding expression to for the odds-ratio between
constant payoff-events is as follows:
log
πf (A|e)
πf (A)
L(e|A)
eψ(f (A|e)−f (A))
= log
+ log
+ log ψ(f (B|e)−f (B))
πf (B|e)
πf (B)
L(e|B)
e
(21)
In Equation 21 the LHS is the posterior log odds-ratio. On the RHS we have
the prior log odds-ratio, followed by the log likelihood ratio, and finally the
log ‘likelihood’ ratio representing the change in payoff-consequences.
If the evidence e is independent of the payoff-function34 belief updating is
purely Bayesian. Even so, it may not appear Bayesian to outside observers.
34
Or, if it has exactly the same effect in all states, as would be the case if the agent is
risk-neutral, and e represents an unconditional gain or loss. Note that the assumption of
risk-neutrality is essential. For example, if the agent is loss-averse then the utility of a
given gain increases following a loss. A loss-averse optimist may therefore come to view
the gain as more likely after she experiences a loss.
29
complements
substitutes
p
I
S 1/4
F 1/4
p
S1
F1
S2
1/4
1/4
U
1/4
1/4
f
I
S +1
F −1
F2
1/4
1/4
f
S1
F1
S2
1
1
U
0
0
F2
1
0
πf
I
S 4/9
F 1/9
U
2/9
2/9
πf
S1
F1
F2
2/7
1/7
S2
2/7
2/7
Figure 2: Complements and substitutes in belief updating. In the complements example, S and F denote whether a merger is a success or failure, and
I and U whether it is important or unimportant to the manager’s promotion. From left to right we have (i) the probability measure representing the
manager’s beliefs if she were indifferent, (ii) the manager’s payoff-function,
and (iii) her actual beliefs assuming she is an optimist with a coefficient of
relative optimism equal to log 2. Learning that the merger is important results in an increase from 2/3 to 4/5 in the subjective probability that the
merger is successful. In the opposite direction, learning that the merger is
successful results in an increase from 5/9 to 2/3 in the subjective probability
that the merger is important. In the substitutes example states indicates
the success or failure of two projects, with the manager needing only one
to get promoted. Learning that one project is a success makes the manager
indifferent about the second. Consequently, the subjective probability that
the other project is a success is reduced from 4/7 to 1/2—the same as the
undistorted probability.
Consider the merger example, and suppose that the success of the merger may
or may not be important to the manager’s promotion. This situation can be
modeled using four states, corresponding to different combinations of two
variables: Success and Failure on the one hand, and Important and Unimportant on the other. Suppose also that a disinterested manager views the
two events as independent, and consider the beliefs of an optimistic manager.
Since payoff is particularly high in the Success/Important state, and particularly low in the Failure/Important state, the manager’s subjective probability
for these two states would be particularly high (low). Consequently, for her
the two variables are not independent, but rather positively correlated. Observing that the merger is important would increase her subjective probability
that it would succeed (Figure 2). However, for a disinterested observer who
sees the two events as independent, the change in the manager’s beliefs may
well appear to be inconsistent with Bayesian updating.
The positive correlation between the two variables goes both ways: the
30
more likely is the merger to be important, the more the manager believes it
would succeed; the more the manager expects the merger to succeed, the more
likely she thinks its success or failure would be a critical factor in determining whether she is promoted. The complementarity in beliefs corresponds to
the complementarity in the utility function.35 When two variables are complements (substitutes) they become positively (negatively) correlated, and
positive news about one increases (decreases) the bias in the other. For an
example of substitutes, suppose the manager’s promotion depends on at least
one of her cost-cutting initiatives bearing fruit, with no additional gain from
success in multiple projects. Good news about one of those initiatives would
make her less anxious—and therefore less biased— about the others. In fact,
if the two initiatives are perfect substitutes the bias could disappear altogether (having earned her promotion she no longer cares about the success
of other initiatives).
Proposition 5. Suppose payoff f is a function of two real-valued random
variables X and Y , such that p(X = x, Y = y) = p(X = x)p(Y = y) for all
x, y ∈ R, and suppose p(e|Y = y) is an increasing function of y, then
1. PX|e LR PX if (i) ψ ≥ 0 and f is supermodular, or (ii) ψ ≤ 0 and f
is submodular.
2. PX LR PX|e if (i) ψ ≤ 0 and f is submodular, or (ii) ψ ≤ 0 and f is
supermodular.
Moreover, if neither X nor Y is a.e. constant, p(e|Y = y) is strictly increasing in y, ψ 6= 0, and f is strictly supermodular/submodular, then the above
relations of stochastic dominance in the likelihood ratio are also strict.
The second channel for belief update involves a change to the payofffunction itself. Consider the beliefs of parents whose child is allocated by
coin-toss either to school A or to school B. Suppose payoff is 1 if the child
goes to the better school and 0 otherwise. There are two subjective states:
A and B, corresponding to whether A or B is the better school. Since the
allocation is decided by a coin-toss, the ordinary likelihood term is the same
in both states. However, learning the outcome of the allocation makes a big
difference to the payoff-function. For example, if the child is allocated to
school A, f (A) increases from 1/2 to 1, and f (B) decreases from 1/2 to 0.
