Information Projection: Model and Applications. Kristóf Madarász London School of Economics and Political Sciencey First Version: April 2007. This Version: May 2009.
Abstract Careful evidence suggests that people underestimate the di¤erence between the information they have and the information available to others. I model such information projection by assuming that after processing a signal, a person overestimates the probability with which this signal is available to others. I show that as a result, a biased person exaggerates the value of the information others have. When learning about an expert’s skill using ex-post information, a biased evaluator exaggerates how much a skilled expert could have known ex ante, and always underestimates the expert on average. Hence even neutral evaluators will come to favor agents who act in stable environments, and discriminate against agents who do not share their informational background. To reduce unfavorable treatments, experts will be too reluctant to produce useful information that will be seen more clearly by the evaluator ex post, and too eager to gather information that the evaluator will independently learn ex post. [Proposition 5]. I show that tighter monitoring might introduce additional noise into evalutaions and in e¤ect uniformly lower e¤ort. A rationale for weak incentives and mandatory communication protocols are provided. Keywords: Discrimination, Monitoring, Medical Malpractice, False Hostility. I am especially grateful to Botond Koszegi and Matthew Rabin for their suggestions and encouragement. I would also like to thank George Akerlof, Vince Crawford, Jerker Denrell, Marina Halac, Daniel Kahneman, Ulrike Malmendier, Marit Rehavi, David Rosenberg, Adam Szeidl, and seminar participants at the London School of Economics, London Business School, ELSE at the University College London, Oxford Nu¢ eld, Stanford, UC Berkeley, UC San Diego, UC Santa Barbara, IZA Bonn, Cambridge, Yahoo Research Labs, Google BAGT, IESE Barcelona, Central European University and the University of Hawaii for comments. This paper is based on Chapter 2 of my dissertation submitted to UC Berkeley. y Contact:
[email protected] or London School of Economics and Political Science, Houghton Street, WC2A 2AE, London, United Kingdom.
1
1
Introduction
People often face di¤erent information regarding a variable they care about; indeed, this insight has inspired a considerable literature in economics e.g. Akerlof (1970) and Spence (1973). While the literature on such informational asymmetries typically assumes that people perceive informational di¤erences correctly, evidence shows that people systematically mispredict informational di¤erences. In particular, people exaggerate the extent to which their own information is available to others, and too often act as if others shared the same information they do. For example, having learned novel information about a patient, a person might well exaggerate the likelihood with which a physician should have diagnosed cancer earlier; by doing so, she will use the wrong standard in judging the physicians performance. Under-appreciating informational di¤erences in this fashion matters not only in performance evaluation, it e¤ects bargaining outcomes, social learning or the way experts respond to incentives. To understand this phenomenon more carefully, this paper introduces a simple, but general model of such information projection. The model identi…es a key economic property of the evidence, and o¤ers a widely applicable tool to explore the consequences of information projection to a broad array of domains. Building on a rich set of experimental and more stylized …eld evidence, the model o¤ers a uni…ed framework to describe well-documented social mispredictions as various instances of information projection. and provides a setup which o¤ers robust predictions to a number of important economic contexts. After deriving some general properties of the model, the bulk of the paper focuses on establishing testable implications of the model to labor markets and organizations. In Section 2, I review both controlled laboratory, and more stylized …eld evidence to support my claim that information projection is a widespread phenomenon. The evidence shows that people typically exaggerate how much of their current information was already available in the past, hindsight bias, Fischho¤ (1975); the extent to which the outcome of a chance event was predictable, outcome bias, Baron and Hershey (1988); or the extent to which their lies or emotions are discernible to others, illusion of transparency, Gilovich, Savitsky, and Medvec (1998); etc. In the context of …nancial markets, Camerer, Loewenstein, and Weber (1989) provide evidence consistent with the claim that better informed traders greatly overestimate how much uninformed traders know, and that such curse of knowledge a¤ects market outcomes. The evidence comes from a variety empirical paradigms, and provides support for the
2
hypothesis that a signi…cant fraction of the people su¤er from considerable information projection. In Section 3, I present the model. I consider an environment where people observe signals about an underlying state, and assume that a person who su¤ers from information projection exaggerates the probability with which the information content of her signals is also available to others. The key property of this de…nition is that people with private information will exaggerate the value of the information others have. This property captures much of the experimental evidence, drives the main predictions of the model and distinguishes this approach from the existing formal intuitions developed in the experimental literature. While the current model builds on important developments in the experimental literature, especially CLW (1989), existing formalizations are incomplete and apply only to a subset of the contexts considered here. Even in such contexts they suggest no robust predictions, and since they are not consistent with the identifying assumptions of the current approach, are unable to generate the economic predictions o¤ered by this model. To preview the results, in Section 4, I specify a broad class of inference problems, where by exaggerating the value of information an actor has, a biased observer, who is also learning about this actor’s type –e.g. his competence, e¤ort, fairness –will underestimate this actor’s type on average. In the context of relative performance evaluation, in Section 5, I show that the above result implies that the evaluations of an otherwise neutral supervisor will exhibit ine¢ cient favoritism. In particular, the model predicts that agents who work in stable environments are more likely to be promoted relative to equally competent agents who work in volatile environments even if the supervisor is more con…dent in her estimate of the latter class of agents. Furthermore, supervisor’s may well discriminate against agents whose information compelements the information revealed by monitoring relative to agents whose information is similar to the information to the latter information. Importantly, I also study who those evaluated by biased supervisor’s might respond strategically to the presence of information projection. The key application of the paper shows that increasing the intensity of monitoring, causes a concerned physician who anticipates being investigated by biased examiners, to ex-ante over-supply medical tests which are substitutes of, and under-supply medical tests which are complements of the ex-post information revealed by monitoring. In both contexts, I provide a rationale why when coarser performance measures replace …ner ones, such distortions in the allocation of labor and the supply of information can diminish, while the welfare of
3
the parties increases. Thus in contrast to the standard Bayesian predictions, more informative monitoring technologies might consistently lead to more discriminatory outcomes, higher production costs, and lower productivity.A similar logic outlined in Section 8, suggests, why particular increases in the information revealed in social interactions can increase perceptions of hostility and reduce cooperation between groups. At the same time, Section 7, the model predicts that a combination of mandating communication and restricting the set of messages a sender might use, can reduce misunderstanding and dominate free-form communication even in pure-coordination problems. To illustrate the logic of the results, consider a radiologist who diagnoses a patient based on an ambiguous X-ray. After the diagnosis is made, the patient returns with novel symptoms and an evaluator is asked to assess the radiologist’s original diagnosis. A biased evaluator projects the ex-post information, and acts as if all radiologists should have guessed the symptoms earlier. Assume that radiologists di¤er in their competence, and competent ones understand the X-ray and uncompetent ones do not. If more exante information increases the chances of an ex-post successful treatment, a biased evaluator exaggerates the success rate for both types of radiologists. In hindsight, a successful treatment becomes the norm and a failed one becomes a surprise. Since the probability of making a failed diagnosis decreases with competence, the evaluator underestimates the radiologist on average. The ’surprisingly’ high failure to success ratio is perceived to be the result of the lack of skill, rather than the lack of su¢ cient ex-ante information. While the evaluator underestimates the agent on average, information projection will typically a¤ect her conditional beliefs as well. Whenever knowing the symptoms ex-ante would have increased the chances of a successful diagnosis for a skilled type than for an unskilled type, the evaluator over-infers skill from performance. For example, if the symptoms alone are uninformative, but combined with the X-ray they are perfectly indicative of cancer, a biased evaluator perceives di¤erences in luck to be di¤erences in skill. If however, knowing the symptoms alone is almost perfectly informative, and hence the probability of a failed treatment depends very little on understanding the X-ray, the evaluator perceives di¤erences in performance to be due to di¤erences in luck. Here, she underinfers skill from performance. Applying these results to CEO turnover, the model predicts an illusion of talent e¤ect where boards blame CEOs after negative events and reward them after positive events that CEO’s could not have predicted if ex-post information is a complement to CEO talent.
4
Given these results, a natural question to ask is how agents might change their behavior to minimize the adverse e¤ects of information projection on their reputation. Evidence from law and medicine suggests that professionals anticipate that those evaluating them are biased, and do alter their behavior in response. For example, it is argued that a signi…cant portion of “defensive medicine”, medical procedures designed to minimize false liability rather than maximize cost-e¤ective health care, is due to the fear of experts that those evaluating them will su¤er from projecting current information to the past. To study such behavior, assume that the radiologist can decide what radiographs to order ex ante. I show that if a radiograph is a substitute of the ex-post information, i.e., it provides information ex-ante that the evaluator will independently learn ex post, then the radiologist has an incentive to over-produce this radiograph. Overly costly MRI’s might be ordered for all patients if such MRI’s produce the same information that the evaluator inevitably learns ex post. At the same time, the radiologist is too reluctant to produce complement information, i.e., radiographs that help him make a good diagnosis but can be interpreted better in light of ex-post information. He will avoid ordering a mammography that helps detect breast cancer if he fears it can be interpreted much better in hindsight than in foresight. Thus as a result of information projection, increasing the likelihood of ex-post evaluations could increase production costs, lower productivity and exacerbate the types of over- and underproduction of information that observers have attributed to medical malpractice regulation. Stylized evidence from medicine parallels these predictions of the model. In particular, it has been repeatedly argued that it is the fear of jurors’ misperceptions of the accuracy of tests that causes physicians to be reluctant to adopt the most e¤ective diagnostic procedures if those can be much better interpreted in hindsight than in foresight.1 If the management of a hospital is aware of evaluators propensity to project information, it might opt for choosing coarser monitoring technologies or adopt di¤erent incentive and liability schemes for the production of information which is a substitute and the production of information that is a complement of the ex-post information. In important situations however, even the perfect anticipation of the evaluator’s bias cannot eliminate ine¢ ciencies. To show this logic more clearly, in Section 6, I turn from a context where the amount of information that the radiologist learns from an X-ray is a function of e¤ort - the case of moral hazard –rather than that of skill –the case of adverse selection. To motivate the radiologist, a hospital may provide incentives to encourage a careful reading of the 1
See Berlin (2003), Studdert et al. (2005), and Jackson, and Righi (2006).
5
X-ray. When the radiologist’s e¤ort is not observed however, he might be rewarded and punished based solely on whether the patient’s condition improved or deteriorated. In cases of limited liability or riskaverse radiologists, no such reward scheme can be …rst-best optimal. A second-best scheme may instead involve monitoring whether the radiologist’s diagnosis accorded with the information that was available to him ex-ante. A biased evaluator, however, is prone to judge the correctness of the diagnosis not on the basis of the ex-ante available information, but on the basis of both the ex-ante and the ex-post information. Thus the radiologist is punished too often for bad luck and rewarded too rarely for good decisions. As a result, the radiologist’s incentive to carefully understand the X-ray is lower than under a Bayesian evaluator. More generally, an agent who is de jure facing a negligence rule is de facto punished and rewarded according to strict liability if he is assessed by a biased judge or jury. I show that the report of a biased evaluator contains too much noise and hence even if the hospital anticipates the bias, it has a reason to monitor less often than in the rational case. I also show that if the hospital does rely on biased reports, it nevertheless decides to induce lower levels of e¤ort to save on incentives that are appropriate in the rational case, but too strong in the biased case. In Section 7, I turn to the in‡uence of information projection on communication. I show that a listener who projects his private information will be too credulous of a speaker’s advice because he overestimates how much the speaker knows. I also show that a speaker who projects information on her non-communicable background knowledge, will mistakenly send messages that are too ambiguous for her audience to interpret. I identify conditions for over- and under-communication. Finally, in Section 8, I conclude with a brief discussion of some further implications and extensions of my model. I discuss how information projection might a¤ect social inferences in networks causing hostility between groups, as well as the possibility of extending my model to capture the related phenomenon of ignorance projection, where a person who does not observe a signal underestimates the probability that this signal is available to others. Although I don’t discuss ignorance projection at much detail in the paper, I discuss how it can be incorporated into the model without much di¢ culty. In fact, although there is much less evidence on this latter phenomenon, information and ignorance projections might well be two-sides of the same coin, that is, a typical person …nds the information she has to be too representative of the information others have.
