A MISATTRIBUTION THEORY OF DISCRIMINATION

KYLE P. CHAUVIN

June 24, 2021

Abstract. This paper analyzes how Fundamental Attribution Error, the tendency to judge others without accounting for their circumstances, warps beliefs about social groups and drives discrim- ination. I model a self-sustaining cycle in which agents observe outcome (e.g. test score) gaps across groups, mistakenly attribute those gaps to trait (e.g. ability) differences, and then discrim- inate on the basis of those beliefs, influencing the observed outcome gaps. In equilibrium, agents of different groups agree about the relative ordering of groups by trait level while systematically disagreeing in absolute terms. The paper further characterizes equilibrium beliefs and compares different de-biasing policy alternatives.

JEL Classification: C72, D84, J15, J16 Keywords: Behavioral Game Theory, Belief Formation, Inequality, Statistical Discrimination

Department of Economics, Harvard University. [email protected]. My sincere thanks to Roland B´enabou, Sylvain Chassang, and Faruk Gul for their support and assistance. This paper has benefited from useful comments from, among others, Dilip Abreu, Katherine Coffman, Mira Frick, Tristan Gagnon-Bartsch, Daniel Gottlieb, Nikhil Gupta, Anna Hopper, Ryota Iijima, Alex Jakobsen, David Laibson, Stephen Morris, Axel Ockenfels, Wolfgang Pesendorfer, Matthew Rabin, Charlie Sprenger, Justin Sydnor, Michael Thaler, and Leeat Yariv, as well as from presentations at various institutions. 1. Introduction

Beliefs about the innate characteristics of gender, racial, and other social groups are common- place and powerful. Explicitly or implicitly, from the notion that men and women have different abilities to assumptions that some racial groups are inherently more intelligent, hard-working, or criminal than others, these beliefs propel discrimination1 and impact public welfare. Although clas- sical economic models predict that group-trait beliefs should be accurate conditional probabilities (Arrow, 1973; Coate and Loury, 1993), such beliefs have been shown to include the influence of psychological biases such as stereotypes (Hilton and von Hippel, 1996; Schneider, 2004; Bordalo et al., 2016), in-group favoritism (Tajfel et al., 1971), or implicit associations (Greenwald et al., 2009).

This paper studies a particular pathway by which cognitive biases influence discriminatory beliefs: people observe outcome gaps across groups, attribute the outcome gaps to trait differences, and then discriminate in ways that contribute to the very outcome gaps they sought to explain. For example, people learn that boys get higher math test scores, infer that boys are ‘more suited to math’ than are girls, and differentially invest in the success of male students. Likewise, employers may discriminate on the basis of race when they observe more examples of success in certain groups relative to others. Police officers may differentially screen suspects from particular groups known to have higher arrest rates.

These examples are cases of what psychologists term Fundamental Attribution Error (FAE) (Ross, 1977),2 the tendency to “underestimate the impact of the situation and overestimate [the