Consequently, the beliefs of optimistic parents would be updated to a more
favorable view of A.36
35
36
Utility is a supermodular function of the two variables.
In practice, the change in beliefs is likely to occur not immediately following the
31
5.1
Crime and punishment
In this section I apply the model to a problem from the economics of crime.
An increase in the severity of the punishment handed to criminals that are
caught increases the stakes in not getting caught, and thus results in greater
optimistic bias than with more lenient sentences. Consequently, an increase
in the severity of punishments can be counter-productive, ending up reducing,
rather than improving deterrence. This contrasts with an increase in the
probability that crime is punished, which does reliably reduce the subjective
utility of crime.
Example 3. Crime and punishment
An optimistic criminal has to choose between Crime and McDonald’s. There
are two subjective states: in B crime is punished and the criminal receives a
payoff of −c, while in state G the criminal manages to stay out of jail, and
receives a payoff normalized at 0. McDonald’s yields −b in both states. If c ≤
b McDonald’s is never optimal, so I assume c > b. Crime is then the attractive
option as long as πf (B) ≤ b/c. Suppose that the undistorted probability of
getting caught is p, and let f denote the criminal’s payoff-function on the
expectation that she chooses Crime37 . The subjective probability of getting
caught is:
p
πf (B) =
(22)
p + (1 − p)eψc
Crime is chosen if πf (B) ≤ b/c, i.e.
pc ≤ pb + (1 − p)eψc b
or
e−ψc ·
p
b
≤
1−p
c−b
(23)
(24)
The LHS of Equation 24 is the subjective odds-ratio between the bad state
(getting caught) and the good state (not getting caught). The RHS is the
odds-ratio between how much better crime is in the good state, and how
much worse it is in the bad state. Consider now the following two changes
the government can try and make. First, the government can try to improve
law enforcement, and thus increase p. Holding c constant, such a change
would reduce the LHS of Equation 24, while leaving the RHS unchanged,
allocation, but rather the next time the parents have an opportunity to reassess the quality
of the schools. For example, they may make a visit to the two schools, and then ascribe
the change in their beliefs to their impressions from the visit.
37
If she is unsure whether to choose Crime or McDonald’s, the bias in her beliefs would
be less.
32
and is therefore guaranteed to increase deterrence for any level of optimism.
Similar results hold more generally, as long as the two payoff distributions
are related via stochastic dominance in the likelihood ratio (Proposition 4).
Second, the government can increase in the severity of punishment c, leaving
the probability of catching criminals unchanged. If ψ = 0 the change would
leave the LHS of Equation 24 unchanged, and would reduce the RHS. Thus,
for realist criminals, any increase in the severity of punishment improves
deterrence.38 Suppose, however, that ψ > 0. In this case the increase in
c reduces the LHS of Equation 24 at the same time as it is reducing the
RHS. There are thus two forces pulling in opposite directions: (i) the utility
effect increases deterrence (conditional on getting caught, crime is a worse
choice), and (ii) the probability effect works to reduce deterrence (a more
severe conditional punishment reduces the subjective probability of getting
caught). Since limc→∞ eψc /c = ∞, making the punishment more severe is
always counter-productive beyond a certain point for any positive level of
optimism (Figure 3).
The empirical evidence on this topic is interesting. For example, Grogger
(1991) looking at the frequency of arrests found a much larger deterrent
effect for the certainty of punishment as compared with severity (-0.562 with
t-score of 8.52 for the probability of conviction vs. 0.017 with t-score of
1.65 for average sentence length). Similarly, Nagin and Pogarsky (2001) also
found that the certainty of punishment was a far more robust a deterrent
than severity. Note that the effects of belief distortion may be particularly
strong if optimists self-select a life of crime over less risky alternatives.
6
Preferences
This section considers the relationship between payoff-dependent belief distortion and choice. The first thing to note is that as a model of beliefs
payoff-dependent belief distortion is not tied to any specific model of choice.
In particular, any model of probabilistically sophisticated choice (Machina
and Schmeidler, 1992; Grant, 1995; Chew and Sagi, 2006) can be readily
combined with the linear distortion model of payoff-dependent beliefs. For
the rest of this section, however, I will assume decision makers are subjective expected utility maximizers. The only change from standard subjective
expected utility maximization (Savage, 1954; Anscombe and Aumann, 1963;
Fishburn, 1970) is that the subjective beliefs underlying the choice are payoff38
This, of course, would also be true for a pessimistic criminal. Of course, with endogenous selection pessimists are unlikely to choose a risky life of crime in the first place.
33
Probability effect
0
πf (not caught) 1
Combined effect
Expected utility
πf (not caught)
Expected utility
Utility effect
0
p(not caught) 1
0
p(not caught) 1
Figure 3: The effect of increasing punishment levels from lenient (solid
blue line) to more severe (dotted red line) on deterring optimistic criminals.
At any given subjective probability more severe punishments reduce utility
(panel 1). At the same time, the increase in the stakes in staying out of
jail results in increased bias in the criminal’s subjective probability of not
getting caught (panel 2). The net effect (panel 3) depends on parameters. If
the punishment is light and capture is likely it would be negative, but if the
punishment is severe and capture occurs only with low probability then the
net effect can be an increase in utility (and hence reduced deterrence).
dependent.
