Priors and Desires: a Model of Optimism,Pessimism, and Cognitive Dissonance

Guy Mayraz

October 29, 2014

Abstract

This paper offers a model of optimism, pessimism, and cognitive dis-sonance. Beliefs—and consequently choices—depend not only on relevantinformation, but also on what makes the decision maker better-off. In anassociated experiment, subjects who stood to gain from an increase in theprice of a financial asset predicted higher prices than subjects who stood togain from a decrease in price. Consistent with the model, better informationresulted in a smaller bias, but incentives for accuracy made no difference.

JEL classification: D01,D03,D80,D81,D83,D84.Keywords: wishful thinking, cognitive dissonance, reference-dependent be-liefs, reference-dependent preferences.

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1 IntroductionWhenever people think of a possible state of affairs, they evaluate it on two di-mensions: good or bad and true or false. Normatively, the two are independent:wanting something to be true doesn’t make it any more or less likely. Nevertheless,when people consider the evidence they are very much aware what conclusion theywant it to support: they want the evidence about their investments to indicate lowrisks and high returns, the evidence about their health to suggest they have little toworry about, the evidence about their actions to imply that they did the right thing,and so on.

If there is only one way of interpreting the evidence, it makes no differencehow people feel about it. But if the evidence is open to multiple interpretations,what people want it to mean can affect what they take it to mean. Optimists aremore likely to believe something is true if it makes them better off, and pessimistsare more likely to believe something is true if it makes them worse off. This paperoffers a formal model of optimism and pessimism, and reports on an experimentaltest of some of the most important predictions of the model.

Whatmakes optimists and pessimists different is not theway theymake choices,but the systematic link between their beliefs and their interests (what they have atstake). The model represents these stakes by a reference mapping 𝑟 from states toutility values and the resulting beliefs by a reference-dependent probability mea-sure 𝜋𝑟. With the help of some simplifying assumptions it is possible to obtaina tractable representation for 𝜋. The formula involves a probability measure 𝑝,representing beliefs under conditions of indifference, and a parameter 𝜓 , whichdetermines how the agent’s interests affect her beliefs. The expression for the logof the relative probability (odds-ratio) between two states 𝑎 and 𝑏 is as follows:

log 𝜋𝑟(𝑎)𝜋𝑟(𝑏) = log 𝑝(𝑎)

𝑝(𝑏) + 𝜓[𝑟(𝑎) − 𝑟(𝑏)]. (1)

The first term on the RHS of Equation 1 represents what the subjective odds-ratio between the two states would have been if the agent were indifferent betweenthe two states, so that 𝑟(𝑎) = 𝑟(𝑏). The second term represents the bias. 𝜓 is aparameter which varies from one person to another. If 𝜓 = 0 the bias term isalways zero. A zero 𝜓 therefore represents realism: whatever the agent has atstake has no effect on her beliefs. If 𝜓 > 0 the bias term has the same sign as thedifference in the reference stakes between the two states, biasing beliefs towards thestate in which the agent is better off. A positive 𝜓 therefore represents optimism.Finally, if 𝜓 < 0 beliefs are biased towards the state that makes the agent worse

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off. A negative 𝜓 represents pessimism. In analogy with constant relative riskaversion, 𝜓 is the the coefficient of relative optimism. The further it is from zero,the stronger the bias.

Holding 𝜓 constant, the magnitude of the bias term is an increasing functionof the reference stakes 𝑟(𝑎) − 𝑟(𝑏): the more a person wants something to be true,the stronger the pull of her optimism or pessimism. The strength of the evidence(represented by the first term in Equation 1) provides resistance. If, for example,there is strong evidence in favor of 𝑎, this term will be highly positive, and the biasterm would have to be strongly negative for the agent to believe that 𝑏 is true.

Optimism and pessimism bias beliefs whenever a person has something atstake. One example is holding an investment in a financial asset, which causesan optimist to underestimate risks and overestimate returns, with the opposite forpessimists. However, ‘having something at stake’ is a much broader concept thanthat, and could also be something like a person’s inherent interest in her health(causing optimists to be overly sanguine about their health, and pessimists to worrytoo much), in her ability (optimists being overconfident about abilities that matterto their future payoff), and even in morality (an optimist who cares about behavingmorally is biased to believe her past actions were morally justified).

Biased beliefs affect subsequent choices: underestimating risks reduces thedemand for insurance, and makes further investment more likely; overestimatingability causes selection into competition, and the pursuit of conflict over compro-mise; believing previous actions were moral makes it more likely that the personrepeats them. Some of the most interesting implications are in dynamic choice,where today’s choice acts as the reference stakes for tomorrow’s beliefs, and therebyinfluences tomorrow’s choices.

The belief patterns predicted by optimism have been observed in many areasof economics,1 consistent with the idea that wishful thinking is a pervasive biasthat affects decisions large and small. However, while this evidence is certainlysuggestive, it is nearly always possible to find alternative explanations for each

1Babcock and Loewenstein (1997) link the low frequency of pretrial bargains to a tendencyby both parties to believe that they would win if the case ends up in court. Olsen (1997) findsevidence for optimistically biased beliefs among professional investment managers. Camerer andLovallo (1999) link excess entry into competitive markets to overconfidence over relative ability.Malmendier and Tate (2008) argue that managerial overconfidence is responsible for corporate in-vestment distortions. Cowgill et al. (2009) find optimistic bias in corporate prediction markets.Park and Santos-Pinto (2010) provide field evidence for overconfidence in tournaments. Hoffman(2011a) and Hoffman (2011b) finds that truck drivers are optimistically biased about their produc-tivity (and hence their pay), resulting in an inefficient failure to switch jobs.

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particular finding.2The economics evidence is much less extensive, but includes a wide variety of

situations.3 Theoretical applications extend into additional areas.4This paper offers a descriptive model of optimism and pessimism, but it does

not explain why they exist in the first place. Economists who tackle this questionassume that biased beliefs have to be somehow utility maximizing. The main ideais that people have preferences not only over the outcome they obtain, but alsoover their anticipatory beliefs when the outcome is still uncertain (Akerlof andDickens, 1982; Carrillo and Mariotti, 2000; Caplin and Leahy, 2001; Benabouand Tirole, 2002; Brunnermeier and Parker, 2005). Since optimism leads to afirst-order improvement in anticipatory beliefs with only a second-order loss inoutcomes, the utility maximizing level of optimism bias is positive (Brunnermeierand Parker, 2005).5

The practical implications of this idea depend on people’s ability to intention-ally bias their beliefs. Obviously, it only has observable implications if peoplehave some sort of ability to bias their beliefs. If people can choose their coef-ficient of relative optimism, we would obtain a preponderance of optimists overpessimists, for which there is good evidence.6 If people can choose their beliefs

2For example, in Babcock and Loewenstein (1997) subjects in the role of plaintiff came toexpect higher penalties than subjects in the role of defendant, even though both groups of subjectswere exposed to the same case materials. However, subjects had to argue their side with the otherparty, which may have caused them to focus their reading on arguments favoring their case. Theirbeliefs could thus have arisen from a failure to correct for this selective attention, rather than froma general wishful thinking bias.

3 Babcock and Loewenstein (1997) find that parties in negotiations are affected bywishful think-ing, resulting in an inefficient failure to reach agreement. Camerer and Lovallo (1999) link excessentry into competitive markets to overconfidence over relative ability. Malmendier and Tate (2008)argue that managerial overconfidence is responsible for corporate investment distortions. Cowgillet al. (2009) find optimistic bias in corporate prediction markets. Mullainathan and Washington(2009) find that voting for a candidate results in more positive views about the candidate. Park andSantos-Pinto (2010) provide field evidence for overconfidence in tournaments. Hoffman (2011a)and Hoffman (2011b) finds that truck drivers are optimistically biased about their productivity (andhence their pay), resulting in an inefficient failure to switch jobs.

4For example, credit markets De Meza and Southey, 1996, banking Manove and Padilla, 1999,corporate finance Heaton, 2002, search Dubra, 2004, savings Brunnermeier and Parker, 2005, in-surance Sandroni and Squintani, 2007, price discrimination Eliaz and Spiegler, 2008, incentives inorganizations Santos-Pinto, 2008, and financial contracting Landier and Thesmar, 2009. Studiesof overconfidence over the accuracy of signals are excluded from this list.

5Biased beliefs may also be instrumental in counteracting the impact of other behavioural bi-ases (Benabou and Tirole, 2002; Compte and Postlewaite, 2004).

6See, for example, Sharot (2011).

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directly we would expect a second optimistic bias that is determined on a case bycase basis. Since the cost of the bias varies with the importance of subsequent de-cisions, the case-by-case bias should be inversely proportional to the importanceof subsequent decisions. This makes it very different from the optimism bias inEquation 1, where the underlying bias (represented by the value of 𝜓) has nothingto do with the importance of subsequent decisions.

This difference is particularly important when a lot depends on the decision athand, which is precisely the sort of situation economists care most about. It wouldbe irrational to choose much of a bias in this situation, but Equation 1 has nothingto do with rational choice. If there is a great deal of uncertainty, and the agentarrives at the decision with a significant existing investment, the model predicts alarge bias regardless of the consequences.

The terms ‘optimism’ and ‘pessimism’ can also refer to a tendency to expectgood (bad) outcomes whatever one does (and regardless of one’s current inter-ests). This very different notion of optimism and pessimism is briefly mentionedby Quiggin (1982), discussed more broadly in Hey (1984), and further developedin a couple of recent papers (Bracha and Brown, 2012; Dillenberger et al., 2014).As noted already by Quiggin (1982) it can be difficult to distinguish from standardrisk preferences (or preferences over ambiguity, if probabilities are only distortedin ambiguous situations), and is perhaps best seen as an alternative interpretationof such preferences. Indeed, Bracha and Brown (2012) show that their model ofoptimism is the mirror image of the variational preferences model of ambiguityaversion (Maccheroni et al., 2006), and Dillenberger et al. (2014) show an equiva-lence between their ownmodel and standard subjective expected utility maximiza-tion with a more convex (concave) utility function.

The empirical part of the paper uses a simple lab experiment to test for thepredictions of wishful thinking. Subjects in the experiment observed a chart ofhistorical wheat prices,7 and their one and only task was to predict what the pricewould be at some future time point. There was random assignment into two treat-ment groups: Farmers, whose payoff was increasing in the future price of wheat,and Bakers, whose payoff was decreasing in this price. Subjects in both groupsalso received a performance bonus as a function of the accuracy of their predic-tion.

Wishful thinking predicts bias whenever decision makers have a stake in whatthe state of the world is. Farmers gain from high prices, and their beliefs shouldtherefore be biased upward as compared to what they would otherwise be. The op-

7Charts were adapted from real asset price data, though not specifically wheat prices.

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posite is true for Bakers. Given the random allocation, there should be a systematicdifference in beliefs between the two groups, with Farmers expecting higher pricesthan Bakers.

The statistic used to identify a systematic difference in beliefs between the twogroups was the difference between the average predictions of Farmers and Bakers.The prediction bonus formula was designed so that truthful reporting maximizessubjective expected payoff. As long as decision makers are risk-neutral over smallamounts of money, the difference in predictions should provide an unbiased es-timate of the difference in beliefs. Risk-averse subjects may, however, seek tointentionally hedge their predictions, so as to smooth their payoff across differentstates. Such hedging would result in Farmers under-reporting their true prediction,and an opposite bias for Bakers. Consequently, the estimated difference in beliefsbetween the two treatment groups may be biased downward.

The null hypothesis was defined as a non-positive difference in beliefs betweenFarmers andBakers. Hedging could plausibly have resulted in a failure to reject thenull when the true difference in beliefs is positive. There were no correspondingreasons to expect a false positive result. The actual observation was a positiveand statistically measurable difference in predictions between Farmers and Bakers(𝑝 < 0.0002).

This result demonstrates that wishful thinking can indeed affect at least somesubjective judgments of likelihood. However, many economically important de-cisions involve much higher stakes, and it is not clear whether wishful thinkingwould remain significant if its cost had been large. According to the Prior andDesires model, the magnitude of the bias is independent of the importance of sub-sequent decisions, so the answer is yes. According to self-deception models suchas Brunnermeier and Parker (2005), the magnitude of the bias is decreasing in itscost, so the answer is no.