1Empirical studies have documented the presence of discrimination in numerous contexts. Examples include audit studies showing discrimination against racial (Bertrand and Mullainathan, 2004; Doleac and Stein, 2013; Edelman, Luca and Svirsky, 2017) and religious (Acquisti and Fong, 2019) minorities and women (Moss-Racusin et al., 2012). Other methodologies have uncovered bias against women (Goldin and Rouse, 2000; Sarsons, 2017) and ethnic sub- groups (Rubinstein and Brenner, 2014). See Bertrand(2011), Bertrand and Duflo(2017), Neumark(2018), Lang and Lehmann(2012) and Fang and Moro(2010) for useful surveys of the literature. Although the degree to which such discrimination is driven by beliefs and how much by preferences remains an open question (Guryan and Charles, 2013), some studies (List, 2004; Levitt, 2004; Pope and Sydnor, 2011; Coffman, Exley and Niederle, 2199) find levels of discrimination that are difficult to attribute to animus, indicating beliefs are a significant factor. 2Also called the ‘correspondence bias’ (Gilbert and Malone, 1995), the FAE has a long history of experimental evidence in social psychology. Lab subjects have been found to overestimate the extent to which a fellow subject’s argument on a debate topic, despite being assigned by the experimenter, reflects fellow subject’s true opinion (e.g. Jones and Harris(1967)). Subjects viewing a silent tape of an anxious interviewee attribute her behavior to innate anxiety to an equal extent when informed either that the interview concerned a contentious or mundane topic (e.g. Snyder and Frankel(1976)). When one among two subjects is randomly selected to pose tricky questions to the other, observers believe that the questioner is smarter than the answering subject (e.g. Block and Funder(1986)). In the infamous Milgram(1963) experiment, many participants were successfully coached into delivering what they believed was a near fatal electric shock on another lab participant. In Bierbrauer(1979), subjects who learned about 1 impact of] the individual’s traits and attitudes” (Myers, 2012) when explaining features about another person. By ignoring situational factors, people internalize a direct correspondence linking one’s internal traits with observable outcomes. In particular, the FAE blinds them to the otherwise obvious fact that different people face different circumstances. Observers may acknowledge that discrimination exists in society, engage in it themselves, and still fail to account for it when judging others. This paper models and analyzes a self-sustaining cycle in which people explain group- outcome differences as group-trait differences, develop beliefs that lead them to discriminate on the basis of group status, and perpetuate the very outcome gaps they sought to explain.

Section2 presents the main model, using the domain of education as a constant example throughout.3 A population of agents are matched together in pairs as students and teachers. Agents belong to one of a set of social groups, such as those defined by race or gender. Students have a privately-observed ability type, and teachers maintain beliefs about the expected abilities of students of each group. In each matched pair, the teacher first observes the student’s group and then chooses a level of investment in the student’s education. The student observes the teacher’s investment level and chooses his own effort level. Together, the teacher’s investment and the student’s effort prepare the student for an exam. As mediated through the two parties’ choice of effort levels, the student’s ultimate score can be understood as a joint function of his actual ability and the teacher’s belief about his group.

All agents in the population infer their beliefs about groups by observing each group’s aggregate exam scores. Because they are subject to the FAE, agents acting as observers incorrectly assume that scores depend only on student abilities. Their reasoning is logically inconsistent, as agents who engage in discrimination themselves simultaneously assume away the effect of any discrimination the earlier experiment attributed the willingness of participants to inflict pain to innate evil and not the situational power of the experiment. See Gawronski(2004) for a survey of psychology experiments on the bias. Pettigrew(1979) in fact proposed an extension of the FAE to explicitly consider social groups, but framed his ‘ultimate attribution error’ as a form of in-group preference, by which negative out-group behaviors are attributed to innate negative characteristics. Economists have long acknowledged that empathy is difficult and that people understand the world through a localized lens of experience. Adam Smith said as much in The Theory of Moral Sentiments: “As we have no immediate experience of what other men feel, we can form no idea of the manner in which they are affected, but by conceiving what we ourselves should feel in the like situation.” Smith(1759). More recently the FAE has been studied within economics. Glaeser and Ponzetto(2017) study the consequences for political economy when voters judge their politicians without accounting for circumstances. In lab experiments, Durell(2001) and Cartwright and Wooders(2014) find subjects rating the performance of other subjects without correctly adjusting for the conditions the other subjects faced. 3Education is a particularly salient example for misattribution. Because children have relatively short track records, teachers and other adults working with them must rely more heavily on expectations. Many of the paper’s results, however, are equally applicable to other example domains, such as employers and workers. 2 when evaluating others; it is precisely this inconsistency to which the FAE blinds them. The supposed mapping between ability and score is calibrated by own-group experiences: each observer presumes that the ability-to-score relationship which in reality holds for the observer’s own group also holds for every student in the population.4 When beliefs successively updated in this way converge, the resulting belief profile is a misattribution equilibrium: teachers maintain a profile of beliefs which, together with students’ actual abilities, lead to scores from which agents, as observers, infer the same profile of beliefs. As in a Perfect Bayesian Equilibrium, teachers and students choose optimal actions on the basis of their beliefs; however, the beliefs updated are subject to the FAE.