Choice is therefore modeled by the combination of subjective expected
utility maximization and the linear distortion model of payoff-dependent beliefs (Definition 2). In Section 6.1 I ask how the model can be used to predict
choice behavior. In Section 6.2 I consider the possibility of using revealed
preferences to recover the parameters of the model.
6.1
Predicting choice
The choice implications of the model are, in principle, straightforward: preferences depend on beliefs via expected utility maximization, and beliefs depend on payoff expectations via Equation 6. As an example of how this
could play out, consider an optimistic investor with a significant preexisting
investment in some asset. The existing investment would result in an overestimation of the probability that this asset would do well, and hence in a
ceteris paribus tendency to invest more in the same asset.39
Choice predictions are, however, made more complicated by the fact that
the outcome the decision maker expects to obtain in a state s depends on
39
If the investor is risk-averse she would benefit from diversifying her investments. Optimism would then result in less diversification than the optimal level.
34
the alternative she expects to choose. For example, an optimistic investor
who anticipates taking on an additional investment would expect a higher
payoff in the event that the market goes up (and a lower payoff in the event
that the market goes down) than an otherwise identical investor who expects
to decline the additional investment. Consequently, the two investors would
end up with somewhat different beliefs, and potentially also with different
choices. Consider the following example:
Example 4. Housing market
A property investor is offered an investment opportunity. Depending on
whether the housing market would go up or down, the investment would
result in a $1 million profit or a $1 million loss. The investor also has a
$10 million existing exposure to the housing market. With obvious notation
her choice is between the acts g = (11, −11) and h = (10, −10). Suppose
that when considering her decision she assigns a probability q to taking on
the additional investment. It follows that if the market goes up (down) she
anticipates a profit of $11m (a loss of $11m) with probability q, and a profit
of $10 (a loss of $10m) with probability 1 − q. Assume for simplicity that she
is risk neutral with u(x) = x, with money amounts measured in units of $1m.
Her payoff-function is thus given by f (up) = 10 + q and f (down) = −10 − q.
Depending on the value of q, therefore, the payoff-difference between the two
states can be anything between 20 to 22. Suppose that, if she were indifferent
about the housing market, her (undistorted) subjective probability would
have been p(up) = 0.5, and suppose that her coefficient of relative optimism
is ψ = 0.04. Using Equation 15 her distorted subjective probability for the
market to go up is between πq=0 (up) = 0.6900 and πq=1 (up) = 0.7068. Since
she is risk neutral and the investment has a symmetric payoff, the investor
will choose to take up the investment as long as πf (up) ≥ 0.5. Thus, the
uncertainty about q turns out to make no difference to predictions. However,
suppose that if the market goes up the investment yields only $0.43m, but still
loses $1m if the market goes down. Since the payoff-difference is less, belief
distortion is also reduced, with πq=1 (up) going down from 0.7068 to 0.7021.
The payoff-difference conditional on not taking up the investment remains
the same, so πq=0 (up) = 0.6900 as before. Since the potential profit is only
$0.43m, the investor would only take up the second investment opportunity
if πf (up) ≥ 0.6993, which is between πq=0 (up) and πq=1 (up). Thus, with
this particular choice of parameters the investor’s decision can be affected by
what she expects to choose when forming her beliefs.
The possibility that unobserved expectations about choice can affect decisions is only an issue in some situations. Nevertheless, it represents a
significant limitation of the model.
35
6.2
Revealed-preferences axiomatization
In this section I focus on the inverse problem of recovering the parameters of
the model from observed choices. The basic datum is whether the decision
maker chooses g or h in a pairwise choice between the two alternatives. The
object is to determine from such observations the undistorted probability
measure p, the utility-function u, and the coefficient of relative optimism ψ.
I start by describing the formal framework. As in Section 3.1 let S denote
the set of subjective states, and let Σ be a σ-algebra of events. The agent’s
underlying preferences are over a set of Z of final outcomes. An act is a Σmeasurable mapping f : S → ∆(Z), associating with each subjective state an
objective lottery over final outcomes.40 I denote by F the set of all such acts.
I use acts to represent two different types of objects. First, acts represent the
alternatives in the agent’s choice set, where the final outcome depends on the
realization of both subjective and objective uncertainty.41 Second, acts also
represent the outcome-function, which describes the agent’s expectations as
to the outcome she would obtain conditional on the realization of subjective
uncertainty: if the state s were to obtain, the agent’s remaining (objective)
uncertainty as to the final outcome is represented by the lottery f (s) that
the outcome-function f associates with s.
Ex-post, the outcome-function coincides with the chosen act. When forming her decision, however, the agent has an additional level of uncertainty, as
she may not yet be certain what alternative she would end up choosing. Her
uncertainty over the outcome obtained in each subjective state is therefore
given by a compound lottery over {g(s) : g ∈ C} with probabilities representing her uncertainty over which choice she would make. The outcomefunction f represents the reduced form of these lotteries. In Example 4 the
agent’s uncertainty is a (q, 1 − q) compound lottery over g = (11, −11) and
h = (10, −10). The outcome-function is therefore f (up) = (11, q; 10, 1 − q)
and f (down) = (−11, q; −10, 1 − q).