6
2
Evidence and Related Literature
Folk wisdom has for a long time recognized the existence of what I call information projection, as noted by common refrains, "hindsight is 20/20" or monday morning quarterbacks. I begin this section by discussing both laboratory and more stylized evidence on the hindsight biass – the phenomenon that people form biased judgements in hindsight relative to foresight –, and the curse of knowledge – the phenomenon that informed people overestimate the information of those uninformed.2 I then turn to a brief summary of some evidence on related biases lending support to the existence of the projection of various forms of private information. Although individual studies are often subject to alternative interpretations, the sum-total of the studies provides a compelling case for the widespread existence of this phenomenon. The research on hindsight bias, initiated by of Fischho¤ (1975) showed however that reporting an outcome of an uncertain historical event increases the perceived ex-ante likelihood of the reported outcome occurring. Fischho¤’s …ndings were replicated by hundrerds of studies, and most of these …nd a strong presence of various forms of such hindsight bias, often signi…cantly larger than the one found in his initial study. These studies and the meta-analyses building on them also show that the presence of hindsight bias is robust to a great number of debiasing techniques. A robust comparative static result is that the more informative the outcome the greater is the bias, Harley et al. (2004). As I demonstrate in Section 3, my model of information projection exhibits the same monotonicity. In an illustrative study, Camerer, Loewenstein, and Weber (1989) asked Wharton and Chicago MBA students to participate in a market, and trade assets of eight di¤erent companies in a double-oral auction. Traders were divided into two groups. In the …rst group, traders were presented with the past earnings history of the companies and traded assets that yielded returns in proportion to the actual earnings of these companies. In the second group, received the same information, and also the actual earnings of the companies. By design, returns for traders in the second group depended on the market price established by the uninformed traders, and to maximize earnings, better-informed traders had to guess this as correctly as possible. CLW …nds that the guesses of better-informed traders were biased by 60% towards the actual 1980 earnings and market prices were biased by 30%.3 The reason why the bias 2 Hindsight bias studies involve both between-subject designs, and within-subject designs, where participants have to recall their own prior estimates after being presented with new evidence. Since my focus in this paper is on interpersonal information projection, I concentrate on the between-subject designs. 3 CLW do not report these numbers explicitly, only graphically, so they are approximate. See CLW (1989) pp.1241.
7
in the market was lower than in judgements is that traders with a smaller bias traded more aggressively. Less biased traders behaved as if they had anticipated that others would project information. Further evidence comes from the experimental study of Loewenstein, Moore, and Weber (2006). They study the curse of knowledge using a set of visual recognition tasks. In these tasks, subjects are presented with two pictures that di¤er in one crucial detail. LMW (2006) divide subjects into three groups: uninformed, informed, and choice. In the uninformed condition, no additional information is available besides the two pictures. In the informed condition, the di¤erence between the pictures is highlighted for the subjects. In the choice condition, subjects could decide whether to obtain additional information for a small fee, or remain uninformed. After looking at the pictures, the subjects in each group are asked to guess what fraction of people in the uninformed group could tell the di¤erence between the two pictures. Subjects are compensated based on how well they predicted this fraction. As Figure 1 indicates, the informed subjects’mean estimate was signi…cantly higher than the uninformed subjects’ mean estimate. Importantly, a signi…cant portion of the people in the choice condition paid for additional information. In this group, the mean estimate was 55:4%, while the mean estimate of subjects who chose to remain uninformed was 34:6%. Hence people not only projected their additional information, but also paid for information that biased their judgements in a way that lower their earnings.4 In an experimental study on what they called the outcome bias, Baron and Hersey (1988) asked subjects to evaluate the quality/correctness of decisions in light of ex-post information. In the most neutral treatment, subjects had to evaluate choices that were the same ex-ante but lead to di¤erent outcomes ex-pots. The choices were between di¤erent gambles where it was common knowledge that the outcomes were determined by the spin of a fair roulette wheel. A typical choice was picking either a gamble with a sure value of $200 or a risky gamble with an 80/20 chance of winning $300 or nothing. Based on 160 of such choice pairs, Baron and Hersey found that the evaluation of the initial correctness of the same choice behavior di¤ered signi…cantly depending on whether the outcome of the risky gamble was low or high. Comparing the same decision with di¤erent outcomes, a higher earning was rated as a more correct choice in 60% of the cases, an equally correct choice in 28% of the cases, and as a less correct choice in 12% of the cases. A similar result was true when the two ex-ante equivalent choices lead to the same earning, but the forgone earnings were di¤erent. An even more direct test would have been 4
The true probability was typically slightly below of the uninformed estimates. Also, to control for curiosity, the authors told all subjects what the di¤erence between the pictures were, at the end of the experiment.
8
to ask people to report what outcome they would have predicted after learning the spin of the roulette wheel. I am not aware of any experiment that tested such a scenario in the case of direct lotteries. Less controlled …eld evidence comes from more explicit studies. In the context of liability judgements, there is a wealth of evidence that juries and experienced judges fail to ignore superior information and instead form judgements as if the defendant had information that was unavailable at the time he acted. Experiments have demonstrated the existence of information projection in the evaluation of ex-ante judgements of various experts. Anderson et al. (1997) documented the existence of the bias in judges deciding on cases of auditors’liability where auditors failed to predict the …nancial problems of their audit clients. Caplan, Posner, and Cheney (1991) conducted a study with 112 practicing anesthesiologists. Here physicians saw identical case histories but were either told that the case ended in minor or were told that it ended in severe damages. Those who were told that a severe damage occurred were more likely to judge the ex-ante care to be negligent. In certain cases, the di¤erence in the frequency of ruling negligence was as great as 51%. Bukszar and Terry (1988) demonstrate hindsight bias in the solution of business case studies, Hastie, Schkade, and Payne (1999) document very serious biases in jurors’judgement of punitive liability. Strong e¤ects were found among others in experiments on the assessment of railroad accidents, legality of search, evaluation of military o¢ cers, etc. For survey articles on the evidence, see e.g., Harley (2007).5 A large set of other psychological …ndings further indicate that people project various types of private information. For example, a study by Gilovich, Medvec, and Savitsky (1998) shows that people greatly overestimate the probability that their lies, once made, are detected by others.6 Such overestimation was also documented in the context of communication. In a set of experiments, Kruger et al. (2005) found that when people communicate through email, they overestimate how well their intent is transmitted through their messages.7 Here, senders had to make serious and sarcastic statements either through email or voice recording, and then guess the probability that receivers would be able to understand their intent. As Figure 2 shows, the mean estimate for both those sending an email and those sending a voice recording was 78%, while the actual probabilities were 73% in the voice condition and 58% in the email 5
The legal profession has long recognized the biasing e¤ects of information projection and developed certain procedures to mitigate its e¤ect. One of such procedures is the bifurcation of trials where ex-post evidence is suppressed at the initial phases of the trial. More on this see Rachlinski (1998). 6 Van Boven, Gilovich, and Medvec (2003), study illusion of transparency and bargaining but here the results are harder to interpret. 7 See also Newton (1990) on tappers and listeners.
9
condition. Kruger et al. (2005) also conduct an experiment where they ask subjects in the email condition to vocalize their messages before sending them. Senders are again randomly divided into two groups; some are asked to vocalize the message in the same tone as the tone of their email, and others are asked to vocalize it in the opposite tone. Senders in both groups overestimate how easy it would be to understand their messages, yet such overestimation decreased signi…cantly in the case where senders vocalize in the opposite tone. While some of these results may be due to general overcon…dence about one’s ability to communicate, the evidence is more consistent with the interpretation of information projection. Two existing papers provide some formalization of parts of what I call information projection. CLW o¤er an incomplete characterization of how a biased but better informed person perceives the mean of another person’s beliefs. Their approach is better suited to illustrating and explaining the experimental evidence rather than If applied to the settings of my model, the CLW assumptions make no robust predictions. Biais, and Weber (2007) follow the assumptions of CLW and study intrapersonal hindsight bias. They derive prediction on whether people underreact to news. They also test their hypothesis using psychometric and investment data from a sample of investment bankers in Frankfurt and London. After presenting the model in Section 5, I compare the properties of my model with the assumptions of CLW and Biais and Weber (2007). Several other papers, with no explicitly developed model or results, argued about the potential importance of various forms of information projection, e.g., Viscusi, and Zeckhauser (2005), Heath, and Heath (2007), Rachlinski (1998), Camerer, and Malmendier (2007), Thaler (200 The model also belongs to the small but growing literature on quasi-Bayesian models of individual biases e.g., Rabin (2002), Rabin and Vayanos (2008), DeMarzo, Vayanos, and Zwiebel (2003), and to the literature on the biases in people’s perception of others, e.g. Eyster and Rabin (2005) and Costas-Gomes, Crawford, and Iriberri (2009). In the context of predicting future changes in one’s taste, the phenomenon of projection has also been studied by Loewenstein, O’Donoghue, and Rabin (2003). In contrast to the projection of taste, the projection of information is most relevant in the interpersonal domain where people think about what others might or might not know, and hence it is primarily a social bias. The evidence summarized in this section is indicative of the fact that people project various forms of information. Although this evidence comes from a diverse set of paradigms that use di¤erent methods of identi…cation, and classify information projection under a variety of rubrics, the model that I present in
10
the next section provides a framework to study this phenomenon in a uni…ed manner. It also provides a setup to make more precise predictions about the implications of information projection in organizations and labor markets and to test such predictions.
3
Model
Consider an environment where people privately observe signals about the payo¤ relevant state such as the fundamental value of a company or the medical conditions of a patient, ! 2
. There is a …nite set
of signals N and the information of person k 2 M is given by the set of signals Ik
N she observes. A
signal is a function from the set of states to the set of lotteries over a realization space, sj (!) : Signals are interpreted given a common prior her utility function uk (y; !) : Y Y and
0 (!)
over
! R; where y 2 Y =
!
Z.
. Finally, let person k’s action set be Yk , and fYk gM k=1 : For simplicity, I assume that both
are …nite.
To characterize the distribution of information, let pjk 2 [0; 1] denote the initial probability that person k observes signal sj . Let us collect these probabilities over signals and across people into a vector M p = ffpjk gN j=1 gk=1 . This vector p describes the over-all distribution of information in this environment as
it characterizes a complete probability distribution over the possible information sets in the population. Note that each sub-vector pk = fpjk gN j=1 is a distribution over all subsets of the N signals, and this distribution assigns probabilities to the events that person k faces information set Ik
N and does so, for all
subsets of N . In turn, the informational environment can be summarized by the tuple f ; ; fsj gN j=1 ; pg.
3.1
De…nition
As long as people have correct rational expectations, the vector p also describes people’s perception of how information is distributed. Information projection introduces a bias in the perception of the vector p: A person who projects information, exaggerates the probability that a signal that is in her information set is also in the information set of others. I introduce a parameter
2 [0; 1] to express the degree of
such information projection. De…nition 1 A person with information set Ik exhibits information project of degree
if her perception
of person i’s information is given by the vector pi where pj; i = (1
)pji +
if sj 2 Ik 11
j and pj; = Ik i = pi if sj 2
(1)
for all j 2 N and i 2 M nfkg. The above de…nition introduces information projection formally. It claims that a biased person misperceives the distribution of information sets in the population. In particular, such a person assigns too high of a probability to the event that another person has the information she has, and too low to the event that this other person does not have the information she does. In the case of full information projection,
= 1, a biased person believes that all her information is available to others. In the case
of partial information projection, 0 <
< 1, she believes that the probability that her information is
available to others is between the truth and the full information projection case. Finally, when
= 0,
she has correct Bayesian expectations.8 The above de…nition is formulated without explicit reference to time, but it naturally extends to projection over time. If person i 2 M is the past or future self of person i0 2 M; then the de…nitio n claims that a biased person projects her current information onto the past and future selves of others. Although the applications in this paper concern only interpersonal projection, evidence from memory tasks indicates the presence of projection onto one own past and future selves. In the conclusion, I return to the possibility of projecting information onto one’s own future self. In above de…nition, the degree of projection
is uniform across signals but the de…nition should be
understood as one where the degree of projection is not a scalar but a vector
= f jk gN j=1 where
degree to which person k projects signal j. The claim of the model thus is that
j k
j k
is the
0. For notational
simplicity, I use uniform projection in the applications, but all results of this paper hold without any loss when projection is considered to be heterogenous. Hence whenever I refer to an increase in the bias , I refer to any increase in any of the components of this vector. 8 The model can be extended by allowing the parameters p to depend on the state !, p(!). In this formulation, after observing a set of signals, a Bayesian person forms a ’posterior estimate pi (!) denoted by pei (!). The de…nition then can be applied to this vector pei (!) in the same way as above. The model thus can be interpreted as one where people have heterogenous priors. Importantly, though relative to postulating the existence of heterogenous priors with no theory of the way these priors will be heterogenous, the current model makes clear directional predictions on people’s con‡icting estimates as a function of the true informational environment.
12
3.2
Projection and the Value of Information
Let me now turn to the key identifying property of the above de…nition, and show that independent of the way information is partitioned or distributed, someone who su¤ers from information projection, overestimates the expected utility of others. To make this statement more precise, consider the utility maximization program of person i given information.Ii .
ui (Ii ) = max E! [ui (y; !) j Ii ]
(2)
yi
where the action of people other than person i is any …xed action y information Ik , let fk (ui ) 2
i
from Y i . Given person k’s
R be the pdf over the real line that person k’s estimate of the expected
utility of person i when the degree of person k’s bias is . The following result, which is based on Blackwell’s theorem on the comparision of information sets, shows that a biased person exaggerates the expected utility of others and that this exaggeration is increasing in her bias. Since a biased person exaggerates the probability with which others know what she knows, she perceives others to be more informed than they truly are. It then follows from Blackwell’s classic theorem on information sets that she exaggerate the expected utility of others.9 Proposition 1 For all environments f ; ; fsj gN j=1 ; pg and von-Neumann-Morgenstern utility ui , fk (ui ) 0
…rst-order stochastically dominates fk (ui ) if and only
<
0
for all i 6= k.
