Wishful Thinking∗

Guy Mayraz†

University of Oxford

September 15, 2011

Abstract

An experiment tested whether and in what circumstances people are more

likely to believe an event simply because it makes them better off. Subjects ob-

served a financial asset’s historical price chart, and received both an accuracy

bonus for predicting the price at some future point, and an unconditional award

that was either increasing or decreasing in this price. Despite incentives for hedg-

ing, subjects gaining from high prices made significantly higher predictions than

those gaining from low prices. The magnitude of the bias was smaller in charts

with less subjective uncertainty, but was independent of the amount paid for

accurate predictions.

JEL classification: D01,D03,D83,D84,G12,G14.

Keywords: Wishful-thinking, asset prices, optimal expectations, priors and de-

sires, payoff-dependent beliefs.

∗I am indebted to many colleagues for helpful comments and discussions, especially Michelle Belot,

Gary Charness, Vince Crawford, Erik Eyster, Paul Heidhues, Luis Miller, Matthew Rabin, Georg

Weizsacker, and Peyton Young. I gratefully acknowledge financial support from the Russell Sage

Foundation and the John Fell OUP Research Fund.†Department of Economics and Nuffield College, University of Oxford, 1 New Road, Oxford OX1

1NF, U.K. Email: guy.mayraz@nuffield.ox.ac.uk.

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1 Introduction

Wishful thinking is the idea that what people want to be true affects what they be-

lieve to be true. When a person’s utility is higher if some even obtains, she is (other

things being equal) more likely to believe that it does obtain. A number of well-

known biases can be viewed as instances of wishful thinking, including over-confidence,

over-optimism, self-serving beliefs, and cognitive-dissonance.1 For example, the well-

known finding that most people believe themselves to be better drivers than most other

people (Svenson, 1981) can be seen as wishful-thinking over the person’s own driving

ability.2

The potential implications for decision making are substantial. In single person de-

cision problems agents would underestimate risks and overestimate uncertain rewards.

In strategic environments the beliefs of agents with different interests would differ sys-

tematically in accordance with their interests. In dynamic environments a change in

the payoff from an event would thereby alter the subjective likelihood that the event

obtains.3 Applications abound in many areas of economic research.4

However, all the above are if-then statements: if wishful thinking affects beliefs in

a given economic environment then it would have certain interesting implications for

the behavior of decision makers in that environment. This raises an obvious question:

under what circumstances wishful-thinking really is a significant phenomenon?

The existing evidence provides a partial answer to this question. There are stud-

ies showing evidence for wishful-thinking in several important economic environments,

including bargaining (Babcock and Loewenstein, 1997),5 market entry (Camerer and

Lovallo, 2000), mergers and acquisitions (Malmendier and Tate, 2008), political econ-

omy (Mullainathan and Washington, 2009), and investment management (Olsen, 1997).

however, it is by no means clear how far we can extrapolate to other environments of in-

terest. Moreover, with the exception of Malmendier and Tate (2008), the cost of getting

beliefs wrong in these studies is small, and it is not obvious whether comparable results

would hold in otherwise similar decision making environments, but with substantially

1See Appendix A for a detailed discussion.2A bias about one’s relative driving skill could result either from wishful-thinking over the event

that one is a better driver than other people (if utility is relative), or simply because of wishful thinkingover one’s own driving ability, but not over the driving skill of others.

3For example, an investment in a financial asset would cause the investor to become more sanguineabout the risks. This can lead the investor to gradually escalate her investment well beyond heroriginal plan.

4For example, finance (Kyle and Wang, 1997; Odean, 1998; Daniel et al., 1998; Hirshleifer and Luo,2001; Hirshleifer, 2001; Heaton, 2002; Brunnermeier and Parker, 2005), savings (Brunnermeier andParker, 2005), search (Dubra, 2004), and insurance (Sandroni and Squintani, 2007).

5See also Loewenstein et al. (1993) and Babcock et al. (1995).

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higher stakes. This is a severe limitation, as much of the interest in wishful-thinking

for economists is precisely in the possibility that it affects high-stakes decisions.

This paper takes a indirect approach to addressing these questions. Instead of

testing for the presence of wishful-thinking in different environments of interest, I test

different theories of wishful-thinking. Different theories have different implications as

to the set of circumstances in which wishful-thinking is significant. If a test is sensitive

enough, so that only one particular hypothesis survives, that hypothesis can then be

used to predict the extent of wishful thinking in environments for which we lack direct

evidence.

For the purpose of this paper we can define theories of wishful-thinking by the set

of circumstances in which they predict a bias would obtain. There are three principal

theories to consider.6 The most general is the universal theory, according to which

any and all subjective judgments are affected by wishful thinking. That is, whenever a

decision maker is better-off if some event is true, she is biased to believe that the event

is true.7 This theory has two distinct variants. In the strategic variant (Akerlof and

Dickens, 1982; Brunnermeier and Parker, 2005), biased beliefs result from trading-off

the enjoyment of positively biased beliefs against the potential cost in poor decisions.

The implication is that the magnitude of the bias depends on the incentives for accu-

racy, so that the bias can only be substantial when incentives for accuracy are weak.

Thus, in high-stakes decision making environments wishful-thinking is present, but is

unlikely to be important. By contrast, in the non-strategic variant (Mayraz, 2011),

wishful thinking is modeled simply as a fact about the subjective judgment of proba-

bilities, rather than as a strategic choice.8 The implication is that the magnitude of the

bias is independent of its costs in biasing subsequent decisions, and that, consequently,

high-stakes decisions may well be affected by wishful thinking.9

While the two variants of the universal theory of wishful thinking differ with respect

to the magnitude of the bias when the cost of getting beliefs wrong is high, both

agree that wishful thinking affects any and all subjective judgments. By contrast,

the two other major theories predict the presence of wishful thinking bias only in

a more restricted set of circumstances. According to the ego-utility theory (Koszegi,

6It is also possible to define different theories by the actual mechanism leading to wishful-thinkingbias. Much of the psychology literature is focused on this question.

7The closest analogue in the psychology literature is the desirability bias (Marks, 1951; Irwin andSnodgrass, 1966). See also Kunda (1990).

8A useful analogy is biases in sensory perception, such as visual illusions.9Note, though, that the magnitude of the bias does depend on the quality of the information

available to the decision maker. To the extent that increased incentives for accuracy result in decisionmakers obtaining substantially better information, the bias may be reduced even if it is non-strategic.See further discussion in the Conclusion.

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2006), wishful thinking only affects subjective judgments over personally relevant events

having to do with the decision maker’s self-image. Finally, the cognitive bias theory

challenges the very idea that there is a causal link between what a person wants to be

true and what she believes to be true, suggesting instead that observed instances of

wishful-thinking are really a manifestation of purely cognitive biases, such as a failure

to correct for information asymmetries, or differences in task.10

The aim of the present paper is to conduct a test of these different theories with the

help of a simple experiment. Subjects in the experiment observed a chart of historical

wheat prices,11 and their one and only task was to predict what the price would be at

some future time point. Subjects were randomly assigned into two treatment groups:

Farmers, whose payoff was increasing in the future price of wheat, and Bakers, whose

payoff was decreasing in this price. Subjects in both groups also received a performance

bonus as a function of the accuracy of their prediction.12

The universal theory of wishful thinking predicts bias whenever decision makers

have a stake in what the state of the world is. Farmers gain from high prices, and

their beliefs should therefore be biased upward. The opposite is true for Bakers. Given

the random allocation, it follows that there should be a systematic difference in beliefs

between the two groups, with Farmers expecting higher prices than Bakers.13

The ego-utility theory predicts such a bias if and only if the decision maker’s self-

image is involved. As this is not the case in the present experiment, the ego-utility

theory predicts no bias. As to the cognitive bias theory, any number of cognitive biases

could potentially affect beliefs in the experiment,14 but since Farmers and Bakers are

given the same information and the same prediction task, there is no plausible reason for

cognitive biases to affect Farmers and Bakers differently. The presence or absence of a

systematic difference in beliefs between Farmers and Bakers therefore makes it possible

to empirically separate the universal theory of wishful thinking (in either variant) from

the ego-utility and cognitive bias theories.

The statistic used to identify a systematic difference in beliefs between the two

groups is the difference between the average predictions of Farmers and Bakers. The

10For example, Weinstein and Lachendro (1982) suggest that people believe negative events aremore likely to happen to others simply because they fail to think much about the effort other peopleput into avoiding such events. See Appendix A for further discussion.

11Charts were adapted from real asset price data, though not specifically wheat prices.12See Section 2.2 for details of the performance bonus formula, Figure 1 for an example of the

interface and Figure 2 for the complete set of price charts.13Note that this prediction depends only on the differential effect of wishful-thinking in the two

treatment groups. There is no need to assume that in the absence of wishful thinking beliefs wouldhave been correct.

14Indeed, it is not obvious what a bias-free prediction would look like.

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prediction bonus formula was designed so that truthful reporting maximizes subjective

expected payoff. As long as decision makers are risk neutral over small amounts of

money, the difference in predictions should provide an unbiased estimate of the differ-

ence in beliefs. Risk-averse subjects may, however, seek to intentionally hedge their

predictions, so as to smooth their payoff across different states. Such hedging would

result in Farmers under-reporting their true prediction, with an opposite bias for Bak-

ers. As a result, the estimated difference in beliefs between the two treatment groups

may be biased downward.

The null hypothesis was defined as a non-positive difference in beliefs between Farm-

ers and Bakers. Hedging could plausibly have resulted in a failure to reject the null

when the true difference in beliefs is positive. There were no corresponding reasons

to expect a false positive result. The actual observation was a positive and statis-

tically measurable difference in predictions between Farmers and Bakers, consistent

with wishful thinking bias. The null hypothesis was rejected (p < 0.0002). This result

is explained by the universal theory of wishful thinking, but by neither the ego-utility,

nor the cognitive bias theories.

Assuming this result is correct, it follows that wishful thinking is a broader phe-

nomenon than would be implied by the ego utility theory, the cognitive bias theory,

or both these theories taken together. Given the range of theories under considera-

tion, the conclusion is that we should provisionally accept that the universal theory

of wishful-thinking is true, and that any and all subjective judgments are subject to

wishful thinking bias.

However, even if any and all subjective judgments are subject to wishful thinking

bias, it does not necessarily follow that the bias remains significant if the cost of getting

beliefs wrong is high. If wishful thinking is non-strategic then the answer is yes. But

if wishful thinking is strategic the answer is no, since in environments with high-stakes

decisions it would not be optimal to choose a bias large enough to significantly affect

choices.

Testing whether wishful-thinking is strategic requires the ability to manipulate the

incentives for holding accurate beliefs. The design of the experiment afforded a simple

way to do so, by varying the scale of the accuracy bonus: the larger the potential

bonus, the more subjects had to lose from holding biased beliefs. If wishful thinking

is strategic, the magnitude of the bias should decrease in the scale of the accuracy

bonus. If, however, wishful thinking is not strategic, there should be no change in the

magnitude of the bias as the scale of the accuracy bonus is increased.

Converting this intuition into a formal test requires quantitative predictions. ‘No

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change’ is a testable hypothesis, but ‘decreasing with the scale of accuracy bonus’ is not.

Consequently, testing the hypothesis that wishful thinking is strategic made it necessary

to focus on some particular strategic model. The best known such model is the Optimal

Expectations model of Brunnermeier and Parker (2005). Agents in this model have

preferences over anticipated consumption, and choose beliefs in order to maximize their

subjective expected utility. The constraint is that, once chosen, beliefs govern future

choices and change only as the result of Bayesian updating. Agents therefore trade-off

the gain from anticipating a high payoff, against the cost in a lower realized bonus:

the more favorable they believe the future price to be, the higher is their anticipatory

utility, but the lower the prediction bonus they can expect to receive. Increasing the

scale of the accuracy bonus increases the cost of biased beliefs and reduces the optimal

level of bias. Assuming risk-neutrality over small stakes, the quantitative prediction

is that the magnitude of the bias would be inversely proportional to the scale of the

accuracy bonus (Section 3.2).

Different sessions were run with different levels of accuracy bonus. The scale of the

bonus was increased five fold, with the maximum bonus amount varying from £1 to £5.

Results showed no decrease in the magnitude of the bias, consistent with the prediction

of non-strategic models. This result is statistically measurable: the prediction of the

Optimal Expectations model was formally rejected (p < 0.0140), while that of non-

strategic models was not (p < 0.4026). The experiment, therefore, corroborates the

universal theory of wishful-thinking in its non-strategic version. This is the version

with the most far-ranging implications, implying that wishful-thinking affects any and

all decisions based on subjective judgment, whatever the cost to the decision maker.

The Brunnermeier and Parker (2005) strategic model and the Mayraz (2011) non-

strategic model make comparative statics predictions not only for how the magnitude

of the bias depends on the cost of holding biased beliefs, but also for its dependence

on the amount of subjective uncertainty and on what subjects have at stake in the

quantity that they form expectations over.15 The prediction of both models is that

the magnitude of the bias increases in both these factors. Testing these predictions

cannot provide a further test of which model is correct, but it can provide some fur-

ther assurance that the experiment is sensitive enough for the main conclusions to be

trusted.

In order to make a test of the comparative statics of subjective uncertainty possi-

ble, subjects were asked to provide a confidence level together with their prediction.

Confidence was provided on a 1-10 scale, calibrated with the help of examples provided

15In the experiment the stakes correspond to the sensitivity of the final payoff to the day 100 price.

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as part of the instructions (Figure 3). By averaging the confidence reports across sub-

jects, it was possible to obtain an estimate of the amount of subjective uncertainty

in different charts. This made it possible to test the prediction that the bias in high

subjective uncertainty charts is greater. Results were consistent with this prediction

(Figure 5), and the null hypothesis that the magnitude of the bias is at least as high in

low subjective uncertainty charts was rejected (p < 0.0142). A robustness test using a

different measure of uncertainty yielded comparable results.

Due to insufficient data, a test of the comparative statics of the stakes was incon-

clusive. Two sessions were run with half the stakes, and the estimated bias was roughly

half what it was in the baseline sessions. However, the null hypothesis that that the

bias is the same could not be rejected.16

One possible concern with interpreting the results of the experiment is that subjects

felt the task of predicting the day 100 price is impossible, and that they may as well

choose whichever number they want to be true. Since Farmers gain from high prices

and Bakers gain from low prices, Farmers would choose high guesses, and Bakers

would choose low ones. If this explanation is correct, we would expect subjects who

are generally confident in their predictions to be less biased than less confident subjects.

Similarly, we would expect subjects who generally believe prices in financial markets are

predictable to be less biased than subjects who do not think prices can be predicted.

I tested the first prediction by defining a subject’s confidence level by the average

confidence rating in her predictions across all charts. I tested the second prediction by

asking subjects in the post experiment questionnaire whether they believe that prices in

financial markets are generally predictable. In both cases I obtained just the opposite

result: subjects who believe prices are predictable and relatively confident subjects are

more biased than those who are less confident. These results suggest that this concern

is misplaced. Moreover, they support the view that over-confidence is a manifestation

of wishful thinking, and that the degree of wishful thinking bias is a stable individual

characteristic.17

The reminder of the paper is organized as follows. Section 2 describes the experi-

ment in detail. Section 3 develops the predictions of the Optimal Expectations (Brun-

nermeier and Parker, 2005) and Priors and Desires (Mayraz, 2011) models. Section 4

16The comparative statics of the stakes have been studied before in a different but related context.In a study of self-deception Mijovic-Prelec and Prelec (2010) found a larger bias when stakes werehigher (the ‘Anticipation Bonus’ treatment) as compared with lower bonus (the ‘Classification Bonus’treatment).

17For example, in Mayraz (2011) each person is characterized by a coefficient of relative-optimism,which is a real-number characterizing her degree of optimism or pessimism. The same coefficient isassumed to determine her bias in all domains of subjective judgment.

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describes how the data were analyzed. Section 5 presents the results, and Section 6

concludes. Appendix A presents a more complete psychology and economics evidence

relating to wishful thinking.

2 Experimental design

This section describes the experimental design. The implementation and protocol are

in Section 2.1, and the specifics of the belief elicitation procedure in Section 2.2.

2.1 Implementation and protocol

The experiment was conducted at the Centre for Experimental Social Science (CESS)

at Nuffield College, Oxford. The subject pool consisted of Oxford students who reg-

istered on the CESS website for participation in experiments. Business, finance, and

economics students were excluded. A week before each session students meeting the

sample restrictions received an email inviting them to participate in an experiment that

would require one hour of their time. Further details were given on-site prior to the

experiment itself. Registration was via an online form, allowing students to select one

of several sessions, up to an upper limit of 14 per session. Students were not allowed

to register to more than one session. Taking no-shows into account, sessions consisted

of between 10 and 13 students. Altogether, 145 students took part in the experiment,

of whom 57 were male and 88 female. The median age was 22.

Sessions were conducted in the afternoon over six days in total. There were 12

sessions altogether. Half the sessions consisted of Farmers, and half of Bakers. The

order of sessions was randomized in order to prevent any consistent relationship between

the time of day in which a session was held, and the role given to the subjects who

took part in that session.18

Each session consisted of 13 periods, the first of which was a training period, and

the remaining 12 were earning periods. A given set of 13 charts was used throughout

the experiment. One of these 13 charts was reserved for the training period, and the

other 12 charts were used for the earning periods (Figure 2). The order of presentation

was randomized independently between subjects. At the end of the experiment, each

subject had one earning period chosen at random, and was paid in accordance with

the payoff in that period.

18This was done in order to minimize the risk of systematic differences between subjects in the twotreatment groups resulting from self-selection into earlier or later sessions.

8

In each period subjects were shown a chart of wheat prices, and were asked to

predict the price of wheat at some future date. Subjects were thus put in a somewhat

similar position to speculators who predict future asset prices on the basis of historical

price charts.19 In order to maximize the realism of the task, prices were adapted from

real financial markets, though not specifically those of wheat.20 Charts were selected to

include a variety of situations. Time was standardized across charts, so that all charts

had space for prices going from day 0 to day 100. Subjects were only shown prices up

to an earlier date, and the task was to predict what the price of wheat would be at

day 100. The price range was also standardized, so that prices were always between

£4,000 and £16,000. Figure 1 shows an example of the interface.

After submitting their prediction, subjects were presented with a waiting screen

until all other subjects had also made their prediction. There was therefore little

or no incentive for speed. The transition to the next period only occurred after all

the subjects in the room had submitted their prediction. A brief questionnaire was

administered following the final period of the experiment. After all subjects completed

the questionnaire, subjects were informed of their earnings, and were called to receive

their payment.