Section3 presents the main results of the paper. There is a unique misattribution equilibrium. In it, the degree to which equilibrium beliefs differ from objective ability hinges on what is termed the effect strength parameter. Effect strength is equal to the marginal effect5 of teacher belief on exam score divided by the marginal effect of student ability on exam score. When the student’s ability and the teacher’s belief push the student’s score in the same direction, the effect strength is positive; when they push in opposite directions, the effect strength is negative. For example, effect strength will be positive if the student and teacher’s effort levels are strategic complements, because higher teacher belief will induce greater teacher effort, which then incentivizes more student effort and leads to a higher test score. With positive effect strength, differences between groups are magnified in equilibrium; with negative effect strength, differences are understated. In terms of population-wide beliefs, which aggregate those of all observers, students of different groups are simultaneously misestimated in different directions. At least one group is overestimated, and at least one group is underestimated. Hence, some groups benefit from misattribution while others suffer from it.

4While there are many potential ability-to-score correspondence which teachers could assume, any realistic correspon- dence would incorporate the actual experiences of students. The extended model in Section4 allows for aggregating the experiences of different groups of students according to arbitrary weights. Section2 specifies own-group calibra- tion as the simplest of such aggregations. One intuitive, if informal, rationale for this conceives of misattribution equilibrium as the steady state of an overlapping generations model in which students of one generation become the teachers of the subsequent generation; in the OLG reading, own-group calibration is equivalent to assuming teachers internalize the treatment they themselves received as students and use that to judge their own students. This resem- bles the ‘cross-situational projection’ discussed in Van Boven and Loewenstein(2005): people reason about others facing particular circumstances by first imagining themselves in the same situation. 5Recall that a student’s score depends on the student’s ability as well as effort choices by both student and teacher. However, their choices can be understood as functions of teacher belief and actual ability, and hence the student’s score can be understood as a function purely of belief and ability. The effect strength parameter is the ratio of the derivative of that function with respect to belief over the derivative with respect to ability.

3 The resulting ranking of groups according to equilibrium beliefs coincides with the ranking according to objective abilities. Observers all share this same relative ranking, but their beliefs systematically disagree in absolute terms. To illustrate, suppose the effect strength is positive. Observers of higher performing groups, basing their inference in own-group experience, presume that all students receive a relatively high level of teacher investment. In explaining observed scores, they therefore make relatively low assessments of student ability. Similarly, observers of lower performing groups presume that all students receive less investment and accordingly infer relatively high assessments. For example, if male students get higher scores than female students, and then teachers invest more in boys, male observers will have intuited that ability translates more easily into scores, leading them to make lower appraisals of all students than do female observers.

When the effect strength is sufficiently large, beliefs successively updated via misattribution do not converge to equilibrium, but rather diverge towards infinity. In such extreme cases of the model, the average belief about some groups increases towards positive infinity while the average belief about other groups tends to negative infinity.6 Moreover, different initial beliefs can induce any group to be positively or negatively misestimated. This illustrates how the FAE can not only warp objective differences but also, in the extreme, completely sever the link between actual trait differences and societal beliefs about them.

Section4 generalizes the model so that the influence that one group has on another is a flexible parameter. Similarly, it also allows for flexible empathy by having observers calibrate their subjective models of outcome production using the experiences of students outside their own social group. Equilibrium existence and uniqueness remain generic outcomes of the general model, but student groups are not necessarily over/underestimated according to their objective rank in the population. Except in knife-edge cases, all groups are misestimated, with at least one group overestimated and at least one group underestimated. I establish conditions under which the relative ranking of students by observers’ beliefs is correct and characterize systematic differences between the beliefs of different observer groups.