Differentiating between these two modelling approaches requires the ability tomanipulate the incentives for holding accurate beliefs. The design of the experi-ment 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 biasedbeliefs. If wishful thinking is strategic (as in self-deception), the magnitude of thebias should decrease in the scale of the accuracy bonus. If wishful thinking is nonstrategic (as in the Priors and Desires model), there should be no change in themagnitude of the bias as the scale of the accuracy bonus is increased.

Converting this intuition into a formal test requires quantitative predictions.‘No change’ is a testable hypothesis, but ‘decreasing with the scale of accuracybonus’ is not. Consequently, testing the hypothesis that wishful thinking is strate-

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gic made it necessary to focus on some particular strategic model. The best knownsuch model is Optimal Expectations Brunnermeier and Parker, 2005. Agents inthis model have preferences over anticipated consumption, and choose beliefs inorder to maximize their subjective expected utility. The constraint is that, oncechosen, beliefs govern future choices and change only as the result of Bayesian up-dating. Agents therefore trade-off the gain from anticipating a high payoff, againstthe cost in a lower realized bonus: the more favorable they believe the future priceto be, the higher is their anticipatory utility, but the lower the prediction bonusthey can expect to receive. Increasing the scale of the accuracy bonus increasesthe cost of biased beliefs and reduces the optimal level of bias. Assuming risk-neutrality over small stakes, the quantitative prediction is that the magnitude ofthe bias would be inversely proportional to the scale of the accuracy bonus (Sec-tion 3.3.2).

Different sessions were run with different levels of accuracy bonus. The scaleof the bonus was increased five fold, with the maximum bonus amount varyingfrom £1 to £5. Results showed no decrease in the magnitude of the bias, consis-tent with the prediction of non strategic models. This result is statistically mea-surable: 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 experi-ment, therefore, corroborates wishful thinking in its non strategic version. This, ofcourse, is the version with the most far-reaching implications, implying that wish-ful thinking affects any and all decisions based on subjective judgment, whateverthe cost to the decision maker.8

Both types of models also prediction that the magnitude of the bias increasein the degree of subjective uncertainty and in what subjects have at stake in thequantity that they form expectations over (the sensitivity of the final payoff to theday 100 price in the context of the present experiment). The prediction of bothmodels is that the magnitude of the bias increases in both these factors. Testingthese predictions cannot provide a further test of which model is correct, but itcan provide some further assurance that the experiment is sensitive enough for themain conclusions to be trusted.

In order to make a test of the comparative statics of subjective uncertaintypossible, subjects were asked to provide a confidence level together with their pre-diction. Confidence was provided on a 1-10 scale, calibrated with the help of

8Consistent with this result, Hoffman (2011a) and Hoffman (2011b) finds substantial overcon-fidence in the trucking industry, and shows that it is highly costly to workers. Prediction accuracyis not reduced by adding monetary incentives.

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examples provided as part of the instructions (Figure 3). By averaging the confi-dence reports across subjects, it was possible to obtain an estimate of the amountof subjective uncertainty in different charts. This made it possible to test the pre-diction that the bias in high subjective uncertainty charts is greater. Results wereconsistent with this prediction (Figure 5), and the null hypothesis that the magni-tude 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 yieldedcomparable results.

Due to insufficient data, a test of the comparative statics of the stakes wasinconclusive. Two sessions were run with half the stakes, and the estimated biaswas roughly half what it was in the baseline sessions. However, the null hypothesisthat that the bias is the same could not be rejected.9

One concern with interpreting the results of the experiment is that subjects mayhave felt the task of predicting the day 100 price is impossible, and that they mayas well choose whichever number they want to be true. Since Farmers gain fromhigh 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 ex-pect subjects who are generally confident in their predictions to be less biased thanless confident subjects. Similarly, we would expect subjects who generally believeprices in financial markets are predictable to be less biased than subjects who donot think prices can be predicted. I tested the first prediction by defining a sub-ject’s confidence level by the average confidence rating in her predictions acrossall charts. I tested the second prediction by asking subjects in the post experimentquestionnaire whether they believe that prices in financial markets are generallypredictable. In both cases I obtained just the opposite result: subjects who be-lieve prices are predictable and relatively confident subjects are more biased thanthose who are less confident. These results suggest that this concern is misplaced.Moreover, they support the view that over-confidence is a manifestation of wish-ful thinking, and that the degree of wishful thinking bias is a stable individualcharacteristic, such as the parameter 𝜓 in Equation 1.

The reminder of the paper is organized as follows. Section 3 describes theexperiment in detail. Section 3.3 develops the predictions of the Optimal Expecta-tions Brunnermeier and Parker, 2005 and Priors andDesiresMayraz, 2011models.Section 3.4 describes how the data were analyzed. Section 4 presents the results,

9The comparative statics of the stakes have been studied before in a different but related con-text. In a study of self-deception Mijović-Prelec and Prelec (2010) found a larger bias when stakeswere higher (the ‘Anticipation Bonus’ treatment) as compared with lower bonus (the ‘ClassificationBonus’ treatment).

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and Section 5 concludes.

2 ModelThe model is intended specifically for decisions under subjective uncertainty. AsKnight (1921) famously noted, this is a very large class of decision problems.10The judgment of probability that decision makers are required to make in suchsituations offers a plausible route for optimism and pessimism to affect beliefs.Moreover, since reasonable people make different probability judgments, opti-mism and pessimism can affect beliefs without this being obvious either to thedecision maker herself or to outside observers.

Subjective uncertainty is represented using a set of states, each of which de-notes some possible set of affairs. The decision maker is assumed to have well-defined probabilities over the set of states. More formally, uncertainty is definedover a measurable-space (𝑆, Σ), where 𝑆 is the set of states, and Σ is a 𝜎-algebraof subsets of 𝑆 called events. The range of possible beliefs is represented by theset Δ of all 𝜎-additive probability measures over (𝑆, Σ). Many decision problemscan be modeled using a finite set of states, in which case Δ is simply the set of allpossible probability distributions.

Although we are principally concerned with choices under uncertainty, the de-cision maker’s world is assumed to contain a source of objective uncertainty (risk).This makes it possible to identify the utility function independently of subjectivebeliefs by observing the decision maker’s choices over objective lotteries. In thefollowing I assume that the utility function is known.

The particular outcomes that the decision maker obtains in different states playno role in the model—all that matters is the utility value associated with these out-comes. This makes it possible to use mappings from states to utility values (payoff-functions) to represent the decision maker’s reference stakes and the choices thatshe has available.

The model can now be formally described for the case of a single decisionproblem. Let 𝐹 = {𝑓 ∶ 𝑆 → ℝ} denote the set of all payoff-functions. At 𝑡 = 0the decision maker starts out with a reference 𝑟 ∈ 𝐹 ; at 𝑡 = 1 she makes a choice 𝑐

10“Business decisions, for example, deal with situations which are far too unique, generallyspeaking, for any sort of statistical tabulation to have any value for guidance. The conception of anobjectively measurable probability or chance is simply inapplicable.” (III.VII.47); “Yet it is true,and the fact can hardly be overemphasized, that a judgment of probability is actually made in suchcases.” (III.VII.40).

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from a choice set 𝐶 ⊆ 𝐹 ; at 𝑡 = 2 uncertainty is resolved: some particular state 𝑠∗

is revealed to be the true state, and the decision maker obtains the outcome 𝑐(𝑠∗).At this point it is worth reviewing how a standard subjective expected utility

maximizer would behave in this setting. Such a decision maker would choose 𝑐to maximize subjective expected utility according to some probability measure𝑞 ∈ Δ. The reference 𝑟 would play no role in her decision.

Decision makers in the present model also maximize subjective expected util-ity, but their beliefs are reference-dependent. Each decision maker is thus char-acterized by a distortion mapping 𝜋 ∶ 𝐹 → Δ, which associates each possiblereference with a probability measure over the set of states. If the reference is 𝑟 thedecision maker chooses 𝑐 to maximize subjective expected utility according to 𝜋𝑟.The standard model with reference-independent beliefs corresponds to the specialcase where 𝜋 is a constant mapping.

Simplifying assumptions (Appendix A) yield a tractable representation for 𝜋.Decision makers are characterized by the combination of (i) a probability measure𝑝 ∈ Δ, which represents the decision maker’s beliefs in the special case that she isindifferent between all states, and (ii) a real-valued parameter 𝜓 , which determineshow beliefs are distorted away from 𝑝 as a function of what she has at stake. Let𝑟 ∈ 𝐹 denote the decision maker’s reference, let 𝑎 and 𝑏 denote any two states,and suppose that 𝑝(𝑏) > 0. The log odds-ratio between the two states is given bythe following expression:

log 𝜋𝑟(𝑎)𝜋𝑟(𝑏) = log 𝑝(𝑎)

𝑝(𝑏) + 𝜓[𝑟(𝑎) − 𝑟(𝑏)]. (2)

In order to understand this expression, note first that if 𝑟(𝑎) = 𝑟(𝑏) the secondterm on the RHS drops out, and 𝜋𝑟(𝑎)/𝜋𝑟(𝑏) = 𝑝(𝑎)/𝑝(𝑏). The probability measure𝑝 therefore represents the beliefs of a disinterested observer for whom every stateis as good as any other.

Suppose now that the decision maker is not indifferent, and 𝑟(𝑎) − 𝑟(𝑏) > 0.If 𝜓 = 0 this makes no difference. A decision maker with 𝜓 = 0 is therefore arealist, and holds the same beliefs regardless of her interests. If 𝜓 > 0 the entireterm is positive, pushing beliefs towards 𝑎. Such a decision maker is an optimist.Finally, if 𝜓 < 0 the distortion term is negative, pushing the decision maker’sbeliefs towards 𝑏. Such a decision maker is a pessimist.

The parameter 𝜓 determines not only whether the decision maker is an opti-mist or pessimist, but also the strength of her bias. In analogy with risk aversion,𝜓 is the coefficient of relative optimism. Note that 𝜓 is defined relative to a given

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utility function representation. Any rescaling of the utility function has to be ac-companied by an inverse rescaling of 𝜓 .11

There are a number of other useful expressions for 𝜋. The expression for theprobability of a state 𝑠 is

𝜋𝑟(𝑠) ∝ 𝑝(𝑠)𝑒𝜓𝑟(𝑠). (3)

In log terms it becomes the equation for a line, with 𝜓 as the slope:

log 𝜋𝑟(𝑠) = log 𝑝(𝑠) + 𝜓𝑟(𝑠) + 𝐶 (4)

The most general expression is the following, where 𝐴 is any event:

𝜋𝑟(𝐴) ∝ ∫𝐴𝑒𝜓𝑟𝑑𝑝. (5)

Consider again Equation 2, and define 𝛿 = 𝑟(𝑎) − 𝑟(𝑏) as a measure of thedecision maker’s stake in 𝑎 rather than 𝑏 obtaining. Other things being equal, thebias scales with 𝛿. As an illustration, suppose 𝑝(𝑎) = 𝑝(𝑏) = 1/2, and consider anoptimist with 𝜓 = log 2. The odds ratio 𝜋𝑟(𝑎)/𝜋𝑟(𝑏) with 𝛿 = 1 would then be 2,corresponding to 𝜋𝑟(𝑎) = 2/3 and 𝜋𝑟(𝑏) = 1/3. If 𝛿 = 2 the odds-ratio would be4, corresponding to 𝜋𝑟(𝑎) = 4/5 and 𝜋𝑟(𝑏) = 1/5. Pessimism is the mirror imageof optimism. For example, a pessimist with 𝛿 = 2 and 𝜓 = − log 2 would have anodds ratio of 1/4, corresponding to 𝜋𝑟(𝑎) = 1/5 and 𝜋𝑟(𝑏) = 4/5.

The magnitude of the bias is also dependent on the weight of the evidence,represented by the indifference beliefs 𝑝. In the above examples the odds-ratio is𝑝(𝑎)/𝑝(𝑏) = 1, corresponding to equal evidence on both sides. The bias ismuch lessif the evidence leans strongly on one side. Suppose, for example, that the evidencefavors 𝑏 with 𝑝(𝑎)/𝑝(𝑏) = 1/4, corresponding to 𝑝(𝑎) = 1/5 and 𝑝(𝑏) = 4/5. With𝛿 = 1 and 𝜓 = log 2, the odds-ratio would double to 1/2, raising the probabilityof the good state only to 1/3. A coefficient of relative optimism larger than log 4would be necessary for 𝜋𝑟(𝑎) to rise above 1/2.