Farmers were instructed that the price of wheat varies between £4,000 and £16,000,

that it had cost them £4,000 to grow the wheat, and that they would be selling their

wheat for the price that would obtain at day 100. Their notional profit was therefore

between zero and £12,000, depending on the day 100 price. The payoff at the end of

the experiment consisted of three parts: an unconditional £4 participation fee, profit

from the sale of the wheat, and a prediction accuracy bonus. In the baseline sessions

subjects received £1 in real money for each £1,000 of notional profit, and could earn

up to an extra £1 from making a good prediction. The prediction procedure and bonus

formula are explained in detail in Section 2.2. Bakers were told that they make bread,

which they would sell for a known price of £16,000, and that in order to make the bread

they would be buying wheat at the price that would obtain at day 100. The notional

profit of Bakers was therefore the same as that of Farmers. All other particulars were

also the same.

19The practice of using historical price charts in making buy and sell decisions is known as TechnicalAnalysis (Murphy, 1999; Edwards and Magee, 2010).

20This is also what subjects were told in the instructions. The actual source was historical stockprices, scaled and shifted to fit into a uniform range. Prices were downloaded from Yahoo Finance,and included data from the following publicly traded stocks: AK Steel Holding (AKS), AlleghenyEnergy (AYE), BB&T (BBT), Cabot Oil & Gas (COG), Cisco Systems (CSCO), Home Depot (HD),Coca-Cola (KO), Lehman Brothers (LEH), Starbucks (SBUX), and Whole Foods (WFMI). In a coupleof cases price data from the same stock (but in non-overlapping periods) was used in more than onechart.

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Table 1: The number of sessions for each combination of bonus scale and stakes.

bonusa stakesb sessionsc subjects

1 1 4 49 (25 Farmers, 24 Bakers)

2 1 2 26 (13 Farmers, 13 Bakers)

5 1 4 44 (23 Farmers, 21 Bakers)

1 0.5 2 26 (12 Farmers, 14 Bakers)a The amount in pounds subjects received for an optimal prediction of the day 100 price. The

larger it was, the more subjects had to gain from holding accurate beliefs. The bonus for lessgood predictions was scaled accordingly.

b The amount in pounds subjects received for each £1,000 of notional profit. The larger thestakes, the more subjects had to gain from the the day 100 price being high (if they wereFarmers), or low (if they were Bakers).

c Sessions were conducted in pairs: one for Farmers and the other for Bakers.

Sessions differed in the scale of the accuracy bonus and in the stakes—or the de-

gree to which payoff depended on the price level at day 100. In the baseline sessions

the maximum obtainable bonus was £1, and the amount received for each £1,000 of

notional profit21 was also £1. Sessions were also conducted with a bonus level of £2

and £5, and with stakes of 50 pence for each £1,000 of notional profit.22 Table 1 lists

the number of sessions in each condition.

2.2 The belief elicitation procedure

The belief elicitation procedure was designed with two goals in mind. The first was to

make it possible to test for the presence or absence of wishful-thinking bias, namely a

systematic difference in beliefs between Farmers and Bakers. The second was to obtain

a measure of the degree of subjective uncertainty in the predictions subjects make. This

was important both for testing whether the magnitude of the bias is greater in charts

with more subjective uncertainty, and for testing whether more confident individuals

are also more biased.

In each period subjects were asked to report two numbers: a prediction and a

confidence-level. The prediction represented the expected day 100 price, and could be

any number in the range of possible prices. The confidence level represented the reverse

of subjective uncertainty, and was reported on a 1-10 scale.

In order to give meaning to the 1-10 confidence scale, the instructions included visual

21The notional profit being the difference between the day 100 price and the £4,000 of growingwheat (for Farmers), or between the £16,000 bread price and the day 100 price (for Bakers).

22In sessions with lower stakes, subjects received an additional £3, so that the average payoff wasthe same as in the baseline sessions.

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examples of distributions with different prediction and confidence levels (Figure 3). The

distributions were the weighted average of a normal distribution and a uniform one,

with almost all the weight given to the normal. The prediction corresponded to the

mean of the normal distribution, and the confidence level was inversely proportional

to its standard deviation. The density corresponding to a prediction of 4, 000 ≤ m ≤16, 000 and confidence level 1 ≤ r ≤ 10 was

q(x) = (1− ε)N (x|m, (λr)−2) + ε (1)

where λ is a scale parameter, used to translate the 1-10 confidence scale into the scale

of prices, and ε is the weight given to the uniform component. The effect of the latter

was to ensure that the density was bounded below by ε, including at prices far from

the prediction.

The scoring rule was logarithmic:23 subjects whose prediction and confidence level

corresponded to a density q received a bonus given by

b(x) = α log(q(x)/ε) (2)

where x is the true day 100 price, and α is a parameter which determines the maximum

bonus level. As q ≥ ε (Equation 1), the bonus was positive for all possible predictions.

The value of α was calibrated for the maximum bonus level in the session (Table 1).

To see under what conditions the scoring rule is incentive compatible, let P denote

the probability measure representing the subject’s true beliefs, and suppose the subject

reports a prediction m and a confidence level r. The subjective expectation of the bonus

is given by the following expression:

EP [b(x)] =

∫p(x)α log

q(x)

εdx = α

(∫p(x) log

q(x)

p(x)dx

+

∫p(x) log p(x) dx− log ε

)= α

(−DKL(P ||Q)−H(P )− log ε

) (3)

where DKL(P ||Q) is the Kullback-Leibler divergence (KL-divergence or relative entropy)

between P and Q, and H(P ) is the entropy of P .24 Maximizing the expected bonus

with respect to Q is thus equivalent to minimizing the KL-divergence DKL(P ||Q).

According to a standard result, DKL(P ||Q) ≥ 0 for all Q, and is minimized if Q = P .25

23The logarithmic scoring rule was introduced in Good (1952). See Gneiting and Raftery (2007) fora recent discussion and comparison to other scoring rules.

24The Kullback-Leibler divergence was introduced in Kullback and Leibler (1951).25This result, known as Gibb’s Inequality, follows directly from the fact that log x is a concave

11

The scoring rule works best if subjects are risk neutral and beliefs are well approx-

imated by a density in the family described by Equation 1. The scoring rule should

then successfully elicit the prediction and confidence level for each subject in each chart.

This, in turn, makes it possible to identify the difference in beliefs between Farmers

and Bakers, as well the average subjective uncertainty in each chart, and the average

confidence for each subject.26

One potential difficulty is hedging.27 Consider a risk-averse Farmer. Her profit is

increasing in the price, and she would therefore prefer to receive the bonus in states

in which the price is relatively low. Consequently, she could increase her subjective

expected utility by reporting a lower number than her true beliefs. By a similar logic,

a risk-averse Baker would be better-off by reporting a higher number. Hedging would,

therefore, result in a downward bias in estimating the difference in beliefs between

Farmers and Bakers.

A second potential problem is the possibility that the beliefs of some subjects are

bi-modal, or otherwise not well approximated by a density in the family described by

Equation 1. There would still be some optimal prediction and confidence level,28 but

it may not be obvious what the optimal prediction is. Consequently, two subjects with

the same beliefs may make different predictions. There is, however, no plausible reason

for Farmers and Bakers to behave differently in this regard. The result, therefore,

would be an increase in prediction noise, rather than a systematic bias.

3 Theory

This section formally develops the relevant predictions of different models of wishful-

thinking, including the standard model, Optimal Expectations (Brunnermeier and

Parker, 2005), and Priors and Desires (Mayraz, 2011). By the standard model I mean

the following two assumptions (i) choices maximize subjective expected utility, and

function. For the proof see any standard information theory text, such as Cover and Thomas (1991),or Kullback (1997).

26Subjects must also believe that truthful reporting will maximize their expected payoff. Subjectsneed not, however, understand the mathematical argument that this is indeed the case. The in-structions explained that the expected bonus is maximized by reporting a prediction and confidencelevel that reflect the subject’s beliefs about the day 100 price. The bonus formula was included in afootnote.

27Blanco et al. (2008) investigate the effects of hedging on belief reporting, and find that thathedging does not significantly affect reporting unless opportunities for hedging are transparent. Inthis experiment, opportunities for hedging are transparent, but are also quite limited. Armantier andTreich (2010) discuss hedging in the context of probability elicitation.

28Moreover, if P and P ′ are related by some shift, so that P ′(x) = P (x − δ), and a prediction mand confidence level r are optimal for P , then m+ δ and r are optimal for P ′.

12

(ii) subjective beliefs are independent of what a person has to gain or lose from an

event being true. Optimal Expectations and Priors and Desires represent the univer-

sal theory of wishful-thinking in its strategic and non-strategic variants respectively.

The ego-utility and cognitive bias theories receive no separate treatment, as in in the

context of this experiment they agree with the assumptions of the standard model.

In developing the predictions of these three models I use the following timing frame-

work: at t = 0 subjects observe a price chart and form their beliefs over the day 100

price; at t = 1 they report their prediction and confidence level, and consume antici-

patory utility; at t = 2 the day 100 price is revealed, and payoffs are realized. In order

to simplify the analysis, I assume that subjects are risk neutral and that their beliefs

about the day 100 price can be represented by a distribution from the family described

by Equation 1. Given these assumptions, the prediction made at t = 1 coincides with

the t = 1 beliefs.

3.1 The standard model

Different individuals in the standard model may end up with different beliefs for un-

modelled (random) reasons. By assumption, however, a person’s subjective beliefs

about the day 100 price are not affected by whether she is assigned the role of Farmer

or Baker. Since the role allocation is random, it follows that the t = 0 subjective

beliefs of Farmers and Bakers are drawn from the same distribution. Since the predic-

tion coincides with the t = 1 beliefs, and since no new information is observed between

t = 0 and t = 1, it follows that predictions are also drawn from the same distribution.

Consequently, there is no systematic difference in predictions between Farmers and

Bakers.

3.2 Optimal Expectations

Optimal expectations agents choose their beliefs in order to maximize their discounted

subjective expected utility, subject to the constraint that—once chosen—beliefs govern

future actions, and change only as the result of Bayesian updating. Instantaneous

utility includes anticipatory utility, as well as standard consumption utility.

The payoff in the experiment is realized at t = 2, and consists of two components:

the profit and the accuracy bonus. The profit is a function of the true price, while the

bonus depends on the accuracy of the t = 1 beliefs. Anticipatory utility is proportional

to the expected value of the profit and bonus, with expectations computed using the

t = 1 beliefs. The more optimistic those beliefs are, the higher is anticipatory utility,

13

but more biased beliefs also result in less accurate predictions. The t = 0 decision

maker choosing her t = 1 beliefs faces a trade-off: increased wishful-thinking bias

increases the anticipatory utility experienced at t = 1, but lowers the expected value

of the t = 2 consumption utility.

Let P and Q denote the probability distributions representing the t = 0 and t = 1

beliefs respectively. At t = 0 the agent maximizes a weighted sum of the t = 1

anticipatory utility and t = 2 realized payoff.29 Let η denote the weight given to

anticipatory utility, so that the weight given to the realized payoff is 1− η. Letting x

denote the true day 100 price, the profit can be written as φκx+ l, where x is true day

100 price, κ represents the stakes (the absolute value of the slope relating the profit to

the day 100 price), and φ denotes the sign of this relationship, with φ = 1 for Farmers

and φ = −1 for Bakers. I denote the bonus by b(x), where b is defined by Equation 2.

The t = 0 maximand can thus be written as follows:

W = ηEQ[φκx+ b(x)] + (1− η)EP [φκx+ b(x)] + l (4)

In order to derive the comparative statics of the bias in closed form I make a couple

of simplifying assumptions. First, I assume that P and Q are normal: P = N (µ0, σ20),

and Q = N (µ1, σ21).30 Second, I assume that only the mean of Q is subject to bias,

i.e. σ1 = σ0 = σ. Given these assumptions and using Equation 3, Equation 4 can be

rewritten as follows:

W = ηEQ[φκx+ b(x)] + (1− η)EP [φκx+ b(x)] + l

= η(φκµ1 − αH(Q)− αDKL(Q||Q)− α log ε

)+ (1− η)

(φκµ0 − αDKL(P ||Q)− αH(P )− α log ε

)+ l

= η(φκµ1 − αH(Q))− (1− η)αDKL(P ||Q) + C

(5)

where C collects factors that are independent of Q. The two terms that depend on Q

represent, respectively, the gain in anticipatory utility from adopting optimistic beliefs,

and the cost in expected realized payoff of adopting such beliefs. The gain term has

two components. The first represents the anticipated profit, and is proportional to

µ1 = EQ[x]. The second represents the anticipated bonus, and is decreasing in the

degree of uncertainty in Q, measured by its entropy H(Q). The gain term is thus

larger the more favorable is the expected day 100 price, and the more certain the

29In principle, the weights are a function of the decision maker’s time preferences, and of the amountof time she would have to enjoy the two types of utility.

30In other words, I let ε = 0 in Equation 1.

14

subject is about her prediction. The cost term represents the reduction in expected

bonus due to the bias in the prediction that follows from the bias in the t = 1 beliefs,

and is proportional to the Kullback-Leibler divergence between the t = 0 beliefs P

and the t = 1 beliefs Q. Thus, if the subject cared only about the realized payoff she

would choose not to bias her beliefs at all (Q = P ). If, instead, she cared only about

her t = 1 instantaneous utility, she would choose to believe that the most favorable

price would be realized,31 and would further choose to assign this prediction as little

subjective uncertainty as possible.

If η is sufficiently small, the optimal choice of µ1 would be an extreme value in

the favorable direction. Otherwise, the optimal value of µ1 would be at an internal

point, where ∂W/∂µ1 = 0. Using the standard formula for the KL-divergence between

two normal distributions (Johnson and Sinanovic, 2001), and noting that H(Q) is

independent of µ1, this derivative can be written as follows:

∂W

∂µ1

= ηφκ+ η∂H(Q)

∂µ1

− (1− η)α∂DKL(P ||Q)

∂µ1

= ηφκ− (1− η)α(µ1 − µ0)

σ2

(6)

Setting the derivative to zero and solving for µ1 we obtain the following expression for

the bias

µ1 − µ0 = φ

(η

1− η

)(κσ2

α

)(7)

where κ represents the stakes, or the degree to which the profit is dependent on the

value of the day 100 price, σ2 corresponds to the degree of subjective uncertainty, and

α corresponds to the scale of the accuracy bonus, or the cost of holding biased beliefs.

Equation 7 describes the bias in the beliefs of one particular individual. The pre-

diction for the average bias in the population of subjects in the same role is

E[µ1 − µ0] = E[µ1]− E[µ0] = φE[

η

1− η

](κσ2

α

)(8)

where I allow for the possibility that η varies between individuals, but assume that it is

independent of σ2.32 Finally, since the allocation into sessions is random, it follows that

the undistorted beliefs of Farmers and Bakers are drawn from the same distribution.

In particular, Eµ0 is the same in both groups. Hence, the expected difference in beliefs

31That is, the highest possible price of £16,000 if a Farmer, and the lowest possible price of £4,000if a Baker.

32It is independent of κ and α by virtue of the random assignment in the experiment.

15

between Farmers and Bakers is given by

boptimal expectations = 2E[

η

1− η

](κσ2

α

)∝ κσ2

α(9)

Thus, the Optimal Expectation model implies a systematic difference in beliefs between

Farmers and Bakers, and further implies that this difference would be proportional to

the stakes and to the degree of subjective uncertainty, and inversely proportional to

the cost of getting beliefs wrong.

3.3 Priors and Desires

Unlike the case with Optimal Expectations, in Priors and Desires (Mayraz, 2011) belief

distortion is not strategic. Instead, the model simply allows for the possibility that what

a person believes to be true may depend on what she wants to be true. The latter

is formalized by a payoff-function, which is a mapping from states to utility values,

representing the dependence of the decision maker’s utility on the state of the world.

In Mayraz (2011) a number of simplifying assumptions are made, and a representation

is derived. The bias in the subjective beliefs of a person with a payoff-function f is

represented by the following equation:

q(s) ∝ p(s)eψf(s) (10)

where s denotes the state, q represents the decision maker’s actual (distorted) beliefs,

and p represents her undistorted beliefs, or the beliefs she would hold if she were indif-

ferent between all states.33 Finally, ψ is a real-valued parameter, called the coefficient

of relative-optimism, which describes how optimistic or pessimistic that particular in-

dividual is. A positive value corresponds to optimism, a negative value to pessimism,

and a zero value to realism.

In the experiment, the payoff-function is the mapping from the day 100 price to

the subject’s payoff.34 Using the same notation as in Section 3.2, the payoff-function is

given by f(x) = φκx + l, where x is the day 100 price, κ represents the stakes, or the

slope relating the profit to the day 100 price, and φ denotes the sign of this relationship,

with φ = 1 for Farmers and φ = −1 for Bakers. Suppose, as in Section 3.2, that undis-

torted beliefs are given by a normally distributed probability measure P = N (µ0, σ2).

The prediction for the actual (distorted) beliefs can be obtained using Equation 10.

33That is, if f(s) = f(s′) for all s and s′.34In principle, it should be the payoff in utility terms, but I am assuming throughout this section

that subjects are risk neutral over small amounts of money.

16

Mayraz (2011, proposition 3) analyzes the case of a normal distribution with a linear

payoff-function, and shows that the distorted beliefs are normally distributed with the

same variance, and that the mean is shifted in proportion to the coefficient of relative

optimism ψ, the stakes, and the variance. In other words, the distorted probability

measure is given by Q = N (µ1, σ2), where

µ1 − µ0 = φψκσ2 (11)

This equation describes the bias in the beliefs of some particular individual, and is the

Priors and Desires analogue of Equation 7. By analogy with Section 3.2, the expected

difference in beliefs between Farmers and Bakers is

bpriors and desires = 2E[ψ]κσ2 ∝ κσ2 (12)

Comparing this result to Equation 9 we see that, as with Optimal Expectations, the

magnitude of the bias is proportional to the stakes κ and the degree of subjective

uncertainty σ2. However, whereas in Optimal Expectations the magnitude of the bias

is inversely proportional to the cost of getting beliefs wrong α, the magnitude of the

bias in Equation 12 is invariant to changes in this cost.35

4 Analysis

This section describes how the data was analyzed. The resulting estimates are presented

in Section 5.

4.1 Minimizing hedging bias

As noted in Section 2.2, hedging could lead to a downward bias in estimating the

difference in beliefs between Farmers and Bakers. In order to minimize this risk, a

questionnaire was administered after the experiment itself was concluded, in which

subjects were asked whether they always reported their best guesses, or whether they

sometimes reported a higher or lower number. Out of a total of 145 students who

talk part in the experiment, 132 claimed to have always reported their best guess, and

35This, of course, is a reflection of the fact that Optimal Expectations models the bias as a strategicchoice, while Priors and Desires is a non-strategic model. Another important difference is that thePriors and Desires model allows, in principle, for pessimistic bias, as well as for wishful-thinking.In the present context it is assumed that optimism is the dominant bias in the population, so thatE[ψ] > 0.