Section5 examines the efficacy of several common policies intended to blunt the effects of dis- crimination. It predicts that ‘affirmative action’ policies which mandate the preferential treatment

6These are directional predictions. As a stylization of reality, the model omits the myriad of factors that would prevent actual beliefs from inflating without bound. The paper’s results about beliefs tending to positive or negative infinity are understood as predictions that beliefs will increase or decrease with succesive iterations of misattribution. 4 of different groups or institute outcome transfers across groups have a multiplier effect on biased beliefs. Intuitively, by artificially inflating some groups’ observed outcomes, a policy maker induces belief shifts that further contribute to outcome differences, and thus a perfectly calibrated policy could completely de-bias the population.7 On the other hand, ‘diversity’ policies which broaden empathy or influence in the population cannot de-bias incorrect beliefs brought about by misattri- bution. Because broadened empathy or influence brings different groups closer to being compared against a common standard without changing the fact that the FAE leads to warped beliefs, these policies cause different observer groups to agree more, but do not lead to beliefs which are more correct overall.

The discussion in Section6 considers how the equilibrium concept featured in this paper com- pares with others in behavioral game theory, particularly the Berk-Nash equilibrium of Esponda and Pouzo(2016). Beyond that connection, this paper contributes primarily to the literatures regarding discrimination and stereotypes. Explanations for discrimination have traditionally been divided between preference-based, e.g. from animus (Becker, 1957) or identity (Akerlof and Kran- ton, 2000; B´enabou and Tirole, 2011), and statistical discrimination (Arrow, 1973; Coate and Loury, 1993), whereby an interaction features multiple equilibria, and group status becomes a coordination device.8

More recent work has highlighted the roles of cognitive biases in producing what Bohren et al. (2019) term ‘inaccurate statistical discrimination.’ Bordalo et al.(2016) study the stereotypes that arise when observers see the trait distributions of different groups but, on account of limited work- ing memory, disproportionately associate each group with its most representative traits (Gennaioli

7This contrasts with the ‘patronizing equilibrium’ phenomenon of Coate and Loury(1993), where policies that lower standards for minority applicants disincentivize human capital investment among those applicants. The key difference is that rational managers who engage in statistical discrimination have no incorrect beliefs to be de-biased. 8For example, in a good equilibrium workers acquire skills and managers hire them, while in a bad equilibrium workers do not invest in human capital and managers do not hire them. Discrimination arises when one group is in a good equilibrium and another is in a bad equilibrium. Managers’ discriminatory beliefs are ‘self-confirming’ in the sense that they help to support the differential treatment of and behavior by the two groups, but not in the sense of self-confirming equilibrium (Battigalli, 1987; Fudenberg and Levine, 1993; Dekel, Fudenberg and Levine, 2004), where the environment fails to generate the feedback needed to dispel incorrect notions. Note that Phelps(1972) illustrates a distinct pathway which is also called ‘statistical’ discrimination: risk-averse managers may favor group A over group B applicants because, conditional on observing only an applicant’s group status, the value of a B worker has higher variance.

5 and Shleifer, 2010).9 Fryer and Jackson(2008) analyze discrimination produced by coarse catego- rization. Unlike these processes, the FAE does not involve distorting or simplifying one’s dataset of observations, but rather imposing an incorrect model. An application of the selective attention model in Schwartzstein(2014) shows how skewed perceptions of group traits follow from people bas- ing their judgements on an unrepresentative sample of personal experiences, and Heidhues, K˝oszegi and Strack(2019) examines a similar pathway for individuals who have stubbornly positive views of their own abilities.10 While attribution error can be understood as a particular instance of inat- tention, the psychology literature (see Footnote 2) suggests it is a ubiquitous tendency rather than an inadvertent mistake.

What most prominently distinguishes beliefs informed by the FAE is the prediction the model makes regarding disagreement across groups. Namely, observers agree about which groups have relatively high traits and which have relatively low traits, but disagree in absolute terms. The discussion in Section6 performs a quick comparison of these predictions with evidence from the General Social Survey. It finds that they are largely borne out in data on beliefs about the intel- ligence, work ethic, and proclivity to violence of four different racial/ethnic groups. This initial exercise provides a foundation for future work to test how much the FAE contributes to assessments of group trait levels.