The above result highlights the fact that the impact of information projection can be expressed and measured in utility terms, and furthermore that projecting information implies exaggerating the value of the information others have. A natural corollary of the above proposition is that the more valuable the projector’s information is, the greater is her overestimation of the value of the information another person has for all utility functions. Corollary 1 An increase in the Blackwell informativeness of Ik , leads to an increase in f (ui ) in the sense of …rst-order stochastic dominance if and only if 9
> 0.
Importantly, while Blackwell informativeness o¤ers only a partial ordering of information sets, the above result shows that the Bayesian and the biased perceptions can always be ordered using this criterion. Furthermore, such an ordering is preserved when comparing a more to a less biased perspective. I reference Blackwell informativeness in more detail, based on Blackwell (1953), in the Appendix.
13
In light of the above results, it is evident that the misperception induced by information projection can be measured in terms of the expected utility of the projectee. Though, expressed in terms of exaggerating the probabilities with which the projector’s signals are available to the projectee, information projection a¤ects the former’s perception of the value of the information of the latter. Importantly, the above result holds independently of the details of the informational environment. In particular, they hold no matter how informational di¤erences are partitioned into various signals, or what the exact characteristics of these signals are. The result follows from the de…nition directly, and do not on depend the exact shape of the participant’s preferences or the nature of how information is broken down into signals. Making additional assumptions on these two adds further structure to the model, and thus leads to a richer set of predictions, but the basic logic of the above result, as well as the result in Proposition 2 present late in this Section, hold generally. As mentioned before, CLW (1989) proposes an incomplete model to explain the experimental data, on the curse of knowledge. The above result allows me to compare my model to their analysis in more detail. In the CLW formulation, a biased person believes that the mean of the other person’s estimate of a random variable is the convex combination of her superior estimate and this person’s true estimate.10 This de…nition applies only to situations where the projector is more informed than the projectee, and speci…es a biased person’s expectation about the mean of another person’s estimate only. Hence it o¤ers little guidance in contexts where moments other than the mean, such as precision of an person’s information matter. Asides from such serious incompleteness, a crucial di¤erence remains between the two approaches. Although these formalizations relate to similar psychological intuitions, a key di¤erence implies that in most economic applications the predictions of the two models will di¤er signi…cantly:
11
Even if one completed their setup by specifying other moment’s of the projector’s estimate, the convex combination of two mean is consistent with both under- and overestimating the value of information and implies neither of them. The prediction on which of these will occur, changes 10
Formally, CLW propose a model where a person who has information set I1 estimates the information of a person with information set I2 I1 according to the following formula E[E[X j I2 ] j I1 ] = E[X j I1 ] + (1 11
)E[X j I2 ]:
Biais and Weber (2009). follow this formulation in the context of indvidual hindsight bias where a person’s recollection of his past beliefs are such but the variance remains the same.
14
with small changes in the informational environment or the degree of the bias. In contrast, the predictions of this model are based on the fact that a biased person overestimates the value of the information others have, and that such a misperception is increasing in the bias and in the informativeness of what the projector knows. I now turn to a distinction between complement and substitute information which could not be made in the CLW framework, but which will drive the comparative static predictions of my model.
3.3
Substitute and Complement Information
A key feature of my model is that the consequences of information projection depend on how the content of the projector ’s information relates to the content of the true information of the projectee. If the two are the same, then even if the signals are di¤erent, information projection is inconsequential when measured in utility terms. At the same time, if the two are di¤erent, then even each alone has little or no value, but their combined content might reveal the true state, information projection has a signi…cant e¤ect. Intuitively, in the …rst case the information contents of the projector’s and the projectee’s signals are substitutes, in the latter they are complements. To characterize these relationships formally, I introduce a distinction between substitute and complement signals. This distinction further highlights the properties of the model, and shows how the e¤ects of information projection depend on the value of the information projected. This distinction is also used later in the paper in establishing the key comparative static results of this paper. De…nition 2 Fix an objective function uk and …xed information Ik . I call two signals sl ; sj 2 = Ik substitutes if uk (sl [ sj )
uk (sj )
uk (sl )
uk (;) and complements if uk (sl [ sj )
uk (sj )
uk (sl )
uk (;):
Note that these relationship are de…ned only relative to a payo¤ function uk since the gain from learning a signal is expressed in expected utility terms. Formally, the above de…nition calls two signals substitutes if uk is submodular in these two signals, and complements if uk is supermodular in them. Crucially, the complement relationship of signals cannot be formulated in the CLW framework because comparing means never allows for non-concavities in the value of information that arise in the case of complement signals.12 12
In Section 7, I illustrate how the projection of complement signals can lead to unintended ambiguity in communication through the non-concavities in the value of information.
15
To conclude the setup of this model, I introduce the possibility that some people might correctly anticipate the bias of others. As mentioned in Section 2, evidence suggests that in important contexts, people do anticipate the bias and respond to them. Since I build on this in the applications, I de…ne such anticipation formally. Let the probability density function
i( k )
concerning the extent to which person k 6= i projects information. If
i
describe the beliefs of person i is not concentrated on 0, person
i believes that there is a non-zero probability that person k is biased.13
4
Underestimation
This section establishes a general consequences of information projection to a class of inference problems. It hows that information projection leads to a systematic violation of the principals of Bayesian learning. Speci…cally, in a class of monotone problems where a person is learning about the characteristics of others, I show that by exaggerates how much information others have, a biased person will underestimate a certain set of characteristics of others. For example, when projecting information about one’s taste, a biased guest will always underestimate the host’s kindness. Similarly, such underestimation follows in the context of assessing skill, Section 5, monitoring e¤ort, Section 7, learning about attentiveness, Section 8. In what follows, I state the result in a general and more abstract manner, and for a more detailed discussion and economic intuition, I direct the reader to Section 4. There, I state a simpler variant of this result in the context of skill assessment. A reader who is not interested in the more general version of this result, can skip this sub-Section and jump directly to Section 4 without loss of content. Consider a family of real valued conditional densities f (x j ; u), where x 2 X. Assume that is drawn from a prior distribution f ( ) and u 2 U from a prior distribution f (u). I interpret
2 as the
hidden characteristic (type) of the agent, and u as the utility value of his information. To establish the next result, I assume that a higher outcome x is positively associated both with a higher expected utility and a higher type. Formally, following the de…nition of Milgrom (1981), I assume that the conditional densities satisfy the monotone likelihood ratio property in
and u. In what follows, f (x j ) stands for
the marginal conditional distribution of x where = ; u. 13 The results of this paper are based on inference problems and do not depend on speci…c assumptions on further higherorder beliefs of the parties. In a related paper, the author studies the e¤ects of information projection on bargaining and bilateral trade extending this model to a full equilibrium type reasoning with biased players. That even with more involved strategic reasoning such in auctions or bargaining assumption on the Crawford and Iribberi (2007)
16
Condition 1 The collection of densities f (x j ; u), satis…es the (strict) monotone likelihood ratio property in both
and u if f (x j u) f (x0 j u0 )
f (x j u0 ) f (x0 j u)
(>)0
whenever x > x0 and u > u0 , and the analogous condition holds for any
>
0.
I now apply Proposition 1 to this setup and assume that the observer misperceives f (u) to be f (u) where the latter is increasing in
in the sense of …rst-order stochastically dominance. Let f ( j x)
denote the marginal conditional distribution of
given beliefs f (u) and observation x. I can now
state Proposition 2 which shows that given the assumptions in Condition 1, an observer who projects information will underestimate
on average.
Proposition 2 Suppose the set of densities f (x j ; u) satis…es the (strict) monotone likelihood ratio property in
2
and u 2 U: For all priors f ( ) and f (u), EX [f ( j x)] (strictly) …rst-order stochastically 0
dominates EX [f ( j x)] if and only if
<
0
where expectations are taken with respect to the true
distribution of signals. Above I assumed that a higher outcome is good news about quality . Since the results depend only on the monotonicity assumption, it follows that when a higher x is bad news about
– in the sense of
Milgrom (1981) –a biased observer will come to overestimate the ’quality’of the agent. Finally, when x is neutral about ;no misestimation is implied by information projection.
5
Performance Evaluation
Let’s now turn to the main application of the paper the impact of information projection on performance evaluation. Consider an environment where a supervisor evaluates an agent, and the latter faces a task to process and act upon information available to him. The role of performance evaluation in this section is to learn about the competence of the agent which then can serve as the bases of optimal allocation of human resources. Such activity is key in organizations and labor markets, and there is ample evidence on the presence of serious information projection in this context. The key feature of evaluations is that the supervisor typically has access to information that was not available to the agent. This creates room for information projection. Below are three illustrative examples of the setup: 17
5.1
Agent
Supervisor
Ex
ante Information
Ex
post information
Radiologist
Medical Examiner/Jury
X-ray, biopsy etc
Reaction to treatment
CEO
Board
value of various assets
Market conditions
Social Worker
Government
Foster family history
Accident to the child14
.
Setup
First the agent (radiologist) receives a noisy signal s0 ;(an X-ray), about the state ! 2
, (the medical
conditions of a patient). He then takes an action ya 2 Y . After this action is taken, the supervisor observes the outcome x along with some novel information, s1 , about the medical condition of the patient. This outcome x can be interpreted either as a success xS or as a failure xF . Skill. Let
2 [0; 1] denote the agent’s skill/competence, where
stands for the probability that the
radiologist understands the information content of the X-ray. With probability 1 from the signal. I denote his posterior on ! by a common prior
0(
1.
I assume that
he infers nothing
is unobservable, and the parties share
) over [0; 1].
Technology. Let A be a technology matrix with dimensions jY j
j j : An entry in this matrix
corresponds to the probability of a successful outcome, xS , for the given action-state combination, (y; !) 2 Y
: I only assume that the elements of the matrix are numbers between 0 and 1, and allow the
probability of success to vary with the agent’s action and the true state.15 Agent’s Choice. The agent always takes an action ya that maximizes the probability of a successful outcome given his beliefs,
1.
This fact is motivated by the assumption that for extrinsic or intrinsic
reasons, the agent maximizes expected utility, and prefers a success to a failure. If ws > wf are the compensations received after a success and a failure respectively, then it su¢ cies that the agent prefers more money to less money. Formally, the agent’s optimal action is given by ya 2 arg max ya0 A ya
1
= arg max Eua (w): ya
(3)
The above assumptions thus imply that both more information and higher skill correspond to a greater probability of success. 15
(y
This assumption covers all the cases where the probability of success depends on some distance between y and !, e.g. !)2 , but is more general than that.
18
Supervisor’s Information Is . The supervisor observes the outcome x and a signal s1
N , distinct
from s0 . This signal s1 can be interpretation both as ex-post information on the task or as the supervisor’s private information/expertise on the task. Importantly, s1 is part of the monitoring technology and it determines the informativeness of the monitoring technology. In other words, since in our setup the probability of a success depends not only on the agent’s competence,
; but potentially also on the
medical conditions of the patient, !, what the supervisor knows about the patient, determines how much she can learn about the competence of the physician. In many situations, observing success or failure alone, might allow for very little inference about competence. I make no assumptions on whether the supervisor observes the information content of s0 or not. The results hold under both scenarios.16 Processing s1 :I assume that had the radiologist knew the content of s1 , the probability of a success following an optimal action, as de…ned by Eq. (3), would have been weakly higher. This assumption nets the cases where no competence is required to understand s1 and where only the most competent agents could have understood it. The only assumption here is that the probability of success is always non-decreasing in the radiologist’s competence.17 Allocation: The purpose of learning about the agent;s competence is to aid allocation decisions, such as matching employees to jobs or investing in training or deciding to dismiss the agent.18 Since the results in this section do not depend on the details of supervisor’s objective, I leave it unspeci…ed and assume only that it takes the form us ( ; ys ) : [0; 1]
Ys ! R where ys is an unspeci…ed action –e.g. promotion,
re-allocation of the agent –that the supervisor might take given her beliefs about the agent’s skill.
5.2
Assessment
Given her perception of the information. that was available to the agent the supervisor updates her prior 0(
) via Bayes rule. Information projection distorts her perception as she exaggerates the probability
with which the agent could have guessed the content of s1 correctly. If
1(
) denotes the posterior of
a -biased supervisor, the following result shows that a biased supervisor underestimates the agent on average. 16
Note that to control for alternative explanations, in the experimental studies informed people always knew the exact information of the uninformed i.e. s0 : Psychological intuition suggests that the degree of information projection however, might be higher when the supervisor does not observe s0 directly. 17 Formally, Pr(xS j s1 ; s0 ; ; !) Pr(xS j s0 ; ; !) for all ; !. One could consider projection of information in a multidimensional competence setting as well, where this assumption might not be always hold, but such a possibility is outside of the scope of the current paper. 18 See the literature on job-mathcing e.g. Jovanovic (1979).