These comparative statics are readily seen in the case of a normal distributionwith linear reference stakes. Consider the effect of holding a financial asset onthe beliefs of a risk neutral investor. States represent the price of the asset, andthe initial stakes are a linear function of the state (the slope corresponding to thesize of the investment). As the following proposition shows, if beliefs are given by

11If we replace the utility function 𝑢 by a different utility function 𝑢′ related to 𝑢 by a positiveaffine transformation (𝑢′ = 𝑎𝑢 + 𝑏 where 𝑎 > 0) we need to replace 𝜓 by a different coefficient ofrelative optimism 𝜓′ = 𝜓/𝑎.

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a normal distribution over asset prices, the effect of the bias is to shift the entiredistribution, the shift being proportional (i) to the size of the investment, and (ii)the variance of the distribution:

Proposition 1 (normal distribution). Let 𝑃 and Π𝑟 denote the cumulative probabil-ity distribution functions that correspond to 𝑝 and 𝜋𝑟, and let 𝜓 denote the decisionmaker’s coefficient of relative optimism. Suppose 𝑆 = ℝ, 𝑃 ∼ 𝒩 (𝜇, 𝜎2) for some𝜇, 𝜎 ∈ ℝ and 𝑟(𝑠) = 𝑎𝑠 + 𝑏 for some 𝑎, 𝑏 ∈ ℝ, then Π𝑟 ∼ 𝒩 (𝜇 + 𝜓𝑎𝜎2, 𝜎2).

Proof. By Equation 5 and the assumption that 𝑃 ∼ 𝒩 (𝜇, 𝜎2),

Π𝑟(𝑠) ∝ ∫𝑠

−∞𝑒𝜓𝑟(𝑠)𝑑𝑝 = ∫

𝑠

−∞𝑒𝜓(𝑎𝑠+𝑏)

(1

√2𝜋𝜎𝑒− (𝑠−𝜇)2

2𝜎2

)𝑑𝑠

= 𝑒𝜓𝑏∫

𝑠

−∞

1√2𝜋𝜎

𝑒− (𝑠−(𝜇+𝜓𝑎𝜎2))2−𝜓2𝑎2𝜎42𝜎2 𝑑𝑠 ∝ ∫

𝑠

−∞

1√2𝜋𝜎

𝑒− (𝑠−(𝜇+𝜓𝑎𝜎2))22𝜎2 𝑑𝑠

= 𝒩 (𝜇 + 𝑎𝜓𝜎2, 𝜎2).

Optimism and pessimism are psychologically very far from Bayesian rational-ity. Nevertheless, it is actually possible to interpret the equations of the modelas Bayesian updating. According to this interpretation, optimists and pessimistsbelieve that Nature has selected the state of the world with their interests in mind,making their interests a valuable source of information as to Nature’s choice. TheBayes Rule analogue of Equation 2 is

log 𝑝(𝑎|𝑒)𝑝(𝑏|𝑒) = log 𝑝(𝑎)

𝑝(𝑏) + [log 𝑝(𝑒|𝑎) − log 𝑝(𝑒|𝑏)], (6)

where the reference stakes play the role of the evidence 𝑒, and the log likelihood in astate 𝑠 is given by log 𝑝(𝑒|𝑠) = 𝜓𝑟(𝑠). Decision makers start with their indifferencebeliefs, observe their reference interests 𝑟, and use Bayesian inference to updatetheir beliefs.12

12It is also possible to give the optimism and pessimism models of Hey (1984), Bracha andBrown (2012), and Dillenberger et al. (2014) a Bayesian interpretation. However, in those modelsthe information is not in the reference stakes, but in the choices that optimists and pessimists make.Moreover, making a choice is not merely informative about the state of the world, but actuallychanges it.

12

In order to identify the parameters of the model, we need to observe the deci-sion maker’s beliefs with two different reference stakes. Once we identify 𝜓 wecan determine how a change of reference would alter beliefs. Consider parentswhose child is to be allocated randomly to one of two schools. The parents wanttheir child to be allocated to the better school, but they do not know which schoolis better. States correspond to the identity of the better school. Ex-ante, the par-ents do not know the allocation, and have the same stake in both states. Since theirreference is constant, their beliefs are represented by the indifference probabilitymeasure 𝑝. Once they learn which school their child would attend, their referencechanges to a higher utility in the state in which that school is better. According toEquation 2, this change in reference causes a change in the parents’ beliefs. Op-timistic (pessimistic) parents come to think more (less) highly of the school theirchild has been allocated to, with the size of this effect being a function of theirutility function and of 𝜓 . Such a change in beliefs in the absence of a change inrelevant information is an example of cognitive dissonance.

Further assumptions are required if we want to be able to say whether the re-sulting beliefs are too high or too low, or even how they compare with the beliefsof other equally informed individuals. One useful assumption is the following:

IPC (interpersonal comparability) The indifference beliefs of all equally informedindividuals coincide.

According to this assumption, differences in subjective beliefs are completelydetermined by information: the normatively relevant information we normallythink of as information, and the normatively irrelevant “information” representedby the reference stakes. Consider a group of equally informed parents, whose chil-dren were allocated to the two schools. Using this assumption we can say that op-timistic parents whose child was allocated to the first school will be more positiveabout that school than optimistic parents whose child was allocated to the secondschool. In the absence of some such assumption we could only make a statisticalprediction about the distribution of beliefs in the two populations.13

Suppose two optimists find themselves in a conflict, in which their interests areopposed. By similar reasoning, optimism creates a gap in the two sides’ evaluationof the situation, with each party more positive about her own chances than theother party’s valuation of her chances. This gap in beliefs can effectively destroy

13This example is analogous to the experiment described in Section 3. The statistical inferenceis only valid if the allocation is independent of ex-ante beliefs.

13

the chances of an efficient compromise.14In many applications it is appropriate to go further, and assume a weak form

of rational expectations:

WRE (weak rational-expectations) Indifferent individuals have rational expecta-tions.

This assumption makes it possible to say that optimistic parents overestimatethe true quality of the school their child was allocated, that optimistic parties to aconflict overestimate their true chances, etc.

Consider an individual who (naturally enough) does not want to fall ill. Anoptimist would overestimate the likelihood of health, and consequently have aninefficiently high reservation price for insurance, skip health checks, and make fewpreparations in case she does fall ill. A pessimist would overestimate the likelihoodof disease, would have an inefficiently low reservation price for insurance, haveinefficiently many health checks, and spend too much time planning for disaster tostrike.

The model applies in an exactly analogous way whenever people prefer onestate to another, whatever the reason. If, for example, people have social prefer-ences, and want their friends to be healthy, they would be biased not only abouttheir own health, but also about that of their friends (though if they don’t care asmuch about their friends’ health as they do about their own, they wouldn’t be asbiased about their friends’ health).15 If a person cares about morality, she wouldbe biased about the morality of her past actions and of her situation in life. If sheis rich, she would be biased to believe that her wealth is deserved.

3 ExperimentSection 3.1 describes the implementation and protocol, Section 3.2 describes specificsof the belief elicitation procedure in Section 3.2, Section 3.3 describes the predic-tions of the standard model, Optimal Expectations, and the Priors and Desiresmodels. Finally, Section 3.4 describes how the data was analyzed.

14See Loewenstein et al. (1993), Babcock et al. (1995), and Babcock and Loewenstein (1997)for related evidence from pretrial bargaining and other disputes.

15See Weinstein (1989) for findings consistent with this.

14

3.1 Implementation and protocolThe experiment was conducted at the Centre for Experimental Social Science(CESS) at Nuffield College, University of Oxford. The subject pool consisted ofOxford students who registered on the CESS website for participation in experi-ments. Business, finance, and economics students were excluded. A week beforeeach session students meeting the sample restrictions received an email invitingthem 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 wasvia an online form, allowing students to select one of several sessions, up to anupper limit of 14 students per session. Taking no-shows into account, sessionsconsisted of between 10 and 13 students. Altogether, 145 students took part in theexperiment, of whom 57 were male and 88 female. The median age was 22.

Sessions were conducted in the afternoon over a total of six days. There were12 sessions altogether. Half the sessions consisted of Farmers, and half of Bakers.The order of sessions was randomized in order to prevent any consistent relation-ship between the time of day in which a session was held, and the role given to thesubjects who took part in that session.

After subjects were seated, they were each given a copy of the instructions,which they were able to refer to until the experiment ended. The instructions werealso read aloud, and there was an opportunity for subjects to ask questions. Theexperiment itself 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 usedthroughout the experiment. One of these 13 charts was reserved for the trainingperiod, and the other 12 charts were used for the earning periods (Figure 2). Theorder of presentation was randomized independently between subjects. At the endof the experiment, each subject had one earning period chosen at random, and waspaid in accordance with the payoff in that period.

The experiment was conducted in a computer lab, and was programmed usingz-Tree Fischbacher, 2007. Figure 1 shows an example of the interface. In eachperiod subjects were shown a chart of wheat prices, and were asked to predict theprice of wheat at some future date. Subjects were thus put in a somewhat similarposition to speculators who ignore fundamental information, and predict future as-set prices on the basis of historical price charts.16 In order to maximize the realismof the task, prices were adapted from real financial markets. The specific sourcewas historical stock prices, scaled and shifted to fit into a uniform range. Charts

16Traders refer to the use of historical price charts in making buy and sell decisions as TechnicalAnalysis Murphy, 1999; Edwards and Magee, 2010.

15

were selected to include a variety of situations. Time was standardized acrosscharts, so that all charts had space for prices going from day 0 to day 100. Sub-jects were only shown prices up to an earlier date, and the task was to predict whatthe price of wheat would be at day 100. The price range was also standardized, sothat prices were always between £4,000 and £16,000.

After submitting their prediction, subjects were presentedwith a waiting screenuntil all other subjects had also made their prediction. There was therefore littleor no incentive for speed. The transition to the next period only occurred afterall the subjects in the room had submitted their prediction. A brief questionnairewas administered following the final period of the experiment. After all subjectscompleted the questionnaire, subjects were informed of their earnings, and werecalled 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 wouldbe selling their wheat for the price that would obtain at day 100. Their notionalprofit 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 accuracybonus. In the baseline sessions subjects received £1 in real money for each £1,000of 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 3.2.Bakers were told that they make bread, which they would sell for a known priceof £16,000, and that in order to make the bread they would be buying wheat at theprice that would obtain at day 100. The range of notion profit was therefore thesame as that of Farmers, and all other particulars were also the same. The onedifference was that that Farmers gained from high wheat prices, whereas Bakersgained from low prices.

Sessions differed in the scale of the accuracy bonus and in the stakes (the degreeto which payoff depended on the price level at day 100). In the baseline sessionsthe maximum obtainable bonus was £1, and the amount received for each £1,000of notional profit 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.17 Table 1lists the number of sessions in each condition.

17In sessions with lower stakes, subjects received an additional £3, so that the average payoffwas the same as in the baseline sessions.

16

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. Thelarger 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.

3.2 The belief elicitation procedureThe belief elicitation procedure was designed with two goals in mind. The firstwas to make it possible to test for the presence or absence of wishful thinking bias,namely a systematic difference in beliefs betweenFarmers andBakers. The secondwas to obtain a measure of the degree of subjective uncertainty in the predictionssubjects make. This was important both for testing whether the magnitude of thebias is greater in charts with more subjective uncertainty, and for testing whethermore confident individuals are also more biased.

In each period subjects were asked to report two numbers: a prediction and aconfidence level. The prediction represented the expected day 100 price, and couldbe any number in the range of possible prices. The confidence level representedthe (inverse of) the uncertainty in the prediction, and was reported on a 1-10 scale.