A preference-relation is a preference-order over pairs of acts. The key
ingredient in the model is a distortion mapping associating with each act f
(in its role as an outcome-function) a preference-relation f . The dependence of f on f is the reduced form of the following causal chain: (i) filtered by her utility function, the outcome-function f determines the agent’s
payoff-function, (ii) filtered by the distortion π, the payoff-function determines the agent’s subjective probability measure πf ,42 and (iii) combined
40
Acts mapping states into objective lotteries over final consequences were first used in
Anscombe and Aumann (1963).
41
Horse and roulette lotteries in the terminology of Anscombe and Aumann (1963).
42
Since the payoff-function is a function of the outcome-function, it is possible to write
36
with the agent’s utility-function, πf determines the agent’s preferences f .
Given this framework there are now two key questions. First, how can we
reveal f when we do not observe the outcome-function f ? Second, how can
we use data of the form {f : f ∈ F } to (i) determine whether the agent’s
preferences are in accordance with the model, and (ii) recover the parameters
of the model from this data? I now address these two problems in turn.
6.2.1
Observing f
A key challenge in recovering the parameters of the model from revealed
choices is that the outcome-function representing the decision maker’s expectations is not directly observable. We want to know whether g f h, but
what we directly observe is only whether it is g or h that is chosen from
{g, h}. In this section I do not aim to offer a general solution to this problem, but rather a more specific solution that is sufficient to the problem of
recovering the parameters of the model from revealed choices.
Suppose therefore that we wish to determine whether g f h, where f, g,
and h are all acts. For ∈ (0, 1] let g denote the mixed act yielding in a
state s the lottery (g(s), ; f (s), 1 − ), and similarly define h . Suppose we
present the decision maker with a pairwise choice between g and h . Let
q denote the (unobserved) probability that she would attach to choosing
g . Her outcome-function would therefore be the act f , yielding in s the
lottery (g (s), q , ; h (s), 1 − q ) = (g, q ; h, (1 − q ); f (s), 1 − ), and so the
corresponding payoff in state s would be (q u(g(s)) + (1 − q )u(h(s))) + (1 −
)f (s). Hence, by Prize Continuity f →f . Thus, whatever q is, for small
enough if g f h (h f g) then g f h (h f g). Moreover, since the agent
is a subjective expected utility maximizer, and g and h are mixtures of g
and h respectively with the same third act and with the same proportions,
then g f h if and only if g f h . Combining these observations we obtain
that for small enough, the choice of g from {g , h } reveals that g f h,
and similarly the choice of h reveals that h f g.
6.2.2
Representation theorem
In this section I offer a revealed-preferences axiomatization of linear distortions. That is, I present a set of axioms on preferences between acts, and
prove that these axioms are necessary and sufficient conditions for preferences to be represented by expected utility functional, in which beliefs are
related to anticipated payoffs in accordance with the definition of a linear
distortion (Definition 2).
πf where f is an act, rather than a payoff-function.
37
I impose 8 axioms altogether. Axioms B1-B4 establish the relationship
between f and πf . Axioms C1-C4 then use this link to translate assumptions A1-A4 in Definition 1 into revealed-preferences assumptions. In the
statement of the axioms f, f 0 , g, h, and fn are any acts, a and b are any
constant acts (yielding the same lottery in all states), s is any state, and E
is any event. Finally, E is f null if for all g and h that differ only on E,
g ∼f h.
B1 (expected-utility). f has a subjective expected-utility representation.
B2 (objectivity). f =f 0 over constant acts.
B3 (indifference). If f (s) ∼ f 0 (s) for all s then f =f 0 43 .
B4 (non-triviality) There exist acts f, g and h for which g f h.
C1 (absolute continuity). E is f null ⇐⇒ E is f 0 null.
C2 (consequentialism). If f = f 0 over E then f =f 0 for acts that differ
only in E.
C3 (shift-invariance). If for some α ∈ [0, 1], f = αg + (1 − α)a and
f 0 = αg + (1 − α)b, then f =f 0 .
C4 (continuity). If fn → f uniformly44 then fn →f .45
B1 is an omnibus axiom, requiring that the preference-relation f have an
expected utility representation.46 Thus, for any act f there exist a probability
measure πf ∈ ∆(S) and a utility function uf : Z → R, such that f ranks
acts according to the following functional:
Z
Z Z
uf (gs (z)) dz dπf
(25)
Vf (g) = (uf ◦ g) dπf =
S
S
Z
where gs ∈ ∆(Z) is the objective lottery over final consequences that results
from the act g in the state s.
43
In this axiom f (s) is identified with the constant Anscombe-Aumann act fs defined
by fs (t) = f (s) for all t ∈ S. The expression f (s) ∼ f 0 (s) is well-defined given B2.
44
For any > 0 there exists n0 ∈ N, such that for all n > n0 , for any state s ∈ S, and
for any outcome z ∈ Z, |fn (s)(z) − f (s)(z)| < (the difference in the probability the two
acts assign to outcome z in state s is less than ).
45
For all acts g and h, if g f h then there exists n0 ∈ N, such that for all n > n0 ,
g fn h.