19
Proposition 3 For all if
0
0
with full support, E[
1]
…rst-order stochastically dominates E[
1
0
] if and only
where expectations are taken with respect to the true distribution of the signals.
The logic of the above result is as follows. By projecting productive information, the supervisor overestimates the overall probability of a successful outcome for all types of radiologists. She is too surprised observing a failure and less surprised observing a success than she should be. Hence, relative to the Bayesian case, she puts too much weight on the information revealed by a failure and too little on the information revealed by a success. Since lower types are more likely to fail than higher types, she underestimates him on average. A next corollary claims that an increase in how much the supervisor knows increases underestimation. The value of information in this setup is measured by its productivity. i.e. the extent to which knowing s1 would have increased the probability of a successful treatment. I call this measure the information gap between the agent and the supervisor and denote it by g. Corollary 2 Let g = ua (s1 [ s0 ) i¤
ua (s0 ), then for all
0,
E[
1]
is decreasing in g in the sense of FOSD
> 0. The more extensive the investigation is the lower is the supervisor’s opinion of the agent on average.
Returning to the example of foster care, if the government increases the depth of investigations, its average opinion of the quality of social workers will decrease. Similarly, if police o¢ cers or physicians are investigated, …ner measures of performance will shift their assessed competence downwards. This will potentially distort allocation decision. A government might withdraw funding or invest too much into the training or oversight of social worker. Recall that success and failure depends not only on the physician’s skill , but also on the patient’s condition !. This means that learning about the patient always allows for better inference under Bayesian rationality. Due to information projection however, more novel information about the patient leads to more undue blame and skepticism about the physician’s competence. Finer performance measures might well back…re, and restricting the supervisor’s information might not only improve her opinion about the agent but improve allocation decisions at the same time. Corollary 3 Consider any two monitoring techniques Is ; Is0 2 N such that Is 20
Is0 and s2 = Is Is0 is
productive. A Bayesian supervisor should always prefer Is0 to Is , but Is0 can lead to lower expected welfare than Is if
> 0.
Note that since Proposition 3 holds for all prior beliefs on
with full support, this result implies
long-term underestimation as well. Hence, if the supervisor samples from a longer sequences of repeated performances, underestimation holds a fortirori.19
5.3
Relative Performance Evaluation
The above results are developed in the contexts of the evaluation of a single agent, but their implications might be more pronounced in the broader context of relative performance measures. In what follows, I show that when agents work in informationally heterogenous environments, information projection provides an unexplored mechanism through which favoritism and discrimination might arise endogenously in labor markets and organizations. Consider a supervisor who evaluates two agents working on separate tasks with independent performance outcomes. Assume that the two agents are equally competent and receive an equally valuable signal s0 and consider changes in the performance measure that is available for the supervisor. The following two results are special cases of Corollary 2. The …rst follows closely from the above discussion, the second is based on the distinction of substitute and complement information. 1. (Adding information to s1 .) Assume that the ex-post information revealed at the …rst agent’s job is a strict subset of the ex-post information revealed at the second agent’s job. A biased supervisor will rank the second agent higher than the …rst agent on average. Intuitively, agents who act in stable environments where there is little ex-post information will be favored to equally competent agents who act in volatile environments with more ex-post information.20 2. (Changing the nature of s1 .) Let’s now hold the value of the supervisor ex-post information constant for both cases. Assume that the …rst agent’s information is a complement of the supervisor’s information, and the second agent’s information is its substitute. The supervisor will rank the …rst 19
It’s important to note that not all addition to an information set Is will lead to further underestimation. For example information about the underlying state might tell the supervisor whether the agent was in a lucky or an unlucky world without supplying information that would have been producative for the agent had it been available. 20 Formally, if s0;1 denotes the information agent 1 gets, and s0;2 the information agent 2 gets, then ua (s1 [ s0;1 ) > ua (s1 [ s0;2 ) follows from De…nition 2 and the assumption that ua (s0;2 ) = ua (s0;1 ).
21
agent lower than the second one on average. This means that the supervisor will favor workers who come from ’di¤erent’informational environments or receive information from di¤erent sources than she does.21 In a classic study, Prendergast and Topel (1996) argue that performance evaluations often exhibit favoritism: a supervisor’s ranking of agents systematically distorts the ranking of equally skilled agents. While Prendergast and Topel model favoritism as an exogenous preference parameter, favoritism arises endogenously even in the absence of an intrinsic preferences for favoritism here. The analysis makes speci…c predictions on the types of agents who will be favored. In models of statistical discrimination, as in Arrow (1973) and Phelps (1972), supervisors discriminate against employee’s belonging to di¤erent groups. Importantly, the source of rational statistical discrimination is imprecise measurement and more precise performance measurement should reduce discrimination and improve e¢ ciency in these models. In contrast, in the current model as long as …ner performance measures increase the information gap, more precise measurement might well increase discrimination and lower e¢ ciency.
5.4
Luck and Talent
Proposition 3 is consistent with the general wisdom that ex-post evaluations involve too much blame. While this result holds on average, it does not follow, as some informal intuition might have suggested, that the supervisor assigns too much blame either after a success or after a failure. In general, in her conditional beliefs, the supervisor might either be too optimistic after a success and not pessimistic enough after a failure. Whether the supervisor over- or underreacts to performance measures depends on the nature rather than the …neness of her information. Let me illustrate the logic by the example of the CEO and the board. Recall that the probability of a success varies positively with the CEO’s talent to assess the pro…tability of the …rm’s projects and that this ranking would have been also preserved, had there been information available to the CEO about market conditions that were revealed only ex-post. The next proposition shows that if having known s1 would have increased the return on the CEO’s talent, then biased members of the board over-infer talent from performance. In contrast, if the information about market conditions is such that it diminishes the role of talent, by making it almost trivial for all types to make the right investment choice in hindsight, 21
Formally, if s0;1 denotes the information agent 1 gets, and s0;2 the information agent 2 gets, then ua (s1 [ s0;1 ) > ua (s1 [ s0;2 ) follows from De…nition 2 and the assumption that ua (s0;2 ) = ua (s0;1 ).
22
then biased board members under-infer talent from performance. Proposition 4 If Pr(xs j s0 ; s1 ; )= Pr(xs j s0 ; ) is increasing in , then ( j xF ) is decreasing in s0 ; ) is decreasing in , then
1(
j xS ) is increasing in , and
in the sense of …rst-order stochastic dominance. If Pr(xs j s0 ; s1 ; )= Pr(xs j 1(
j xS ) is decreasing in
in the sense of …rst-order stochastic dominance:
A special case of the above proposition is when CEO always understand the ex-post news about market conditions independent of his type. Thus competence is only required to process the information about the pro…tability of the various projects of the …rm. Here the above proposition implies that if the ex-post information on market conditions tells much of what could learned ex-ante,i.e. the two are substitutes, then biased members of the board perceive the task of identifying a successful investment as a trivial one, and attribute performance di¤erences too much to luck rather than talent. Since success is perceived to be the norm both for the low types and the high types, members of board will revise their beliefs too little after a success.22 In contrast, if the two signals are complements, the board will greatly exaggerate the extent to which a success is good news and a failure is bad news about the CEO’s talent. Hence they will misattributes di¤erences in luck to di¤erences in ability. In the limit, even if the actual performance di¤erences are entirely due to luck, board members will perceive them as re‡ections of di¤erences in talent. Corollary 4 Assume that s1 is understood independently of . If s1 and s0 are complements for ua , 1(
in
j xS ) increases in
in the sense of FOSD. If s1 and s0 are substitutes for ua ,
1(
j xS ) decreases
in the sense of FOSD. The above results demonstrate that information projection is consistent with excess optimism after a
success as well as too little pessimism after a failure. The key to whether observers will over- or under-infer skill from the performance data is determined by in principle observable features of the informational environment.23 22
Their reaction after a failure is more subtle. Under-inference will cause them to attribute failure to excessively bad luck rather than lack of talent. At the same time, underestimation of Proposition 3, shifts average beliefs downwards. Whether the assessment after a failure will be too low or too high depends on the relative strengths of the opposing forces of underestimation and under-inference. In the case where the marginal return to skill is the same both in the true and in the biased perception, only underestimation has an e¤ect. 23 In setups very di¤erent from mine, Rabin (2002) and Spiegler (200?) also obtain very interesting results on how inferential mistakes may well lead to over-inference and the illusion of talent.
23
5.5
Behavioral Response: The Supply of Information
The analysis so far has con…rmed the adverse e¤ect of information projection on allocation decision, but also suggests that the reputation of agents is a¤ected. I now turn to what is the key behavioral prediction of the model: the optimal response of agents to biased investigations. Evidence from law, Rachlinski (1998), and medicine, Studdert et al (2005), suggests that many professionals do anticipate the presence of what I call information projection, and respond to it strategically. To study such responses, I introduce two necessary extensions of the setup: I assume that agents have some discretion over the task they work on, and care about the supervisor’s assessments i.e. have career concerns.24 To preview the results, I show that the agent wants to stay away from the producing useful information if it is a complement of the ex-post information revealed by the investigation of the supervisor. In contrast, the agent wants to over-produce information that is a substitute of the information revealed ex-post. The reason for both under- and overproduction is the same: the agent’s reputation depends negatively on the information gap between the information available ex-ante and the information available ex-post. In what follows I try to show this result in the simplest possible way. Assume that in addition to s0 the agent can now produce a di¤erent signal s00
N; order an additional
radiograph or biopsy. The bene…t of this is the increased probability of success if more information is processed.25 The cost depends on a variety of factors: exposing the patient to more radiation, the value of the radiologist’s time, the delay in treatment etc. The net e¤ ect of the bene…t and social costs is captured by the realization of a random variable a 2 ( 1; 1). It is socially optimal to produce s00 i¤ a
0. To abstract away from any direct con‡ict of interest between the principal and the agent, I assume
that the latter fully internalizes this variable a. I also assume that a is observed only by the agent. To capture the fact that the agent cares about his reputation, I assume upon being investigated the agent receives a ’bonus’ b( assessments
and
0,
if
1 ).
This bonus is increasing in the supervisor’s assessment; given two
…rst-order stochastically dominates
0,
then b( ) > b( 0 ): Crucially, this bonus
payment plays no direct incentive role in this setup since the physician fully internalizes a, rather it re‡ects the fact that the physician’s future earnings are in‡uenced by whether he is assessed to competent or incompetent. This payment is short for the increased earning/promotion/status which follows from a 24
Similarly to the classic models of career-concerns, e.g. Holmström (1999), i maintain the assumption that the supervisor and the agent have symmetric information about the agent’s competence. 25 The bene…t measured in the agent’s objective is ua (s0 [ s00 ) ua (s0 ) 0:
24
high assessment or the …ring/loss of prestige associated with a low assessment. To isolate all other considerations, I assume that the agent’s concern for good reputation is additively separable from his concern about the optimality of producing s00 . Thus the agent’s payo¤ is
a + b(
1)
where I follow the convention that when s00 is not produced a equals 0, and when the agent is not investigated; b(
1)
= b(
0 ).
Furthermore, for clarity of exposition, I assume that the agent’s payo¤
depend on the mean of the supervisor’s assessment i.e. b(
1)
= E[ j
1]
where
measures the relative
importance of assessments in the agent’s utility. This assumption, which in e¤ect implies risk-neutrality over assessments is again to isolate the role of risk-preferences from the analysis. After presenting the results, I discuss below how the result generalize for the case where the agent is risk-averse over assessments. The agent’s choice whether to produce s00 is then determined by the following condition:
E[a + E[ j
1 ]]
0
(4)
Since the agent is uninformed about his type, his competence has no in‡uence on whether its optimal for him to produce s00 , in the Bayesian case. Here the supervisor’s expected assessment is independent of the information available to the agent, E[E[ j decision to produce s00 reduces to a
0 1 ]]
= E[ j
0]
always holds. Hence for
= 0, his
0; he produces s0 whenever its socially optimal to do so.
It follows from Proposition 3, that investigations are no longer neutral in the case of information projection, and the gap between the ex-ante and ex-post information will have a direct e¤ect on his assessed competence. Let a( ; m) denote the cuto¤ level such that if a
a( ; m) then the agent decides
to produce s0 based on Eq. (4). The following proposition speci…es the directions in which the agent will deviate from the socially optimal production. Proposition 5 1. For all ; m 2 [0; 1] a(0; m) = a( ; 0) = 0. 2. If s00 and s1 are substitutes, then a( ; m) is increasing in m if and only if
> 0.
3. If s00 and s1 are complements, then a( ; m) is decreasing in m if and only if 25
> 0.
A radiologist has additional incentives to undertake diagnostic procedures that substitute for ex-post information. The reason is that such diagnostic procedures reduce the probability of unwarranted ex-post blame. Even when if such procedures are socially ine¢ cient, because they are too costly or have bad potential side-e¤ects, the radiologist will undertake them to appear competent. As a result the more he is monitored, the more expensive and potentially more harmful his activities will be on such tasks. At the same time, a radiologist will produce too little complement information. The reason for this result is that he has incentives to avoid information that can be interpreted much better in hindsight than in foresight.