In order to give meaning to the 1-10 confidence scale, the instructions includedvisual examples of distributions with different prediction and confidence levels(Figure 3). The distributions were the weighted average of a normal distributionand a uniform one, with almost all the weight given to the normal. The predictioncorresponded to the mean of the normal distribution, and the confidence level wasinversely proportional to its standard deviation. The density corresponding to aprediction of 𝑚 ∈ [4000, 16000] and confidence level 𝑟 ∈ [1, 10] was

𝑞(𝑥) = (1 − 𝜖)𝒩 (𝑥|𝑚, (𝜆𝑟)−2) + 𝜖 (7)

where 𝒩 (⋅|𝜇, 𝜎2) represents a normal distribution with a given mean and variance,𝜆 is a scale parameter, translating the 1-10 confidence scale into the scale of prices,

17

and 𝜖 is the weight given to the uniform component. The effect of the latter was toensure that the density was bounded below by 𝜖, including at prices far from theprediction.

The scoring rule was logarithmic: subjects whose prediction and confidencelevel corresponded to a density 𝑞 received a bonus given by

𝑏(𝑥) = 𝛼 log (𝑞(𝑥)/𝜖) (8)

where 𝑥 is the true day 100 price, and 𝛼 is a parameter which determines the max-imum bonus level.18 As 𝑞 ≥ 𝜖 (Equation 7), the bonus was positive for all possiblepredictions. The value of 𝛼 was calibrated for the maximum bonus level in thesession (Table 1).

To see under what conditions the scoring rule is incentive compatible, let 𝑃denote the probability measure representing the subject’s true beliefs, and sup-pose the subject reports a prediction 𝑚 and a confidence level 𝑟. The subjectiveexpectation of the bonus is given by the following expression:

𝔼𝑃 [𝑏(𝑥)] = ∫ 𝑝(𝑥)𝛼 log 𝑞(𝑥)𝜖 d𝑥 = 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)(9)

where 𝐷KL(𝑃 ||𝑄) is the Kullback-Leibler divergence (KL-divergence or relativeentropy) between 𝑃 and 𝑄, and 𝐻(𝑃 ) is the entropy of 𝑃 . Maximizing the ex-pected bonus with respect to 𝑄 is thus equivalent to minimizing the KL-divergence𝐷KL(𝑃 ||𝑄). According to a standard result, 𝐷KL(𝑃 ||𝑄) ≥ 0 for all 𝑄, and is min-imized if 𝑄 = 𝑃 .19

The scoring rule works best if subjects are risk neutral and beliefs are well ap-proximated by a density in the family described by Equation 7. The scoring ruleshould then successfully elicit the prediction and confidence level for each subjectin each chart, making it possible to identify the difference in beliefs between Farm-ers and Bakers, the average subjective uncertainty in each chart, and the averageconfidence for each subject.

18The logarithmic scoring rule was introduced in Good (1952). See Gneiting and Raftery (2007)for a recent discussion and comparison to other scoring rules.

19This result, known as Gibb’s Inequality, follows directly from the fact that log 𝑥 is a concavefunction Cover and Thomas, 1991. The instructions explained that the expected bonus is maxi-mized by reporting a prediction and confidence level that reflect the subject’s beliefs about the day100 price. The bonus formula itself was included in a footnote.

18

One potential difficulty is hedging.20 Consider a risk-averse Farmer. Her profitis increasing in the price, and she would therefore prefer to receive the bonus instates in which the price is relatively low. Consequently, she could increase hersubjective expected utility by reporting a lower number than her true beliefs. By asimilar logic, a risk-averse Baker would be better-off by reporting a higher number.The result would be a downward bias in the estimated difference in beliefs betweenFarmers and Bakers.

A second potential problem is the possibility that the beliefs of some subjectsare bi-modal, or otherwise not well approximated by a density in the family de-scribed by Equation 7. This could make it harder for subjects to see what pre-diction would maximize their payoff, making predictions within each group morevariable than they would be otherwise. This increase in variance would trans-late into more noise in the estimated difference in beliefs between the two groups,though it should not result in bias.

3.3 PredictionsThis section develops the predictions of the standard model, the Priors and Desiresmodel, and the Optimal Expectations model. The following timing frameworkis used: at 𝑡 = 0 subjects observe a price chart and form their beliefs over theday 100 price; at 𝑡 = 1 they report their prediction and confidence level, andconsume anticipatory utility; at 𝑡 = 2 the day 100 price is revealed, and payoffs arerealized. Subjects are assumed to be risk neutral. Beliefs about the day 100 priceare represented by a distribution from the family described by Equation 7. Giventhese assumptions, the prediction made at 𝑡 = 1 coincides with the 𝑡 = 1 beliefs.

3.3.1 The standard model

Subjects are allocated into the Farmer and Baker roles randomly. The 𝑡 = 0 beliefsof beliefs of Farmers and Bakers are therefore drawn from the same distribution.Since the prediction coincides with the 𝑡 = 1 beliefs, and as no new informationis observed between 𝑡 = 0 and 𝑡 = 1, it follows that predictions are also drawnfrom the same distribution. Consequently, there is no systematic difference inpredictions between Farmers and Bakers.

20Blanco et al. (2008) find evidence of hedging in belief reporting when opportunities are trans-parent and incentives are strong. Armantier and Treich (2010) discuss hedging in probability elic-itation.

19

3.3.2 Optimal Expectations

Optimal expectations agents choose their prior beliefs in order to maximize theirdiscounted subjective expected utility, where each period’s instantaneous utilityincludes anticipatory utility as well as standard consumption utility.

The payoff in the experiment is realized at 𝑡 = 2, and consists of two compo-nents: 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 util-ity is proportional to the expected value of the profit and bonus, with expectationscomputed using the 𝑡 = 1 beliefs. The more optimistic those beliefs are, the higheris anticipatory utility, but the less accurate the prediction is likely to prove. The𝑡 = 0 decision maker choosing her 𝑡 = 1 beliefs therefore faces a trade-off: morebias increases the anticipatory utility experienced at 𝑡 = 1, but lowers the expectedvalue of the 𝑡 = 2 consumption utility.

Let 𝑃 and 𝑄 denote the probability distributions representing the 𝑡 = 0 and𝑡 = 1 beliefs respectively. At 𝑡 = 0 the agent maximizes a weighted sum of the𝑡 = 1 anticipatory utility and 𝑡 = 2 realized payoff. Let 𝜂 denote the weightgiven 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, where𝑥 is true day 100 price, 𝜅 represents the stakes (the absolute value of the sloperelating the profit to the day 100 price), and 𝜙 denotes the direction, with 𝜙 = 1for Farmers and 𝜙 = −1 for Bakers. I denote the bonus by 𝑏(𝑥), where 𝑏 is definedby Equation 8. The 𝑡 = 0 maximand can thus be written as follows:

𝑊 = 𝜂𝔼𝑄[𝜙𝜅𝑥 + 𝑏(𝑥)] + (1 − 𝜂)𝔼𝑃 [𝜙𝜅𝑥 + 𝑏(𝑥)] + 𝑙 (10)

In order to derive the comparative statics of the bias in closed form I make acouple of simplifying assumptions. First, I assume that 𝑃 and 𝑄 are normal:𝑃 = 𝒩 (𝜇0, 𝜎2

0), and 𝑄 = 𝒩 (𝜇1, 𝜎21). Second, I assume that only the mean of

𝑄 is subject to bias, i.e. 𝜎1 = 𝜎0 = 𝜎. Given these assumptions and using Equa-tion 9, we can rewrite Equation 10 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

(11)

where 𝐶 collects factors that are independent of 𝑄. The two terms that depend on𝑄 represent, respectively, the gain in anticipatory utility from adopting optimistic

20

beliefs, and the cost in expected realized payoff of adopting such beliefs. Thegain term has two components. The first represents the anticipated profit, and isproportional to 𝜇1 = 𝔼𝑄[𝑥]. The second represents the anticipated bonus, andis 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, andthe more certain the subject is about her prediction. The cost term represents thereduction in expected bonus due to the bias in the prediction that follows from thebias in the 𝑡 = 1 beliefs, and is proportional to the Kullback-Leibler divergencebetween the 𝑡 = 0 beliefs 𝑃 and the 𝑡 = 1 beliefs 𝑄. Thus, if the subject cared onlyabout 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 chooseto believe that the most favorable price would be realized,21 and would furtherchoose 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 inthe favorable direction. Otherwise, the optimal value of 𝜇1 would be at an internalpoint, where 𝜕𝑊 /𝜕𝜇1 = 0. Since we do not observe subjects making extreme pre-dictions I assume that 𝜂 is large enough that the optimal value of 𝜇1 is at an internalpoint. Using the standard formula for the KL-divergence between two normal dis-tributions Johnson and Sinanovic, 2001, and noting that 𝐻(𝑄) is independent of𝜇1, the 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

(12)

Setting the derivative to zero and solving for 𝜇1 we obtain the following expressionfor the bias:

𝜇1 − 𝜇0 = 𝜙 (𝜂

1 − 𝜂 ) (𝜅𝜎2

𝛼 ) (13)

where 𝜅 represents the stakes, or the degree to which the profit is dependent on thevalue of the day 100 price, 𝜎2 represents the degree of subjective uncertainty, and𝛼 represents the scale of the accuracy bonus, or the cost of holding biased beliefs.

Equation 13 describes the bias in the beliefs of one particular individual. The21That is, the highest possible price of £16,000 if a Farmer, and the lowest possible price of

£4,000 if a Baker.

21

prediction 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 ) (14)

where I allow for the possibility that 𝜂 varies between individuals, but assume thatit is independent of 𝜎2 (because of the random assignment 𝜂 is independent of 𝜅 and𝛼). Finally, it also follows from the random allocation that the undistorted beliefsof Farmers and Bakers are drawn from the same distribution, and that in particular𝔼𝜇0 is the same in both groups. The expected difference in beliefs between the twogroups is therefore given by

𝑏optimal expectations = 2𝔼 [𝜂

1 − 𝜂 ] (𝜅𝜎2

𝛼 ) ∝ 𝜅𝜎2

𝛼 (15)

Optimal Expectations thus implies a systematic difference in beliefs betweenFarm-ers and Bakers that is proportional to the stakes and to the degree of subjectiveuncertainty, and inversely proportional to the cost of getting beliefs wrong.

3.3.3 Priors and Desires

The payoff-function in the experiment is the mapping linking the subject’s payoffto the day 100 price.22 Using the same notation as in Section 3.3.2, the payoff-function is given by 𝑟(𝑥) = 𝜙𝜅𝑥 + 𝑙, where 𝑥 is the day 100 price, 𝜅 represents thestakes, or the slope relating payoff to the day 100 price, and 𝜙 denotes the direction,with 𝜙 = 1 for Farmers and 𝜙 = −1 for Bakers. Suppose, as in Section 3.3.2, thatundistorted beliefs are given by a normal distribution 𝑃 = 𝒩 (𝜇0, 𝜎2). Accordingto Proposition 1 the distorted probability measure is given by 𝑄 = 𝒩 (𝜇1, 𝜎2),where

𝜇1 − 𝜇0 = 𝜙𝜓𝜅𝜎2 (16)This equation describes the bias in the beliefs of some particular individual, andis the Priors and Desires analogue of Equation 13. By analogy with Section 3.3.2,the expected difference in beliefs between Farmers and Bakers is

𝑏priors and desires = 2𝔼[𝜓]𝜅𝜎2 ∝ 𝜅𝜎2 (17)

Comparing this result to Equation 15, we see that—aswithOptimal Expectations—themagnitude of the bias is proportional to the stakes 𝜅 and the degree of subjective

22In principle, it should be the payoff in utility terms, but I am assuming throughout this sectionthat subjects are risk neutral over small amounts of money.

22

uncertainty 𝜎2. However, whereas in Optimal Expectations the magnitude of thebias is inversely proportional to the cost of getting beliefs wrong 𝛼, the magnitudeof the bias in Equation 17 is independent of 𝛼.

3.4 AnalysisAs noted in Section 3.2, hedging could lead to a downward bias in estimating thedifference in beliefs between Farmers and Bakers. In order to minimize this risk, aquestionnaire was administered after the experiment itself was concluded, in whichsubjects were asked whether they always reported their best guesses, or whetherthey sometimes reported a higher or lower number. Out of a total of 145 studentswho took part in the experiment, 132 claimed to have always reported their bestguess, and 13 admitted to an intentional bias in their predictions. Observationsfrom these 13 subjects were excluded from the main analysis.