46
Anscombe and Aumann (1963) axiomatize such a representation for a finite statespace.
38
The representation in Equation 25 allows not only for reference-dependent
beliefs (πf ), but also for reference-dependent preferences over final outcomes
(uf ). Axiom B2 rules out the latter possibility by imposing the requirement that the ranking of objective lotteries (which are constant AnscombeAumann acts) does not depend on the outcome-function f . Since the ranking of constant Anscombe-Aumann acts is sufficient to determine the utility
function, it follows that uf in Equation 25 can be replaced by an referenceindependent utility function u:
Z
Z Z
Vf (g) = (u ◦ g) dπf =
u(gs (z)) dz dπf
(26)
S
S
Z
Equation 26 establishes a mapping between acts and payoff-functions, where
an act f : S → R∆(Z) is associated with the payoff-function f : S → R,
defined by f(s) = Z u(fs (z)). Axiom B3 ensures that the probability measure
πf depends on an act f only via the associated payoff-function. Axiom
B4 rules out the trivial case in which the agent is indifferent between all
acts. Non-triviality ensures that it is possible to back out πf from observing
f , and hence that statements that link f to f 0 have the corresponding
implications for πf and πf 0 :47
Lemma 3. Suppose B1-B4, then f =f 0 if and only if πf = πf 0 .
Given Lemma 3 the axioms C1-C4 effectively restate assumptions A1-A4
in the language of revealed preferences:
Lemma 4. Suppose B1-B4, then axioms C1-C4 respectively are equivalent
to A1-A4.
Thus, the combination of B1-B4 and C1-C4 meets the conditions of the
representation theorem of Section 3 (Theorem 1). Combined with the standard expected utility representation (Equation 26) we obtain the following
theorem:
Theorem 4 (payoff-dependent preferences). Suppose B1 − B4 and C1 − C4
hold then there exist a probability measure p over (S, Σ) (the undistorted
measure), a mapping u : Z → R (the utility function), and a real number ψ
(the coefficient of relative optimism), such that for any act f ∈ F , f ranks
acts according to the following functional:
Z
Z Z
Vf (g) = (u ◦ g) dπf =
u(gs (z)) dz dπf
(27)
S
S
47
Z
Non-triviality is a necessary condition, since if g ∼f h for all acts, then the choice data
can be explained by a constant utility function and any probability measure πf .
39
where πf is a probability measure over (S, Σ) that is related to p via the
following equation:
Z
eψu(f (s)) dp
(28)
πf (A) ∝
A
0
Moreover, if the triplet (p , u , ψ ) also represents {f }f ∈F then p0 = p, and
there exist a positive real-number a, and a real-number b, such that u0 = au+b
and ψ 0 = ψ/a.
7
0
0
Conclusion
This paper introduces a model of Bayesian decision making where a person’s
beliefs about the likelihood of different outcomes depend upon the anticipated
payoff consequences of those outcomes. This dependence is modeled as a
distortion mapping linking the actual beliefs of a person to the beliefs the
person would have held if he or she were indifferent what the true state is.
The distortion is defined by a real-valued coefficient of relative optimism,
which is positive for optimists, negative for pessimists, and is zero for people
whose beliefs are independent of payoffs.
This simple model can account for a number of phenomena. The predicted
relationship between beliefs and payoff is consistent with optimistic and pessimistic bias. Moreover, comparative statics for the level of uncertainty and
the stakes in the outcome are consistent with available evidence. When new
information results in a change in anticipated payoff the model predicts belief
change, even if the new information is normatively irrelevant. These predictions are consistent with evidence on cognitive dissonance (Cooper and Fazio,
1984), as well as evidence from studies of “motivated cognition” (Kunda,
1990).
The representation derived in the model is mathematically convenient,
and can be readily adapted for use in applications. Optimistic bias has been
applied to a variety of economic areas, such as financial markets, corporate
finance, bargaining, and insurance. One would hope that the model of this
paper can lead to better and stronger predictions in some of these areas. In
financial markets, for example, the model predicts not only an overall underestimation of risk by optimists, but more specifically how this bias depends
on the level of subjective uncertainty and the degree to which investors are
exposed to the risk. Moreover, because past decisions affect current payoffs, beliefs exhibit path dependence, offering a mechanism to explain how
significant belief heterogeneity may develop.
In certain situations choices may depend on the decision maker’s unobserved expectations at the time of forming her decision. Although this is
40
only an issue in some situations, it is undeniably a significant limitation.
One way of solving this problem would be to model explicitly the development of expectations over time. If this is done successfully, then determining
hat expectations are at some an initial point in time would be sufficient in
order to predict choices from that point onwards.
Another area for expanding the model is to model strategic interaction
explicitly. Yildiz (2007) has developed a model of strategic interaction in the
limit of extreme optimism, but no such model presently exists for moderate
optimism. One goal in particular would be to model a failure by players to
appreciate the information value in the choices made by other players.
Last but not least, the model makes a number of novel testable predictions, and one would very much like to see these predictions tested.
A
A.1
Proofs
Proof of Theorem 3
In all three parts the proof that the requirements are necessary is trivial. I
thus prove only that the requirements are sufficient.