Even if the production of such information increases productivity more than it increases
costs, the radiologist is better-o¤ without producing such information because this way he can avoid developing a bad reputation. For example, a social worker might avoid making a phone call to a foster family even if the phone call provides valuable but noisy information. If ex-post when the supervisor learns that an accident has occurred to the child, this information alone might not clarify the cause of the accident. Yet if signals s00 and s1 together reveal ex-post whether the child was abused or not, then not making the phone call reduces the information gap between the ex-ante and the ex-post stage. This way, the social worker can prevent such future insight and that the supervisor employs the wrong standard in her assessment. There are reasons other than the ones above why an agent might want to stay away from the production of information. In particular in a similar setting, Holmström (1999) and Hermalin (1993) argued that a risk averse agent, but not a risk-neutral, facing career concerns will have an incentive to choose tasks that allow for the least amount of inference about his talent in order to avoid future wage ‡uctuations. Note however that the mechanism here is di¤erent. The fear of information projection will amplify such aversion for complement signals but reduce it for substitute signals. Proposition 5 provides a distinct prediction on over- and under-production of information as a function of the ex-post information s1 independent of risk-aversion. Distinct forms of ine¢ ciencies arise even when optimal risk-sharing is not an issue.26 The result on the under-production of complement information rests crucially on the assumption that the supervisor observes that s00 was or was not produced. If the radiologist could produce s00 secretly, i.e. in a way that supervisor is not aware of the production of s00 , then the radiologist’s willingness to 26
The assumption that s0 is always available to the agent guarentees that the supervisor makes inferences about his talent independent of whether s00 is produced or not.
26
produce depends solely on what future wages she expects. To see these e¤ects let’s return for a moment to Proposition 4 and assume again that b(
1)
= E[ j
1]
and assume that
1
is formed as if only s0 had
been produced. In case where the supervisor over-infers skill from performance, wages are too high after a success and too low after a failure hence the radiologist has incentives to secretly over-produce all sorts of ex-ante information. In the case of under-inference, the radiologist might have an incentive to secretly under-produce ex-ante information for similar reasons. If these over- or under-productions are discovered however, the incentive to engage in these disappears unlike in the case of Proposition 5. Although I am not aware of any direct test on the above result, evidence documents that both assurance behavior, ordering tests in clinically unnecessary situations, and avoidance behavior, avoiding various e¤ective procedures to avoid negative assessments are widespread.27 Importantly, the medical profession often attributes these behavior to the logic emphasized above. For example Berlin (2003) and Jackson, and Righi (2006) argue that in the case of mammograms, the public and juries su¤er from a ’serious misconceptions about the ability of mammograms to detect cancers’ and that they make the ’assumption that radiologists who read mammograms should be able to see what is discernible only in hindsight’. The authors assert that due to this reason, radiologist stay away from ordering mammograms, which is otherwise believed to be the e¤ective way to detect breast cancer.28 Consistent with the above setup, Kessler and McClellan (1996) argue that the main motivation for such practices is not peculiary loss of malpractice litigation rather the loss of reputation, adverse publicity resulting from low assessments. Kessler and McClellan (1996) …nds that reducing liability reduces costs of care in the treatment of heart diseases without worsening outcomes. Supportive of my predictions, Kessler and McClellan (2000) also show that the main e¤ect of cost reduction is on diagnostic rather than therapeutic treatments. 27
For example, Studdert et al. (2005) interviewed physicians in Pennsylvania and report that 93% of those surveyed engage in such defensive medicine. 28 See also the NYTimes interview http:// www.nytimes.com/1999/11/02/health/how-good-can-mammograms-be-suitsseek-to-set-a-standard.html
27
6
Reward and Punishment
In the pervious section, I showed that a biased supervisor underestimates the agent’s skill on average. A principal who makes employment and compensation decisions can to some extent correct the supervisor’s mistake if she anticipates that the supervisor’s reports are too negative on average. In most situations, however, a principal does not have as detailed information about the agent’s task as the supervisor. Hence such corrections might introduce other forms of ine¢ ciencies and might not eliminate the incentives of the agent to act against underestimation. In this section, I turn from a context where the amount of information that the agent learns from a signal is a function of his skill, to situations where it is a function of how much e¤ort he exerts. How often the radiologist understands X-rays depends on how carefully he evaluates them. A careful evaluation is costly because it requires the radiologist to exert e¤ort. To provide incentives for the radiologist, the principal o¤ers him a contract which rewards the radiologist for a good health outcome and punishes for a bad one. If a health outcome is only a noisy measure of the correctness of the radiologist’s diagnosis better incentives can be provided if the principal hires a supervisor to monitor the radiologist. This way the principal can tie reward and punishment closer to whether the radiologist made the correct diagnosis based on the information available to him.29 The main result of this section shows that if the supervisor projects ex-post information, the e¢ ciency gains from monitoring are decreased. I show that if the supervisor believes that the agent could have learned the true state, the radiologist is punished too often and exerts less e¤ort than in the Bayesian case. I also show that when the principal designing incentives anticipates the supervisor’s bias, he wants to monitor less often. Even if she decides to monitor, she induces less e¤ort on the part of the agent than in the Bayesian case. The reason is that information projection, even if anticipated by the principal, introduces noise in the supervisor’s reports and hence decreases the e¢ ciency of monitoring. 29
For the classic insight that increasing observability leads to increased e¢ ciency see Holmström (1979) and Shapiro and Stiglitz (1984).
28
6.1
E¤ort
Assume that the level of the e¤ort the agent exerts determines the probability with which he reads signal s0 . Let p(a) be the probability that s0 is read when the agent exerts e¤ort a, and let 1
p(a) be
the probability that she does not understand it. I assume decreasing returns to e¤ort in terms of the processing probability. Formally, p0 (a) > 0 and p00 (a) < 0. I also assume that lima!0 p0 (a) = 1 and lima!1 p0 (a) = 0. Let s0 be given by Pr(s0 = ! j !) = e h. Assume that the probability of a success conditional on the
fact that the agent’s action equals the state, y = !, is k. Assume that the probability of success for actions di¤erent from the state, y 6= !, is z where k > z. Finally, assume that if the agent does not process s0 , he is equally likely to take any action y 2
and the probability that such a random action
matches the state is b where b < e h.
For simplicity assume that both the agent and the principal are risk neutral. Let the agent’s utility
function again be U (w; a) = w0
a and the principal’s utility function be V (r; w) = r
w0 , where r is
the revenue to the principal from the task. Let the revenue of the principal be 1 after a success and 0 after a failure.
6.2
Performance Contract
As the benchmark, I characterize the …rst-best e¤ort level where the marginal social bene…t from exerting e¤ort equals the marginal social cost. The …rst best e¤ort level, af , is then de…ned implicitly by the following equality qp0 (af ) = 1 where q = (h
b)(k
z) and it measures the productivity gain from processing signal s0 .30 This
productivity increases in h; the precision of the agent’s signal, and in k, the probability of success conditional on an optimal action. The productivity decreases in b; the probability of making the right choice by chance, and in z; the probability of success conditional on a non-optimal choice. With a slight abuse of notation, let the vector q denote the collection of the parameters, h; b; k; z. Let’s now turn to the case where the agent’s e¤ort is unobservable. Assume that the agent is protected I assume that the solution is always interior. Furthermore, h = e h + (1 action space. 30
29
e h) j j j j 1 b where j j is the cardinality of the
by limited liability and w0
0 has to be true in all contingencies. Let the agent’s outside option be
0. Given the assumption of risk-neutrality, the principal’s optimal contract is one that o¤ers the lowest compensation possible after a failure. This implies that the compensation after a failure is wF = 0.
31
Let wS denote the compensation o¤ered to the agent upon a success. In light of these considerations, the principal’s problem is to maximize his expected utility
max V (r(a; q); w) = [p(a)q + bk + (1 a;wS
b)z](1
wS )
(5)
b)z]wS
a
(6)
subject to the agent’s incentive compatibility constraint
a(q; w) = arg max[p(a)q + bk + (1 a
Given the agent’s utility function, we can replace this incentive compatibility constraint with its …rst-order condition.32 To guarantee that there is a unique equilibrium, I assume in what follows that p000 (a)
0 for all a. The optimal e¤ort level, an (q); which solves this constrained maximization problem
is de…ned implicitly by following equation: qp0 = 1
p00 (p + (bk + (1 (p0 )2
b)z)=q)
(7)
Let wn (q) denote the corresponding optimal wage. Here the subscript n refers to no monitoring. Note that an (q) is always smaller than af (q). The reason is that the principal faces a trade-o¤: implementing a higher level of e¤ort is only feasible at the cost of leaving a higher rent for the agent. Thus e¤ort is lower and the agent’s rent is higher than in the …rst-best. A simple comparative static result follows from Eq. (11). Increasing h or k increases the productivity of processing information and thus generates higher utility for the principal given a contract. Since p0 > 1 is always true in equilibrium, a higher h or k allows for cheaper incentives and thus the principal wants to induce more e¤ort, implying that e¤ort is increasing in h and k. 31
On the use of limited liability contracts see Innes (1990), Dewatripont and Bolton (2005). I believe that the results of this section hold a fortiori given a risk-averse radiologist. 32 In characterizing optimal incentives in this section, I can ignore the individual participation constraints since the agent’s outside option is a wage of 0 and by limited liability she cannot receive a lower wage under a performance contract either.
30
Lemma 1 An increase in h or k increases the equilibrium e¤ ort level an (q) and the payo¤ of the principal.
6.3
Bayesian Monitoring
The e¤ort level characterized by Eq. (11) is optimal given that the supervisor observes a performance measure that consists only of success and failure. Information about the agent’s e¤ort choice, however, reduces the ine¢ ciency of the above simple incentive contract. In my setup this means that obtaining more precise reports about the agent’s action, allows the principal to induce the same level of e¤ort at a lower cost. Consider that the principal can monitor the agent by learning the agent’s action and the information that was available to him. In case of such monitoring, the optimal contract rewards the agent if his action is the one suggested by the information available to him and punishes the agent otherwise. Since whether a success happens or not does not contain additional information, it is easy to see that such a compensation scheme is optimal. Given such a reward scheme, the agent’s incentive compatibility constraint can now be expressed by the following …rst-order condition:
a(q; wS ) = arg max p(a)(1 a
b)wS + bwS
a
(8)
and the optimal contract induces an equilibrium e¤ort level, am (q); de…ned implicitly by the following condition: p0 q = 1
p00 (p + b=(1 (p0 )2
b))
(9)
and let wm (q) denote the corresponding optimal wage. Here the subscript m refers to unbiased monitoring. As I prove in the appendix, the equilibrium e¤ort under monitoring, am (q); is always greater than equilibrium e¤ort without monitoring an (q). The reason is that monitoring improves the trade-o¤ between providing incentives and leaving a positive rent for the agent because it rewards good decisions rather than good luck. As a result, if the principal monitors the agent he can induce the same level of e¤ort at a lower cost and hence for any given level of e¤ort, he realizes a greater expected pro…t. The fact that it becomes cheaper for the principal to induce e¤ort means that the principal is willing to pay for monitoring. Lemma 2 The equilibrium under monitoring induces a higher e¤ ort, an (q) < am (q) and the principal is 31
better-o¤ with monitoring the agent.
6.4
Biased Monitoring
Let the supervisor’s ex-post signal be s1 and assume that the projected information is such that along with s0 it perfectly reveals the state but alone its uninformative. This means that a biased supervisor perceives the true problem as if h = 1 for all h
1. Furthermore, it also implies that upon not processing s0 the
supervisor still believes that the probability that the agent can take the right action is b :The consequence of such information projection is that the supervisor makes wrong ’inferences’from the agent’s choice.33 Whenever y 6= !, the supervisor takes this as evidence that the agent did not successfully read the information available to her. Hence, if the agent did read and follow s0 but this information turned out to be ’incorrect’ex-post, the supervisor mistakenly infers that the agent did not read s0 . The probability of this mistake is p(a)(1
h); the probability that s0 is processed times the probability that s0 did not
suggest the right action. Assume that the agent correctly predicts the bias of the supervisor. In this case, the agent’s e¤ort is given by the solution of the following maximization problem: a1m (q; w) = arg max p(a)h(1 a
b)w + bw
a
(10)
Here the superscript refers to fully biased monitoring. Comparing this condition with that of Eq. (12), it is immediately visible that the return to e¤ort is smaller in the biased case than in the Bayesian one. Note also that while in the Bayesian case the precision of s0 does not enter into the agent’s maximization it does enter in the biased case. The reason for the former is that an unbiased supervisor can distinguish – up to probability (1
b) – between a bad decision that is due to wrong ex-ante information and a
bad decision that results from not processing a signal. In contrast, a biased supervisor mistakes a bad decision due to wrong ex-ante information to a bad decision that is due to not having processed the available information. This implies that for any given compensation wS the agent exerts less e¤ort in the biased case. Proposition 6 Suppose h < 1: Then a1m (h; w) < am (h; w),and a1m (h; w) is increasing in h with a1m (1; w) = 33
Note that the ’inference’ of the supervisor is only about whether the agent’s e¤ort was successful or not. In a moral hazard context there is no inference about the agent’s e¤ort a.