The raw data from the experiment consist of the predictions and confidencelevels reported by individual subjects in individual charts. The primary goal inanalyzing the data was to determine whether predictions were affected by wishfulthinking. 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. Wewant to knowwhether𝑦𝑖𝑗 is systematically higher if 𝑡𝑖 = 1. In order to answer this question formally Iused the following regression model:

𝑦𝑖𝑗 = 0.5𝛽𝑡𝑖 + ∑𝑗

𝛾𝑗𝑑𝑗 + 𝜖𝑖𝑗 (18)

where 𝑑𝑗 is a dummy for chart 𝑗, and 𝜖𝑖𝑗 is the error term. The value of 𝛽 representsthe contribution of wishful thinking. The null hypothesis is that 𝛽 ≤ 0.

The second goal in analyzing the data was to investigate the comparative stat-ics of the bias. This required estimating the bias separately in different subsamplesof interest. Let 𝐾 denote a partition of the sample, indexed by 𝑘, and let 𝑐𝑖𝑗𝑘 denotea dummy for whether the prediction of subject 𝑖 in chart 𝑗 belongs to subsample𝑘. Assuming wishful thinking is the only systematic source of difference in pre-dictions between subjects, we can generalize Equation 18 as follows:

𝑦𝑖𝑗 = 0.5 ∑𝑘∈𝐾

𝛽𝑘𝑐𝑖𝑗𝑘𝑡𝑖 + ∑𝑗

𝛾𝑗𝑑𝑗 + 𝜖𝑖𝑗 (19)

In this equation 𝛽𝑘 represents the average difference in predictions between Farm-ers and Bakers in class 𝑘, and can be used to define formal comparative staticshypotheses.

23

Unobserved factors may result in a correlation in the predictions made by thesame subject in different charts, so that 𝜖𝑖𝑗 may be correlated with 𝜖𝑖𝑘 for 𝑗 ≠ 𝑘.In order to allow for this possibility, standard errors are clustered by subject in allregressions.

4 ResultsThis section presents the results of the experiment, starting with the overall dif-ference in predictions between Farmers and Bakers, and continuing with the com-parative statics of the bias. Parameter estimates and statistical test results are pre-sented in summary form in Table 3. Figures 4 and 5 provide a graphical illustrationof the results.

4.1 Wishful thinking biasThe overall magnitude of the wishful thinking bias corresponds to the systematicdifference in predictions between Farmers and Bakers across the entire sample,represented by the value of 𝛽 in Equation 18. The estimate for this number is£452, measured with a robust standard error of £123. The null-hypothesis that itis non-positive is rejected with a 𝑝-value of 0.0002.

This estimate excludes observations from the 13 subjects who admitted in thepost experiment questionnaire to biased reporting of their beliefs (Section 3.4).If these subjects are nonetheless included, the estimate goes down to £390. Thisdifference is consistent with the prediction that risk-averseFarmers (Bakers) wouldintentionally understate (overstate) their estimates of the day 100 price.

The observed difference in predictions between Farmers and Bakers can beexplained by wishful thinking, but not by ego-utility or by a cognitive bias.

4.2 Incentives for accuracySelf deception predicts that the magnitude of the bias would be decreasing in theincentives for accuracy, while Priors and Desires predicts that it would remain thesame. In order to determine whether higher incentives for accuracy result in lowerbias, Equation 19 was used to estimate the difference in beliefs between Farmersand Bakers separately in sessions with different levels of accuracy bonus (Table 1).

The results in Table 3 are that the estimated bias is actually greater in sessionswith a higher bonus level, the point estimates being 298, 560, and 645, respectively.

24

On the face of it, these results are consistent with neither type of model. Formaltesting, however, reveals that the apparent increase in the magnitude of the biasmay well be random (𝑝 < 0.4026). The data is, therefore, consistent with theprediction of Priors and Desires that the magnitude of the bias would be invariantto changes in the incentives for holding accurate beliefs.

The same is not true, however, for Optimal Expectations. The prediction of thismodel is that the magnitude of the bias would be inversely proportional to the scaleof the accuracy bonus (Section 3.3.2). That is, the bias in £2 bonus sessions shouldbe half the size of the bias in £1 bonus sessions, and the bias in £5 bonus sessionsshould be one fifth the size. This prediction is rejected by the data (𝑝 < 0.0140).23

The first panel of Figure 5 shows these results graphically. Though the pointestimates are increasing in the maximum level of the accuracy bonus, a horizontalparallel line can be comfortably fitted within the confidence intervals. The sameis not true, however, for a hyperbolic curve.

4.3 Subjective uncertaintyAccording to both Optimal Expectations (Section 3.3.2) and Priors and Desires(Section 3.3.3), the magnitude of the bias should be increasing in the degree ofsubjective uncertainty. In order to test this prediction, I divided the 12 charts usedin the paying periods into two equal sized groups by the degree of subjective uncer-tainty in the chart, and used Equation 19 to estimate the bias separately in the twosubsamples.24 I used two different measures of subjective uncertainty. The firstwas based on the confidence ratings that subjects provided: charts were classifiedinto the high (low) subjective uncertainty group if the mean (across all subjects) ofthe confidence rating for the chart was below (above) median. The second measureof uncertainty was the within group variance of predictions: charts were classifiedinto the high (low) subjective uncertainty group if the within group variance of pre-dictions for that chart was above (below) median. In practice, the two measuresresulted in nearly identical classifications.

Depending on the measure used, the estimated bias was 635 or 677 in the groupof high subjective uncertainty charts, and 269 or 227 in the low subjective uncer-tainty group. The null hypothesis—that the magnitude of the bias in high subjec-tive uncertainty charts would be less than or equal to the magnitude of the bias in

23This is the a joint hypothesis test. The hypothesis that the magnitude of the bias in £5 sessionsis one fifth that of £1 sessions is rejected with a 𝑝-value of 0.0069.

24Each subsample consists of observations from all subjects, but in only half the charts.

25

low subjective uncertainty charts—was rejected with a 𝑝-value of 0.0142 when us-ing the first classification 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 biasis increasing in the degree of subjective uncertainty. Given that the qualitativeprediction of the two models fits the data, it is interesting to try and test the spe-cific functional form predicted by the two models. The quantitative prediction isthat 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 thanEquation 18:

𝑦𝑖𝑗 = 0.5𝛽′𝜎2𝑗 𝑡𝑖 + ∑

𝑗𝛾𝑗𝑑𝑗 + 𝜖𝑖𝑗 (20)

In this equation the 0.5𝛽𝑡𝑖 term in Equation 18 is replaced by 0.5𝛽′𝜎2𝑗 𝑡𝑖, where 𝜎2

𝑗is the variance of subjective uncertainty in chart 𝑗.

Testing this quantitative prediction requires a good proxy for the variance ofsubjective uncertainty. Using the above measures of subjective uncertainty, wecan identify 𝜎2

𝑗 either with the square of the inverse mean confidence rating inchart 𝑗, or with the mean within group prediction variance for chart 𝑗.25 Table 2shows the resulting regression fit when estimating the two equations using bothproxies for the variance of subjective uncertainty, as well the results of fitting amodel which includes both the 0.5𝛽𝑡𝑖 term of Equation 18 and the 0.5𝛽′𝜎2

𝑗 𝑡𝑖 ofEquation 20. The results show that Equation 20 indeed provides a better fit to thedata, consistent with the prediction that the magnitude of the bias is linear in thedegree of subjective uncertainty.

The same results can also be seen graphically in the second and third panels ofFigure 5. Panel 2 plots the estimated wishful thinking bias in the 12 charts againstthe mean prediction confidence in the chart, and panel 3 plots the same data againstthe within group prediction variance. In both panels a curve is fitted to the datausing Equation 20.

4.4 StakesOptimal Expectations and Priors and Desires also predict that the magnitude of thebias is increasing in the stake subjects have in what the day 100 price would be.Payoff depends on the day 100 price via the notional profit, which is linear in the

25This assumes a representative agent approximation.

26

Table 2: Testing whether the magnitude of the bias increases with the varianceof subjective uncertainty. Column 1 fits a model in which the bias is independentof subjective uncertainty (Equation 18). Column 2 and 4 fit a model in which themagnitude of the bias is linear in the variance (Equation 20). Columns 3 and 5 fita model which allows for both regressors. Method 1 and method 2 refer to the twoproxies for subjective uncertainty (Section 4.3). The 𝑡𝑖𝜎2

𝑗 variable is normalizedto have the same standard deviation as 𝑡𝑖, so that the regression coefficients arecomparable in size. Robust standard errors are in parentheses. The regression R2

is computed after netting out the contribution of the chart dummies. Statisticalsignificance indicators: *** 𝑝 < 0.01, ** 𝑝 < 0.05, * 𝑝 < 0.1.

method 1 method 2𝑡𝑖 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

day 100 price with a slope of 1. The amount of money received for each £1,000 ofnotional profit was £1 in 10 sessions and 50p in the remaining 2 sessions (Table 1).

I estimated themagnitude of the bias separately in these two subsamples (Equa-tion 19). The magnitude of the bias was 260 in the low stakes subsample, and 495in the standard stakes subsample. These results are consistent with the predictionthat the magnitude of the bias is linear in the stakes (𝑝 < 0.9668). The modestvariance in the stakes between sessions was, unfortunately, insufficient to producestatistically measurable results, and the hypothesis that the bias is not any smallerin the low stakes subsample could not be rejected (𝑝 < 0.2313). See also Table 3and panel 4 of Figure 5.

4.5 Over-confidenceSection 4.1 demonstrates the existence of a systematic difference in predictionsbetween Farmers and Bakers. This difference in predictions is interpreted as ev-idence of wishful thinking bias affecting subjects’ judgment about the day 100price. A key assumption is that subjects believe they have better than random oddsof making a good prediction, so it is in their interest to report their true beliefs. If

27

this assumption is not true, subjects could very well choose whichever predictionthey 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 ob-serve a systematic difference in predictions between Farmers and Bakers that hasnothing to do with wishful thinking.

If this alternative explanation is correct, we would expect subjects who lackconfidence in their predictions to be more biased than confident subjects, sincesuch subjects have less to lose from biasing their prediction. Similarly, we wouldexpect subjects who generally believe prices in financial markets are unpredictableto be more biased than subjects who believe prices can be predicted.

In order to test the first prediction I defined a proxy for a subject’s confidenceby the average prediction confidence for that subject across all charts. I then splitthe sample into more and less confident subjects, and estimated the bias separatelyin the two subsamples. In order to test the second prediction I included a questionin the post experiment questionnaire about the predictability of prices in financialmarkets, and divided subjects into two groups by whether they thought prices cangenerally be predicted. The bias was then estimated separately in the two subsam-ples.26

The result was just the opposite: subjects who believe prices are predictableand relatively confident subjects are more biased than those who are less confi-dent. Specifically, the estimated bias among relatively confident subject is 628,compared with 276 among less confident subjects. The hypothesis that more con-fident subjects are less biased is rejected with 𝑝-value of 0.0732. Similarly, theestimated bias among subjects who believe prices in financial markets to be gen-erally predictable was 613, as compared with 292 among subjects who believedprices cannot be predicted. The hypothesis that subjects who believe prices to bepredictable 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 pre-dict the day 100 price, and the stronger this belief is, the more biased they are. Thisresult is consistent with the wishful thinking interpretation, and further suggeststhat over-confidence is a manifestation of wishful thinking, and that the degree ofwishful thinking bias is a stable individual characteristic.27

26The question was “We are interested in what people believe about financial markets. Howpredictable are themovements of prices in financial markets in your opinion?” The possible choiceswere: “Prices can be predicted to a significant extent”, “Prices can rarely be predicted”, and “Theidea that prices can be predicted is an illusion”. The first choice was defined as yes, and the othertwo as no. The distribution of answers was 66, 58, and 8, respectively.

27This explains why individuals with more than average wishful thinking bias also tend to be

28

4.6 GenderThough the psychology evidence is mixed Lundeberg et al., 2000, certain behav-ioral differences betweenmen andwomen, such as a propensity to overtrade amongmen, have been interpreted as evidence of gender differences in confidence Barberand Odean, 2001. Subjects in the experiment included 62 percent females and 38percent males, and there was therefore sufficient variation to test for gender differ-ences in wishful thinking. The estimated bias is 411 for males and 477 for females,and the hypothesis of no difference cannot be rejected (𝑝 < 0.7956).