First part
Proof. Let p = πa for some payoff-value a. Let S ∗ denote the support of p,
which by Absolute Continuity is also the support of πf for any f ∈ F . For
s ∈ S ∗ and x ∈ Z, let f (s, x) be the payoff-function mapping s to x and all
other states to a. Let t1 , . . . , tn denote the other states in S ∗ . f (s, x) and
the constant payoff-function a agree on ti and tj for all i and j, and hence
by Consequentialism with E = ti ∨ tj we obtain that πf (s,x) (ti )/πf (s,x) (tj ) =
p(ti )/p(ti ) for all i and j. Consequently,
1 − πf (s,x) (s) =
X
πf (s,x) (ti ) =
i
πf (s,x) (tj )
πf (s,x) (tj ) X
p(ti ) =
(1 − p(s))
p(tj )
p(t
)
j
i
(29)
Define µs (x) = [πf (s,x) (s)/(1−πf (s,x) (s))]/[p(s)/(1−p(s))]. Using Equation 29
we obtain that for any state v ∈ S ∗ \ {s},
p(s)µs (f (s)) = (1 − p(s))
πf (s,f (s)) (s)
πf (s,f (s)) (s)
= p(v) ·
1 − πf (s,f (s)) (s)
πf (s,f (s)) (v)
(30)
Let f be an payoff-function, and let s and t be any two states in S ∗ . Let f 0
be a payoff-function that coincides with f on s and t, and with a on all other
41
states. Let v be any third state. Then,
πf (s,f (s)) (s)/πf (s,f (s)) (v)
πf (s)
πf 0 (s)
πf 0 (s)/πf 0 (v)
=
=
=
πf (t)
πf 0 (t)
πf 0 (t)/πf 0 (v)
πf (t,f (t)) (t)/πf (t,f (t)) (v)
πf (s,f (s)) (s)
πt(t,f (t)) (t)
p(s)µs (f (s))
= p(v) ·
/ p(v) ·
=
πf (s,f (s)) (v)
πf (t,f (t)) (v)
p(t)µt (f (t))
(31)
where the first and third steps follows from Consequentialism, and the final
step on Equation 30. Since Equation 31 holds for all s, t ∈ S ∗ we conclude
that Equation 9 holds for all s ∈ S ∗ . For s ∈
/ S ∗ define µs (x) = 1 for all x, and
Equation 9 then holds trivially for s ∈
/ S ∗ , since for such s, πf (s) = p(s) = 0.
Combining these results, Equation 9 holds for all f ∈ F and s ∈ S.
Second part
Proof. Using the result from the first part, for all s, t ∈ S ∗ ,
πf (s)
p(s) µs (f (s))
=
·
πf (t)
p(t) µt (f (t))
(32)
Pick some state s∗ ∈ S ∗ and define the mapping ν : Z → R+ by ν(z) = µs∗ (z).
For z ∈ Z let z ∈ F denote the constant payoff-function yielding z in all
states. Plugging f = z and t = s∗ in Equation 32 we obtain that for all
s ∈ S ∗ and z ∈ Z,
p(s) µs (z)
πz (s)
=
·
(33)
∗
πz (s )
p(s∗ ) ν(z)
By Indifference πz = πa = p. Hence, it follows from Equation 33 that
µs (z) = ν(z). Thus, πf (s) ∝ p(s) · ν(f (s)) for all s ∈ S ∗ . Finally, this is also
trivially true for s ∈
/ S ∗ , since πf (s) = p(s) = 0 for s ∈
/ S ∗.
Third part
Proof. Let x, y ∈ Z be any two payoff-values, let s, t ∈ S ∗ be two states,
let f denote the payoff function mapping s to x and all other states to
0, and let g = f + y. By Shift-Invariance, πg = πf . Plugging this into
Equation 10 we obtain that for all x, y ∈ Z, ν(x)ν(y) = ν(x + y)ν(0). Define
σ(x) = log(ν(x)/ν(0)) to obtain that for all x, y ∈ R,
σ(x + y) = σ(x) + σ(y)
(34)
For m ∈ N let y = mx. By induction we obtain that σ(mx) = mσ(x). For
n ∈ N let y = x/n, then σ(x) = σ(ny) = nσ(y), and hence σ(x/n) = σ(x)/n.
Let y = −x to obtain that σ(−x) = −σ(x). Combining these results, and
42
defining ψ = σ(1), we obtain that for all q ∈ Q, σ(q) = ψq. Thus, for q ∈ Q,
ν(q) = ν(0)eψq . Finally, for x ∈ Z define fx by fx (s) = x and fx (t) = 0,
and let {qn }n∈N be a sequence of rational numbers converging to x. By prizecontinuity πqn → πx . By Equation 10 it follows that ν(qn ) → ν(x). Moreover, by the result just established for rational numbers, ν(qn ) = ν(0)eψqn ,
and hence ν(qn ) → ν(0)eψx . Being the limit of the same sequence, the two
limits must coincide, and so we conclude that ν(x) = ν(0)eψx . Finally, since
Equation 10 is invariant to multiplying ν by a positive number, we can set
ν(0) = 1 to obtain that ν(x) = eψx for all x ∈ R, thereby completing the
proof that π is a linear distortion.