17

13 admitted to an intentional bias in their predictions. Observations from these 13

subjects were excluded from the main analysis.

4.2 Main regression

The raw data from the experiment consist of the predictions and confidence levels

reported by individual subjects in individual charts. The primary goal in analyzing the

data was to determine whether predictions were affected by wishful thinking. Let yi,j

denote the prediction made by subject i in chart j, and let ti ∈ {1,−1} denote whether

subject i is a Farmer or a Baker. We want to know whether yij is systematically higher

if ti = 1. In order to answer this question formally I used the following regression model:

yij = 0.5βti +∑j

γjdj + εij (13)

where dj is a dummy for chart j, and εij is the error term. The value of β represents

the contribution of wishful thinking. The null hypothesis is that β ≤ 0.

4.3 Comparative statics

The second goal in analyzing the data was to investigate the comparative statics of the

bias. This required estimating the bias separately in different subsamples of interest.

Let K denote a partition of the sample, indexed by k, and let cijk denote a dummy

denoting whether the prediction of subject i in chart j belongs to subsample k. Assum-

ing wishful thinking is the only systematic source of difference in predictions between

subjects, we can generalize Equation 13 as follows:

yij = 0.5∑k∈K

βkcijkti +∑j

γjdj + εij (14)

In this equation βk represents the average difference in predictions between Farmers

and Bakers in class k, and can be used to define formal comparative statics hypotheses.

4.4 Standard errors

Unobserved factors may well result in a correlation in the predictions made by the same

subject in different charts. In other words, εij may be correlated with εik for j 6= k. To

correct for this possibility, standard errors are clustered by subject in all regressions.

18

5 Results

This section presents the results of the experiment, starting with the overall difference in

predictions between Farmers and Bakers, and continuing with the comparative statics

of the bias. Parameter estimates and statistical test results are presented in summary

form in Table 3. Figure 5 provides a graphical illustration.

5.1 Wishful thinking bias

The overall magnitude of the wishful thinking bias corresponds to the systematic differ-

ence in predictions between Farmers and Bakers across the entire sample, represented

by the value of β in Equation 13. The estimate for this number is £452, measured with

a robust standard error of £123. The null-hypothesis that it is non-positive is rejected

with a p-value of 0.0002.

This estimate is based on observations from the 132 subjects who claimed to have

always reported their true beliefs (Section 4). As noted in Section 2.2, hedging could

have led to a downward bias in the estimate. In order to minimize this potential prob-

lem, the 13 subjects who reported intentional bias in their predictions were excluded

from the sample (Section 4). If these subjects are nonetheless included, the estimate

goes down to £390. The reduction in the estimate is consistent with the prediction

that risk-averse Farmers (Bakers) would intentionally understate (overstate) their true

estimates of the day 100 price.

The observed difference in predictions between Farmers and Bakers is consistent

with the hypothesis that any and all subjective judgments are affected by wishful

thinking. The data cannot be explained by the ego utility or cognitive bias theories of

wishful thinking, neither of which predicts bias in these circumstances.

5.2 Incentives for accuracy

Strategic models of wishful thinking predict that the magnitude of the bias would be

decreasing in the incentives for accuracy, while non-strategic models predict that it

would remain the same. In order to determine whether higher incentives for accuracy

result in lower bias, Equation 14 was used to estimate the difference in beliefs between

Farmers and Bakers separately in sessions with different levels of accuracy bonus. The

maximum level of the accuracy bonus was £1 in six sessions, £2 in two other, and

£5 in the remaining four, and the bonus for any given level of accuracy was scaled

19

accordingly.36

The results in Table 3 show that the estimated bias is actually larger in sessions

with a higher bonus level, the point estimates being 298, 560, and 645, respectively.

On the face of it, this result appears consistent with neither type of model. Formal

testing, however, reveals that the apparent increase in the magnitude of the bias may

well be random (p < 0.4026). The data is, therefore, consistent with the prediction of

non-strategic models that the magnitude of the bias would be invariant to changes in

the incentives for holding accurate beliefs.

The same is not true, however, for strategic models. The prediction of the Optimal

Expectations model is that the magnitude of the bias would be inversely proportional

to the scale of the accuracy bonus (Section 3.2). That is, the bias in £2 bonus sessions

should be half the size of the bias in £1 bonus sessions, and the bias in £5 bonus sessions

should be one fifth the size. This prediction is rejected by the data (p < 0.0140).37

The first panel of Figure 5 shows these results graphically. Though the point esti-

mates are increasing in the maximum level of the accuracy bonus, a horizontal parallel

line can be comfortably fitted within the confidence intervals. The same is not true,

however, for a hyperbolic curve.

5.3 Subjective uncertainty

According to both Optimal Expectations (Section 3.2) and Priors and Desires (Sec-

tion 3.3), the magnitude of the bias should be increasing in the degree of subjective

uncertainty. In order to test this prediction, I divided the 12 charts used in the pay-

ing periods into two equal sized groups by the degree of subjective uncertainty in the

chart, and used Equation 14 to estimate the bias separately in the two subsamples.38

I used two different measures of subjective uncertainty. The first was based on the

confidence ratings that subjects provided: charts were classified into the high (low)

subjective uncertainty group if the mean (across all subjects) of the confidence rat-

ing for the chart was below (above) median. The second measure of uncertainty was

the within group variance of predictions: charts were classified into the high (low)

subjective uncertainty group if the within group variance of predictions for that chart

was above (below) median. In practice, the two measures resulted in nearly identical

classifications.

Depending on the measure used, the estimated bias was 635 or 677 in the group

36See Equation 2, and the discussion in Section 2.2.37Ignoring the £2 sessions, and testing only the prediction that the magnitude of the bias in £5

sessions would be one fifth that of £1 sessions results in a p value of 0.0069.38Each subsample consists of observations from all subjects, but in only half the charts.

20

of high subjective uncertainty charts, and 269 or 227 in the low subjective uncertainty

group. The null hypothesis—that the magnitude of the bias in high subjective uncer-

tainty charts would be less than or equal to the magnitude of the bias in low subjective

uncertainty charts—was rejected with a p-value of 0.0142 when using the first classifi-

cation method, and a p-value of 0.0034 when using the second (Table 3).

These results support the qualitative prediction that the magnitude of the bias is

increasing in the degree of subjective uncertainty. Given that the qualitative prediction

of the two models fits the data, it is interesting to try and test the specific functional

form predicted by the two models. The quantitative prediction is that the magnitude

of the bias is linear in the variance of subjective uncertainty. The following equation

should thus prove to be a better model of the data than Equation 13:

yij = 0.5β′σ2j ti +

∑j

γjdj + εij (15)

In this equation the 0.5βti term in Equation 13 is replaced by 0.5β′σ2j ti, where σ2

j is

the variance of subjective uncertainty in chart j.

Testing this quantitative prediction requires a good proxy for the variance of subjec-

tive uncertainty. Using the above measures of subjective uncertainty, we can identify

σ2j either with the square of the inverse mean confidence rating in chart j, or with

the mean within group prediction variance for chart j.39 Table 2 shows the resulting

regression fit when estimating the two equations using both proxies for the variance of

subjective uncertainty, as well the results of fitting a model which includes both the

0.5βti term of Equation 13 and the 0.5β′σ2j ti of Equation 15. The results show that

Equation 15 indeed provides a better fit to the data, consistent with the prediction

that the magnitude of the bias is linear in the degree of subjective uncertainty.

The same results can also be seen graphically in the second and third panels of

Figure 5. Panel 2 plots the estimated wishful thinking bias in the 12 charts against

the mean prediction confidence in the chart, and panel 3 plots the same data against

the within group prediction variance. In both panels a curve is fitted to the data using

Equation 15.

5.4 Stakes

Optimal Expectations and Priors and Desires also predict that the magnitude of the

bias is increasing in the stake subjects have in what the day 100 price would be. In the

experiment the day 100 price affects the payoff via the notional profit, which is linear

39This assumes a representative agent approximation.

21

Table 2: Testing whether the magnitude of the bias increases with the variance ofsubjective uncertainty. Column 1 fits a model in which the bias is independent ofsubjective uncertainty (Equation 13). Column 2 and 4 fit a model in which the mag-nitude of the bias is linear in the variance (Equation 15). Columns 3 and 5 fit a modelwhich allows for both regressors. Method 1 and method 2 refer to the two proxies forsubjective uncertainty (Section 5.3). The tiσ

2j variable is normalized to have the same

standard deviation as ti, so that the regression coefficients are comparable in size. Ro-bust standard errors are in parentheses. The regression R2 is computed after nettingout the contribution of the chart dummies, and represents the part of the variance thatis explained by wishful thinking. Statistical significance indicators: *** p < 0.01, **p < 0.05, * p < 0.1.

method 1 method 2

ti 452∗∗∗ −473∗ −458∗

(122) (259) (272)

tiσ2j 497∗∗∗ 955∗∗∗ 503∗∗∗ 945∗∗∗

(129) (309) (130) (319)

R2 0.0181 0.0218 0.0230 0.0224 0.0237

in the day 100 price with a slope of 1. In 10 sessions subjects received £1 for each

£1,000 of notional profit, and in the remaining two session they received only 50p for

each £1,000 of notional profit. There are thus ten sessions with standard stakes and

two sessions with low stakes.

I estimated the magnitude of the bias separately in these two subsamples (Equa-

tion 14). The magnitude of the bias was 260 in the low stakes subsample, and 495 in

the standard stakes subsample. These results are consistent with the prediction that

the magnitude of the bias is linear in the stakes (p < 0.9668). This modest variance

in the stakes between sessions was, unfortunately, insufficient to produce statistically

measurable results, and the hypothesis that the bias is not any smaller in the low stakes

subsample could not be rejected (p < 0.2313). See also Table 3 and panel 4 of Figure 5.

5.5 Over-confidence

Section 5.1 demonstrates the existence of a systematic difference in predictions between

Farmers and Bakers. This difference in predictions is interpreted as evidence of wishful

thinking bias affecting subjects’ judgment about the day 100 price. A key assumption in

this interpretation is that subjects believe they have better than random odds of making

a good prediction, so it is in their interest to report their true beliefs. If this assumption

22

is not true, subjects could very well choose whichever prediction they enjoy making,

without having to worry about losing the prediction bonus. As long as subjects prefer

reporting a price that would benefit them, we could observe a systematic difference in

predictions between Farmers and Bakers that has nothing to do with wishful thinking.

In this section I test for whether this alternative explanation of the data may be

true. If so, we would expect subjects who are generally confident in their predictions

to be less biased than less confident subjects, as subjects who believe their prediction

is good have more to lose from reporting a biased number. Similarly, we would expect

subjects who generally believe prices in financial markets are predictable to be less

biased than subjects who do not think prices can be predicted.

In order to test the first prediction I defined a proxy for a subject’s confidence by

the average prediction confidence for that subject across all charts. I then split the

sample into more and less confident subjects, and estimated the bias separately in the

two subsamples. In order to test the second prediction I included a question in the

post experiment questionnaire about the predictability of prices in financial markets,

and divided subjects into two groups by whether they thought prices can generally be

predicted. The bias was then estimated separately in the two subsamples.40

The result was just the opposite: subjects who believe prices are predictable and

relatively confident subjects are more biased than those who are less confident. Specif-

ically, the estimated bias among relatively confident subject is 628, compared with 276

among less confident subjects. The hypothesis that more confident subjects are less

biased is rejected with p-value of 0.0732. Similarly, the estimated bias among subjects

who believe prices in financial markets to be generally predictable was 613, as compared

with 292 among subjects who believed prices cannot be predicted. The hypothesis that

subjects who believe prices to be predictable are less biased was rejected with a p-value

of 0.0997.

By and large, therefore, subjects believe they have at least some ability to predict

the day 100 price, and the stronger this belief is, the more biased they are. This

result is consistent with the wishful thinking interpretation, and further suggests that

over-confidence is a manifestation of wishful thinking, and that the degree of wishful

thinking bias is a stable individual characteristic.41

40The question was “We are interested in what people believe about financial markets. How pre-dictable are the movements of prices in financial markets in your opinion?”. The possible choices were:“Prices can be predicted to a significant extent”, “Prices can rarely be predicted”, and “The idea thatprices can be predicted is an illusion”. The first choice was defined as yes, and the other two as no.The distribution of answers was 66, 58, and 8, respectively.

41This explains why individuals with more than average wishful thinking bias also tend to be over-confident. The tendency to be more or less biased can be identified with the coefficient of relativeoptimism in Mayraz (2011). See also Appendix A for further discussion of over-confidence.

23

5.6 Demographic characteristics

None of the models under consideration make predictions about the degree of bias in

different demographic groups. I nevertheless report here the results of estimating the

bias separately in groups defined by gender and age. I find essentially no difference by

gender, and very little difference by age. The estimated bias is 411 for males and 477

for females. The hypothesis that the bias does not vary by gender is consistent with

the data (p < 0.7956). The estimated bias for subjects at or below the median age

of 22 is 517, and the estimate for subjects older than 22 is 375. The hypothesis of no

difference cannot be rejected (p < 0.5747). It is worth noting that the sample variation

with respect to gender was pretty good, with 38% males and 62% females, but there

was only a small variation in age.

6 Conclusion

This paper describes an experimental test of wishful-thinking bias in predictions of asset

prices. Subjects received an accuracy bonus for their predictions of the future price

of an asset, and at the same time received an unconditional payment that was either

increasing or decreasing in the price of the asset. Both groups of subjects had the same

information, and faced the same incentives for accuracy. Nevertheless, and despite

incentives for hedging, subjects in the group benefiting from high prices predicted

systematically higher prices than subjects in the group benefiting from low prices.

These results are consistent with the universal interpretation of wishful-thinking, and

cannot be accounted for by either the ego-utility or cognitive bias approaches.42

By varying the scale of the accuracy bonus it was possible to test whether the

magnitude of the bias decreases with the incentives to hold accurate beliefs, but no such

decrease was found. Moreover, the quantitative prediction of the Optimal Expectations

model (Brunnermeier and Parker, 2005)—that the magnitude of the bias is inversely

proportional to the incentives for accuracy—was formally rejected. These results are

hard to square with a strategic model of wishful-thinking, and support instead the

non-strategic modelling approach (Mayraz, 2011), according to which the magnitude

of the bias is independent of its effect on subsequent decisions.

Other comparative statics results include good evidence that wishful thinking bias is

stronger when subjective uncertainty is high, evidence that over-confidence and wishful-

42Though the results imply that the ego utility and cognitive bias accounts of wishful thinking areincomplete, the experiment in no way implies that they are irrelevant. In situations in which theego utility or cognitive bias theories predict the existence of bias, the size of the bias could well besignificantly more than what would be predicted by the universal account of wishful thinking.

24

thinking bias go together, and some evidence of greater bias when payoff is more

strongly dependent on the state of the world.

Taken together, these results suggest that any and all subjective beliefs are affected

by wishful-thinking bias, and that the bias may well be sufficiently strong to materially

affect economically important decisions. High stakes decisions in financial markets

are a case in point, as they involve probability assessments in situations characterized

by high stakes and high subjective uncertainty—both of which are conducive to the

presence of an economically significant bias.

In interpreting this conclusion, it is important to bear in mind that decision makers

in high stakes situations have an incentive to purchase quality information in order

to reduce the uncertainty in their beliefs.43 Since the strength of the bias depends

on the degree of subjective uncertainty, quality information will not only reduce the

variance in beliefs, but would also (perhaps unintentionally) reduce the magnitude of

the bias.44 The degree to which wishful thinking is likely to affect high stakes decisions

is therefore strongly dependent on the availability of quality information when the

relevant decisions are taken.

One way to determine whether quality information is available is to examine the

beliefs of informed experts. In many important decision making environments (financial

markets, corporate decision making, politics, war) informed experts commonly disagree.

The failure of experts to come to anything approaching consensus suggests the existence

of a substantial level of subjective uncertainty that cannot be reduced by purchasing

information. When that is the case, there is evidently significant potential for wishful

thinking to materially affect decisions.

The present paper describes one particular experiment on one particular group of

subjects. While the main conclusions are strongly statistically significant, it would

clearly be important to see whether the results can be replicated by other researchers

and in other decision making environments. Another important limitation for interpret-

ing the results of the experiment is the limited range of theories under consideration.

While I am not aware of any other non ad-hoc theory that can explain the results of

the experiment, it is important to emphasize that if such a theory were to be offered,

it could well significantly change the interpretation of the experiment’s results.

43In the experiment all the information was available at zero cost, so the amount of informationsubjects had did not vary with their incentives.

44‘Sophisticated’ decision makers may seek information, in part, with the intention of reducing thebias in their beliefs. ‘Naive’ decision makers may falsely believe that they do not suffer from wishfulthinking bias at all. Nevertheless, as long as they try to reduce the uncertainty in their beliefs, andfind means to do so, they will also end up reducing the magnitude of their wishful thinking bias.

25

A Existing Evidence

This appendix describes some of the most important psychology and economics evi-

dence for over-confidence, over-optimism, self-serving beliefs, and cognitive dissonance,

and explains how the different theories of wishful thinking discussed in the introduction

can account for this evidence. The theories considered include (i) the universal inter-

pretation of wishful thinking, according to which any and all subjective judgments of

likelihood are affected by wishful thinking bias: whenever a decision maker is better-off

if some event is true, she is biased to believe that the event is true, (ii) the ego-utility in-

terpretation, according to which wishful-thinking applies only to subjective judgments

having to do with self-image or ‘ego utility’, and (iii) the cognitive-bias interpretation,

according to which failure to correct for information asymmetries, or differences in task

is the underlying bias.

A.1 Over-confidence

Over-confidence describes a tendency by people to overestimate their skill level in ac-

tivities that are important to them. Best known, perhaps, is the finding that most

people believe themselves to be better drivers than most other people (Svenson, 1981),

but there are many other examples. In economics the term is sometimes used specifi-

cally to refer to the finding that people are over-confident about the accuracy of their

predictions (Alpert and Raiffa, 1982). Camerer and Lovallo (2000) find that, consistent

with over-confidence, subjects in a lab experiment enter more into a competitive game

if the result is decided by relative skill than by luck. They argue that this provides an

explanation for the high rate of failure of new businesses. Malmendier and Tate (2008)

is an empirical study of over-confidence in merger and acquisition decisions. It identifies

over-confident CEOs as ones who choose concentrated holdings in their own company’s

stock,45 and shows that such CEOs make more and worse quality acquisitions than

other CEOs.

According to the universal interpretation of wishful thinking, whenever a decision

maker is better-off if some event is true, she is biased to believe that the event is true.

Over-confidence is explained simply by the fact that (i) people want to have high skill

level, and that (ii) judgments of skill level involve subjective judgment. The ego-utility

explanation of wishful thinking applies only for personally relevant judgments that

reflect on the decision maker’s ego. A person’s skill level is part of her self-image,

making it possible to explain over-confidence using the ego-utility theory of wishful

45Another test is the CEO’s press portrayal.