5 ConclusionThis paper describes an experimental test of wishful thinking bias in predictionsof asset prices. Subjects received an accuracy bonus for their predictions of thefuture price of an asset, and an unconditional payment that was either increasingor decreasing in this price. Both groups of subjects had the same information,and faced the same incentives for accuracy. Nevertheless, and despite incentivesfor hedging, subjects in the group benefiting from high prices predicted systemat-ically higher prices than subjects in the group benefiting from low prices. Theseresults are consistent with wishful thinking, and cannot be accounted for by suchalternative explanations as ego-utility or cognitive bias.

By varying the scale of the accuracy bonus it was possible to test whether themagnitude of the bias decreases with the incentives to hold accurate beliefs. Nosuch decrease was found, and the prediction of Optimal Expectations Brunner-meier and Parker, 2005 that the magnitude of the bias is inversely proportional tothe incentives for accuracy, was formally rejected. This result is hard to squarewith strategic models of wishful thinking, but is consistent with Priors and De-sires. The implication is that wishful thinking can significantly affect beliefs evenif the costs are high.

Other comparative statics results include good evidence that wishful thinkingbias is stronger when subjective uncertainty is high, evidence that over-confidenceand wishful thinking bias go together, and some evidence of greater bias whenpayoff is more strongly dependent on the state of the world.

Taken together, these results suggest that any and all subjective beliefs are af-fected by wishful thinking bias, and that the bias may well be sufficiently strong

over-confident. The tendency to be more or less biased can be identified with the coefficient ofrelative optimism in the Priors and Desires model.

29

to materially affect economically important decisions. High stakes decisions infinancial markets are a case in point, as they involve probability assessments insituations characterized by high stakes and high subjective uncertainty—both ofwhich are conducive to the presence of an economically significant bias.

In interpreting this conclusion, it is important to bear in mind that decisionmakers in high stakes situations have an incentive to invest in quality informationin order to reduce the uncertainty in their beliefs. Since the strength of the biasdepends on the degree of subjective uncertainty, quality information will not onlyreduce the variance in beliefs, but would also (perhaps unintentionally) reduce themagnitude of the bias. The degree to which wishful thinking is likely to affecthigh stakes decisions is therefore dependent on decision makers’ ability to reduceuncertainty before making their choices.

Oneway to asses the degree of uncertainty is to examine the beliefs of informedexperts. In many important decision making environments (financial markets, cor-porate decision making, politics, war) informed experts commonly disagree. Thefailure of experts to come to anything approaching consensus suggests the exis-tence of a substantial level of irreducible uncertainty. When that is the case, thereis evidently significant potential for wishful thinking to materially affect decisions.

The present paper describes one particular experiment on one particular groupof subjects. While the main conclusions are strongly statistically significant, itwould clearly be important to see whether the results can be replicated by otherresearchers and in other decision making environments. Another important limi-tation in interpreting the results of the experiment is the limited range of theoriesunder consideration. While I am not aware of any other non ad hoc theory thatcan explain the results of the experiment, it is important to emphasize that if sucha theory were to be offered, it may significantly change the interpretation of theexperiment’s results.

A Axiomatic foundationThis appendix provides an axiomatic foundation for the representation of 𝜋 in Equations 2–5. In the following definitions 𝑟 and 𝑟′ stand for any reference stakes, 𝑎 for any real num-ber, and 𝐸 for any event. The first definition states the properties we want 𝜋 to satisfy,and the second describes the logit formula. The theorem says that the two definitions areequivalent.

Definition 1. 𝜋 ∶ 𝐹 → Δ is a well-behaved distortion if the following conditions aresatisfied:

30

A1 (absolute continuity) 𝜋𝑟′(𝐸) = 0 ⟺ 𝜋𝑟(𝐸) = 0.

A2 (consequentialism) If 𝑟 = 𝑟′ over a non-null28 event 𝐸 then 𝜋𝑟′(⋅|𝐸) = 𝜋𝑟(⋅|𝐸).

A3 (shift-invariance) If 𝑟′ = 𝑟 + 𝑎 then 𝜋𝑟′ = 𝜋𝑟.

A4 (prize-continuity) If 𝑟𝑛 → 𝑟 then 𝜋𝑟𝑛(𝐸) → 𝜋𝑟(𝐸).

These properties should be understood as simplifying assumptions, the purpose ofwhich is to obtain as simple as possible a representation, while retaining the ability torepresent the phenomenawewish tomodel. Absolute Continuity defines the scope of beliefdistortion as the set of events that the agent is uncertain about. Consequentialism requiresthat beliefs (and therefore any consequences for choices) depend only on the referencestakes in states that are consistent with the available evidence. Beliefs conditional on anevent 𝐸 cannot depend on the reference stakes in states not in 𝐸.

The idea behind Shift Invariance is that different reference stakes may result in differ-ent beliefs only if they differ in what a person wants to be true, or how strongly she feelsabout it. Shift-Invariance embodies this idea on the assumption that the agent only wantssomething to be true if her reference stakes yield a higher utility if it is true, and that equaldifferences in utility correspond to equal degrees of ‘wanting to be true’.

Definition 2 (Logit distortion). 𝜋 ∶ 𝐹 → Δ is a logit distortion if there exists a probabil-ity measure 𝑝 (the indifference measure), and a real-number 𝜓 (the coefficient of relativeoptimism), such that for any reference 𝑟 ∈ 𝐹 and any event 𝐴,

𝜋𝑟(𝐴) ∝ ∫𝐴𝑒𝜓𝑟 d𝑝. (21)

It is easy to see that every logit distortion is well-behaved. The opposite requires thetechnical assumption that there exist at least three events with positive probability:29

Definition 3 (Minimally complex distortion). 𝜋 ∶ 𝐹 → Δ is minimally complex if thereexists three disjoint events 𝐴, 𝐵, and 𝐶 , and reference stakes 𝑟 such that 𝜋𝑟(𝐴), 𝜋𝑟(𝐵), and𝜋𝑟(𝐶) are all positive.

Theorem 1 (Representation theorem). A minimally complex distortion is a logit-distortionif and only if it is well-behaved.

28That is, both 𝜋𝑟(𝐸) > 0 and 𝜋𝑟′(𝐸) > 0. Absolute Continuity ensures that these two require-ments coincide.

29If there are only two disjoint events, there exist well-behaved distortions that are not logitdistortions.

31

A.1 Intermediate representation resultsIn this section I prove the theorem for the special case where there are only finitely manyevents. That is, I assume that there exists a finite partition 𝒮 of the state-space, such thatΣ is the algebra generated by 𝒮 . In addition, I prove a sequence of partial representationresults requiring only a subset of the assumptions. In order to state the necessary andsufficient conditions for these representations I define a new property, Indifference, whichis related to Shift Invariance, but is considerably weaker:

A3’ (Indifference). 𝜋𝑟 = 𝜋𝑟′ if both 𝑟 and 𝑟′ are constant payoff-functions.

Note that unlike Shift Invariance, Indifference does not require the set of payoffs to havecardinal (or even ordinal) meaning.

Lemma 1. Suppose that there exists a finite partition 𝒮 of the state-space, such that Σ isthe 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 prob-ability distribution 𝑝 ∈ Δ and a function ℎ ∶ 𝐹 × 𝒮 → ℝ+, such that for anyreference 𝑟 and any event 𝐴 ∈ 𝒮 ,

𝜋𝑟(𝐴) ∝ 𝑝(𝐴) ℎ𝑟(𝐴). (22)

2. Assume Absolute Continuity. Consequentialism is a necessary and sufficient condi-tion for there to exist a probability distribution 𝑝 ∈ Δ, and a mapping 𝜇 ∶ 𝒮 ×𝑋 →ℝ+, such that for any reference 𝑟 and any event 𝐴 ∈ 𝒮 ,

𝜋𝑟(𝐴) ∝ 𝑝(𝐴) 𝜇𝐴(𝑟(𝐴)). (23)

3. Assume Absolute Continuity and Consequentialism. Indifference is a necessaryand sufficient condition for there to exist a probability distribution 𝑝 ∈ Δ, anda mapping 𝜈 ∶ 𝑋 → ℝ+, such that for any reference 𝑟 and any event 𝐴 ∈ 𝒮 ,

𝜋𝑟(𝐴) ∝ 𝑝(𝐴) 𝜈(𝑟(𝐴)). (24)

4. Assume Absolute Continuity and Consequentialism. Shift-Invariance and Prize-Continuity are necessary and sufficient conditions for there to exists a probabilitydistribution 𝑝 ∈ Δ, and a parameter 𝜓 ∈ ℝ, such that for any reference 𝑟 and anyevent 𝐴 ∈ 𝒮 ,

𝜋𝑟(𝐴) ∝ 𝑝(𝐴) 𝑒𝜓𝑟(𝐴). (25)

32

Note that while the representation in Equations 22–25 is defined with respect to eventsin 𝒮 , the implication for general events is straightforward.30 The following simple exampledemonstrates that Minimal Complexity is a necessary assumption. Let 𝒮 = {𝐴, 𝐵}, let𝜋𝑟(𝐴) ∝ 𝑝(𝐴)(1 + (𝑟(𝐴) − 𝑟(𝐵))2) and 𝜋𝑟(𝐵) ∝ 𝑝(𝐵). This distortion is well-behaved(Definition 1), but it cannot even be given the representation of Equation 23, let alone thatof a logit distortion (Definition 2).

A.2 Completing the proofThis section concludes the proof of Theorem 1 for the general case. The first step is togeneralize Equation 25 to any reference and any constant-payoff events:

Lemma 2. Suppose 𝜋 ∶ 𝐹 → Δ is a minimally complex well-behaved distortion, thenthere exist a probability measure 𝑝 and a parameter 𝜓 ∈ ℝ, such that for any reference 𝑟and any events 𝐴 and 𝐵 such that 𝑝(𝐵) > 0 and 𝑟 is constant on 𝐴 and on 𝐵,

𝜋𝑟(𝐴)𝜋𝑟(𝐵) = 𝑝(𝐴)

𝑝(𝐵)𝑒𝜓𝑟(𝐴)

𝑒𝜓𝑟(𝐵) . (26)

Theorem 1 for references that are simple payoff-functions is an immediate corollary.31 Thefollowing claim is a little more general, allowing for functions that are almost everywheresimple:

Definition 4. A payoff-function 𝑓 ∈ 𝐹 is almost everywhere simple if there exists apayoff-function 𝑔 ∈ 𝐹 and an event 𝐸 such that 𝑟 obtains only finitely many values on 𝐸and 𝜋𝑔(𝐸) = 1.

Corollary 1. Theorem 1 holds when restricted to payoff-functions that are almost every-where simple.

The remaining case involves functions which are not almost everywhere simple. If suchpayoff-functions exist, theremust also exist an infinite sequence of non-null events {𝐴𝑛}𝑛∈ℕ.But then, as long as 𝜓 ≠ 0 and the set of feasible payoffs is unbounded, it is possible toconstruct a reference 𝑟 such that lim𝑛→∞

𝜋𝑟(𝐴𝑛)𝜋𝑟(𝐴1) = ∞. But this implies that 𝜋𝑟(𝐴1) = 0, in

contradiction to Absolute Continuity. Hence, if 𝜓 ≠ 0 the set of feasible payoffs must bebounded.

Lemma 3. Suppose 𝜋 ∶ 𝐹 → Δ is a minimally complex well-behaved distortion, and thatthere exists a reference 𝑟 that is not everywhere simple, then there exist an upper bound𝑀 ∈ ℝ, such that for any feasible payoff-value 𝑥, 𝑒𝜓𝑥 ≤ 𝑀 .

30Any event in Σ is the finite union of events in 𝒮 .31A payoff-function 𝑟 is simple if 𝑟(𝑆) is finite.