32
am (h; w): Our next result is a corollary to the above proposition. It shows that in contexts where information projection is particularly severe, monitoring decreases rather than increases e¤ort and total output. The reason is that the contract that is optimal in the Bayesian case o¤ers a wage that is lower than the wage under the performance contract without monitoring. This has the e¤ect of decreasing e¤ort. The fact that the agent is monitored has the e¤ect of increasing e¤ort. In the Bayesian case the latter e¤ect is always dominates the former. In the biased case though if the probability of mistaken punishment is high enough, the wage e¤ect might be stronger than the monitoring e¤ect. Corollary 5 There exists h > 0 such that if h
0 ) < a (q; w ) and surplus is smaller h then a1m (q; wm n n
under monitoring and wS = wm than under no monitoring and wS = wn . This result claims that if ex-ante information is su¢ ciently noisy, then monitoring will back…re and induce less e¤ort than the simple performance contract without monitoring. It implies that more information, even if it is costless to acquire, can hurt all parties. This result depends on the fact that it is only the agent but not the principal anticipates the bias. As a …nal scenario, consider the case where the bias of the supervisor is common knowledge between the principal and the agent. The result here shows that information projection still introduces an ine¢ ciency in the contractual relationship. If the principal is aware of the supervisor’s bias then he knows that the supervisor will come to the wrong conclusion with positive probability. Since the principal can only determine the probability of this mistake and not whether the supervisor’s report is actually wrong or right, information projection still adds noise to the supervisor’s reports. Thus, the data obtained by monitoring contains more noise than in the Bayesian case, which decreases the e¢ ciency of monitoring. As a result, the principal decides to induce less e¤ort than he would had he believed that the supervisor had perfect Bayesian perception. Let the e¤ort level induced be denoted by a1m;b (q) , where subscript b refers to the case that the principal anticipates the full bias of the supervisor, and implicitly de…ned by: p0 q =
(p0 )2
p00 (p + b=h(1 (p0 )2
33
b))
Proposition 7 If the principal anticipates the bias, he induces e¤ ort a1m;b (q) < am (q) and a1m;b (q) is increasing in h. I assumed so far that the projected information leads to an exaggeration of the precision of s2 . It might happen though that the projected information leaves h una¤ected but leads to an exaggeration of b, the probability of taking the right action without processing information. In the above framework such a mistake does not a¤ect output because it does not change the probability with which an agent is rewarded conditional on exerting e¤ort. In a more general setup, however, where h and b are known ex-ante only to the agent, and where compensation is conditional on the supervisor’s assessment of these parameters, exaggerating b might also lower the agent’s e¤ort. The analysis in this section has implications to the e¤ect of hindsight bias on tort liability. It claims that whenever there is unobservable e¤ort involved information projection reduces rather than increases an injurer’s incentive to exercise due care. This observation is in contrast with the common conjecture that the agent anticipating hindsight bias, takes too much care to avoid ex-post blame, see Rachlinski (1998).34
7
Communication
In the previous sections, I focused on the problem of performance evaluation but surely information projection might a¤ect other aspects of organizational life as well. Importantly, in a Bayesian setting, informational di¤erences might be overcome through e¢ cient communication.
The analysis in this
section shows that even in the absence of strategic considerations distorting communication, information projection can lead to the break-down of communication relationship and to lower welfare to the receiver. It also suggests ways communication can be improved between the parties by for example the use of communication protocols. 34 One key di¤erence between the setup in this section and the one where optimal liability is typically studied is that the same action is the right action in one state and the wrong action in another. In the contexts typically studied, e.g. Shavell (1980), adverse outcomes are decreasing in the agent’s action and there is no unobservable component. Even in those environments however, if e¤ort is unobservable, the agent’s e¤ort might be lower in the biased case.
34
7.1
Ambiguity, Over- and Under-Communication
Consider an advisor (doctor, professor) who wants to send a message to the advisee (patient, student). Assume that the two parties have a common objective and their utility is maximal if the advisee’s action ye matches the state !. Formally, uk (ye ; !) = 1 if ye = ! and 0 otherwise for all k. Consider an environment analogous to the Examples in Section 4.4. Let ! = $1 $2 and $1 ; $2 2 f 1; 1g along with a symmetric prior on ! and on $1 and $2 as well. Assume for simplicity that only the advisor has private information and her private information is given by the following three signals: s1 = $; s2 = $2 , and s3 where s3 2 f 1; 1g and Pr(s3 = ! j !) = h < 1: The information in s1 is the advisor’s background knowledge and cannot be communicated to the advisee. The information in s2 can be communicated but given the information structure it only conveys valuable information only to the extent that someone knows the value of s1 . As an example, a medical term (s2 ) is only understood given some background knowledge of the language of medicine (s1 ):The value of the third signal does not require the knowledge of the medical language and hence a patient can understand it even if she does not share the expertise of the doctor. Assume that advisor can send at most one signal and a cost of sending a signal is c: The advisor can also decide not to talk to the advisee. Formally, if Yr is the set of actions available for the advisor, then Yr = fs2 ; s3 , silenceg. The table below summarizes the advisor’s perceived payo¤ in the unbiased case and as a function of her bias
2 [0:1]:
silence
send s2 1 2
1 2 2
+ (1
2 )(
h+
1 2
1 2
); (1
) h+
send s3
c
h 2 +1
2
c;
2
+ (1
c 2 )h
c:
An unbiased advisor realizes that the medical term (s2 ) alone conveys no information and hence never sends this as a message. Furthermore, she only communicates when h
c > 12 , that is when the expected
bene…t of sending s3 is greater than the cost of not remaining silent. Let’s turn to the biased case. First note that a fully biased advisor never talks. Since she believes that all her private information is shared with others, she attaches no value to communication and hence unwilling to incur cost c. Provided the advisor …nds it worthwhile to talk, information projection also a¤ects her choice of message. In particular, because she projects her background knowledge she 35
overestimates the value of communicating the specialized description over the lay description, and might come to prefer s2 to s3 . The fact that she exaggerates the informativeness of communication s3 also means that she might talk at times when an unbiased advisor would remain silent. Hence when remaining silent is preferred to telling the lay description, but the specialized description is perceived to be informative for the advisee, the advisor might communicate too often.
35
Proposition 8 If
k2 ( ; h) and is silent otherwise. The function
< 2h
1; the advisor sends s3 i¤ c
k2 ( ; h) is increasing in h and decreasing in : If
> 2h
1, the advisor sends s2 i¤ c
silent otherwise. Furthermore, if h = 0:5, the advisor sends s1 i¤ c
0:5 (1
k1 ( ; h) and is
).
The above proposition shows that a biased advisor might send a message that is dominated in the rational case because its always more ambiguous than another available message. This proposition also o¤ers some comparative static results with respect to the bias. If the advisor is only moderately biased, and the lay description is su¢ ciently informative, she communicates too rarely. Here underestimating the return to communication dominates her overestimation of how informative the medical description is. If the advisor is su¢ ciently biased, then depending on how informative the lay description is, she might communicate too often. Since she overestimates the probability that following the medical description the advisee will take the right action, she engages in costly communication even if it conveys no information to the advisee. Hence adding dominated communication options might decrease e¢ ciency in the presence of information projection. Results in the above proposition are consistent with the intuition that the curse-of-knowledge leads to too much ambiguity. For example, many argue that this is true for computer manuals written by experts but targeted to lay people. While in the case of computer manuals, hiring a lay person rather than an expert to proof-read the manuscript could decrease the curse, in many other situations more explicit communication protocols restricting the set of messages that can be sent could improve the quality of communication. Note that in accord with Proposition 1, a biased advisor always overestimates the value of the advisee’s information. Consider for a moment that the advisee attends to the advisor’s message only with probability 35
2 [0; 1] where parameter
This happens when
2h
is the advisee’s unobservable type. A corollary of Proposition 2
1.
36
is that upon observing the advisee’s action ye , the advisor will underestimate the advisee’s attentiveness on average. Since the advisor misattributes the lack of information or the ambiguity of the information to the lack of attentiveness on the part of the advisee, she might well decide not to communicate with him anymore and terminate the relationship even when communication would be mutually bene…cial.
7.2
Credulity
Consider a situation where an advisee has to take an action ye that is as close as possible to an unknown state ! on which the shared prior is N (0;
0 ).
This state could describe the optimal solution of a research
problem, the best managerial solution on the organization of production or the diagnosis of a patient. The advisee has some private information about ! that is given by se = !+"e where "e is a mean zero Gaussian noise term such that the posterior on !, given the prior and se , is N (b se ; be ). The advisor also has some private information about ! given by sr = ! + "r where "r is a mean zero Gaussian noise term such that the posterior on !, given the prior and sr , is N (b sr ; br ).36 The advisor makes a recommendation yr equal
to her posterior mean. The advisor cannot communicate the full distribution or the true signal directly. Such limits on communication might arise due to complexity considerations, or because it’s prohibitively costly to explain this private information. Instead, she can give a recommendation regarding the best action she would follow. Let the advisee’s and the advisor’s objective be
max E! (ye ye
!)2
(11)
thus the advisee’s goal is to take an action that minimizes the distance between his action and the state. Given the advisor’s recommendation yr , and the advisee’s private information se , a rational advisee takes action ye0 such that: ye0 = E[N (!; c0 ; v 0 )] where c0 =
yr be2 be2 +br2
+
sbe br2 br2 +be2
(12)
and N (! ; c; v) is a short form for a normally distributed random variable
with mean c and variance v. This action is based on the correct perception of how information is 36
Formally, if "e
ce =
2 2 0 r 2+ 2 r 0
.
N (0;
e)
then sbe =
2 0 2 2 0+ e
se and ce =
37
2 2 0 e 2 2 0+ e
. Similarly, if "r
N (0;
r)
then sbe =
2 0 2 2 0+ r
sr and
distributed between the advisor and the advisee. This action e¢ ciently aggregates the information in the recommendation yr and the advisee’s private information se . Consider now the case where the advisee exhibits full information projection. Here, he believes that the advisor’s recommendation is based not only on the realization of sr , but also on se , and thus it already incorporates all information available to the parties. As a result, he reacts to the advice yr by taking action ye1 such that:
ye1 = E[N (!; c1 ; v 1 )]
(13)
where c1 = yr and v 1 = v 0 . It follows that if the advisee exhibits full information projection, he puts all the weight on what the advisor says and no weight on his private information. This way, his private information is lost. The following proposition shows that a biased advisee follows the recommendation of his advisor too closely. Proposition 9 E jyr
ye j is decreasing in
and E yr
ye1 = 0 where expectations are taken with
respect to the true distribution of signals. This proposition follows from the discussion above. Note that the more precise the advisee’s private information is, the greater is the loss relative to the unbiased case. In the biased case, information aggregation fails because the advisee fails to su¢ ciently update the advisor’s recommendation given his private information. One way to eliminate this information loss is to invest in a technology that allows the advisor to communicate her posterior distribution. Another option is to block communication between the advisor and the advisee. Assuming full information projection, the advisee is ex-ante better-o¤ without a recommendation if and only if his signal is more precise than the advisor’s signal. More generally, the following corollary is true: Corollary 6 There exists an indicator function k( ; with a recommendation if k( ; k( ;
e;
r)
e;
= 1. The function k( ;
e;
r)
2 f0; 1g such that the advisee is better-o¤
r)
= 0; and the advisee is better-o¤ without a recommendation if
e;
r)
is increasing in
38
and
r
and decreasing in
e.
8
Conclusion
In many economic situations people have to predict what others might know or believe. In this paper I argued that such prediction are in‡uenced too much by a person’s own information. I developed a simple but generally applicable framework to model such information projection. Although the current application as well as the results of this paper can be extended in various ways, let me conclude by mentioning other settings where information projection might lead to novel predictions about behavior. Hostility and Group Formation. In Section 3, I considered a simple dinner example that can be extended to an analysis of group formation and intergroup con‡ict. Recall that a biased guest underestimated both the probability that her host is kind and the probability that her host shared the same taste as she does. This means that people who attend to di¤erent information channels will exaggerate the extent to which others are incompatible with them, and also the extent to which others have unkind or hostile intentions towards them. Furthermore, such an analysis might o¤er novel predictions on the type of social interaction that reduces con‡ict and the type of social interaction that exacerbates it. Biases in Prospective Memory In Section 2, I mentioned the possibility of projecting information onto one’s own future selves. Such projection might explain documented biases in prospective memory. In particular, a person who projects information onto her future selves, will be overcon…dent about the probability that she will be able to recall her current information in the future. This particular form of overcon…dence can have e¤ects on the optimal design of deadlines, reminders and intertemporal incentives. Ignorance Projection: Another direction to extend the ideas presented in this paper is to consider the related phenomenon of ignorance projection. Ignorance projection happens when someone who does not observe a signal underestimates the probability with which this signal is available to others. Though evidence on ignorance projection is not as strong as the evidence on information projection, it might still be a phenomenon worth studying, both empirically and theoretically.