26

thinking. Finally, since each individual is better informed about the efforts she makes

to be skillful than she is about the efforts made by others, over-confidence can also

result from a cognitive bias in correcting for this information asymmetry (Weinstein

and Lachendro, 1982).46

A.2 Over-optimism

Over-optimism describes a tendency by people to be biased about the likelihood of

personally relevant events, overestimating the likelihood of desirable events, and un-

derestimating that of undesirable events. For example, Weinstein (1980) found evi-

dence of over-optimism over life-events such as finding a good job after graduation,

or (in the opposite direction) having a divorce at some point in life. Explanations for

over-optimism parallel those of over-confidence. Some of the literature uses the terms

‘over-confidence’ and ‘over-optimism’ interchangeably.

A.3 Self-serving belief bias

Self-serving belief bias refer to a tendency to interpret ambiguous evidence as sup-

porting a desirable conclusion.47 For example, in Babcock and Loewenstein (1997)48

subjects allocated to the role of plaintiff or defendant examined the evidence in a com-

pensation trial. They then had an opportunity to reach a compromise, or else their

payoff was determined by a judge’s decision less ‘court costs’. Prior to the bargaining

phase subjects made an incentive compatible prediction of the judge’s decision, and the

result was that plaintiffs predicted significantly higher award amounts than defendants.

Self-serving belief bias can be readily explained by the universal interpretation of

wishful thinking. Some example can also be explained by the ego-utility interpretation,

but in other examples, including Babcock and Loewenstein (1997), self-image plays

no meaningful role. However, since subjects read the evidence in preparation for a

bargaining task, they plausibly focused selectively on arguments favoring their case.

The observed bias could therefore have resulted from a cognitive bias in correcting for

this focus asymmetry.49

46The cognitive bias explanation is, however, less successful in accounting for evidence that peopleare over-confident against an objective standard, as in the case of over-confidence about predictionaccuracy (Alpert and Raiffa, 1982), or in the Malmendier and Tate (2008) study of overconfidenceover the success of mergers and acquisitions.

47The term is sometimes used specifically in reference to a tendency by people to attribute successesto themselves, and failures to various situational factors outside their control.

48See also Loewenstein et al. (1993) and Babcock et al. (1995).49For example, if subjects formed their prediction of the judge’s ruling on the basis of the arguments

they remembered, the selective reading of the evidence could plausibly account for the observed bias

27

A.4 Cognitive dissonance

Cognitive dissonance studies document a change in beliefs following a change in what

people what to be true, typically brought about by an action on their part. For

example, Knox and Inkster (1968) found that the act of placing a bet on a horse leads to

increased confidence that the horse would win the race. In economics, Mullainathan and

Washington (2009) use voting age restrictions to show that, consistent with cognitive

dissonance, the act of voting leads to polarization in beliefs.

Cognitive dissonance can be readily explained by the universal interpretation of

wishful-thinking, and also by the ego-utility explanation (since whether or not a person

is taking good actions is part and parcel of that person’s self-image). The cognitive

bias interpretation is rarely a good explanation, since the relevant information typically

remains the same during the cognitive-dissonance change in beliefs.

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31

Table 3: Wishful-thinking bias and comparative statics. The table reports the esti-mated bias in different sub-samples and statistical tests of related hypotheses.

Sample Estimated biasa Observationsb

All subjects 452∗∗∗ (s.e. 123) 1584 (132)

negative ? p < 0.0002

Cost of bias

Accuracy bonus: low (£1) 298∗∗ (s.e. 164) 816 (68)

Accuracy bonus: medium (£2) 569∗∗ (s.e. 328) 300 (25)

Accuracy bonus: high (£5) 645∗∗∗ (s.e. 210) 468 (39)

low = medium = high ? p < 0.4026

low = 2 ·medium = 5 · high ? p < 0.0140c

Degree of

subjective

uncertainty

Chart uncertainty: low 269∗∗ (s.e. 127) 792 (66)

Chart uncertainty: high 635∗∗∗ (s.e. 166) 792 (66)

low > high ? p < 0.0142

Within chart variance: low 227∗∗ (s.e. 113) 792 (66)

Within chart variance: high 677∗∗∗ (s.e. 175) 792 (66)

low > high ? p < 0.0034

Stakes in the

value of the

day 100 price

Stakes: low (50p) 260 (s.e. 289) 288 (24)

Stakes: standard (£1) 495∗∗∗ (s.e. 135) 1296 (108)

standard ≤ 2 · low ? p < 0.2313d

standard = 2 · low ? p < 0.9668

Confidence in

ability to

predict prices

Average confidence: low 276∗ (s.e. 174) 792 (66)

Average confidence: high 628∗∗∗ (s.e. 169) 792 (66)

low > high ? p < 0.0732

Prices predictable? no 292∗∗ (s.e. 174) 792 (66)

Prices predictable? yes 613∗∗∗ (s.e. 174) 792 (66)

no > yes ? p < 0.0997

Demographics

Males 411∗∗ (s.e. 187) 600 (50)

Females 477∗∗∗ (s.e. 166) 984 (82)

same ? p < 0.7956

a Robust standard errors in parentheses. Statistical significance indicators: *** p < 0.01, **p < 0.05, * p < 0.1.

b An individual observation refers to the prediction of a given subject in a given chart. Clusteringis by subjects. The number of clusters is in parentheses.

c If the regression is restricted to the sessions with standard stakes the test p-values are 0.5094and 0.0171 respectively.

d If the regression is restricted to the sessions with a low maximum bonus the test p-values are0.7620 and 0.4269 respectively.

32

Figure 1: The interface of the Farmers treatment with a maximum accuracy bonusof £5. The interface of the Bakers treatment was the same with the following twoexceptions: (a) the first three lines were: “You have a buyer for £16,000 worth ofbread from your bakery. At day 100 you will get the money from the order, and willhave to use some of it to buy wheat at the market. Your profit is whatever you wouldhave left after paying for the wheat you need.”, and (b) instead of an arrow on thechart pointing to £4,000 with the label “Wheat production costs”, there was an arrowpointing to £16,000 with the label “The price you would get for your bread”. Subjectsin both treatment groups get paid £1 for every £1,000 of notional profit, an accuracybonus of up to £5, and an unconditional £4 participation fee.

33

Figure 2: The charts used in the 12 earning periods of the experiment to representwheat prices. The x-axis represents time, ranging from day 0 to day 100, and they-axis represents price, ranging from £4,000 to £16,000. Subjects were asked for theirprediction for the price at day 100 using these charts as their source of information.The data for the charts were adapted from historical equity price charts, shifted andscaled to fit into a uniform range. Figure 1 illustrates how these charts were presentedto subjects in the experiment.

34

0.0

5.1

.15

.2D

ensi

ty

4000 6000 8000 10000 12000 14000 16000Price (in £)

£10,000 confidence level 1

0.1

.2.3

.4D

ensi

ty

4000 6000 8000 10000 12000 14000 16000Price (in £)

£6,400 confidence level 3

0.2

.4.6

.8D

ensi

ty

4000 6000 8000 10000 12000 14000 16000Price (in £)

£12,600 confidence level 5.5

0.5

11.

5D

ensi

ty

4000 6000 8000 10000 12000 14000 16000Price (in £)

£10,000 confidence level 10

Figure 3: The examples of distributions used in the instructions. Each distributionis characterized by a prediction and a confidence level. These examples were usedin explaining the prediction elicitation procedure. They were particularly useful inestablishing a reference for the 1-10 scale that was used in reporting the subject’sconfidence in her prediction.

35

7000 10102 13000

Farmers

7000 9650 13000

Bakers

Figure 4: Histogram of the mean predictions made by Farmers and Bakers. A normaldistribution curve was fitted to both histograms. The mean prediction was 10102 and9650 respectively. 16 of the 20 subjects making the highest (lowest) mean predictionswere Farmers (Bakers).

36

050

010

0015

00T

reat

men

t effe

ct

0 1 2 3 4 5 6Maximum accuracy bonus (£)

Accuracy Bonus

010

0020

00T

reat

men

t effe

ct

4 5 6 7 8Mean prediction confidence in chart

Prediction Confidence

010

0020

00T

reat

men

t effe

ct

0 2 4 6Prediction variance in chart (within treatment)

Prediction Variance−

400

040

080

0T

reat

men

t effe

ct

0 .5 1Payoff for each £1,000 of notional profit

Stakes

Figure 5: The comparative statics of wishful thinking bias. The panels show a 95%confidence interval for difference in predictions between Farmers and Bakers (the treat-ment effect) in different subsamples. The first panel shows the comparative statics ofthe cost of holding wrong beliefs, represented by the maximum accuracy bonus. Thesolid hyperbolic line represents the best fit for the Optimal Expectations model, andthe dashed horizontal line that of Priors and Desires. The second panel plots the biasin a chart against the mean confidence in predictions for that chart. The curve is fittedto the inverse of the square of the mean confidence level. The third panel plots thebias in a chart against the mean within group predictions variance. The dashed line isa linear fit through the origin. Finally, the fourth panel shows the comparative staticsof the stakes, the x-axis representing the amount in pounds that a subject receives foreach £1,000 of notional profit. The dashed line is a linear fit through the origin.

37

Priors and DesiresA Model of Payoff-Dependent Beliefs∗

Guy Mayraz†

March 2011‡

Abstract

Whatever a person wants to be true affects what she believes to be true,

and consequently the decisions she makes. This paper introduces an axiomatic

model of decision making that allows for this possibility, and uses a number of

simplifying assumptions to derive a generally applicable formal representation.

In the resulting representation the payoff in an event affects beliefs as if it were

part of the evidence about its likelihood. A single parameter determines both

the direction and weight of this ‘evidence’, with positive values corresponding

to optimism, and negative values to pessimism. Changes to the payoff conse-

quences of an event amount to new ‘evidence’, and can alter beliefs even in the

absence of new information. The magnitude of the bias is greatest in situations

that combine high stakes and great uncertainty, and is only indirectly related

to the cost in poor decisions. If uncertainty cannot be readily reduced a sub-

stantial bias may remain regardless of its consequences. The model can account

for a wide range of psychology evidence, including wishful thinking, cognitive

dissonance, and unrealistic pessimism. Economic consequences are explored in

various settings, such as the economics of crime, where increased punishment

may have little or no deterrent value.

JEL classification: D01,D03,D80,D81,D83,D84.

Keywords: payoff-dependent beliefs, reference-dependent preferences, optimism,

wishful-thinking, cognitive-dissonance.

∗I am grateful to Erik Eyster for his guidance and support, to Wolfgang Pesendorfer and Matthew

Rabin for stimulating discussions early in the writing, and to Vincent Crawford, Peter Klibanoff, and

Sujoy Mukerji for helpful suggestions at the end stages of writing. I am grateful also to seminar and

conference participants at Bar-Ilan, Ben-Gurion, Berkeley, CalTech, Collegio Carlo Alberto, Essex,

Hebrew University, LSE, Maastricht, MIT, Oxford, Royal Holloway, Tel-Aviv, UBC, UCL, UCSB,

Warwick, Gerzensee, RUD, and SAET.†Department of Economics and Nuffield College, Oxford, and Centre for Economic Performance,

London School of Economics.‡First version: November 2008.

1

1 Introduction

Understanding decisions frequently boils down to understanding beliefs, and in par-

ticular subjective beliefs.1 Relevant information is the most important ingredient that

goes into determining what a person believes to be true, but decades of psychology

research suggest another: what the person wants to be true.2 In this paper I intro-

duce a model of decision making that allows for this possibility, and use a number of

simplifying assumptions to derive a generally applicable formal representation.

At any point in time a decision maker is associated with a state-dependent endow-

ment, describing what outcome she obtains given each possible realization of subjec-

tive uncertainty. The utility function further associates each of these outcomes with

a utility-value. The desire for states to be true can be identified with the resulting

mapping from states to utility-values or payoff-function. The premise of this paper is

that the decision maker’s beliefs may depend on her payoff-function.3

The dependence of beliefs on the payoff-function creates a corresponding depen-

dence for preferences. Staying as close as possible to the standard model I assume

that preferences maximize subjective expected utility, and that while subjective be-

liefs may depend on the payoff-function, the utility-function does not. Additional

assumptions characterize the mapping from payoff-functions to beliefs. The resulting

model of preferences is only a small step from the standard model, and includes only

one additional parameter. A revealed-preferences axiomatization is provided using

the Anscombe-Aumann model of uncertainty.4

The model provides a unified account of a diverse body of psychology and eco-

nomics evidence with such varied names as over-optimism, wishful-thinking, cognitive-

dissonance, and unrealistic pessimism.5 At the same time, it provides a theoretical

1This observation was first made by Knight (1921): “Business decisions, for example, deal withsituations which are far too unique, generally speaking, for any sort of statistical tabulation tohave any value for guidance. The conception of an objectively measurable probability or chance issimply inapplicable.” (III.VII.47); “Yet it is true, and the fact can hardly be overemphasized, thata judgment of probability is actually made in such cases.” (III.VII.40).

2See Appendix A for an overview of the relevant psychology and economics evidence.3Denoting the endowment by e and the utility-function by u the payoff-function is f = u◦e. One

example of an endowment is an investment portfolio, associating stock prices with profit and lossfigures. Another is the team a sports fan supports, associating match results with victory or defeatfor the supported team. The origin of the endowment can be a previous choice, some exogenousallocation, or even an accident of birth (as may be the case for the sports fan).

4An endowment e is associated with a preference-relation �e. Axioms ensure (i) that each �e hasa subjective expected-utility representation, (ii) that there exists a utility-function u such that each�e can be represented by u and a subjective probability measure πe, (iii) that πe may depend on eonly via the associated payoff-function f = u ◦ e. The remaining axioms characterize the mappingπ.

5For example, in a study of motivated-cognition Sherman and Kunda (1989) found that a woman’s

2

foundation for modeling the implications of payoff-dependent belief bias in applica-

tions. In particular, the model predicts under what circumstances should we expect

significant bias, and how changes to the environment affect its magnitude.

There is now experimental evidence for payoff-dependent belief bias over such eco-

nomically important variables as the outcome of litigation (Babcock and Loewenstein,

1997) and the future price of financial assets (Mayraz, 2011). Observed bias has been

linked empirically to bargaining impasse (Babcock and Loewenstein, 1997), excess

entry into competitive markets (Camerer and Lovallo, 2000), and failed acquisition

deals by CEOs (Malmendier and Tate, 2008). Theoretical applications abound in

many areas of economic research.6

The reminder of the introduction describes the assumptions used in developing

the model, the resulting representation, and some of the most important implications.

This is followed by a comparison with other models of reference-dependent preferences,

as well as with other models of similar phenomena—Brunnermeier and Parker (2005)

in particular.

Subjective uncertainty is modeled by a set of states. Each state is associated with

two real numbers: one representing the decision maker’s subjective probability for

the state, and the other representing her payoff in utility terms if the state obtains.

Letting f denote the payoff-function, I let πf denote the decision maker’s subjective

beliefs. The central object in the model is the mapping π. To derive a representation

for π I make some simplifying assumptions, specifying special circumstances in which

different payoff-functions result in the same beliefs. The key assumptions are shift-

invariance, and consequentialism.7 Shift-invariance is the assumption that subjective

beliefs are affected only by utility differences—how the decision maker values different

states relative to each other.8 Consequentialism requires that the probability ratio

between events depends only on the consequences of those events.9

coffee drinking habits are a good predictor of whether she would believe a study claiming a linkbetween breast cancer and caffeine consumption; in a study of cognitive-dissonance Knox and Inkster(1968) compared the beliefs of bettors before or after placing a bet on a horse, and found that placingthe bet increases the subjective probability that the horse would win the race. In both studies thedifference in beliefs can be explained by the difference in the payoff-function.

6For example, finance (Kyle and Wang, 1997; Odean, 1998; Daniel et al., 1998; Hirshleifer andLuo, 2001; Hirshleifer, 2001; Heaton, 2002; Brunnermeier and Parker, 2005), savings (Brunnermeierand Parker, 2005), search (Dubra, 2004), and insurance (Sandroni and Squintani, 2007).

7I also assume continuity : small changes in payoffs lead only to small changes in beliefs, andabsolute continuity : the set of zero probability events is the same for all payoff-functions.

8f ′ = f + a ⇒ πf ′ = πf . Note that π is defined relative to a particular choice of utility-function. Shift-Invariance is therefore a substantial assumption, rather than a consequence of thenon-uniqueness of the utility-function in expected-utility preferences.

9f ′ = f over an event E ⇒ πf ′(·|E) = πf (·|E). Needless to say, the absolute probability of anevent can be affected by the payoff in outside states. Note that ‘consequentialism’ is used by different

3

The key result of the paper is that these assumptions are necessary and sufficient

conditions for the existence of a probability measure p and a real-valued parameter

ψ, such that for any payoff-function f and any event A,10

πf (A) ∝∫A

eψf dp (1)

To understand this representation note first that if f is constant then πf = p. The

probability measure p therefore represents what subjective beliefs would be if the

decision maker were an impartial observer with no stake in what the true state is.

More generally, of course, f is not constant, and πf depends on f and on ψ. If ψ is

positive (negative) πf is higher in states in which payoff is higher (lower). A positive

value of ψ represents optimistic bias, and a negative value represents pessimistic

bias.11 If ψ = 0 then πf = p regardless of f . A zero value of ψ therefore represents

realism. Moreover, the larger ψ is in absolute terms the greater the distortion. In

analogy with relative risk-aversion, ψ is the coefficient of relative optimism.

A simple linear expression is obtained for the odds-ratio between two events.

Suppose A and B are two non-null events with a well-defined payoff. The logarithm

of the odds-ratio between the two events can be written as follows:12

logπf (A)

πf (B)= log

p(A)

p(B)+ ψ · [f(A)− f(B)] (2)

The payoff-difference between two events is thus a sufficient statistic for their relative

probability. If it is zero there is no belief distortion. Otherwise, the relative probability

of the event in which payoff is higher is biased upwards if ψ > 0, and downwards if

ψ < 0.

One of the most important properties of these equations is that they are formally

identical to Bayes Rule. The Bayesian update formula that corresponds to Equation 2

authors as the name for quite different assumptions.10In other words, πf is absolutely continuous w.r.t. p and there exist real numbers ψ and C such

that the Radon-Nikodym derivative is dπf/ dp = Ceψf .11In ordinary language we say that someone is an optimist (pessimist) in a given situation if they

hold relatively positive (negative) beliefs, whatever the reason. In this paper I reserve the terms‘optimism’ and ‘pessimism’ to the ceteris paribus effect of payoffs on beliefs. Note also that twooptimists (pessimists) with opposing interests would be biased in opposite directions.