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Lemma 3 ensures that 𝑒𝜓𝑋 is bounded from above. If it is also bounded from below, alimit argument based on simple payoff-functions can be used to extend the claim further:

Lemma 4. Suppose 𝜋 ∶ 𝐹 → Δ is a minimally complex well-behaved distortion thenthere exists a probability measure 𝑝 and a parameter 𝜓 ∈ ℝ, such that for any events 𝐴and 𝐵 for which 𝑝(𝐵) > 0, and any reference 𝑟 for which there exist a number 𝑚 > 0 suchthat 𝑟(𝑠) ≥ 𝑚 for all 𝑠 ∈ 𝐴 ∪ 𝐵,

𝜋𝑟(𝐴)𝜋𝑟(𝐵) =

∫𝐴 𝑒𝜓𝑟 d𝑝∫𝐵 𝑒𝜓𝑟 d𝑝

. (27)

The final step in the proof of Theorem 1 uses a limit argument whereby a general event𝐴 is approached by events of the form 𝐴𝑛 = {𝑠 ∈ 𝐴 ∶ 𝑒𝜓𝑟(𝑠) ≥ 2−𝑛}, and Lemma 4 isapplied on each of these events separately.

A.3 ProofsLemma 1

In all the four parts of Lemma 1 the proof that the requirements are necessary is trivial. Ithus prove only that the requirements are sufficient:

Part 1. Let 𝑎 denote some arbitrary constant payoff-function. Define 𝑝 = 𝜋𝑎, and let𝒮 ∗ = {𝐴 ∈ 𝑆 ∶ 𝑝(𝐴) > 0}. Define ℎ𝑟(𝐴) = 𝜋𝑟(𝐴)

𝑝(𝐴) for 𝐴 ∈ 𝒮 ∗ and ℎ𝑟(𝐴) = 0 for𝐴 ∉ 𝒮 ∗. For 𝐴 ∈ 𝒮 ∗ the claim follows from the definition of ℎ𝑓 . By Absolute Continuity𝑝(𝐴) = 0 ⇒ 𝜋𝑟(𝐴) = 0, and hence the claim holds also for 𝐴 ∉ 𝒮 ∗.

Part 2. Let 𝐴 ∈ 𝒮 ∗ and 𝑥 ∈ 𝑋, let 𝑟(𝐴, 𝑥) be the payoff-function mapping 𝐴 to 𝑥 and allstates outside 𝐴 to 𝑎. Let 𝐸1, … , 𝐸𝑛 denote the other events in 𝒮 ∗. By Minimal Complex-ity and Absolute Continuity 𝒮 ∗ includes at least two events other than 𝐴. 𝑟(𝐴, 𝑥) and theconstant payoff-function 𝑎 agree on 𝐸𝑖 and 𝐸𝑗 for all 𝑖 and 𝑗. Hence, by Consequentialismwith 𝐸 = 𝐸𝑖 ∪ 𝐸𝑗 ,

𝜋𝑟(𝐴,𝑥)(𝐸𝑖)𝜋𝑟(𝐴,𝑥)(𝐸𝑗 ) = 𝑝(𝐸𝑖)

𝑝(𝐸𝑗 ) .Thus,

1 − 𝜋𝑟(𝐴,𝑥)(𝐴) = ∑𝑖

𝜋𝑟(𝐴,𝑥)(𝐸𝑖) = ∑𝑖

𝜋𝑟(𝐴,𝑥)(𝐸𝑗)𝑝(𝐸𝑗) 𝑝(𝐸𝑖) =

𝜋𝑟(𝐴,𝑥)(𝐸𝑗)𝑝(𝐸𝑗) (1 − 𝑝(𝐴)). (28)

Define 𝜇𝐴(𝑥) = (1−𝑝(𝐴)

𝑝(𝐴) )(𝜋𝑟(𝐴,𝑥)(𝐴)

1−𝜋𝑟(𝐴,𝑥)(𝐴) ). By Equation 28,

𝑝(𝐴)𝜇𝐴(𝑟(𝐴)) = (1 − 𝑝(𝐴))𝜋𝑟(𝐴,𝑟(𝐴))(𝐴)

1 − 𝜋𝑟(𝐴,𝑟(𝐴))(𝐴) = 𝑝(𝐸𝑗)𝜋𝑟(𝐴,𝑟(𝐴))(𝐴)𝜋𝑟(𝐴,𝑟(𝐴))(𝐸𝑗) . (29)

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Let 𝑟 be any payoff-function, and let 𝐴 and 𝐵 be any two events in 𝒮 ∗. Let 𝑟′ be a payoff-function that coincides with 𝑟 on 𝐴 and 𝐵, and with 𝑎 elsewhere, and let 𝐶 be any thirdevent in 𝒮 ∗. Inserting 𝐸𝑗 = 𝐶 in Equation 29 we obtain that

𝜋𝑟(𝐴)𝜋𝑟(𝐵) = 𝜋𝑟′(𝐴)

𝜋𝑟′(𝐵) = 𝜋𝑟′(𝐴)/𝜋𝑟′(𝐶)𝜋𝑟′(𝐵)/𝜋𝑟′(𝐶) = 𝑝(𝐶)

𝑝(𝐶)𝜋𝑟(𝐴,𝑟(𝐴))(𝐴)/𝜋𝑟(𝐴,𝑟(𝐴))(𝐶)𝜋𝑟(𝐵,𝑟(𝐵))(𝐵)/𝜋𝑟(𝐵,𝑟(𝐵))(𝐶)

= 𝑝(𝐴)𝜇𝐴(𝑟(𝐴))𝑝(𝐵)𝜇𝐵(𝑟(𝐵))

(30)

where the first and third steps follows from Consequentialism, and the final step fromEquation 29. Since Equation 30 holds for all 𝐴, 𝐵 ∈ 𝒮 ∗ it follows that Equation 23holds for any event 𝐴 ∈ 𝒮 ∗. For an event 𝐴 ∉ 𝒮 ∗, define 𝜇𝐴(𝑥) = 1 for all 𝑥. Since𝜋𝑟(𝐴) = 𝑝(𝐴) = 0 for 𝐴 ∉ 𝒮 ∗ Equation 23 holds however 𝜇𝐴 is defined. Combining theseresults Equation 23 holds for reference 𝑟 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 30 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, 𝜋𝑟(𝐴) ∝ 𝑝(𝐴)𝜈(𝑟(𝐴)) for all 𝐴 ∈ 𝒮 ∗. Finally,this is also trivially true for 𝐴 ∉ 𝒮 ∗, since 𝜋𝑟(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, and𝑥 + 𝑦 are in 𝑋. Define the payoff-functions 𝑓𝑥 and 𝑔𝑥,𝑦 as follows: 𝑓𝑥(𝑠) = 𝑥 for 𝑠 ∈ 𝐴and 𝑓𝑥(𝑠) = 0 for 𝑠 ∉ 𝐴, and 𝑔𝑥,𝑦 = 𝑓𝑥 + 𝑦. By Shift-Invariance, 𝜋𝑔𝑥,𝑦

= 𝜋𝑓𝑥, and in

particular𝜋𝑔𝑥,𝑦 (𝐴)𝜋𝑔𝑥,𝑦 (𝐵) = 𝜋𝑓𝑥 (𝐴)

𝜋𝑓𝑥 (𝐵) . By Equation 24 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 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, for𝑛 ∈ ℕ let 𝑦 = 𝑥

𝑛 to obtain that 𝜎(𝑥) = 𝜎(𝑛𝑦) = 𝑛𝜎(𝑦), and hence 𝜎( 𝑥𝑛 ) = 𝜎(𝑥)

𝑛 . Let𝑦 = −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 . Letnow 𝑥 ∈ 𝑋 be any feasible payoff-value, and let {𝑞𝑛}𝑛∈ℕ be a sequence of rational feasiblepayoff-values converging to 𝑥. By prize-continuity 𝜋𝑓𝑞𝑛

→ 𝜋𝑓𝑥, which given Equation 24

implies that 𝜈(𝑞𝑛) → 𝜈(𝑥). By the result for rational numbers, 𝜈(𝑞𝑛) = 𝜈(0)𝑒𝜓𝑞𝑛 , and hence𝜈(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 24 is invariant to multiplying 𝜈by a positive number, we obtain that Equation 25 holds for all 𝑥 ∈ ℝ.

35

Lemma 2

Proof. Let 𝑎 ∈ 𝐹 denote some constant payoff-function, and define 𝑝 = 𝜋𝑎. By MinimalComplexity and Absolute Continuity there exists a finite partition 𝒮 of the state-spaceconsisting of at least three events, such that 𝜋𝑟(𝐴) > 0 for any 𝑟 ∈ 𝐹 and 𝐴 ∈ 𝒮 . LetΣ(𝒮 ) ⊆ Σ denote the algebra generated by 𝒮 , and let 𝑅(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 25 holds any probability measure𝑓 ∈ 𝐹 (𝒮 ) and any event 𝐴 ∈ 𝒮 . In particular 𝑎 ∈ 𝐹 (𝒮 ) (any constant payoff-functionis), and hence for any 𝐴 ∈ 𝒮 , 𝑝(𝐴) = 𝜋𝑎(𝐴) ∝ 𝑝𝒮 (𝐴)𝑒𝜓𝒮 𝑎. Thus, 𝑝(𝐴) = 𝑝𝒮 for any event𝐴 ∈ 𝒮 , and hence also for any event 𝐴 ∈ Σ(𝒮 ). Define 𝜓 = 𝜓𝒮 . It follows that for anypayoff-function 𝑓 ∈ Σ(𝒮 ) and any event 𝐴 ∈ 𝒮 , 𝜋𝑟(𝐴) ∝ 𝑝(𝐴)𝑒𝜓𝑟(𝐴).32

Let now 𝐴 and 𝐵 denote any events such that 𝑝(𝐵) > 0, and let 𝑟 be any payoff-function. I need to show that 𝜋𝑟(𝐴)

𝜋𝑟(𝐵) = 𝑝(𝐴)𝑝(𝐵) 𝑒

𝜓(𝑟(𝐴)−𝑟(𝐵)). To simplify notation let 𝛿𝑓 (𝐴,𝐵) =log 𝜋𝑟(𝐴)

𝜋𝑟(𝐵) − log 𝑝(𝐴)𝑝(𝐵) . With this notation I need to prove that 𝛿𝑓 (𝐴, 𝐵) = 𝜓(𝑟(𝐴) − 𝑟(𝐵)).

Let 𝐸1, 𝐸2, … 𝐸𝑛 denote the events in 𝒮 . Without limiting generality suppose 𝐴 ∩ 𝐸1 isnot-null. Define a payoff-function 𝑔 ∈ 𝐹 by 𝑔 = 𝑟(𝐴) on 𝐴 ∩ 𝐸1 and 𝑔 = 𝑟(𝐵) elsewhere,and a payoff-function ℎ ∈ 𝐹 (𝒮 ) by ℎ = 𝑟(𝐴) on 𝐸1 and ℎ = 𝑟(𝐵) elsewhere. Withthese 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) − 𝑟(𝐵)), where the last stepuses the fact that ℎ is in 𝑅(𝒮 ), and the other steps use Consequentialism and the fact thatby Shift-Invariance 𝜋𝑟(𝐴) = 𝜋𝑟(𝐵) = 𝑝.

Corollary 1

Proof. The proof that a logit distortion is well-behaved is trivial. I thus prove only that if𝜋 is well-behaved then it is a logit-distortion. The conditions of Lemma 2 are met. Let 𝑝and 𝜓 be parameters for which the claim in Lemma 2 holds. Suppose 𝑟 is a.e. simple thenthere exist a finite set of disjoint events {𝐸1, … , 𝐸𝑛} such that 𝑟 is constant on each ofthese events, and for some payoff-function 𝑔, 𝜋𝑔(∪𝑖𝐸𝑖) = 1. By Absolute Continuity also𝜋𝑟(∪𝑖𝐸𝑖) = 1, and so 𝜋𝑟(𝐴 ∩ ∪𝑖𝐸𝑖) = 𝜋𝑟(𝐴). Given that the events are disjoint it followsthat 𝜋𝑟(𝐴) = ∑𝑖 𝜋𝑟(𝐴∩𝐸𝑖). Using Lemma 2we obtain that 𝜋𝑟(𝐴) ∝ ∑𝑖 𝑝(𝐴∩𝐸𝑖)𝑒𝜓𝑟(𝐴∩𝐸𝑖).By Absolute Continuity 𝑝(𝑆 ⧵ ∪𝑖𝐸𝑖) = 0, and hence ∫𝐴 𝑒𝜓𝑟 d𝑝 = ∑𝑖 𝑝(𝐴 ∩ 𝐸𝑖)𝑒𝜓𝑟(𝐴∩𝐸𝑖).Combining these observations we obtain that 𝜋𝑟(𝐴) ∝ ∫𝐴 𝑒𝜓𝑟 d𝑝.