39
9
Appendix
Proof of Proposition 1. To prove this proposition, I show that information projection shifts probabilistic weight from less to more informative information sets. It then follows from Blackwell’s theorem that at a more informative information set a decision-maker’s expected utility is higher than at a less informative one. This then implies that a biased person’s beliefs about the expected utility of the decision-maker will be higher in the sense of FOSD. Consider two information sets I; Ib 2 N such that I
b It follows that a biased person always assigns I.
a weakly higher probability toIb than a Bayesian person. Let’s denote the posterior on ! induced by
b by information I ( I)
and Z are …nite,
I
as well and have …nitely many di¤erent realizations. Let’s collect the realizations of
I
I
(
). Ib
Given the assumption that both
possible signal realizations fsj gN j=1 into matrices absent observing the signals in Ib n I - is equal to there exists a Markov-matrix A such that theorem that if
and bI are …nite
and bI given all
and b :Since the expected posterior of I
given Ib
I –i.e.
by the law of iterated expectations, it follows that
= A b . The next step of the proof follows from Blackwell’s
= A b where A is a non-negative matrix with columns summing up to 1, then for any
ui (y; !) and a particular realization of the signals fsj gN j=1 , it is true that E!;I [ui ]
E!;Ib[ui ]:37 :Since
information projection corresponds to shifting probabilistic weight from I to Ib where I
b and the I;
weight shifted is increasing in the degree of projection , the proposition follows from the fact that information sets in the comparison are ordered by their Blackwell informativeness. Note also that the Corollary to this Proposition follows from the fact that projecting a more informative information set, leads to shifting probabilistic weight to information sets with greater expected utility. Proof of Proposition 2. Let’s the cardinality of the outcome set X be M and let’s index the outcomes by xi . Recall that by the law of conditional probability, when prior f ( ) and f (u). Consider now the case where
E[f ( j x)] = 37
M P
i=1
= 0, EX [f 0 ( j x)] = f ( ) holds for all
> 0. Here, the expected belief about
f ( j xi )f 0 (xi ) =
M f (x j )f ( ) P i f 0 (xi ) f (xi ) i=1
For a proof based on Blackwell (1953) see Hirschleifer and Riley (1999).
40
is given by
(14)
Fixing a value for
we get that the expected density of M P
E[f ( j x)] = f ( ) where
i=1
is equal to
f (xi j )
f 0 (xi ) = f ( )[ f (xi )
( )]
(15)
( ) is a short-hand form for the term in the square-brackets. The next step of the proof is to 0
show that whenever have to show that
>
it follows from the assumptions that
( )>
( 0 ) for all : To see this, we
( 0 ) > 0, i.e. that
( )
M P
i=1
[f (xi j )
0
f (xi j
)]
f 0 (xi ) >0 f (xi )
Note that for the lowest outcome x1 it is always true that [f (x1 j ) 0)
because f (x1 j ) > f (x1 j
(16) f (x1 j
0 )][f 0 (x )=f 1
(x1 )] < 0
by virtue of the strict MLRP:(The inequality is weak if we assume the
weak version of the MLRP.) To show that the inequality holds for the two lowest outcomes we have to show that [f (x1 j ) If f (x2 j
0)
If f (x2 j
f (x1 j
0
)]
f 0 (x1 ) > [f (x2 j f (x1 )
0
)
f (x2 j )]
f 0 (x2 ) f (x2 )
(17)
< f (x2 j ), we are done since the RHS of the inequality is negative and the LHS is positive.
0)
> f (x2 j ) it follows that f (x2 j
0)
f (x2 j ) < f (x1 j )
f (x1 j
0)
since we
assumed that f (x j u; ) satis…es the MLRP in . In addition, if f (x j u; ) also satis…es the MLRP in u, we know that for all xi and xj such that i > j it follows that ff inequality.
38
i=1
Since [f (xM j )
f (xM j
( )>
[f (xi j )
0 0 )] f (xM ) f (xM )
( 0 ) whenever
before, in the Bayesian case
0(
f 0 (xi ) f 0 (xj )
and this proves the above
)]
f 0 (xi ) >0 f (xi )
(18)
< 0 it then follows that Eq (?) is satis…ed and thus we have shown <
0.
) = 1 for all Z
0
f (xi j
Let’s …rst compare the Bayesian case when
38
>
Based on a similar logic one can show that for all k < M k P
that for all
(xi ) (xj )
f( )
2
= 0 with the biased case when :It follows then that for any
( )d >
<
See also Milgrom (1981).
41
Z
f( ) <
0
( )d
> 0. As indicated 2
hence E[f 0 ( j x)] …rst-order stochastically dominates E[f ( j x)]. To show that the same relation holds for
<
0
note that it follows from Proposition 1 that
0
f (xi ) 0 f (xj )
>
f (xi ) f (xj ) .
This concludes the proof.
Proof of Proposition 3.. The proof of this result follows from Proposition 2. To see this we need to show that (xS j ; p1 ) is increasing in both
and in p1 where p1 is the probability that s1 is available
to the agent. Consider the agent’s optimal action ye 2 arg maxye ye0 A
1
and consider the value of this
program ye 0 A
1.
increasing in
and in the probability that s1 is processed. Hence given the agent’s action (xS j ; p1 )
is increasing in
Since both s0 and s1 are informative, it follows that this value of this program is
and in p1 , and the biased perception of p1 , p1 , is increasing in : (xS j ; g 0 ) for all .
Proof of Corollary 2. To show this corollary we need to show that (xS j ; g)
whenever g > g 0 but this follows trivially from the de…nition of g and our assumption that productivity of the agent is still weakly increasing in
even if knowing s1 .
Proof of Corollary 3. Note that given the assumption that pt1 e > 0 for all t, a biased assessments follows a supermartingale as this assumption guarantees that the biased updating process is adapted for any t < 1. Hence the Proposition 3 follows from the properties of supermartingales see Durrett (2005) pp 229. Proof. De…nition 3 This follows directly from the proof of Proposition 2 when M = 2:
De…nition 4 Proof. De…nition 5 To show that for all
1 (S)
FOSD
0 1 (S)
we have to show that
R
0 1(
0
j S)d
< 1. One can re-write this inequality to obtain the following one:
Pr (S)= Pr(S)
Z
0
Pr (S j )
0(
)d
! 42
=
Z
0
Pr(S j )
0(
)d
!
for all
R
0
1(
< 1:
j S)d
(19)
R1 If Pr (S j )= Pr(S j ) is increasing in , then this inequality holds since 0 1 ( j S)d = 1 R1 R1 and hence 0 Pr (S j ) 0 ( )d = 0 Pr(S j ) 0 ( )d = Pr (S)= Pr(S). If Pr (S j )= Pr(S j ) is decreasing in
then the reverse inequality holds, and then
0 1 (S)
FOSD
1 (S).
De…nition 6 Proof of Proposition 5. Note …rst that in the Bayesian case the RHS of Eq.(4) is zero for all m independently of s1 . In the biased case
1
depends on s1 and hence the same is true for E[w1 ]:Furthermore,
it follows from Proposition 3 that E[w1 ] is decreasing in . The decision to produce sb0 more often or less often than in the Bayesian case depends on whether the following expression is positive or negative
E[E[w1 ]]
E[E[w1 ] j sb0 ]
(20)
If this term is positive, the agent has a non-Bayesian incentive to abstain from the production of sb0 . If this term is positive, the agent has a non-Bayesian incentive to engage in the production of s00 . It follows from the proof of Proposition 3 that if for all ( )<
0
( )>
( ) then
S
>
0 S
and E[E[w1 ]] > E[E[w1 ] j sb0 ]
> 0. This is true because underestimation is decreasing in Pr(S)= Pr (S) = 0
( ) then
S
<
0 S
and E[E[w1 ]] < E[E[w1 ] j sb0 ] for all
given by Eq. (4): It follows that if
( )>
0
S.
Similarly, if
> 0. The critical value a(m; ) is
( ); then a(m; ) is decreasing in m and if
( )<
0
( )
then a(m; ) is increasing in m. Proof of Lemma 1. First let’s derive the optimal contract as given by Eq. (11). The principal’s maximization problem yields the following Lagrangian:
L(wS ; a; ) = (p(a)q + bk + (1 The FOC with respect to a is given by p0 q(1 bk + (1
b)z) + p0 (a)q = 0. Solving for
b)z)(1
wS ) + (p0 (a)qwS
1)
wS ) + p00 qw = 0 and with respect to wS it is
(p(a)q +
and substituting for wn = 1=p0 (a)q the equilibrium e¤ort level
is given by qp0 = 1
p00 (p + (bk + (1 (p0 )2
b)z)=q) _
43
=1
p00 (p + b=(h b) + z=q) (p0 )2
(21)
Let the solution of this equation be denoted by an (q). Note that the second-order conditions are satis…ed as long as p000 (an (q))
0.
An increase in k or h increases q and hence increases the LHS of Eq.(28). An increase in k or h decreases the RHS of Eq.(28). Since p is increasing and concave and p000
0, it follows that this leads to
a higher equilibrium e¤ort level. To see the e¤ects of an increase in k and h on the principal’s welfare note that for a given wS , (p(a)q + bk + (1
b)z)(1
wS ) is increasing in a since wS < 1. Furthermore the
optimal wS given k and h cannot be larger than the original wS because h b < 1 < p0 and k z < 1 < p0 :
Proof of Lemma 2. Let’s …rst derive the optimal contract given monitoring as given by Eq. (13). The principal’s maximization problem yields the following Lagrangian:
L(wS ; a; ) = (p(a)q + bk + (1
b)z)
b) + b)wS + (p0 (a)(1
(p(a)(1
The …rst-order condition with respect to a is given by p0 q …rst-order condition with respect to wS is given by substituting wm = 1=p0 (1
b)wS + p00 (1
b) + b) + p0 (1
1)
b)wS = 0 and the
b) = 0. Solving for
and
b) we get that the equilibrium e¤ort level am is determined by p00 (p + b=(1 (p0 )2
p0 q = 1
Note …rst that (bk + (1
(p(1
p0 (1
b)wS
b)z)=(h
b)(k
z) > b=(1
b))
(22)
b) () b(k
z) + z(1
b) > bh(k
z) which is
always true if h < 1. If we compare Eq. (29) with Eq. (28), it follows then that e¤ort is greater under monitoring because for any a the LHS’s of these two equations are the same and the RHS of Eq. (29) is smaller than the RHS of Eq.(28). Given the assumption that p000
0 the result follows.
To show the increase in the principal’s welfare note that
EVn = p(an )q + bk + (1
b)z
(p(an ) + b=(h
b) + z=q)=p0 (an )
and EVm = p(am )q + bk + (1
b)z 44
(p(am ) + b=(1
b))=p0 (am )
Since (1 b=(h
1=p0 (am )) and (1
b) + z=q > b=(1
1=p0 (am )) are both positive because p0 (an ); p0 (am ) > 1, and because
b) if h < 1; it follows that EVm > EVn :
Proof of Proposition 6. Let’s …x a wage wS . It follows from the discussion in the text that the agent’s e¤ort choice is given by the following maximization problem a1m (q; wS ) = arg max p(a)h(1
b)wS + bwS
a
The …rst-order condition is then given by p0 h(1
a
(23)
b)wS = 1. It is easy to see that for any given wS ,
a1m (q; wS ) < a0m (h; wS ) as long as h < 1 and also that a1m (q; wS ) is increasing in h.
Proof of Corollary 3. To see this corollary note that a0m (q; wS ) does not depend on h directly. It 0 ) satis…es p0 (a) = p0 (a0 )=h. Hence for any follows that a(wn ; h) is such that p0 (a) = p0 (an ) and a1m (q; wm m 0 ) < p0 (an ) < 1 there exists h such that if h < h then p0 (am )=h > p0 (an ). This implies that a1m (q; wm
an (q; wn ). Since social surplus is increasing in a as long as qp0 > 1 it follows that monitoring decreases social surplus Proof of Proposition 7. To prove this Proposition we have to consider the principal’s problem when the principal knows that the agent’s action is given by a1m (q; w). Here the principal’s Lagrangian is given by L(wS ; a; ) = p(a)q + (bk + (1
b)z)
p(a)(h
the …rst-order condition with respect to a is given by p0 q …rst-order condition with respect to wS is given by substituting wm;b = 1=p0 (h
bwS + (p0 (a)(h
b)wS
p(h
p0 (h b)
b)wS + p00 (h
b + p0 (h
b)wS
1)
b)wS = 0 and the
b) = 0. Solving for
and
b) we get that am;b (q) is given by: p0 q = 1
p00 (p + b=(h (p0 )2
b))
(24)
Comparing Eq. (31) with Eq.(29) it follows am;b < am as long as h < 1 because the RHS of (31) is always greater than the RHS of Eq.(29): Furthermore since the RHS of (31) is decreasing in h, am;b is decreasing in h. 45
Proof of Proposition 8.. Simple calculations show that s3 dominates s2 i¤ 2h 2 )(h
advisor sends s3 i¤ c < (1 decreasing in . If 2h
+ 0:5)(1
1 > . Here the
) = k2 ( ; h): It follows that k2 ( ; h) is increasing in h and
1 < , the advisor sends s2 iif c <
3 (h
0:5) + (0:5
h) = k1 ( ; h).