12An even simpler linear expression is obtained if the state-space is discrete, and we allow fornon-normalized probabilities. The distortion can then be expressed by the following linear vectorequation:

logπf = log p+ ψf

(In this expression it is assumed that all states have positive probabilities. Alternatively, the con-vention is assumed whereby log 0 = −∞ and −∞+ x = −∞ for all x ∈ R).

4

is the following:

logp(A|E)

p(B|E)= log

p(A)

p(B)+ log

p(E|A)

p(E|B)(3)

where p(A)/p(B) is the prior odds ratio, p(A|E)/p(B|E) is the posterior odds ratio

after observing some event E, and p(E|A)/p(E|B) is the likelihood ratio. A compar-

ison of Equation 2 with Equation 3 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 observes the same payoff ‘evi-

dence’, but arrives at the opposite conclusion. Both optimists and pessimists make

their judgments of likelihood as if they believed that nature took their interests into

consideration when selecting the state of the world, so that their payoff-function is

relevant information as to what that state is. Where they differ is in the intentions

they ascribe to nature: optimists see nature as benevolent, whereas pessimists see it

as malevolent.

Combining Equations 2 and 3 we obtain the following expression for distorted

posterior beliefs:

logπf (A|E)

πf (B|E)= log

p(A)

p(B)+(

logp(E|A)

p(E|B)+ ψ · [f(A)− f(B)]

)(4)

This expression itself has the same structure as Bayes Rule, and can be interpreted

as Bayesian updating on an extended state-space. An obvious implication of this

expression is that changes in the payoff-function can lead to changes in beliefs in the

absence of any new relevant information. For example, when parents learn the result of

a random school allocation process, their payoff from the state in which that school is

the best school increases, while their payoff from states in which other schools are best

goes down. Learning which school their child has been allocated to would therefore

cause optimistic parents to think more highly of that school.13 Less obviously, changes

in relevant information may also result in a change in belief that seem inconsistent

with Bayesian updating. Consider an optimistic manager whose promotion may or

may not depend on some particular deal. Her subjective probability for the state in

which the deal is both successful and important for her promotion would therefore

be particularly high. Consequently, the Bayesian update given the information that

the deal is successful would have the effect of increasing the subjective probability

13Example 2 in Section 5.1.

5

that its success would determine her promotion.14 To outside observers this change

in beliefs may well seem inconsistent with Bayesian updating.

In order to use these formulas in applications it is necessary to introduce model-

ing assumptions for p and for ψ. One option is to reinterpret rational expectations

as applying to the undistorted probability measure p. As for ψ, a reasonable first

assumption is simply that it is positive, consistent with the fact that studies in the

general population find evidence for optimistic bias.15 As an example, consider the

beliefs of investors during an asset bubble. The implication of rational expectations is

that investors with no exposure to the asset hold unbiased beliefs about the prospect

of market collapse. The assumption that ψ > 0 implies that investors who hold the

asset underestimate this probability. The model can also be readily used to determine

the implications of payoff-dependent belief distortion in an existing application, by

identifying the probability measure p with the one currently used to represent the

decision maker’s beliefs, and using the distorted probability measure πf to predict

choices.

Supposing that payoffs affect beliefs in some particular situation of interest, there

remains the question of how important this effect is. The comparative statics of the

bias are that it is increasing in the degree of subjective uncertainty, the degree of

optimism or pessimism (the absolute value of ψ), and in the degree that the decision

maker cares about the state of the world (the absolute value of the payoff-difference

f(A) − f(B)). The case of normally distributed random variables provides a par-

ticularly clear illustration. If the undistorted distribution of a random variable X is

normal, and utility is linear (f = aX), then the distorted distribution is also normal,

and is characterized by the same variance, and a mean shifted by ψaσ2. The magni-

tude of the bias is increasing in the absolute value of ψ, in uncertainty (σ2), and in

the stakes (the absolute value of a).

As an example of these comparative statics consider again the case of investors

in an asset bubble. The value of ψ is presumably positive for heavy investors. Such

investors hold a significant proportion of their wealth in the asset, and hence from

their perspective the stakes are high for the continued health of the market. Finally,

asset bubbles are characterized by disagreement among experts, some of whom argue

that prices would continue to rise, while others caution that collapse is imminent.

Subjective uncertainty is therefore also high. Consequently, the comparative statics

imply that asset bubbles would be characterized by a particularly large belief bias

14Example 3 in Section 5.2.15An analogy may be drawn with attitudes to risk, where the assumption of risk-aversion is

commonly added to subjective expected utility theory.

6

among investors.

Asset bubbles are also situations in which biased beliefs can prove very costly.

When the bubble bursts, investors who continue to believe that prices would recover

risk having their investments wiped out. It is thus very important to note that in the

model of this paper the cost of biased beliefs does not affect the magnitude of the

bias, except indirectly by increasing the motivation to acquire information. If such

information cannot substantially reduce uncertainty, a large bias may persist despite

the ruinous consequences.16

This last feature of the model contrasts sharply with the predictions of the strategic

belief distortion model of Akerlof and Dickens (1982) and Brunnermeier and Parker

(2005). In the latter model decision makers choose beliefs to maximize subjective

expected utility, subject to the constraint that in the future beliefs would be governed

by Bayesian updating. They thus trade-off the anticipation of desirable outcomes

against the reduced likelihood of obtaining such outcomes if they adopt distorted

beliefs. The important comparative statics in such a model are the length of time

in which decision makers can enjoy biased beliefs, and the degree to which future

outcomes depend on these beliefs—neither of which is a factor in the model of this

paper. A recent study of wishful-thinking bias conducts a statistical test of these

predictions. Mayraz (2011) controls the cost of holding distorted beliefs by varying

the size of the prediction bonus, and finds no change in the observed bias. The

predictions of Brunnermeier and Parker (2005) are rejected.

There are many models of decision making in which the weight put on different

states of nature varies with the act that the decision maker is evaluating.17 Prefer-

ences are stable, but cannot be represented by a subjective expected utility functional.

In this model of this paper, by contrast, preferences are not stable (they vary with

the endowment). However, holding the endowment constant preferences can be rep-

resented by a subjective expected utility functional.

The model stands in an interesting relationship to other models of reference de-

pendent preferences, and in particular to Koszegi and Rabin (2006, 2007, 2009). In

both models preferences depend on a reference act. In Koszegi and Rabin (2006, 2007,

2009) the utility of different outcomes is dependent on the probability in which these

outcomes are obtained. In the model of this paper the probability of different states

is dependent on the utility in those states. In both models there is also an aspect of

16Needless to say, choices are affected by the utility-function as well as by beliefs. Highly risk-averseindividuals may avoid risky choices even if they underestimate the true risk.

17Models with probability-weights fall under this category, as do many models of ambiguity-aversion.

7

the reference act that can be ignored. In Koszegi and Rabin (2006, 2007, 2009) it

doesn’t matter in which particular states a given consumption outcome is obtained

(only the overall probability matters). In the model of this paper it doesn’t matter

which particular outcome is obtained in a given state (only its utility matters).

Many of the most interesting implications of the model have to do with how the

bias varies with changes to the environment. For example, a well-known puzzle in

the economics of crime is the relative ineffectiveness of increasing the severity of pun-

ishment as compared with the deterrent effect of increasing the likelihood that crime

is punishment (Grogger, 1991; Nagin and Pogarsky, 2001; Durlauf and Nagin, 2010,

2011). This puzzle can be resolved if we assume criminals are optimistically biased.

The key is the comparative statics prediction that the bias increases with the stakes.

Severe punishments increase the stakes in not getting caught, and therefore make

criminals more biased than they would be if punishments were lighter. Thus, while

jail is worse, it is also subjectively less likely. By contrast, increasing the likelihood

that criminals are brought to justice leaves the bias in their beliefs unchanged, and

unambiguously improves deterrence.

The reminder of this paper is organized as follows. Section 2 describes the formal

model and representation theorem. Section 3 provides a revealed axiomatization on

the assumption that preferences maximize subjective expected utility. Section 4 is

devoted to the implications of the model for beliefs in static situations, and Section 5

to the implications for belief updating. Section 6 presents the deterrence application.

Section 7 concludes. Appendix A reviews some of the relevant psychology and eco-

nomics evidence for payoff-dependent beliefs. Proofs not in the body of the paper are

in Appendix B.

2 Representation Theorem

In this section I state and prove a representation theorem relating a person’s sub-

jective beliefs to her payoff-function. This is followed in Section 3 by a revealed

preferences axiomatization on the assumption that preferences maximize subjective

expected utility. The development in this section requires fewer assumptions, and is

therefore simpler and more general. In particular, since subjective expected utility

maximization is not assumed, the resulting representation of subjective beliefs can

also be combined with non-expected utility decision criteria. The cost of this simplic-

ity and generality is that the primitives in this section (beliefs and payoff-functions)

are not directly observable.

8

Section 2.1 describes the framework for representing the relationship between be-

liefs and the payoff-function, the properties that are assumed to characterize it, and

the formal statement of the representation theorem. Section 2.2 demonstrates the

role of the individual assumptions by presenting the partial representation results

that can be obtained with only a subset of the assumptions. The proof is described

for the special case in which the algebra contains only finitely many events, making

it possible to focus on the key ideas, while avoiding the technical complications that

arise in the more general case. Section 2.3 concludes the proof of the representation

theorem by extending this result to a generic measurable-space.

2.1 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, called events. Beliefs are represented

by σ-additive probability measures over (S,Σ). I let X = [m,M ] ⊆ R denote the set

of all feasible payoffs, which I assume to be an interval which includes 0.18 A payoff-

function is a Σ-measurable mapping f : S → X.19 Let F denote the set of all such

functions, 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 → ∆, associating

with each payoff-function a probability measure over (S,Σ). The interpretation is

that the payoff-function represents the utility that is obtained in each state of nature,

and the distortion π represents the possibility that subjective probabilities are payoff-

dependent.

In the following definitions f and f ′ stand for any payoff-functions, a for any con-

stant payoff-function, and E for any event. The first definition states the properties we

want the distortion mapping to satisfy, and the second describes the logit-distortion

formula. The theorem says that the two definitions are equivalent.

Definition 1. π : F → ∆ is a well-behaved distortion20 if the following conditions

are satisfied:

18The interpretation is that the payoff in a state is the utility of the objective lottery the endowmentyields in that state. Since objective lotteries are a mixture space, the set of feasible payoffs is convex.Moreover, a utility-function representation can always be chosen such that 0 is a possible payoff-value. The assumption that the set of feasible payoffs is bounded can be avoided (see footnote atthe end of Section 2.3).

19f−1(B) ∈ Σ for any Borel set B ⊆ X.20The term ‘well-behaved distortion’ is used in the reminder of the paper as a shorthand to

assumptions A1-A4, which ensure the existence of a simple representation. There is no suggestionthat these are the only reasonable assumptions that can be made, nor that decision makers whosesubjective beliefs are subject to such a distortion are more rational than decision makers whosebeliefs are subject to other distortions.

9

A1 (absolute continuity) πf ′(E) = 0 ⇐⇒ πf (E) = 0.

A2 (consequentialism) If f = f ′ over a non-null21 event E then πf ′(·|E) = πf (·|E).

A3 (shift-invariance) If f ′ = f + a then πf ′ = πf .

A4 (prize-continuity) If fn → f then πfn(E)→ πf (E).

Absolute Continuity limits belief distortion to events that the decision maker is

uncertain about. Consequentialism requires that if two payoff-functions coincide over

some event E then the corresponding probability measures conditional on E also

coincide. Equivalently, the probability ratio between two events depends only on the

consequences of those events.22 Shift Invariance requires subjective probabilities to

depend only on payoff differences between states.23 Finally, Prize Continuity requires

that small differences in payoffs have only a small effect on beliefs.

Definition 2 (Logit distortion). π : F → ∆ is a logit 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,

πf (A) ∝∫A

eψf dp (5)

Consequentialism only has bite when there are at least three events with positive

probability. This (mild) condition is therefore necessary for the equivalence between

the two definitions to hold.

Definition 3 (Minimally complex distortion). π : F → ∆ is minimally complex if

there exists three disjoint events A, B, and C, and a payoff-function f such that

πf (A), πf (B), and πf (C) are all positive.

Theorem 1 (Representation theorem). A minimally complex distortion is a logit-

distortion if and only if it is well-behaved.

21That is, both πf (E) > 0 and πf ′(E) > 0. Absolute Continuity ensures that these two require-ments coincide.

22For example, if A, B and C are disjoint events then the payoff in A may affect the relativeprobability between A and B or between A and C, but not the relative probability between B andC.

23The interpretation is that the distortion of subjective probabilities depends on the degree towhich an event is desirable relative to other events, and that equal differences in desirability cor-respond to equal differences in utility (the notion of equal differences is independent of the utility-function representation, as all representations are related by an affine transformation). Scale-invariance (to a positive scaling factor) is an alternative assumption leading to a representationin which the eψf term in Definition 2 is replaced by fψ. In this alternative representation the payoffin a state is identified with the exponent of the utility of the outcome in that state, rather than withthe utility value itself.

10

2.2 Intermediate representation results

In this section I prove Theorem 1 for the special case of an algebra containing finitely

many events. That is, I assume that there exists a finite partition S of the state-space,

such that Σ is the algebra generated by S (this includes—but is not limited to—the

case where the state-space is itself finite). In addition, I prove a sequence of partial

representation results requiring fewer than the four assumptions in the definition of

a well-behaved distortion. In order to state the necessary and sufficient conditions

for these representations I define a new property, Indifference, which is related to

shift-invariance, but is considerably weaker:

A3’ (Indifference). πf = πf ′ if both f and f ′ are constant payoff-functions.

Note that Indifference does not require the set of payoffs to have cardinal (or even

ordinal) meaning. With Indifference defined, the claim consisting of the partial rep-

resentation results can be stated:

Lemma 1. Suppose that there exists a finite partition S of the state-space, such that

Σ is the algebra generated by S, and that π is minimally complex, then:

1. Absolute Continuity is a necessary and sufficient condition for there to exist a

probability distribution p ∈ ∆ and a function h : F × S → R+, such that for

any payoff-function f and any event A ∈ S,24

πf (A) ∝ p(A) · hf (A) (6)

2. Consequentialism is a necessary and sufficient additional condition for there to

exist a probability distribution p ∈ ∆, and a mapping µ : S × X → R+, such

that for any payoff-function f and any event A ∈ S,

πf (A) ∝ p(A) · µA(f(A)) (7)

3. Indifference is a necessary and sufficient additional condition for there to exist

a probability distribution p ∈ ∆, and a mapping ν : X → R+, such that for any

payoff-function f and any event A ∈ S,

πf (A) ∝ p(A) · ν(f(A)) (8)

24This result follows from the Radon-Nikodym theorem, and is included for completeness.

11

4. Shift-Invariance and Prize-Continuity are necessary and sufficient additional

conditions for there to exists a probability distribution p ∈ ∆, and a parameter

ψ ∈ R, such that for any payoff-function f and any event A ∈ S,

πf (A) ∝ p(A) · eψf(A) (9)

Note that while the representations in Equations 6–9 are stated specifically with

respect to events in S, the implication for general events in Σ is straightforward, as

any such event is the finite union of events in S.

To see that Minimal Complexity is a necessary assumption let S = {A,B}, and

define a distortion π by πf (A) ∝ p(A)(1 + (f(A) − f(B))2) and πf (B) ∝ p(B).

This distortion is well-behaved, but it cannot even be given the representation of

Equation 7, let alone that of a logit distortion.

2.3 Completing the proof

The first step in completing the proof of Theorem 1 for general payoff-functions and

general events is a generalization of Equation 9 to constant-payoff events on a general

algebra:

Lemma 2. Suppose π : F → ∆ is a minimally complex well-behaved distortion, then

there exist a probability measure p and a parameter ψ ∈ R, such that for any payoff-

function f and any events A and B such that (i) p(B) > 0 and (ii) f is constant on

A and on B,πf (A)

πf (B)=p(A)

p(B)· e

ψf(A)

eψf(B)(10)

The restriction of Theorem 1 to simple payoff-functions is an immediate corollary.

As the set of payoffs is assumed to be bounded it is possible to approach any payoff-

function uniformly by a sequence of simple payoff-functions. A limit argument using

Prize Continuity and the dominated convergence theorem completes the proof of

Theorem 1.25

25This proof can be replaced by a more complicated argument which does not require the assump-tion that the set of feasible payoffs is bounded. There are several cases to consider. If there are onlyfinitely many non-null events the theorem can be proved without a limit argument. Otherwise, let pand ψ be parameters for which Lemma 2 holds. If the set eψX is not bounded from above it is easy toconstruct a contradiction to Absolute Continuity. Thus, if there are infinitely many non-null eventseψX must be bounded from above, making it possible to use the dominated convergence theoremwithout a separate assumption that set of payoffs is bounded.

12

3 Revealed Preferences

In this section I state and prove the revealed-preferences analogue of the representa-

tion theorem of Section 2, relating a person’s preferences over acts to her endowment

on the assumption that preferences maximize subjective expected utility with a ref-

erence independent utility function. The resulting representation comprises three

elements that are co-determined: a utility-function, a probability-measure, and a

real-valued parameter, corresponding to the coefficient of relative-optimism in Sec-

tion 2.

The axioms can be divided into three groups. B1-B3 ensure that preferences have a

subjective expected-utility representation, in which the utility function is independent

of the endowment,26 and that preferences depend on an endowment only via the

associated payoff-function. B4-B5 are technical assumptions ensuring that the choice

data is sufficiently rich to reveal the subjective probability measure associated with

each endowment, and requiring there to be a best and a worst act. B6-B9 are a

revealed-preferences restatement of the assumptions in Definition 1.

3.1 Framework

Let S denote the set of subjective states, and let Σ be a σ-algebra of events. I let Z

denote the set of final outcomes. An act is a Σ-measurable mapping a : S → ∆(Z),

associating with each subjective state an objective lottery over final outcomes.27 I

denote by A the set of all acts. The key object is a distortion mapping �: A → A×A,

associating with each act e ∈ A a preference relation �e. The interpretation of g �e his that the decision maker prefers g over h if her endowment is e. The use of second-

order acts makes it possible to represent objective as well as subjective uncertainty

(important in some applications). A second advantage is that the utility-function

of an expected-utility maximizing decision maker is revealed independently of her

subjective-beliefs.

In the following a constant-act is an act that yields the same lottery in all states.

I use g, h, e, e′ and en for general acts, and a and b for constant acts. I use s for a

general state, and E for a general event. For an act e and a state s I let es denote

the constant act yielding e(s) in all states. Finally, an event E is �e null if for all g

and h that differ only on E, g ∼e h.

26Hence, reference-dependence in preferences must originate in a corresponding reference-dependence in beliefs.