2. Model

There is a continuum population of agents, each of whom has a privately-known ability θ ∈ R and a publicly-observable group identity g ∈ G = {1,...,N}. At issue is how the conditional distribution of traits varies across groups. In a Perfect Bayesian Equilibrium of any game played by this population, the agents’ beliefs about groups’ trait distributions would be, by definition, correct.

9This produces several predictions, such as exaggerated perceptions of group differences and a ‘kernel of truth’ in incorrect beliefs, which are common to the present paper. It also contrasts in various ways, including in predicting underestimation of within-group heterogeneity, where misattribution leads to correct estimation of higher order distribution moments, and predicting that an observer’s beliefs are independent of her own group status, while in the present paper differential empathy across observer groups produces systematic differences in beliefs. 10More broadly, the consequences of the FAE correspond with other cases of mis-inference in which an agent’s failure to understand his own circumstances leads him to misunderstand others. For example, B´enabou and Tirole(2006) model how agents can sustain their belief in the value of effort as a means to motivate themselves. Consequently, people who believe that effort readily produces wealth accordingly infer that the poor must be lazy. Frick, Iijima and Ishii(2019) study members of a population who interact through assortative matching but mistakenly believe that they interact with a representative sample; in settings with coordination, this bias produces more extreme beliefs about others. 6 The model here lays out an alternative process of belief updating that incorporates Fundamental Attribution Error and leads to systematically incorrect beliefs.

In the interest of simplicity, I focus attention on what agents think is the mean trait of each group. The true mean trait of group g is denoted µg. Each agent maintains a point belief about i µg; the average belief of group i members isµ ˜g. (Indices i and j denote observer groups; g and h denote observed groups.) Group i comprises fraction wi of the population, so the population-wide P i belief about group g agents isµ ˜g = i∈G wiµ˜g. I assume that there is some non-trivial variation among groups: µg 6= µh for some g, h ∈ G.

i Belief updating proceeds as follows. Starting with an arbitrary initial profile of beliefs (˜µg)i,g∈G, the agents are randomly paired together to play an interaction game. Using education as the key example, each pair involves one agent acting as teacher and the other as student for the purpose of preparing the student for an exam. Agents face a uniform probability of matching with any other agent and, independently, an equal chance of acting as teacher or student.11 Once matched, the teacher chooses a level of effort at, and then the student selects his own effort level as. When choosing, the teacher can observe the student’s group identity but not his ability; the student observes the teacher’s effort level before selecting his own. Together with the student’s ability, their decisions jointly determine the student’s exam score x(as, at, θ). Both parties are expected- S T utility maximizers with ex-post payoffs u (as, at, θ) and u (as, at, θ). In order to ensure that the agents’ beliefs about mean traits are sufficient statistics to determine their optimal actions and guide their belief updating, I impose the following functional form assumption.

Assumption 1. x(·) is affine and strictly increasing in each argument; uS(·) and uT (·) are qua- dratic and strictly concave.

∗ As a consequence of Assumption1, the student’s optimal effort as is an affine function of his ability θ and the teacher’s effort at. The teacher, unable to directly observe the student’s effort, ∗ instead relies on her belief about the average ability of the student’s group. Her action at is accordingly an affine function of her beliefµ ˜. (See the end of this section for an explicit derivation

11See Section4 for an extension of the model that allows for non-uniform matching. 7 and discussion.) All together, the student’s exam score is

T ∗ ∗ ∗ X(θ, µ˜g ) = x(as(at (˜µ), θ), at (˜µ), θ),

T an affine function of θ andµ ˜g . At the population level, repeated matching of students and teachers produces leads to average exam scores for each group, and the averages are known publicly.