Lemma 3

Proof. The case of 𝜓 = 0 is trivial. Henceforth I assume 𝜓 ≠ 0. By Corollary 1 there ex-ist a probability measure 𝑝 and a parameter 𝜓 such that Equation 5 holds for any reference

32Note that 𝑝 = 𝜋𝑎 is a probability measure over all the events in Σ—not just the events in Σ(𝒮 ).

36

𝑟 that is almost everywhere simple. If there exists a reference 𝑟 that is not almost every-where simple then there exists an infinite sequence {𝐴𝑛}𝑛∈ℕ of disjoint non-null events.33I need to prove that in this case there exists a number 𝑀 ∈ ℝ such that 𝑒𝜓𝑥 ≤ 𝑀 forall 𝑥 ∈ 𝑋. Suppose otherwise, then it is possible to choose from 𝑋 a sequence {𝑥𝑛}𝑛∈ℕ,s.t. for all 𝑛, 𝑝(𝐴𝑛)𝑒𝜓𝑥𝑛 ≥ 𝑝(𝐴1)𝑒𝜓𝑥1 . Define a reference 𝑟 by 𝑟(𝐴𝑛) = 𝑥𝑛 for 𝑛 ∈ ℕ,and 𝑟(𝑠) = 𝑥1 outside ∪𝑛𝐴𝑛. For 𝑛 ∈ ℕ define also a simple payoff-function 𝑟𝑛 by𝑟𝑛(𝐴𝑛) = 𝑥𝑛 and 𝑟𝑛(𝑠) = 𝑥1 for 𝑠 ∉ 𝐴𝑛. By construction 𝑟 and 𝑟𝑛 agree on 𝐴1 and 𝐴𝑛.Thus, for all 𝑁 ∈ ℕ, 1 ≥ ∑𝑛≤𝑁 𝜋𝑟(𝐴𝑛) = 𝜋𝑟(𝐴1) ∑𝑛≤𝑁

𝜋𝑟(𝐴𝑛)𝜋𝑟(𝐴1) = 𝜋𝑟(𝐴1) ∑𝑛≤𝑁

𝜋𝑟𝑛 (𝐴𝑛)𝜋𝑟𝑛 (𝐴1) =

𝜋𝑟(𝐴1) ∑𝑛≤𝑁𝑝(𝐴𝑛)𝑒𝜓𝑥𝑛𝑝(𝐴1)𝑒𝜓𝑥1 ≥ 𝜋𝑟(𝐴1) ∑𝑛≤𝑁 1 = 𝑁𝜋𝑟(𝐴1) where the second equality follows

from Consequentialism, and the third from Corollary 1. Letting 𝑁 → ∞ we obtain that𝜋𝑟(𝐴1) = 0, in contradiction to the assumption that 𝐴1 is not null.

Lemma 4

Proof. By Corollary 1 there exist a probability measure 𝑝 and a parameter 𝜓 such thatEquation 27 holds for any reference 𝑟 that is almost everywhere simple. I show that theclaim holds with the same 𝑝 and 𝜓 also for a reference 𝑟 that is not everywhere simple. Ifsuch payoff-functions then by Lemma 3 there exists a number 𝑀 , such that 𝑒𝜓𝑥 ≤ 𝑀 forall 𝑥 ∈ 𝑋. Assume first that 𝜓 ≠ 0. For any 𝑛 ∈ 𝑁 divide the interval [𝑚, 𝑀] into 2𝑛 non-overlapping intervals of length 𝑀−𝑚

2𝑛 . For any state 𝑠 let 𝐼𝑛(𝑠) denote the interval to which𝑒𝜓𝑟 belongs, and let 𝐼𝑚𝑖𝑛

𝑛 (𝑠) denote its lower endpoint. Define a simple payoff-function 𝑟𝑛by 𝑟𝑛(𝑠) = log 𝐼𝑚𝑖𝑛

𝑛 (𝑠)𝜓 . With this definition 𝑒𝜓𝑟(𝑠) − 𝑀−𝑚

2𝑛 ≤ 𝑒𝜓𝑟𝑛(𝑠) ≤ 𝑒𝜓𝑟(𝑠) for all 𝑠, andso 𝑒𝜓𝑟𝑛(𝑠) ↗ 𝑒𝜓𝑟(𝑠) for all 𝑠. Moreover, since 𝑒𝜓𝑥 is a monotonic continuous function also𝑟𝑛 → 𝑓 . Thus,

𝜋𝑟(𝐴)𝜋𝑟(𝐵) =

lim 𝜋𝑟𝑛(𝐴)

lim 𝜋𝑟𝑛(𝐵) = lim

𝜋𝑟𝑛(𝐴)

𝜋𝑟𝑛(𝐵) = lim

∫𝐴 𝑒𝜓𝑟𝑛 d𝑝∫𝐵 𝑒𝜓𝑟𝑛 d𝑝

=lim ∫𝐴 𝑒𝜓𝑟𝑛 d𝑝lim ∫𝐵 𝑒𝜓𝑟𝑛 d𝑝

=∫𝐴 𝑒𝜓𝑟 d𝑝∫𝐵 𝑒𝜓𝑟 d𝑝

(31)

where the first step follows from Prize-Continuity, the second and fourth since 𝑝(𝐵) > 0and 𝜋𝑟𝑛

(𝑠) ∈ [𝑚, 𝑀] on 𝐴∪𝐵, the third from Corollary 1, and the fifth from the monotoneconvergence theorem.

33If 𝑟 has infinitely many atoms these atoms can form the sequence. Otherwise, let 𝐸 denotethe event outside the set of atoms (if any). 𝐸 cannot be null, or else 𝑟 is almost always simple.Since 𝑟 has no atoms on 𝐸 it follows that there exists a value 𝑦 (the median of 𝑟 on 𝐸) such that𝑝(𝑠 ∈ 𝐸 ∶ 𝑟(𝑠) ≤ 𝑦) = 𝑝(𝐸)

2 . Thus 𝐸 includes two non-null events on which 𝑟 has no atoms: 𝐸(𝑦)and 𝐸 ⧵ 𝐸(𝑦). This process can be repeated recursively, where in the 𝑛’th stage 𝐸 is split into 2𝑛

disjoint non-null events. An infinite sequence of disjoint non-null events can therefore be formed.

37

Theorem 1

Proof. I prove that if 𝜋 is a well-behaved distortion then it is a logit distortion. The oppo-site direction is trivial. By Lemma 4 there exist a probability measure 𝑝 and a parameter 𝜓such that Equation 27 holds for any events 𝐴 and 𝐵 for which 𝑝(𝐵) > 0 and any reference 𝑟for which there exists a number 𝑚 > 0 such that 𝑒𝜓𝑟 ≥ 𝑚 on 𝐴∪𝐵. To complete the proof Ineed to show that Equation 27 holds even if no such number 𝑚 exists. Let 𝑟 be any payoff-function and let 𝐴 and 𝐵 be any events such that 𝑝(𝐵) > 0. If 𝑟 is almost everywheresimple the claim follows from Corollary 1. Otherwise, by Lemma 3 there exists a number𝑀 ∈ ℝ such that 𝑒𝜓𝑥 ≤ 𝑀 for all 𝑥 ∈ 𝑋. For 𝑛 ∈ ℕ let 𝐴𝑛 = {𝑠 ∈ 𝐴 ∶ 𝑒𝜓𝑟 ≥ 2−𝑛}, andsimilarly define 𝐵𝑛. By construction lim𝑛→∞ 𝐴⧵𝐴𝑛 = ∅ and similarly lim𝑛→∞ 𝐵⧵𝐵𝑛 = ∅.Moreover, since 𝑝(𝐵) > 0 there exists 𝑛0 ∈ ℕ such that for all 𝑛 ≥ 𝑛0, 𝑝(𝐵𝑛) > 0. Theconditions for Lemma 4 therefore hold for 𝐴𝑛 and 𝐵𝑛 for all 𝑛 ≥ 𝑛0. Combining theseobservations we obtain that

𝜋𝑟(𝐴)𝜋𝑟(𝐵) = lim

𝑛→∞𝜋𝑟(𝐴𝑛)𝜋𝑟(𝐵𝑛) = lim

𝑛→∞

∫𝐴𝑛𝑒𝜓𝑟 d𝑝

∫𝐵𝑛𝑒𝜓𝑓 d𝑝

=lim𝑛→∞ ∫𝐴𝑛

𝑒𝜓𝑟 d𝑝lim𝑛→∞ ∫𝐵𝑛

𝑒𝜓𝑓 d𝑝=

∫𝐴 𝑒𝜓𝑟 d𝑝∫𝐵 𝑒𝜓𝑓 d𝑝

(32)

where step 3 holds since the integrals are bounded from below and above: (i) 𝑒𝜓𝑥 ≤ 𝑀for all 𝑥 ∈ 𝑋, so the integrals are bounded from above by 𝑀 , and (ii) 𝑝(𝐵𝑛0

) > 0 and𝑓 ≥ 2−𝑛0 on 𝐵𝑛0

, and hence there exists some 𝑚 > 0 such that for all 𝑛 ≥ 𝑛0, ∫𝐵𝑛𝑒𝜓𝑟 d𝑝 ≥

∫𝐵𝑛0𝑒𝜓𝑟 d𝑝 ≥ 2−𝑛0𝑝(𝐵𝑛0

) ≥ 𝑚.

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Figure 1: The interface of the Farmers treatment with a maximum accuracy bonusof £5. The interface of the Bakers treatment was similar, except: (a) the first threelines were: “You have a buyer for £16,000 worth of bread from your bakery. Atday 100 you will get the money from the order, and will have to use some of it tobuy wheat at the market. Your profit is whatever you would have left after payingfor the wheat.”, and (b) instead of an arrow on the chart pointing to £4,000 withthe label “Wheat production costs”, there was an arrow pointing to £16,000 withthe label “The price you would get for your bread”.

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Figure 2: The charts used in the 12 earning periods. The 𝑥-axis represents time,ranging from day 0 to day 100, and the 𝑦-axis represents price, ranging from £4,000to £16,000. The data for the charts were adapted from historical equity price data,shifted and scaled to fit into a uniform range. Figure 1 shows how these chartswere presented to subjects.

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0.0

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

£10,000 confidence level 1

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£6,400 confidence level 3

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£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.

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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 ofsubjectiveuncertainty

Chart uncertainty: low 269∗∗ (s.e. 127) 792 (66)Chart uncertainty: high 635∗∗∗ (s.e. 166) 792 (66)

low > high ? 𝑝 < 0.0142Within 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 thevalue of the day100 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 inability topredict prices

Average confidence: low 276∗ (s.e. 174) 792 (66)Average confidence: high 628∗∗∗ (s.e. 169) 792 (66)

low > high ? 𝑝 < 0.0732Prices predictable? no 292∗∗ (s.e. 174) 792 (66)Prices predictable? yes 613∗∗∗ (s.e. 174) 792 (66)

no > yes ? 𝑝 < 0.0997

DemographicsMales 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.Clustering is 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.

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7000 10102 13000

Farmers

7000 9650 13000

Bakers

Figure 4: Histogram of the mean predictions made by Farmers and Bakers. Anormal distribution curve was fitted to both histograms. The mean prediction was10102 and 9650 respectively. 16 of the 20 subjects making the highest (lowest)mean predictions were Farmers (Bakers).

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Strength of interests

Figure 5: The comparative statics of wishful thinking bias. The panels show a95 percent confidence interval for difference in predictions between Farmers andBakers (the treatment effect) in different subsamples. The first panel shows thecomparative statics of the cost of holding wrong beliefs, represented by the max-imum accuracy bonus. The solid hyperbolic line represents the best fit for theOptimal Expectations model, and the dashed horizontal line that of Priors and De-sires. The second panel shows the bias in a chart against the mean confidencein predictions for that chart. The curve is fitted to the inverse of the square ofthe mean confidence level. The third panel shows the bias in a chart against themean within group predictions variance. The dashed line is a linear fit through theorigin. Finally, the fourth panel shows the comparative statics of the stakes, the𝑥-axis representing the amount in pounds that a subject receives for each £1,000of notional profit. The dashed line is a linear fit through the origin.

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