27Acts mapping states into objective lotteries over final consequences were first used in Anscombeand Aumann (1963).

13

Definition 4. � are reference-dependent preferences with payoff-dependent beliefs if

the following conditions are satisfied:

B1 (Anscombe-Aumann) �e has an Anscombe-Aumann expected utility represen-

tation.

B2 (objectivity) �e=�e′ over constant acts.28

B3 (indifference) If es ∼ e′s for all s then �e=�e′ .29

B4 (non-triviality) For any act e there exist constant acts a and b such that a �e b.

B5 (best and worst act) For any act e there exist constant acts a and a such that

for any act g, a �e g �e a.

B6 (absolute continuity) E is �e null ⇐⇒ E is �e′ null.

B7 (consequentialism) If e = e′ over E and g = h outside E then g �e h ⇐⇒g �e′ h.

B8 (shift-invariance) If for some α ∈ [0, 1], e = αg+(1−α)a and e′ = αg+(1−α)b,

then �e=�e′ .30

B9 (continuity) If en → e uniformly31 then �en→ �e.32

As in Section 2 the proof requires the existence of at least three disjoint non-null

events:

Definition 5. � is minimally complex if there exist an act e and disjoint events A,

B, and C that are not null with respect to �e.

Theorem 2 (reference-dependent preferences). Suppose � are reference-dependent

preferences with payoff-dependent beliefs, and that � is minimally complex, 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),

28If a and b are constant acts then for any e and e′, a �e b ⇐⇒ a �e′ b.29That is, for e and e′ to result in different preferences it is not enough that e(s) 6= e′(s) for some

state s—it is also necessary that one of these outcomes is strictly preferred to the other (formally,the decision maker has strict preferences between the constant acts es and e′s. B2 ensures that thesepreferences are well-defined).

30This condition implies that the utility difference between e and e′ is the same in all states.31For any ε > 0 and any state s ∈ S there exists n0 ∈ N, such that for all n > n0 and for any

outcome z ∈ Z, |en(s)(z) − e(s)(z)| < ε (the difference in the probability the two acts assign tooutcome z in state s is less than ε)

32For all acts g and h, if g �e h there exists n0 ∈ N, such that for all n > n0, g �en h.

14

such that for any act e ∈ A, �e ranks acts according to the following Anscombe-

Aumann expected utility functional:

Ve(g) =

∫S

(u ◦ g) dπf(e) =

∫S

∫Z

u(z)gs(z) dz dπf(e) (11)

where g ∈ A is any act, f(e) = u ◦ e, F = {u ◦ e : e ∈ A}, and π : F → ∆(S) is

a logit distortion (Definition 2). Moreover, if the triplet (p′, u′, ψ′) also represents �then p′ = p and there exist a positive real-number α and a real-number β, such that

u′ = αu+ β and ψ′ = ψ/α.

3.2 Proof

B1 is an omnibus axiom, requiring that �e have an Anscombe-Aumann expected

utility representation for all e. Given B1 for any endowment e there exists a prob-

ability measure µe ∈ ∆(S) and a utility function ue : Z → R, such that �e ranks

acts according to the following functional: Ve(g) =∫S(ue ◦ g) dµe where (ue ◦ g)(s) =∫

Zue(z)gs(z) dz. This representation allows for both beliefs and tastes to vary with

the endowment e. B2 rules out the latter possibility by imposing the requirement

that the ranking of constant Anscombe-Aumann acts does not depend on e. Since

the ranking of constant acts determines the utility function up at a positive affine

transformation the same utility-function u can used in representing �e for all e.

Given the subjective expected utility representation implied by B1 and B2, the

condition in B3 that es ∼ e′s for all s is equivalent to u ◦ e = u ◦ e′, and the statement

that �e=�e′ implies that µe = µe′ . Let F = {u ◦ e : e ∈ A} denote the set of payoff-

functions consistent with u and A. For consistency of notation with Section 2 define

a mapping f : A → F by f(e) = u ◦ e. Given B3 the mapping µ : A → ∆(S) can be

expressed as the composition of a mapping π : F → ∆(S) and the utility function u,

so that for any e, µe = πf(e).33

B4 is a technical assumption ruling out the trivial case in which the decision maker

is indifferent between all acts. Non-triviality ensures that it is possible to back out

πf(e) from observing �e. Hence, �e=�e′ if and only if πf(e) = πf(e′). We thus obtain

the following Lemma:

Lemma 3. Suppose B1-B4, then (i) there exists a utility function u : Z → R, and a

mapping π : F → ∆(S) where F = {u ◦ e : e ∈ A}, such that for any e ∈ A, �e ranks

33π is formally defined by choosing for any payoff-function f some particular act e(f) to representthe equivalence class of all the acts having f as their payoff-function, and defining π(f) = µe(f).

15

acts in accordance with the following subjective expected utility functional:

Ve(g) =

∫S

(u ◦ g) dπf(e) =

∫S

∫Z

u(z)gs(z) dz dπf(e) (12)

and (ii) for any two acts e and e′, �e=�e′ if and only if πf(e) = πf(e′).

B5 is a second technical assumption, ensuring that there exist a best and a worst

lottery (and therefore also a best and a worst outcome). B6-B9 effectively restate the

axioms A1-A4 of Definition 1. The proof of the following Lemma is in the appendix:

Lemma 4. Suppose B5-B9 hold in addition to B1-B4 then the mapping π in Lemma 3

is a well-behaved distortion (Definition 1).

The main claim in Theorem 2, namely the existence of a triplet (p, u, ψ) representing

� in accordance with Equation 11 and Definition 2, is an immediate corollary of Lem-

mas 3 and 4, and of Theorem 1. The proof of the uniqueness part is in Appendix B.

4 Belief distortion

This section examines the belief distortion formula and its implications for subjec-

tive beliefs in a static setting. The implications for belief updating are considered

in Section 5. An analysis of the belief distortion formula (Section 4.1) is followed

by a particular interpretation of this formula, according to which optimists and pes-

simists behave as if the payoff in an event is relevant evidence about its likelihood

(Section 4.2). Section 4.3 looks at the relationship between logit-distortions and

model-independent notions of optimism and pessimism. Section 4.4 investigates the

comparative statics of the belief distortion formula. Finally, Section 4.5 shows that

the ranking of distributions by first-order stochastic dominance does not survive be-

lief distortion. Hence, a distribution that is unambiguously better than another may

nonetheless be perceived as worse.

4.1 The belief distortion formula

Let π denote the mapping from payoff-functions to subjective beliefs. According

to Theorem 1 if π is a logit distortion there exists a probability measure p, and a

real-valued parameter ψ, such that for any payoff-function f , and any event A,

πf (A) ∝∫A

eψf dp (13)

16

To understand this result consider first the case where f is constant, representing a

situation in which the decision maker is completely indifferent as to what the state of

the world is. The eψf term can then be dropped out, and we obtain that πf = p. The

probability measure p can therefore be identified with a decision maker’s undistorted

beliefs, or the beliefs she would have held if she were indifferent about the state of the

world.

Equation 5 allows for payoff to vary arbitrarily between the different states in

A. If we restrict attention to constant-payoff events we can rewrite the equation as

follows:

πf (A) ∝ p(A) · eψf(A) (14)

The factor eψf(A) is increasing in the payoff in A if ψ is positive, decreasing in the

payoff if ψ is negative, and independent of it if ψ = 0. Payoff-dependent belief

distortion therefore makes desirable events subjectively more probable if ψ > 0, and

subjectively less probable if ψ < 0. If ψ = 0 the desirability of an event has no effect

on its subjective likelihood. A positive value of ψ represents optimistic bias, while a

negative value of ψ represents pessimistic bias. A zero value for ψ represents realism.

The magnitude of belief distortion increases when moving away from zero, whether

in the optimistic or pessimistic direction. In analogy with the coefficient of relative

risk aversion, ψ can therefore be thought of as the coefficient of relative optimism.

Further insight can be obtained by comparing the probability of two events in

relation to each other. Suppose that f is constant over two events A and B, and

that B is not-null.34 The log-odds ratio between the two events is then given by the

following linear equation:

logπf (A)

πf (B)= log

p(A)

p(B)+ ψ ·

(f(A)− f(B)

)(15)

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. If a decision maker is indifferent between two

events, their relative probability is unchanged (their overall probability may affected

by the payoff in other events). Consider now the case where f(A) 6= f(B), assuming

without loss of generality that f(A) > f(B). If ψ = 0 there is again no bias, but if

ψ > 0 (ψ < 0) then A becomes subjectively more (less) likely relative to B.

The same subjective probabilities may result from different combinations of in-

34Absolute Continuity guarantees that the term“not-null” is well-defined without having to specifythe payoff-function.

17

f(A)− f(B)

log p(A)p(B)

0 1 2−1−2

logπf (A)

πf (B)

optimist believes A

optimist believes B

Figure 1: Iso-belief lines for an optimist as a function of the undistorted log odds-ratio on the x-axis and the payoff-difference between the two events on the y-axis.Iso-belief lines are straight lines sloping down and to the right with a slope of 1/ψ.The Iso-belief lines for pessimists slope upwards and to the right. Those of a realistare vertical.

formation (represented by the undistorted log odds-ratio log p(A)/p(B)), and what

the person has at stake in what the state of the world is (represented by the payoff-

difference f(A) − f(B)). Since the Equation is linear, the resulting iso-belief lines

are also linear (Figure 1).

4.2 Payoffs as information

The equations of the model have a close analogue in Bayes Rule. For Equation 14

the analogous equation is the following:

p(A|e) ∝ p(A) · L(e|A) (16)

where e represents new evidence, p represents beliefs prior to observing the new

evidence, p(A|e) represents posterior beliefs, and L(e|A) the likelihood of the new

evidence. Similarly, the analogue of Equation 15 is

logp(A|e)p(B|e)

= logp(A)

p(B)+ log

L(e|A)

L(e|B)(17)

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 or prior beliefs, πf for

18

distorted or posterior beliefs, and ψf(A) for the log-likelihood of the evidence in A,

with an analogous expression for B.

It is thus possible to view payoff-dependent belief distortion as a Bayesian update

on an expanded state-space. Starting with undistorted beliefs (represented by p) as

her prior, the decision maker observes the payoff-function f , and updates her beliefs

to arrive at the posterior πf . The payoff in an event functions as relevant evidence

about its likelihood: an optimist (pessimist) takes high payoff to be evidence that an

event is more (less) likely. It is as if optimists (pessimists) believe that nature is not

an indifferent force, but is instead well-disposed (ill-disposed) towards them. Given

this belief the observation that they would be better off if A is the case is for optimists

(pessimists) evidence in favor (against) A being true.35

4.3 Optimism and pessimism

This section relates the coefficient of relative optimism ψ to a model-independent def-

inition of optimism and pessimism. Section 4.3.1 shows that decision makers whose

beliefs are represented by the model are optimists whenever ψ is positive, and pes-

simists if it is negative. Section 4.3.2 shows that the limit of ψ → ±∞ captures the

notion of infinite optimism and pessimism. Section 4.3.3 shows that if payoffs are

normally distributed the bias results simply in a shift to the mean of the distribution,

the shift being proportional to ψ.

4.3.1 Optimism, pessimism, and ψ

Suppose a decision maker’s beliefs are characterized by an undistorted probability

measure p and a distortion-mapping π. Let Pf (x) = p(f ≤ x) denote the undistorted

cumulative distribution function (CDF) of payoff, and let Πf (x) = πf (f ≤ x) denote

the corresponding CDF for πf . For two distributions F and G I write F �1 G if F

first-order stochastically dominates G, and F �LR G if F stochastically dominates

G in the likelihood ratio.36 With these definitions the natural model-independent

definition of optimism and pessimism is the following:

35Compare the malevolent nature interpretation of certain models of ambiguity aversion, where(unlike in this paper) the state of the world is not pre-determined, and can be altered by nature inresponse to the decision maker’s choice. Note also that optimists and pessimists may well try toform the most objectively accurate view of the world that they can, agreeing that payoffs provide norelevant information. The claim is that their subjective judgment is nonetheless biased as if theyviewed payoffs as informative.

36That is, if there exists a non-decreasing function h : R→ R+, such that F (x) ∝∫ x−∞ h(x)dG(x).

19

Definition 6. A decision maker is an optimist (pessimist) if Πf �1 Pf (Pf �1 Πf ).

A decision maker who is both an optimist and a pessimist is a realist.

The following proposition establishes the relationship between these definition and

the coefficient of relative optimism ψ:

Proposition 1. Suppose a decision maker’s beliefs are characterized by a logit distor-

tion with a coefficient of relative optimism ψ, then the DM is an optimist (pessimist) if

and only if ψ ≥ 0 (ψ ≤ 0). Moreover, ψ ≥ 0⇒ Πf �LR P and ψ ≤ 0⇒ Pf �LR Πf .

Logit distortions therefore represent a special class of optimistic and pessimistic belief

distortions, much as the class of constant relative risk-aversion (CRRA) preferences

is a special class of risk-averse and risk-seeking preferences.

4.3.2 Extremes of optimism and pessimism

The higher (lower) ψ is, the more probability shifts towards the states with the highest

(lowest) possible payoff. If there are only finitely many payoff values the limit is always

well-defined, and takes a particularly simple form: an extreme optimist (pessimist) is

certain she would obtain the best (worst) possible payoff:37

Proposition 2 (extreme optimism/pessimism). Let f be a simple payoff-function,

and let Amin and Amax denote respectively the event that the minimal (maximal) payoff

is obtained, then limψ→∞ πf (Amax) = limψ→−∞ πf (Amin) = 1.

4.3.3 Normally distributed payoffs

Optimistic and pessimistic bias typically distorts the shape of the distribution of pay-

offs. For example, if the undistorted distribution is uniform over some interval then

the distorted distribution would be exponential. A normal distribution is an excep-

tion: if payoff is linear in 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 the coefficient of relative optimism:

Proposition 3 (normally distributed payoffs). Suppose X : S → R is a random

variable with undistorted distribution PX ∼ N (µ, σ2), and that there exist a, b ∈ R,

such that the payoff-function is f = aX + b, then ΠX ∼ N (µ+ ψaσ2, σ2).

37Extreme optimism coincides with Leibniz’s concept of the “best of all possible-worlds” (Leibniz,1710) if ‘best’ is understood to mean best for the optimist. Yildiz (2007) explores a model in whichdecision makers behave like extreme optimists in this sense.

20

4.4 Comparative Statics

The intuition for the comparative statics can be obtained by writing Equation 15

qualitatively:

beliefs = undistorted beliefs + ψ · stakes (18)

The magnitude of the bias is thus greater the more is at stake, and is decreasing

in the strength of undistorted beliefs (increasing in the degree of uncertainty). This is

seen most clearly if payoff is proportional to a normally distributed random variable

X, i.e. f(s) = aX(s), where X ∼ N (µ, σ2). According to Proposition 3 the distorted

probability density function is also normal with the same variance, with the mean

biased by aψσ2. The bias is thus increasing both in the stakes (a) and in uncertainty

(σ2). The comparative statics for the normal case are illustrated in Figure 2.

Another interesting is when payoff is binary. For example, f = v over some

event E and f = 0 elsewhere. Using Equation 14 the bias in expected utility is

(πf (E) − p(E))v =(

p(E)eψv

1−p(E)+p(E)eψv− p(E)

)v = (eψv−1)p(E)(1−p(E))

1+p(E)(eψv−1) v. The magnitude

of the bias increases in the stakes |v| and decreases as undistorted beliefs approach

certainty (p(E)→ 0 or p(E)→ 1).

There is evidence for both comparative statics. Kunda (1990) shows that uncer-

tainty is essential to belief distortion, and suggests that more uncertainty results in a

greater bias. Weinstein (1980) and Sjoberg (2000) elicit beliefs over events which vary

in how desirable or undesirable they are, and find more biased beliefs over events that

are either strongly desirable or strongly undesirable. Mayraz (2011) elicits predic-

tions of future prices in different price charts, and finds stronger bias in price charts

in which subjective uncertainty is high.

Note that the potential consequences of holding distorted beliefs do not affect the

magnitude of the bias. Decision makers may, of course, react to important decision

problems by seeking as much information as possible, and to the extent that such

information reduces subjective uncertainty it would also reduce the bias. However,

controlling for information the magnitude of the bias is independent of its costs. The

implication is that payoff-dependent belief bias may well be an important factor in

high-stakes situations in which the cost of biased beliefs is substantial.38

38Compare visual illusions, which are similarly independent of their consequences for decisions.See also the discussion of Brunnermeier and Parker (2005) in the introduction.

21

payoff

payoff

Figure 2: Belief distortion with normally distributed payoffs. The effect of beliefdistortion is to shift the payoff distribution to the right or left depending on whetherthe decision maker is optimistic or pessimistic. The shift is proportional to the degreeof uncertainty, as measured by the variance of the distribution.

4.5 Can a better distribution be perceived as worse?

Suppose one environment is unambiguously ‘better’ than another, as perceived by dis-

interested (and therefore unbiased) observers, and consider the subjective perception

of a person whose payoff is determined by the two environments. Would this person

necessarily agree that the first environment is better than the second? As I show in

this section the answer depends on how ‘better’ is defined. If ‘better’ is defined via

stochastic dominance in the likelihood ratio then the answer is yes, but if it is defined

as first-order stochastic dominance then the answer is no, or rather not necessarily.

Consider first two events in which payoff is x and x′ respectively. Following belief

distortion, the odds-ratio between the two events is multiplied by eψ(x−x′) (Equa-

tion 14). 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 distributions. Generalizing this observation it is

straightforward to show that the relationship of stochastic dominance in the likelihood

ratio is preserved under payoff-dependent belief bias:

Proposition 4. Suppose π is a logit distortion, let F and G denote two payoff dis-

tributions, and let F ′ and G′ denote the corresponding distorted distributions, then

F ′ �LR G′ whenever F �LR G.

As the following example shows, first-order stochastic dominance is in general not

invariant to payoff-dependent belief distortion:39

39The deterrence application in Section 6 is another example. Increasing the severity of punishmentreplaces the payoff-distribution with a first-order stochastically dominated distribution, which (asshown in Section 6) may nonetheless be perceived as subjectively better by an optimistic criminal.

22

Example 1. Violation of first order stochastic dominance

Let A, B and C denote three events with a payoff of 2, 1 and 0 respectively. Let

F be a payoff-distribution defined by pF (A) = pF (B) = 0.5 and pF (C) = 0, and

let G be 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.

According to Equation 14 the distorted payoff distribution of F would be πFf (A) =

2/3, πFf (B) = 1/3, and πFf (C) = 0, and the distorted payoff-distribution of G would

be πGf (A) = 4/5, πGf (B) = 0, and πGf (C) = 1/5. Thus, F first-order stochastically

dominates G, but F ′ does not first-order stochastically dominate G′.