Given the initial profile of beliefs and the average exam scores they produce, agents in the population now update their beliefs. At this point, the Fundamental Attribution Error blinds ob- servers to the fact that a student’s exam score depends both on his ability as well as his teacher’s treatment of him. Biased observers instead make inferences under the incorrect assumption that the student’s ability is the only determinant of his score. Although observers ignore teacher effort, the assumption that there exists a single ability-to-score correspondence for all students is mathe- matically equivalent to maintaining that there exists a single distribution of teacher beliefs – and hence, teacher investments – faced by all students. As a base specification for the main model, agents are assumed to engage in inter-personal projection, using their own experiences to calibrate their subjective models.12

An observer from group i therefore understands the relationship between average ability and average score to be µi 7−→ X(µi, µ˜i). This implicitly bakes in the population-wide belief about group i students,µ ˜i. When evaluating group g, the group i observer neglects how group g students will receive a different level of teacher investment when the population-wide beliefµ ˜g is different fromµ ˜i. Hence, while group g’s average score Xg is actually generated as X(µg, µ˜g), the group i i0 i0 observer seeks to account for Xg by searching for a beliefµ ˜g that satisfies satisfies Xg = X(˜µg , µ˜i). i0 On account of the affine functional form of X(·), there is a unique profile of updated beliefs (˜µg )i,g∈G i 13 pinned down by the original profile (˜µg)i,g∈G. When agents in the population iteratively update their beliefs in this fashion, the resulting sequence is termed a misattribution belief sequence: i[t] i[t+1] [t] [t] (˜µg )i,g∈G;t=0,1,... where X(˜µg , µ˜i ) = X(µg, µ˜g ) for all i, g ∈ G and t ≥ 1.

12Section4 relaxes this assumption with a general framework by which observers calibrate their subjective models. 13Although the model specifies a single population that is iteratively rematched and updated, one can also understand the belief sequence as tracking the opinions of overlapping generations, whereby the students of generation t become the teachers of generation t + 1, forming beliefs on the basis of time t scores and then acting on those beliefs in time t + 1. In this interpretation, the assumption that teachers calibrate their subjective models using the experiences of their own group is equivalent to assuming teachers in time t + 1 recall their personal experiences as former students in time t.

8 As X(·) is an affine function with constant derivatives dX/dµ˜ and dX/dθ, it is straightforward to derive an explicit formula for updated beliefs as a function of original beliefs. The updating equation is dX 0 dX dX dX X + µ˜i + µ˜ = X + µ + µ˜ , 0 dθ g dµ˜ i 0 dθ g dµ˜ g yielding the following Fact. (See the end of this section for further details.)

i Fact 1. Given original beliefs (˜µg)i,g∈G, the profile of updated beliefs is

i0 µ˜g = µg + η(˜µg − µ˜i), (1) where η = (dX/dµ˜)/(dX/dθ).

The belief updating equation in Fact1 has an intuitive functional form. Group i’s updated

i0 belief about group g,µ ˜g , is equal to the true value µg plus an offset term. The offset depends on the gap between the original population beliefs about g and i as well as an effect strength parameter η that depends on the fundamentals of the interaction game. To illustrate these pathways, suppose group g was estimated to be higher than group i,µ ˜g > µ˜i, so that group g students were treated according to a higher level of teacher belief than were group i students. If η > 0, say because higher teacher beliefs as well as higher actual ability both lead to better scores, then the originally higher belief in group g students leads to a boost in scores relative to group i students. The group i observers neglect to account for the boost, so they end up overestimating group g students’ abilities. For example, if male students are initially deemed to have higher ability than female students, and receive better teacher treatment because of that, then female observers’s will overlook that advantage when updating their beliefs, and as a consequence overestimate male students’ abilities. The degree of overestimation hinges on the magnitude of η, which for technical reasons is assumed to be non-zero throughout the paper.

Averaging over all observer groups delivers the corresponding belief updating equation for population-wide beliefs: 0 µ˜g = µg + η(µg − µ¯), (2) P whereµ ¯ = h∈G µh is the population-wide average trait. This shows that the population-wide average belief for any given group similarly evolves by misestimating µg according to the gap 9 between µg and a benchmark value. Whereas the benchmark for a single observer group is the group’s own mean trait, the benchmark for the population is the average traitµ ¯.

The rest of this section traces out how the effect strength parameter η emerges from model primitives. Readers eager for the main results should skip to the equilibrium analysis in Section3.