A.2
Proof of Theorem 1 using Theorem 3
Proof of Lemma 1
Proof in text.
Proof of Corollary 1
Obvious given Lemma 1.
Proof of Lemma 2
Proof. The claim is trivial if there are only finitely many disjoint non-null
events. Assume therefore that there exists a sequence {An }n∈N of disjoint
non-null events. It is sufficient to show that in this case the set eψZ = {x ∈
R : ∃z ∈ Z, x = eψz } is bounded. I show first that it is bounded from
above. Suppose otherwise, then it is possible to choose from Z a sequence
of values {zn }n∈N , s.t. for all n, p(An )eψzn ≥ p(A1 )eψz1 .48 Define a simple
payoff-function f ∈ F by setting f (An ) = zn , and f (s) = z1 outside ∪n An .
Since f is simple and π is a linear distortion, the following inequality holds
for all N ∈ N,
X
1 ≥ πf (S) ≥
n≤N
πf (An ) = πf (A1 )
X p(An )eψzn
X πf (An )
= πf (A1 )
πf (A1 )
p(A1 )eψz1
n≤N
n≤N
≥ N πf (A1 )
(35)
48
Choose z1 arbitrarily. Since by assumption eψZ is unbounded from above and none
of the events in {An } is null, it is possible to choose for all n > 1 a value zn so that the
condition holds.
43
Since Equation 35 holds for all N ∈ N it follows that πf (A1 ) = 0 in contradiction to the assumption that A1 is not null. The proof that eψZ is bounded
from below is similar, reversing the roles of πf and p. If the claim is false, a
sequence {zn0 }n∈N can be constructed, s.t. for all n, p(A1 )eψzn ≤ p(An )eψz1 ,
f 0 defined in analogy to f , and then the following inequality holds for all
n ∈ N,
X p(An )
X πf 0 (An )eψz1
1 ≥ p(S) ≥ p(A1 )
= p(A1 )
≥ N p(A1 )
(36)
p(A1 )
πf 0 (A1 )eψzn
n≤N
n≤N
and since Equation 35 holds for all N ∈ N it follows that p(A1 ) = 0, in
contradiction to the assumption that A1 is not null.
Proof of Theorem 1
Proof. This is the proof of Theorem 1 for general payoff-functions using
the result for simple payoff-function (Corollary 1 and Lemma 2) as the
starting point. LetR f be any payoff-function. I need to show that for all
πf (A)
∝ A eψf (s) dp. I prove the equivalent claim that πf (A) =
RA ⊆ψfS,
R
e (s) dp / S eψf (s) dp. By Lemma 2 there exist positive real-numbers m
A
and M , such that eψf (s) ∈ [m, M ] for all s except perhaps on a null-set
E. With loss of generality suppose that E is empty.49 Define a sequence
of simple payoff-functions as follows. For any n ∈ N divide [m, M ] into 2n
non-overlapping intervals of length (M − m)/2n . For any state s let In (s)
denote the interval to which f (s) belongs, let Inmin (s) be its lower endpoint
and let Inmax (s) be its upper endpoint. Define fn as follows:
(
Inmin (s) s ∈ A
(37)
fn (s) =
/A
Inmax (s) s ∈
Thus, fn % f point-wise. Since eψx is monotonically increasing, it follows
that eψfn (s) % eψf (x) . Thus, by the monotone convergence theorem,
Z
Z
ψfn (s)
lim
e
dp =
eψf (s) dp
(38)
n→∞
A
A
and similarly,
Z
lim
n→∞
e
ψfn (s)
Z
dp =
S
eψf (s) dp
(39)
S
Choose some value a ∈ Z, and define a payoff-function f 0 by f 0 = f on S \ E and
f = a on E. By Absolute Continuity πf (E) = πf 0 (E) = 0, and by Consequentialism
πf (A|S \ E) = πf 0 (A|S \ E) for all A ⊆ S \ E. If E is any event then πf (E) = πf (E ∩ E) +
πf (E ∩ (S \ E)) = 0 + πf (E \ E ∩ E) = πf 0 (E \ E ∩ E) = πf 0 (E). Hence πf = πf 0 , and so
if E is non-empty we can prove the claim instead on f 0 .
49
44
Moreover, by Lemma 2 both these integrals are strictly positive and bounded.
Hence, using the result for simple payoff-functions (Corollary 1),
R ψf (s)
R ψf (s)
n
e
dp
e
dp
πfn (A)
lim πfn (A) = lim
= lim RA ψfn (s)
= RA ψf (s)
(40)
n→∞
n→∞ πfn (S)
n→∞
e
dp
e
dp
S
S
Finally, fn → f uniformly, and hence by Prize-Continuity, limn→∞ πfn (A) =
πf (A). Combining
with Equation 40 we obtain that for all
R ψf (s) this
R observation
ψf (s)
πf (A) = A e
dp/ S e
dp as required.
A.3
Other proofs
Proof of Proposition 1. By assumption ψH ≥ RψL , and so e(ψH −ψL )x is an
x
increasing function. By Equation 6, ΠψXL (x) = −∞ eψL f (x) p(X = x) dx, and
Rx
Rx
similarly Πψf H (x) = −∞ eψH f (x) p(X = x) dx = −∞ e(ψH −ψL )f (x) eψL f (x) p(X =
x) dx. Therefore, Πψf H LR Πψf L with h(x) = e(ψH −ψL )f (x) .