5 Belief Updating

Subjective beliefs are a function of undistorted beliefs, the payoff-function, and the

coefficient of relative optimism ψ (Equations 13–15). This section considers belief

updating on the assumption that ψ is constant. Given this assumption there are two

possible sources for belief change: a change in undistorted beliefs, and a change in

the payoff-function.

5.1 Changes to the payoff-function

Changes to the payoff-function can result in belief change in the absence of any new

relevant information, providing a possible model for cognitive-dissonance.40 Both

choice and the resolution of objective uncertainty can result in a change to the payoff-

function. For example, the finding in Knox and Inkster (1968) that placing a bet on a

horse leads to increased confidence that the horse would win the race can be explained

by the change in the payoff-function that follows from the placing of the bet. The

following example illustrates a change in beliefs following the resolution of objective

uncertainty:

Example 2. School allocation

A child is allocated by coin-toss to one of schools: A or B. Parents’ 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. Learning the outcome

of the allocation alters the payoff-function: prior to learning the allocation the utility

in both states is 0.5, whereas after learning the allocation it is 1 in one state and 0 in

40See Appendix A.

23

the other.41 Suppose that the undistorted beliefs are symmetric: p(A) = p(B) = 0.5,

and consider the beliefs of optimistic parents on the assumption that the allocation

is to school A. Since the ex-ante payoff is the same in both states, the ex-ante

distorted subjective probabilities are also symmetric: πfpre(A) = πfpre(B) = 0.5.

Ex-post, however, the payoff-difference becomes positive: fpost(A) − fpost(B) = 1.

Supposing that ψ = log 2, the ex-post subjective probabilities are πfpost(A) = 2/3 and

πfpost(B) = 1/3 (Equation 14). Thus, learning that their child has been allocated to

school A alters the beliefs of the parents, so that they come to believe that A is the

better school.42

5.2 New relevant information

Relevant information affects distorted beliefs via the Bayesian updating of undistorted

beliefs. However, since belief distortion is itself equivalent to Bayesian updating it

is also possible to view the new information as affecting distorted beliefs directly

(assuming there is no concurrent change in the payoff-function). Inserting Bayes

Rule in Equation 15 we obtain the following:

logπfpost(A)

πfpost(B)=(

logppre(A)

ppre(B)+ log

L(E|A)

L(E|B)

)+ ψ · [f(A)− f(B)]

=(

logppre(A)

ppre(B)+ ψ · [f(A)− f(B)]

)+ log

L(E|A)

L(E|B)

= logπfpre(A)

πfpre(B)+ log

L(E|A)

L(E|B)

(19)

The updating of subjective beliefs is therefore Bayesian in spite of the payoff-dependent

belief distortion. Nevertheless, it may not appear so to outsiders. Consider the fol-

lowing example:

Example 3. Merger

A merger deal can succeed (S) or fail (F ), and its outcome may be important (I) or

unimportant (U) to the manager’s promotion. Payoff depends on success only if the

merger is important. Consequently, an optimistic manager’s subjective probability

for the (S, I) state would be biased upwards, and her probability for (F, I) would be

biased downwards. As a result, the two variables would become positively correlated

41The outcome in each subjective state is an objective lottery over final consequences. The payoffin the state is the utility of this lottery.

42In practice, the change in beliefs may occur the next time the parents have an opportunity toreassess the quality of the schools. For example, they may make a visit to the two schools, andascribe the change in their beliefs to their impressions from the visit.

24

in her beliefs, and consequently news about one variable would affect beliefs about

the other. The following is a numeric example, assuming the undistorted probability

measure p is symmetric and ψ = log 2:

p I U

S 1/4 1/4

F 1/4 1/4

f I U

S +1 0

F −1 0

πf I U

S 4/9 2/9

F 1/9 2/9

In this example the prior probability of success is 2/3, but the probability of success

conditional on the merger being important is 4/5.43 Similarly, learning that the

deal has failed would decrease the subjective probability that it would be important

for promotion from 5/9 to 1/3. An outside observer who thinks of the two events as

independent may well see the change in beliefs as inconsistent with Bayesian updating.

In this example the two variables are complements in the utility-function, and are

consequently positively correlated in the distorted prior beliefs. Substitutes would

become negatively correlated. For example, if a company pursues two research ap-

proaches in parallel, a failure in one would increase the confidence of optimistic man-

agers that the other approach would succeed. Conversely, success would decrease the

subjective probability that the other approach could have worked.

The following proposition shows formally that when two variables are complements

(substitutes) in the utility-function they become positively (negatively) correlated in

beliefs, and that consequently positive news about one increases (decreases) the bias

in the other:

Proposition 5. Suppose the payoff-function 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 that E is an event such that p(X = x, Y = y|E) = p(Y = y|E) for all x

and y, and that p(E|Y = y) is an increasing function of y, then

1. ΠX|E �LR ΠX if (i) ψ ≥ 0 and f is supermodular, or (ii) ψ ≤ 0 and f is

submodular.

2. ΠX �LR ΠX|E if (i) ψ ≥ 0 and f is submodular, or (ii) ψ ≤ 0 and f is

supermodular.

Moreover, the above relations of stochastic dominance in the likelihood ratio are strict

whenever ψ 6= 0, f is strictly supermodular/submodular, p(E|Y = y) is strictly in-

creasing in y, and neither X nor Y is almost everywhere constant.

43πf (S) = 4/9 + 2/9 = 2/3, πf (S|I) = 4/9/(4/9 + 1/9) = 4/5.

25

In the examples I assumed each variable can take only two values. Proposition 5

makes it possible to derive implications in more complicated situations. For example,

in the R&D example we can let X and Y denote respectively progress in two possible

research approaches, and allow progress to be measured on a continuum. Using

Proposition 5 we can conclude that any good news about one research approach

would make the manager more realistic (less optimistically biased) about the prospects

for the second approach. Conversely, any bad news about one of the two research

approaches would make the manager less realistic about the other. A pessimistic

manager would exhibit the opposite bias.

6 Deterrence

In this section I apply the model to a problem from the economics of crime: does

more severe punishment deter crime? I assume that criminals are optimistic, and

that they therefore underestimate the probability that they would end up in jail. As

I show in a formal model, an increase in the severity of punishment increases the

stakes in not ending up in jail, which in turns leads to an increase in the bias. Thus,

a more severe punishment has an ambiguous effect on deterrence: on the one hand

jail is subjectively worse (the sentence is more severe), but on the other hand it is

subjectively less likely (because of the greater optimistic bias). Consequently, an

increase in the severity of punishment can potentially be counter-productive, ending

up decreasing, rather than increasing deterrence. This contrasts with an increase in

the probability that crime is punished, which unambiguously reduces the subjective

utility of crime.

6.1 Model

An optimistic criminal has to choose whether to continue a life of crime or to take

up a job at McDonald’s. There are two states corresponding to whether or not crime

would land the criminal in jail. The payoff from crime is f(B) = −c in the bad

state and f(G) = 0 in the good state. The payoff from a job at McDonald’s is −bin both states, with 0 < b < c. I let p denote the probability measure representing

the beliefs the criminal would have had if she were indifferent as to whether crime

would land her in jail, and assume for simplicity that p coincides with the objective

probabilities.44 Since crime is the status-quo subjective beliefs are represented by the

44The minimal assumption is that p is monotonic with objective probabilities, so that improvedlaw enforcement would increase p(B).

26

probability measure πf , related to p via Equation 14. Thus,

πf (B)

πf (G)=p(B)

p(G)· e−ψc (20)

Deterrence is successful if the expected gain from quitting crime is more than the

expected loss, i.e. πf (B)(c− b) ≥ πf (G)b, or

b

c− b≤ πf (B)

πf (G)=p(B)

p(G)· e−ψc (21)

Consider now the following two potential policy changes. First, the government can

improve law enforcement, thereby increasing p(B)/p(G). Holding c constant, such a

change would increase the RHS of Equation 21, while leaving the LHS unchanged, and

would therefore increase deterrence for any level of optimism. Second, the government

can increase in the severity of jail c, leaving its probability unchanged. If ψ = 0

the change would reduce the LHS, and leave the RHS unchanged. Thus, for realist

criminals any increase in the severity of punishment improves deterrence. However, for

optimistic criminals ψ > 0, and so the increase in c reduces the RHS of Equation 21

at the same time as it is reducing the LHS. There are thus two forces pulling in

opposite directions: (i) the utility effect works to increase deterrence (jail is worse),

and (ii) the probability effect works to decrease deterrence (jail is subjectively less

likely). Since limc→∞(c− b) · e−ψc = 0 for ψ > 0, making the punishment more severe

is always counter-productive beyond a certain point (Figure 3). Note that the key to

these results is the assumption that crime is the status-quo. A decision maker who is

not engaged in crime would be deterred by increased punishment.45

The empirical evidence is consistent with these predictions. Grogger (1991) stud-

ied the frequency of arrests and 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 probabil-

ity of conviction vs. 0.017 with t-score of 1.65 for average sentence length). Nagin and

Pogarsky (2001) also found that the certainty of punishment was a far more robust

a deterrent than severity. In a recent review Durlauf and Nagin (2011) argue along

similar lines that increased punishment does not lead to better deterrence.

45However, the necessary assumption is that the DM is completely indifferent as to whether crimeresults in jail, and there may be many reasons for this strong assumption to fail (e.g. family orfriends who are engaged in crime). Moreover, since most criminals enter crime before adulthood,the deterrent effect on non-criminals is relatively insignificant to the overall level of crime.

27

πf (G)

Exp

ecte

du

tility

0 1

Utility effect

p(G)

πf(G

)

0 1

Probability effect

p(G)

Exp

ecte

du

tility

0 1

Combined effect

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 givensubjective probability more severe punishments reduce utility (panel 1). At the sametime, the increase in the stakes in staying out of jail results in increased bias in thecriminal’s subjective probability of not getting caught (panel 2). The net effect (panel3) depends on parameters. If the punishment is light and capture is likely it wouldbe negative, but if the punishment is severe and jail occurs only with low probability,the net effect can be an increase in the subjective expected utility of crime (and hencereduced deterrence).

7 Conclusion

The model developed in this paper can be readily used to incorporate payoff-dependent

beliefs bias into any existing application in which decision-makers maximize subjec-

tive expected utility. In principle, all that is necessary is to identify the subjective

probability measure used in the existing application with the undistorted probability

measure p. The distorted probability measure πf can then be used in the revised

application in place of the original probability measure. In particular, an existing

assumption of rational expectations can be replaced with the assumption that the

undistorted probability measure p corresponds to rational expectations. Decision

makers in the revised model would hold biased beliefs, but the bias in their beliefs

would be due entirely to payoff-dependent belief bias

One would hope that the model of this paper may lead to better and stronger

predictions in many areas of economic research. In financial markets, for example,

the model predicts not only that optimists would underestimate the risks that they

face, but more specifically how the degree of bias depends on the investor’s subjective

uncertainty and exposure to the risk in question. A particularly important implication

of the model is that a significant bias in belief may obtain in high-stakes economic

environments in which the cost of holding biased beliefs is substantial.

One weakness of the model is the need to specify exactly what elements of uncer-

28

tainty are subjective, and are therefore subject to payoff-dependent belief distortion.

For example, the predictions in the deterrence application (Section 6) depend cru-

cially on the assumption that the severity of punishment is taken as given, whereas

the likelihood of getting caught is subjective. If these assumptions were reversed the

predictions would also be very different.46

A second weakness is that in some situations the decision maker’s payoff-function

depends on her plans, which may not be readily observable. Consider two investors

who are holding some stock. A short-term speculator stands to gain if the stock does

well over the near future, and would therefore be biased about this possibility. A

long-term investor would be biased, instead, about the long-term future of the stock.

The usefulness of the model may therefore depend on the ability to identify the type

of the investor, as well as the stocks she owns.

This paper assumes a static decision environment. Additional issues arise in dy-

namic choice. Do decision makers take into account the effect of their decisions on

the bias in the beliefs (and hence preferences) of their future selves? Another issue

is the possibility that the coefficient of relative optimism is not constant, and may

increase (decrease) following good (bad) events. There is some evidence suggestive of

this possibility.47 Such dynamics may be important during the popping of an asset

bubble, when decreasing prices lead to losses, which in turn (if this hypothesis is true)

reduce the optimistic bias among investors. A reduction in optimistic bias would lead

to further selling and further drops in prices.

Strategic interaction is another important area for extending the model. Yildiz

(2007) has developed a model of strategic interaction in the limit of extreme optimism,

but no such model presently exists for realistic levels of bias. Strategic interaction

may also provide an evolutionary rationale for the existence of payoff-dependent belief

bias.48

46In particular, an increased likelihood of punishment would result in increased bias as to the howserious the punishment would be.

47For example, Saunders (1993) and Hirshleifer and Shumway (2003) find that changes in stockprices are correlated with the weather, while Edmans et al. (2007) find similar correlations with theoutcome of recently concluded international soccer games.

48This is the same idea as in an entry game with ‘crazy’ types. Where a realist player wouldaccommodate an intruder, an over-optimistic player may choose to fight. Knowing this, other playersmay choose not to enter, rewarding the over-optimistic player for the bias in her beliefs. See alsoCompte and Postlewaite (2004).

29

A Evidence

In this appendix I describe some of the most relevant empirical findings in psychology

and economics.

A.1 Psychology evidence

Studies of over-optimism (also wishful-thinking or over-confidence.)49 demonstrate a

tendency to overestimate the probability of desirable events, and underestimate the

probability of undesirable events. The best known studies are probably 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) found that students see desirable (un-

desirable) life events as more (less) likely to occur to them than to other students.

Sjoberg (2000) and Weinstein (1989) report comparable results in risk perception.

Weinstein et al. (2005) shows that smokers underestimate the risks of smoking.

Closely related are studies of self-serving beliefs. Instead of demonstrating a bias

vs. rational expectations, studies in this group compare the beliefs of two groups of

subjects that differ in their payoff-function on the assumption that any systematic

differences in beliefs can be attributed to the difference in the payoff-function. For

example, Hayes (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 in other elections, making this a very robust

finding (Dolan and Holbrook, 2001; Granberg and Brent, 1983). The Sherman and

Kunda (1989) study described in the introduction also falls into this category.

Some of best evidence is obtained in studies in which subjects are randomly as-

signed into groups with different payoff-functions, and are then tested for differences

in their 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).

Cognitive-dissonance studies document a change in beliefs following a change in

the payoff-function, typically brought about by an action on the part of the subject.

The Knox and Inkster (1968) study described in the introduction is one example,

49The term ‘over-confidence’ is also used to describe the finding that people select too smallconfidence intervals to represent the accuracy of their predictions (Alpert and Raiffa, 1982).

30

documenting an increase in the subjective probability that a horse would win the

race following the placing of a bet on that horse. In a classic experiment by Festinger

and Carlsmith (1959) students took part in a boring task, following which some of

the students were put in a position where they had to tell another student that the

task is interesting. The finding was that subjects in the treatment group came to

believe the task to have been more interesting than subjects in the control group.

Such results are often explained by reference to a dissonance between an action

(e.g. recommending the task to another student) and a belief (e.g. that the task was

boring). The idea is that the dissonance is resolved by a change in belief (coming

to believe the task wasn’t so boring after all). Optimistic payoff-dependent belief

distortion provides an alternative explanation: recommending the task lowers the

payoff in the state in which the task was boring, which then reduces the subjective

probability of this state. Similarly, in Knox and Inkster (1968) placing the bet on

the horse increases the payoff from the horse winning the race (and also decreases

the payoff from the horse not winning), with the effect of increasing the subjective

odds-ratio between the two outcomes.

Whereas the previously mentioned studies are performed by social psychologists

looking to understand belief biases, studies of unrealistic pessimism are typically

conducted by researchers interested in depression. The focus is not on the average

bias in the population, but on differences between individuals. Seligman (1998) views

a number of studies documenting that some people are biased in the pessimistic

direction. In particular, such people tend to interpret failures as broad, long-lasting,

and personal, and successes as narrow, fleeting, and likely due to favorable external

circumstances.

A.2 Economics evidence

In Babcock and Loewenstein (1997)50 subjects allocated to the role of plaintiff and

defendant examined the evidence in a compensation trial. Subjects had an oppor-

tunity to reach a compromise, or else their payoff was determined by the judge’s

decision less ‘court costs’. Prior to the bargaining phase subjects made an incentive

compatible prediction of the judge’s decision. Consistent with belief bias, plaintiffs

predicted significantly higher award amounts than defendants. Moreover, the differ-

ence in predictions between each pair of subjects was predictive of their success in

reaching compromise.

In Mayraz (2011) subjects observed the price chart of a financial asset and received

50See also Loewenstein et al. (1993) and Babcock et al. (1995).

31

an accuracy bonus for predicting the price at a future point, as well as an uncondi-

tional award that was either increasing or decreasing in this price. Consistent with

wishful thinking, and despite incentives for hedging, subjects gaining from high prices

predicted significantly higher prices than those gaining from low prices. Increasing

the incentives for accuracy had no effect on the magnitude of the bias.

Malmendier and Tate (2005) argue that over-confidence would lead CEOs to over-

estimate the return to investments, and so over-invest when they have abundant

internal funds. They identify over-confident CEOs as those CEOs 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 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 (2007) 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.

B Proofs

Lemma 1

In all the four parts of Lemma 1 the proof that the requirements are necessary is

trivial. I thus prove only that the requirements are sufficient:

Part 1. Let a denote some arbitrary constant payoff-function. Define p = πa, and let

S∗ = {A ∈ S : p(A) > 0}. Define hf (A) = πf (A)/p(A) for A ∈ S∗ and hf (A) = 0

for A /∈ S∗. For A ∈ S∗ the claim follows from the definition of hf . By Absolute

Continuity p(A) = 0⇒ πf (A) = 0, and hence the claim holds also for A /∈ S∗.

Part 2. Let A ∈ S∗ and x ∈ X, let f(A, x) be the payoff-function mapping A to x and

all states outside A to a. Let E1, . . . , En denote the other events in S∗. By Minimal

Complexity and Absolute Continuity S∗ includes at least two events other than A.