How Effect Strength Depends on Best Response Functions. To more thoroughly unpack what is happening in the belief updating process, we can explicitly write out the involved functions:

 dx dx dx x(a , a , θ) = x + a + a + θ  s t 0 s t  das dat dθ   ∗ das das as(at, θ) = as0 + at + θ  dat dθ   dat  a∗(˜µ) = a + µ.˜  t t0 dµ˜

From this we can regroup X(·) into similar notation:

         dx dx das dx dx das dx dx das dx dat X(θ, µ˜) = x0 + as0 + + at0 + + θ + + µ,˜ das das dat dat das dθ dθ das dat dat dµ˜ | {z } | {z } | {z } X0 dX/dθ dX/dµ˜ and, using this, we can obtain an explicit formula for the effect strength parameter η:   dx das + dx dat dX/dµ˜ das dat dat dµ˜ η = = . dX/dθ dx das + dx das dθ dθ

As x(·) is assumed to be strictly increasing in each argument, all of the x partials are positive. If we further assume that das/dθ > 0, i.e. that students with higher ability also invest more effort in their own success, then higher student ability unambiguously leads to higher scores, and the sign of η hinges on two critical partials. First, dat/dµ˜ measures how the teacher adjusts her effort as her belief in the student’s group increases. Second, das/dat captures the student’s response to an increase in teacher effort. When both of these factors are positive, the interaction game features a positive feedback loop: higher teacher belief inspires more teacher effort, which in turn spurs more student effort, leading to higher scores. In this case, η > 0. When both are negative and sufficiently extreme, a similarly positive feedback loop ensues: higher teacher belief dampens teacher effort, but lower teacher effort leads the student to compensate with more effort, again leading to higher scores. Alternatively, if the factors have opposite signs and are sufficiently extreme, the feedback loop is negative, and η < 0.

10 How Best Response Functions Depend on Model Primitives. Without loss of generality we can express S S u (as, at, θ) = x(as, at, θ) − c (as, at, θ), where cS(·) is a (strictly convex) quadratic cost function:

T T  S      S S S    cs as as css cst csθ as     1       cS(a , a , θ) = cS +  S   +    S S S    . s t 0 ct  at at cst ctt ctθ  at     2       S S S S cθ θ θ csθ ctθ cθθ θ

From this we can compute

( dx − cS) − (cS a + cS θ) ∗ das s st t sθ as(at, θ) = S . css

S S This shows that the partial das/dat is equal to −cst/css, a very intuitive expression: when higher S S teacher effort lowers the cost of student effort, i.e. cst < 0, it follows that das/dat > 0, since css > 0 by cS(·) convex. When teacher effort increases the cost of student effort, the student’s response is accordingly negative: das/dat < 0. A similar derivation using corresponding notation yields h i dx T das dx T T das T T das das T T das ( − ct ) + ( − cs ) − (cst + )as0 + (c + cst + (c + css ))˜µ ∗ dat dat das dat tθ dθ dat sθ dθ at (˜µ) = , cT + 2 das cT + ( das )2cT tt dat st dat ss so da cT + cT das + das (cT + cT das ) t = − tθ st dθ dat sθ ss dθ . dµ˜ cT + 2 das cT + ( das )2cT tt dat st dat ss The numerator of this expression captures the aggregate change in cost to teacher effort as the

T student’s projected ability rises. This includes both the direct cost cross-partial ctθ as well as the indirect channel by which increased student ability impacts the student’s choice of effort, and the corresponding effect of student effort on the teacher’s costs. The denominator is assumed to be

∗ positive by convexity. Hence, dat /dµ˜ is positive if and only if the aggregate effect of higher student ability serves to lower the teacher’s cost of effort.

3. Long-Run Beliefs

This section examines the long-run behavior of beliefs successively updated according to the procedure in Section2. Depending on the value of the effect strength parameter η, updated beliefs either converge to finite equilibrium values or diverge towards positive or negative infinity. 11 3.1. Equilibrium

Moderate values of η lead beliefs towards a finite equilibrium. The reason is straightforward: with η low in absolute value, initial misestimations are mitigated through each round of updating, so iterating that process steers the sequence towards the (unique) fixed point of the belief updating equation. Furthermore, the structure of the belief updating equation is simple enough that exact formulas can be obtained.

i[t] Proposition 1. If η ∈ (−1, 1), then each misattribution sequence (˜µg )i,g∈G;t=0,1,... converges to the misattribution equilibrium: for all groups i and g,

η µ˜i[t] −→ µ˜i∗ = µ + (µ − µ ). g g g 1 − η g i

Proof in the appendix.