Proof of Proposition 2. Let x1 > x2 > · · · > xn denote the payoffs in the
range of f . Thus, Amax = f −1 (x1 ), and by Equation 15,
lim πf (Amax )
ψ→∞
−1
−1
X p(f −1 (xi ))
p(f −1 (x1 ))eψf x1 )
=
1
+
· eψ(xi −x1 )
= P
−1 (x ))eψf xi
1 (x ))
p(f
p(f
i
1
i
i>1
X p(f −1 (xi ))
=1+
·0=1
1 (x ))
p(f
1
i>1
(41)
I omit the proof for Amin and ψ → −∞ as it is very similar.
Proof of Proposition 3. By Equation 6 and the assumption that Pf ∼ N (µ, σ 2 ),
Z x
Z x
(x−µ)2
1
−
ψf (x)
ψ(ax+b)
√
Πx (x) ∝
e
p(X = x) dx =
e
e 2σ2
dx
2πσ
−∞
−∞
Z x
(x−µ)2 −2ψaσ 2 x
1
ψb
2σ 2
√
e−
=e
dx
2πσ
−∞
Z x
Z x
(x−(µ+ψaσ 2 ))2 −ψ 2 a2 σ 4
(x−(µ+ψaσ 2 ))2
1
1
−
ψb
2σ 2
2σ 2
√
√
=e
e
dx ∝
e−
dx
2πσ
2πσ
−∞
−∞
(42)
completing the proof that ΠX ∼ N (µ + aσ 2 ψ, σ 2 ).
45
Proof of Proposition 4. I need to showR that there exist a non-decreasing funcx
tion h0 , such that for all x, F 0 (x) ∝ −∞ h0 dG0 . By assumption there exist
Rx
a non-decreasing function h such that F (x) ∝ −∞ h dG, and so dF ∝ hdG.
Rx
Rx
Moreover, by Equation 6, F 0 (x) ∝ −∞ eψf (x) dF and G0 (x) ∝ −∞ eψf (x) dG.
From the latter we obtain that dG0 ∝ eψf (x) dG, and so
Z x
Z x
Z x
Z x
0
ψf (x)
ψf (x)
ψf (x)
−ψf (x)
0
F (x) ∝
e
dF ∝
e
hdG ∝
e
he
dG =
hdG0
−∞
−∞
−∞
−∞
(43)
0
0
0
and so F LR G with h = h.
R
Proof of Proposition
5.
For
any
x
∈
R,
π
(X
=
x|e)
=
πf (X = x, Y =
f
R
R
y|e) dy ∝ p(X = x, Y = y|e)eψf (x,y) dy = p(X
=
x)
p(Y
=
y|e)eψf (x,y) dy,
R
0
and similarly πf (X = x|Y = y 0 ) ∝ p(X = x) p(Y = y 0 )eψf (x,y ) dy 0 . Hence,
for any two values xH , xL ∈ R,
R
p(X = xH ) p(Y = y|e)eψf (xH ,y) dy
πf (X = xH |e)
R
=
(44)
πf (X = xL |e)
p(X = xL ) p(Y = y|e)eψf (xL ,y) dy
and similarly,
R
0
πf (X = xH )
p(X = xH ) p(Y = y 0 )eψf (xH ,y ) dy 0
R
=
πf (X = xL )
p(X = xL ) p(Y = y 0 )eψf (xL ,y0 ) dy 0
(45)
ΠX|e LR ΠX (ΠX LR ΠX|e ) if and only if whenever xH ≥ xL the expression in Equation 44 is greater than or equal (smaller than or equal) to the
expression in Equation 45. Moreover, the stochastic dominance is strict if
the inequalities are strict. The sign of this expression equals the sign of
R
R
0
p(Y = y 0 )eψf (xH ,y ) dy 0
p(Y = y|e)eψf (xH ,y) dy
R
−R
(46)
p(Y = y|e)eψf (xL ,y) dy
p(Y = y 0 )eψf (xL ,y0 ) dy 0
which is the same sign as
Z Z
0
ψ(f (xH ,y)+f (xL ,y 0 ))
ψ(f (xH ,y 0 )+f (xL ,y))
p(Y = y|e)p(Y = y ) e
−e
dy 0 dy
y
y0
(47)
combining terms in which y < y with terms in which y > y we obtain that
the expression in Equation 47 equals the following:
Z Z
(p(Y = y|e)p(Y = y 0 ) − p(Y = y 0 |e)p(Y = y)) ·
0
y y <y
(48)
ψ(f (xH ,y)+f (xL ,y 0 ))
ψ(f (xH ,y 0 )+f (xL ,y))
0
e
−e
dy dy
0
0
46
In this expression the first term is (strictly) positive if p(e|Y = y) is (strictly)
increasing in y, and y is not a.e. constant. Since y > y 0 and log is a strictly
increasing function, the second term has (strictly) the same sign as ψ if f
is (strictly) supermodular, and (strictly) the opposite sign if f is (strictly)
submodular. The claim follows by combining these observations.
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