32

f(A, x) and the constant payoff-function a agree on Ei and Ej for all i and j. Hence,

by Consequentialism with E = Ei ∪Ej, πf(A,x)(Ei)/πf(A,x)(Ej) = p(Ei)/p(Ej). Thus,

1− πf(A,x)(A) =∑i

πf(A,x)(Ei) =∑i

πf(A,x)(Ej)

p(Ej)p(Ei) =

πf(A,x)(Ej)

p(Ej)(1− p(A))

(22)

Define µA(x) =(

1−p(A)p(A)

)(πf(A,x)(A)

1−πf(A,x)(A)

). By Equation 22,

p(A)µA(f(A)) = (1− p(A))πf(A,f(A))(A)

1− πf(A,f(A))(A)= p(Ej) ·

πf(A,f(A))(A)

πf(A,f(A))(Ej)(23)

Let f be any payoff-function, and let A and B be any two events in S∗. Let f ′ be a

payoff-function that coincides with f on A and B, and with a elsewhere, and let C

be any third event in S∗. Inserting Ej = C in Equation 23 we obtain that

πf (A)

πf (B)=πf ′(A)

πf ′(B)=πf ′(A)/πf ′(C)

πf ′(B)/πf ′(C)=p(C)

p(C)·πf(A,f(A))(A)/πf(A,f(A))(C)

πf(B,f(B))(B)/πf(B,f(B))(C)=p(A)µA(f(A))

p(B)µB(f(B))

(24)

where the first and third steps follows from Consequentialism, and the final step from

Equation 23. Since Equation 24 holds for all A,B ∈ S∗ it follows that Equation 7

holds for any event A ∈ S∗. For an event A /∈ S∗, define µA(x) = 1 for all x. Since

πf (A) = p(A) = 0 for A /∈ S∗ Equation 7 holds however µA is defined. Combining

these results Equation 7 holds for any payoff-function f and any event A ∈ S.

Part 3. Let A∗ ∈ S∗ be some event. Define the mapping ν : X → R+ by ν(x) =

µA∗(x). For x ∈ X let x denote also the constant payoff-function yielding the payoff x

in all states. Inserting f = x and B = A∗ in Equation 24 we obtain that for all A ∈ S∗

and x ∈ X, πx(A)πx(A∗)

= p(A)p(A∗)

· µA(x)ν(x)

. Since x is a constant payoff-function it follows from

Indifference that πx = πa = p. Hence, µA(x) = ν(x). Thus, πf (A) ∝ p(A) · ν(f(A))

for all A ∈ S∗. Finally, this is also trivially true for A /∈ S∗, since πf (A) = p(A) = 0

for A /∈ S∗.

Part 4. Let A,B ∈ S∗ be two events, and let x and y be real-numbers such that x, y,

and x + y are in X. Define the payoff-functions fx and gx,y as follows: fx(s) = x for

s ∈ A and fx(s) = 0 for s /∈ A, and gx,y = fx + y. By Shift-Invariance, πgx,y = πfx ,

and in particular πgx,y(A)/πgx,y(B) = πfx(A)/πfx(B). By Equation 8 it follows that

ν(x + y)/ν(y) = ν(x)/ν(0). Hence, defining σ(x) = log(ν(x)/ν(0)) we obtain that σ

33

is linear, i.e. for all x and y, σ(x + y) = σ(x) + σ(y). For m ∈ N let y = mx. By

induction we obtain that σ(mx) = mσ(x). Similarly, for n ∈ N let y = x/n to obtain

that 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 defining ψ = σ(1), we obtain that for

any rational number q ∈ X, σ(q) = ψq, and so ν(q) = ν(0)eψq. Let now x ∈ X be any

feasible payoff-value, and let {qn}n∈N be a sequence of rational feasible payoff-values

converging to x. By prize-continuity πfqn → πfx , which given Equation 8 implies

that ν(qn) → ν(x). By the result for rational numbers, ν(qn) = ν(0)eψqn , and hence

ν(qn) → ν(0)eψx. Thus, ν(x) and ν(0)eψx are both the limit of the same sequence

of real-numbers, and so ν(x) = ν(0)eψx. Finally, since Equation 8 is invariant to

multiplying ν by a positive number, we obtain that Equation 9 holds for all x ∈ R.

Lemma 2

Proof. Let a ∈ F denote some constant payoff-function, and define p = πa. By

Minimal Complexity and Absolute Continuity there exists a finite partition S of the

state-space consisting of at least three events, such that πf (A) > 0 for any f ∈ F and

A ∈ S. Let Σ(S) ⊆ Σ denote the algebra generated by S, and let F (S) ⊆ F denote

the set of Σ(S)-measurable payoff-functions. By Lemma 1 there exists a probability

measure pS over (S,Σ(S)) and a parameter ψS ∈ R such that Equation 9 holds any

probability measure f ∈ F (S) and any event A ∈ S. In particular a ∈ F (S) (any

constant payoff-function is), and hence for any A ∈ S, p(A) = πa(A) ∝ pS(A)eψSa.

Thus, p(A) = pS for any event A ∈ S, and hence also for any event A ∈ Σ(S). Define

ψ = ψS . It follows that for any payoff-function f ∈ Σ(S) and any event A ∈ S,

πf (A) ∝ p(A)eψf(A).51

Let now A and B denote any events such that p(B) > 0, and let f be any payoff-

function. I need to show that πf (A)/πf (B) = (p(A)/p(B))eψ(f(A)−f(B)). To simplify

notation let δf (A,B) = log πf (A)/πf (B)− log p(A)/p(B). With this notation I need

to prove that δf (A,B) = ψ(f(A) − f(B)). Let E1, E2, . . . En denote the events in

S. Without limiting generality suppose A ∩ E1 is not-null. Define a payoff-function

g ∈ F by g = f(A) on A ∩ E1 and g = f(B) elsewhere, and a payoff-function

h ∈ F (S) by h = f(A) on E1 and h = f(B) elsewhere. With these definitions,

δf (A,B) = δf (A ∩ E1, B) = δg(A ∩ E1, B) = δg(A ∩ E1, B ∪ E2) = δg(A ∩ E1, E2) =

δh(A ∩ E1, E2) = δh(E1, E2) = ψ(f(A) − f(B)), where the last step uses the fact

that h is in F (S), and the other steps use Consequentialism and the fact that by

Shift-Invariance πf(A) = πf(B) = p.

51Note that p = πa is a probability measure over all the events in Σ—not just the events in Σ(S).

34

Theorem 1

Proof. Note first that the conditions of Lemma 2 obtain. Let p and ψ be parameters

for which the claim of the Lemma hold. For n ∈ N divide the interval [m,M ] into 2n

non-overlapping intervals I1n, I2n, . . . , I

2n

n of equal length, and define a simple payoff-

function fn by mapping each state s to the lower endpoint of the interval to which f(s)

belongs. With this definition 0 ≤ f−fn ≤ (M−m)/2n, and so fn → f uniformly. By

Prize Continuity, the assumption that p(B) > 0, and Absolute Continuity we obtain

thatπf (A)

πf (B)= lim

n→∞

πfn(A)

πfn(B)(25)

Since fn is simple there exists at least one non-null event on which fn is constant. Let

Cn denote such an event. Using Cn and Lemma 2 we can write πfn(A) as follows,

πfn(A)

πfn(B)=

∑kπfn (A∩Ikn)πfn (Cn)∑

kπfn (B∩Ikn)πfn (Cn)

=

∑kp(A∩Ikn)eψfn(Ikn)

p(Cn)eψfn(Cn)∑kp(B∩Ikn)eψfn(Ikn)

p(Cn)eψfn(Cn)

=

∑k p(A ∩ Ikn)eψfn(I

kn)∑

k p(B ∩ Ikn)eψfn(Ikn)=

∫Aeψfn dp∫

Beψfn dp

(26)

Since eψx is a continuous function it follows that eψfn → eψf . Define a constant-

payoff function g by g = M if ψ ≥ 0 and g = −m if ψ < 0. With this definition

|eψfn| ≤ eψg, and eψg is integrable, and hence by the dominated convergence theo-

rem limn→∞∫Eeψfn dp =

∫Eeψf for any event A. Combining this observation with

Equations 25 and 26 we obtain that

πf (A)

πf (B)= lim

n→∞

πfn(A)

πfn(B)= lim

n→∞

∫Aeψfn dp∫

Beψfn dp

=

∫Aeψf dp∫

Beψf dp

(27)

Lemma 4

A1. Let f and f ′ be payoff-functions and E an event, and suppose that πf (E) = 0. I

need to prove that πf ′(E) = 0. Let e and e′ be acts such that f = u◦e and f ′ = u◦e′.Since πf (E) = 0 it follows from Equation 12 that for any acts g and h that differ only

in E, Ve(g) = Ve(h), and so E is e-null. By B6 it follows that E is also e′-null. Let a

and b be constant acts such that a �e′ b, and let g be an act defined by g = b on E

and g = a outside E. By construction a and g differ only on E, and hence (since E

is e′-null) Ve′(a) = Ve′(g). Using Equation 12 it follows that πf ′(E)(u(a)− u(b)) = 0.

By assumption a �e′ b, and hence u(a)−u(b) > 0. Thus, πf ′(E) = 0 as required.

35

A2. Suppose f = f ′ over a non-null event E. I need to prove that for any event

A ⊆ E, πf (A|E) = πf ′(A|E). Let e and e′ be acts such that f = u ◦ e and f ′ = u ◦ e′.By B4 there exist constant acts a and b such that a � b. Let c be a constant act

such that u(c) = πf (A|E)u(a) + (1 − πf (A|E))u(b). Define c by c = c on E and

c = a outside E. Let g = b on E \ A and g = a elsewhere. By Equation 12,

g ∼e c. Let e′′ = e on E and e′′ = e′ outside E. Since f ′ = f on E it follows that

u ◦ e′′ = f ′. Since e′′ = e on E and g = c outside E it follows from B7 that g ∼e′′ c.By Equation 12, πf ′(A)u(a)+πf ′(E \A)u(b) = πf ′(E)u(c), and hence πf ′(A|E)u(a)+

(1−πf ′(A|E))u(b) = u(c). Since u(c) = πf (A|E)u(a)+(1−πf (A|E))u(b) we conclude

that πf (A|E) = πf ′(A|E).

A3. Suppose f ′ = f + x for some real number x. I need to show that πf = πf ′ .

Without limiting generality assume that x ≥ 0. If f is constant then there exist

constant acts a and b such that f = u ◦ a and f ′ = u ◦ b, and by B8 with g = a

and α = 0 we obtain that �a=�b, which by Lemma 3 implies that πf = πf ′ as

required. Suppose, therefore, that f is not constant, and let µ and ν denote the

minimum and maximum values of f respectively. Let α = (ν − µ)/(x + ν − µ),

and define a payoff-function f ′′ by f ′′ = µ + (f − µ)/α. Since f − µ ≤ ν − µ it

follows that µ ≤ f ′′ ≤ ν + x. Hence, there exists an act g such that f ′′ = u ◦ g.

Define acts c and c, such that u(c) = µ and u(c) = ν + x. Let e = αg + (1 − α)c

and e′ = αg + (1 − α)c. With this definition u ◦ e = αf ′′ + (1 − α)µ = f and

u ◦ e′ = αf ′′ + (1− α)(ν + x) = f + (1− α)(x+ ν − µ) = f + x = f ′. Finally, by B8

we obtain that �e=�e′ , which by Lemma 3 implies that πf = πf ′ as required.

A4. Suppose that fn → f uniformly. I need to prove that πfn → πf . Let e be any

act. By B5 there exist constant acts a and a, such that a �e g �e a for any act g. In

particular there exists mappings β : S → [0, 1] and (for any n) βn : S → [0, 1] such

that f = (1− β)u(a) + βu(a) and fn = (1− βn)u(a) + βnu(a). Define acts e′ and (for

any n) en by e′ = (1−β)a+βa and en = (1−βn)a+βna. By construction, f = u ◦ e′

and for any n, fn = u ◦ en. Since fn → f uniformly u(e′) → u(en) uniformly. By

construction, u(e′)− u(en) = (β− βn)(u(a)− u(a)) and hence βn → β uniformly, and

so en → e′ uniformly. By B9 it follows that �en→ �e′ .Suppose, by way of contradiction, that for some ε > 0 and any N > 0 there

exists n > N such that πfn(E) < πf (E) − ε. In particular, there exists a constant

act b such that for any N > 0 there exists n > N such that πfn(E) < (u(b) −u(a))/(u(a) − u(a)) < πf (E). Define an act h by h = a on E and h = a outside E.

Using Equation 12 we obtain that h �e′ b but for any N > 0 there exists n > N such

that b �en h. But this cannot be the case, since �en→ �e′ .

36

Theorem 2 (uniqueness part)

Proof. Note first that the utility function is determined up to a positive affine trans-

formation by preferences over constant Anscombe-Aumann acts. Hence, if the triplet

(p′, u′, ψ′) represents � there exists real numbers α > 0 and β such that u′ = αu+ β.

Next, let e be any act and A any event. By B4 there exist constant acts a and b

such that a �e b. Let gA be an act defined by gA = a on A and gA = b on Ac.

Let cA be a constant act defined by cA = πu◦e(A)a + (1 − πu◦e(A))b. By Equa-

tion 11 gA ∼e cA. Similarly, let c′A = πu′◦e(A)a + (1− πu′◦e(A))b then also gA ∼e c′A,

and so cA ∼e c′A. Since a �e b it follows that u(a) > u(b), and hence by Equa-

tion 11 πu′◦e(A) = πu◦e(A). Since this is true for all A it follows that for any act

e, πu′◦e = πu◦e. I now use this observation to show first that p′ = p and then that

ψ′ = ψ/α. First, let e be a constant act. By Equation 5 πu◦e = p and πu′◦e = p′. Hence

it follows from the above observation that p′ = p. Finally, by Minimal Complexity

and B6 there exists an event A such that neither A nor Ac is �e null for any act e.

Applying the above observation to gA we obtain that πu′◦gA = πu◦gA . By Equation 5,

log πu◦gA(A)/πu◦gA(Ac) = log p(A)/p(Ac) + ψ(u(a) − u(b)). Using the corresponding

equation for πu′◦gA , the result that p′ = p, and the fact that u′ = αu + β, we obtain

that ψ(u(a)− u(b)) = ψ′(u′(a)− u′(b)) = αψ′(u(a)− u(b)). Hence ψ′ = ψ/α.

Proposition 1

Proof. By Equation 5, Πf (x) ∝∫ x−∞ h(x) dPf (x) with h(x) = eψx. If ψ ≥ 0, eψx is

non-decreasing, and hence Πf �LR Pf . If ψ ≤ 0, h′ = e−ψx is non-decreasing, and

Pf (x) ∝∫ x−∞ dPf (x) =

∫ x−∞ h

′(x)eψxdPf (x) =∫ x−∞ h

′(x) dΠf , and so Pf �LR Πf .

The first part of the claim follows since stochastic dominance in the likelihood ratio

implies first-order stochastic dominance.

Proposition 2

Proof. I prove only that limψ→∞ πf (Amax) = 1, as the proof that limψ→−∞ πf (Amin) =

1 is very similar. Let x1 > x2 > · · ·xn denote the payoffs in the range of f . Thus,

Amax = f−1(x1), and by Equation 14,

limψ→∞

πf (Amax)−1 = lim

ψ→∞

(p(f−1(x1))e

ψx1∑i p(f

−1(xi))eψxi

)−1= 1 +

∑i>1

p(f−1(xi))

p(f−1(x1))· limψ→∞

eψ(xi−x1)

and since xi < x1 for i > 1, the last term on the RHS is zero. Hence, limψ→∞ πf (Amax) =

1 as required.

37

Proposition 3

Proof. By Equation 5 and the assumption that PX ∼ N (µ, σ2),

ΠX(x) ∝∫ x

−∞eψf(x)p(X = x) dx =

∫ x

−∞eψ(ax+b)

(1√2πσ

e−(x−µ)2

2σ2

)dx

= eψb∫ x

−∞

1√2πσ

e−(x−µ)2−2ψaσ2x

2σ2 dx = eψb∫ x

−∞

1√2πσ

e−(x−(µ+ψaσ2))2−ψ2a2σ4

2σ2 dx

∝∫ x

−∞

1√2πσ

e−(x−(µ+ψaσ2))2

2σ2 dx = N (µ+ aψσ2, σ2)

Proposition 4

Proof. I need to show that there exist a non-decreasing function h′, such that for all x,

F ′(x) ∝∫ x−∞ h

′ dG′. By assumption there exist a non-decreasing function h such that

F (x) ∝∫ x−∞ h dG, and so dF ∝ hdG. Moreover, by Equation 5, F ′(x) ∝

∫ x−∞ e

ψxdF

and G′(x) ∝∫ x−∞ e

ψxdG. From the latter we obtain that dG′ ∝ eψxdG, and so

F ′(x) ∝∫ x

−∞eψxdF ∝

∫ x

−∞eψxhdG ∝

∫ x

−∞eψxhe−ψxdG′ =

∫ x

−∞hdG′ (28)

and so F ′ �LR G′ with h′ = h.

Proposition 5

Proof. For any x ∈ R, πf (X = x|E) =∫πf (X = x, Y = y|E) dy ∝

∫p(X =

x, Y = y|E)eψf(x,y) dy = p(X = x)∫p(Y = y|E)eψf(x,y) dy. Hence, for any two values

xH , xL ∈ R,

πf (X = xH |E)

πf (X = xL|E)=p(X = xH)

p(X = xL)

∫p(Y = y|E)eψf(xH ,y) dy∫p(Y = y|E)eψf(xL,y) dy

(29)

Similarly, πf (X = x) ∝ p(X = x)∫p(Y = y′)eψf(x,y

′) dy′, and so

πf (X = xH)

πf (X = xL)=p(X = xH)

p(X = xL)

∫p(Y = y′)eψf(xH ,y

′) dy′∫p(Y = y′)eψf(xL,y′) dy′

(30)

ΠX|E �LR ΠX (ΠX �LR ΠX|E) if and only if whenever xH ≥ xL the expression in

Equation 29 is greater than or equal (smaller than or equal) than the expression

in Equation 30. Equivalently, if the following expression is weakly positive (weakly

38

negative): ∫p(Y = y|E)eψf(xH ,y) dy∫p(Y = y|E)eψf(xL,y) dy

−∫p(Y = y′)eψf(xH ,y

′) dy′∫p(Y = y′)eψf(xL,y′) dy′

(31)

Moreover, the stochastic dominance is strict if this expression is strictly positive(negative).

The expression in Equation 31 has the same sign as∫y

∫y′p(Y = y|E)p(Y = y′)

(eψ(f(xH ,y)+f(xL,y

′)) − eψ(f(xH ,y′)+f(xL,y)))

dy′ dy (32)

combining terms in which y < y′ with terms in which y > y′ the expression in

Equation 32 equals the following:∫y

∫y′<y

(p(Y = y|E)p(Y = y′)− p(Y = y′|E)p(Y = y)) ·(eψ(f(xH ,y)+f(xL,y

′)) − eψ(f(xH ,y′)+f(xL,y)))

dy′ dy

(33)

In this expression the first term is (strictly) positive if p(E|Y = y) is (strictly) in-

creasing in y, and y is not a.e. constant. Since y > y′ and log is a strictly increasing

function, the second term has (strictly) the same sign as ψ if f is (strictly) supermod-

ular, and (strictly) the opposite sign if f is (strictly) submodular. The claim follows

by combining these observations.

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