Proof of Proposition 9. Since se and sr are independent it follows that the joint distribution of !, se and sr is given by a multivariate normal distribution with mean vector (0; 0; 0) and the corresponding covariance matrix C. Given the assumptions on C, it follows that E[! j sr ] = is given by 2 0;
E[! j se ; sr ] =
2 6 4
2 0
Straightforward calculation shows that
E[! j se ; yr ] = where sbe =
2 0 2 2 0+ e
se , be2 =
2 2 0 e 2 2 0+ e
and br2 =
Consider now the biased case where
takes an action ye1 = yr . For
2 2 0 r 2 2 0+ r
.
2 0
+ 2 0
2 e
2 0 2 0
2 r
+
3 7 5
12
sr and E[! j se ; sr ]
3
6se 7 4 5 sr
(25)
yr be2 sbe br2 + be2 + br2 br2 + be2
= 1. Here the advisee believes that yr = E[! j se ; sr ] and hence
< 1 the advisee believes that with probability
E[! j se ; sr ] and with probability 1
2 0 2+ 2 e 0
it is the case that yr =
it is the case that yr = E[! j sr ]. Hence it is always true that
ye 2 [minfye0 ; yr g ; maxffye0 ; yr g]. Furthermore as the probability Proof of Corollary 4. Note …rst that
E(ye
increases jye
!)2 is decreasing in
yr j decreases.
by virtue of Proposition 8 since the
estimate of ! has the lowest variance given s1 and s2 if ye = E[! j se ; yr ]. Also for a …xed , E jyr decreasing in r
such that
small that
r
and increasing in
E(b se E(b se
!)2 > !)2 >
E(ye E(ye
e.
Hence if we …x
e
< M < 1, there always exists su¢ ciently large
!)2 . Similarly, for a …xed !)2 . It follows that k( e ;
and increasing in .
46
ye j is
r r;
> 0 there always exists ) is increasing in
e
e
su¢ ciently
decreasing in
r
References [1] Akerlof, George. 1970 "The Market for Lemons: Quality Uncertainty and The Market Mechanism." Quarterly Journal of Economics, Vol. 84, 488-500. [2] Anderson, J. C., Jennings M. M., Lowe D. J. and Reckers P. M. J. 1997. "The mitigation of hindsight bias in judges’evaluation of auditor decisions." Auditing: A Journal of Practice and Theory, 16(2), 20–39. [3] Arrow, Kenneth. 1973. “The Theory of Discrimination.”in Orley Ashenfelter and Albert Rees, eds., Discrimination in Labor Markets Princeton NJ: Princeton University Press. 3-33. [4] Baron, Jonathan, and John Hershey. 1988. "Outcome Bias in Decision Evaluation." Journal of Personality and Social Psychology, Vol. 54, 569-579. [5] Baker George, Michael Gibbs, and Bengt Holmstrom. 1994. "The Wage Policy of a Firm." Quarterly Journal of Economics, Vol. 109, No. 4., 921-955. [6] Biais, Bruno, and Martin Weber. 2007. "Hindsight bias and investment performance." Mimeo IDEI Toulouse. [7] Becker, Gary. 1971. The Economics of Discrimination, 2nd ed. Chicago: University of Chicago Press. [8] Berlin, Leonard. 2003. Statement to the U.S. Senate Committe on Health, Education Labor and Pensions: Re Mammography Quality Standards Act Reauthorization. [9] Blackwell David 1953. "Equivalent Comparisons of Experiments." Annals of Mathematical Statistics, 24, 265-272. [10] Brody, Jane. 1999. "How Good Can Mammograms Be?
Suits Seek to Set a Standard" New
York Times http://www.nytimes.com/1999/11/02/health/how-good-can-mammograms-be-suits-seekto-set-a-standard.html [11] Bukszar, Ed, and Connolly Terry. 1988. "Hindsight Bias and Strategic Choice: Some Problems in Learning From Experience." Academy of Management Journal, Vol. 31, No. 3., 628-641.
47
[12] Camerer, Colin, George Loewenstein, and Martin Weber. 1989. "The Curse of Knowledge in Economic Settings: An Experimental Analysis." Journal of Political Economy, Vol. 97, No. 5., 1234-1254. [13] Camerer Colin, and Ulrike Malmendier. 2007. "Behavioral Economics of Organizations" in: P. Diamond and H. Vartiainen (eds.), Behavioral Economics and Its Applications, Princeton University Press, Princeton. [14] Camerer Colin, [15] Caplan RA, Posner KL, Cheney FW. 1991. "E¤ect of outcome on physician judgments of appropriateness of care." Journal of the American Medical Association, 265, 1957-1960. [16] Crawford [17] Costas-Gomes Miguel, Vincent P. Crawford, and Nagore Iriberri 2009. "Comparing Models of Strategic Thinking in Van Huyck, Battalio, and Beil’s Coordination Games," Journal of the European Economic Association, in press. [18] DeMarzo, Peter, Dimitri Vayanos, and Je¤rey Zwiebel. 2003. "Persuasion bias, social in‡uence, and uni-dimensional opinions." Quarterly Journal of Economics, Vol. 18, 909-968. [19] Eyster, Erik and Matthew Rabin. 2005. "Cursed Equilibrium." Econometrica, Vol. 73 No. 5., 16231672. [20] Eyster Erik and Matthew Rabin. 2009. Naive Inference
Mimeo UC Berkeley.
[21] Fischho¤, Baruch. 1975. "Hindsight / foresight: The e¤ect of outcome knowledge on judgement under uncertainty." Journal of Experimental Psychology: Human Perception and Performance, 1, 288-299. [22] Gilovich, Thomas, Kenneth Savitsky, and Victoria Medvec. 1998. "The illusion of transparency: Biased assessment of other’s ability to read our emotional states." Journal of Personality and Social Psychology, 76, 743-753.
48
[23] Jolls, Christine, Cass Sunstein, and Richard Thaler. 2000. "Behavioral Approach to Law and Economics" in C. Sunstein (ed.), Behavioral Law and Economics, Cambridge University Press, Cambridge. [24] Jakiela, Pamela, and Kristof Madarasz. 2009. Projection and Con‡ict: Theory and Experiments, mimeo LSE and U Wash. St Louis. [25] Kessler, Daniel and Mark McClellan.1996. "Do Doctors Practice Defense Medicine?," Quarterly Journal of Economics, Vol 111, No. 2., 353-390. [26] Kessler, Daniel, and Mark McClellan. 2000. "How Liability Law A¤ects Medical Productivity." NBER Working Papers No. 7533. [27] Kruger, Justin, Epley Nicholas, Jason Parker, and Zhi-Wen Ng. 2005. "Egocentrism over e-mail: Can people communicate as well as they think?" Journal of Personality and Social Psychology, Vol. 89, No. 6., 925-936. [28] Harley, Erin, Keri Carlsen, and Geo¤rey Loftus. 2004. "The “Saw-It-All-Along” E¤ect: Demonstrations of Visual Hindsight Bias." Journal of Experimental Psychology: Learning, Memory, and Cognition, Vol. 30, No. 5., 960-968. [29] Harley, Erin. 2007. "Hindsight Bias in Legal Decision Making." Social Cognition, Vol. 25, No 1., 48-63. [30] Hastie, Ried, David Schkade, and John Payne. 1999. "Juror Judgments in Civil Cases: Hindsight E¤ects on Judgments of Liability for Punitive Damages." Law and Human Behavior, Vol. 23, No. 5., 597-614. [31] Heath, Chip, and Dan Heath. 2007. Made to Stick: Why Some Ideas Survive and Others Die. Random House. [32] Heath, Chip, and Nancy Staudenmayer [33] Hermalin, Benjamin. 1993. "Managerial Preferences Concerning Risky Projects." Journal of Law, Economics, & Organization, Vol. 9 No.1., 127-35. 49
[34] Hirschleifer, Jack and Riley John.1999. The Analytics of Uncertainty and Information. Cambridge University Press, Cambridge. [35] Holmström, Bengt. 1979. "Moral Hazard and Observability." Bell Journal of Economics, 13, 324-40. [36] Holmström, Bengt, and Paul Milgrom. 1991. Multitask principal-agent analyses: Incentive contracts, asset ownership and job design. Journal of Law, Economics and Organization Vol. 7, :24-51. [37] Holmström, Bengt. 1999. "Managerial Incentive Problems - A Dynamic Perspective." Review of Economic Studies, Vol. 66, No. 1, 169-182. [38] Innes, Robert. 1990. "Limited Liability and Incentive Contracting with Ex-ante Action Choices." Journal of Economic Theory, Vol. 52(1), 45-67. [39] Jackson, Rene, and Alberto Righi. 2006. Death of Mammography: How Our Best Defense Against Cancer Is Being Driven to Extinction. Caveat Press. [40] Jakiela, Pamela, and Kristof Madarasz. 2009. False Con‡ict: Theory and Experiments mimeo U. Wash St Louis and LSE [41] Jenter, Dirk, and Fadi Kanaan. 2009. "CEO Turnover and Relative Performance Evaluation" mimeo Stanford GSB, http://papers.ssrn.com/sol3/papers.cfm?abstract_id=885531 [42] Jovanovic, Boyan. 1979. Job Matching and the Theory of Turnover. Journal of Political Economy, No. 5, Vol 87, 972-990. [43] Lazear, Edward. 2000. "Performance Pay and Productivity." American Economic Review, Vol. 90, No 5., 1346-61. [44] Kahneman, Daniel, and Amos Tversky. 1996. On the reality of cognitive illusions. Psychological Review, 103, 582-591. [45] Loewenstein, George, Ted O’Donoghue, and Matthew Rabin. 2003. "Projection Bias in Predicting Future Utility." Quarterly Journal of Economics, No. 118, Vol. 4., 1209-1248. [46] Loewenstein, George, Don Moore, and Roberto Weber. 2006. "Misperceiving the Value of Information in Predicting the Performance of Others." Experimental Economics, 3., 281-295. 50
[47] Madarasz, Kristof. 2009. Bargaining with Biased Beliefs mimeo LSE. [48] Milgrom, Paul. 1981. Good News and Bad News: Representation Theorems and Applications Bell Journal of Economics, Vol. 12, No 2. 380-391. [49] Newton, Elisabeth. 1990. "Overcon…dence in the Communication of Intent: Heard and Unheard melodies." Unpublished doctoral dissertation, Stanford University, Stanford, CA. [50] Prendergast, Canice, and Robert H. Topel. 1996. "Favoritism in Organizations." Journal of Political Economy, Vol. 104, No. 5. 958-978. [51] Phelps, Edmund. 1972. “The Statistical Theory of Racism and Sexism.”American Economic Review Vol. 62, 659 - 61. [52] Pronin Emily, Puccio Carolyn, and Ross Lee. 2002. "Understanding Misunderstanding: Social Psychological Perspectives" in Gilovich, Gri¢ n, and Kahneman eds. Heuristics and Biases: The Pscychology of Intuitive Judgment. Cambridge University Press, Cambridge. [53] Rabin Matthew 2002. Inference by the Believer’s of the Law of Small Numbers Quarterly Journal of Economics [54] Rabin, Matthew, and Dimitri Vayanos. 2008. "The Gambler’s and Hot-Hand Fallacies: Theory and Applications." Review of Economic Studies forthcoming [55] Rachlinski, Je¤rey. 1998. "A Positive Psychological Theory of Judging in Hindsight." The University of Chicago Law Review, Vol. 65, No. 2., 571-625. [56] Shapiro, Carl, and Joseph Stiglitz. 1984. "Equilibrium Unemployment as a Worker Discipline Device." American Economic Review, Vol. 74, No. 3., 433-444. [57] Shavell, Steven. 1980. "Strict Liability Versus Negligence." Journal of Legal Studies, Vol. 9., 1-25. [58] Spence, Michael. 1973."Job Market Signaling." Quarterly Journal of Economics, Vol. 87, 355-374. [59] Studdert, David, Michelle Mello, William Sage, Catherine DesRoches, Jordon Peugh, Kinga Zapert, and Troyen Brennan. 2005. "Defensive Medicine Among High-Risk Specialist Physicians in a Volatile Malpractice Environment." Journal of the American Medical Association Vol. 293., 2609-2617. 51
[60] Tversky, Amos, and Daniel Kahneman. 1974. "Judgment under uncertainty: Heuristics and biases." Science, 185, 1124-113. [61] Viscusi, Kip, and Richard Zeckhauser. 2005. "Recollection Bias and the Combat of Terrorism." Journal of Legal Studies, Vol. 34, No. 1., 27-55. [62] Van Boven, Leaf, Gilovich Thomas, and Victoria Medvec. 2003. "The Illusion of Transparency in Negotiations." Negotiation Journal, April, 117-131.
52