The simplicity of the expression for equilibrium beliefs allows us to read off a number of characterizing features, organized below as Facts2 to7. The first set of results concerns population- wide beliefs, which are the determinant of the treatment that groups face. The second set concerns differences between the beliefs of individual observer groups.

Fact 2. Population beliefs in equilibrium are

η µ˜ = µ + (µ − µ¯). g g 1 − η g

At least one group is overestimated, and at least one group is underestimated. If two groups share the same mean trait, equilibrium beliefs about them are the same.

The structure of equilibrium beliefs mirrors that of the belief updating equation (Equation2 for population beliefs), with misestimations from successive rounds of updating summed to a finite level. In equilibrium, all groups are effectively evaluated with respect to the common benchmark µ¯. The gap between a group’s true mean trait and the benchmark, scaled by factor η/(1 − η), is the equilibrium degree of bias at the population level. Since it is assumed not all groups share the same mean trait, this implies that at least one group must be overestimated while another is underestimated, illustrating the symmetric nature of distorted beliefs following the Fundamental Attribution Error. Finally, it is worth noting that equilibrium beliefs are equal for any two groups with the same actual mean traits. 12 Corollary. At least one group’s students are strictly harmed by biased beliefs, and, for sufficiently low values of η, at least one group’s students are made strictly better off.

The fact that groups are simultaneously misestimated implies what is perhaps the most signifi- cant welfare consequence of misattribution, namely that for students the bias creates both winners and losers. When both student and teacher choose effort levels optimally and the teacher’s belief is correct, the derivative of student welfare is

S  S S  du du ∗ ∗ ∗ das du ∗ ∗ ∗ dat = (as(at (θ), θ), at (θ), θ) · + (as(at (θ), θ), at (θ), θ) , dµ˜ µ˜=θ das dat dat dµ˜ which is non-zero (except for in knife-edge cases). Owing to the quadratic form of student utility, misestimation in the opposite direction of this derivative is guaranteed to create welfare loss. Con- versely, for small η, this first-order effect dominates for misestimation in the same direction, and hence at least one group must enjoy a welfare boost. By contrast, all teachers lose from misesti- mation. Hence, for sufficiently large values of η, overall population welfare is strictly harmed by bias.

Fact 3. The severity of population misestimation, |µ˜g − µg|, increases with |η| and with |µg − µ¯|.

There are two factors which determine how badly a group is misestimated in equilibrium. The first is the absolute value of η. A more extreme effect strength parameter means that each round of belief updating incorporates larger errors, converging to a larger equilibrium bias. The second factor is the gap between the group’s actual trait µg and the population averageµ ¯, which we can re-express as follows:   1 X µg − µ¯ = (1 − wg) µg − P whµh . wh h6=g h6=g

This is the product of the non-g population size and the gap between µg and the average trait among non-g agents. We see that groups more unlike the others are more severely misestimated, but also that numeric minority groups (those with smaller population shares wg) are likewise more severely misestimated.

Fact 4. When η > 0, differences between groups are exaggerated: |µ˜g − µ˜h| > |µg − µh| for all g, h ∈ G. When η < 0, differences are understated: |µ˜g − µ˜h| < |µg − µh| for all g, h ∈ G. 13 The sign of η is critical in determining whether misattribution leads to equilibrium beliefs exaggerating or understating differences between groups. From Fact2, it follows

1 µ˜ − µ˜ = (µ − µ ). g h 1 − η g h

When student ability and teacher belief push the student’s score in the same direction, η is positive, so factor 1/(1 − η) is greater than 1, and differences between groups are exaggerated. When they push the student’s score in opposite directions, a negative value of η means that factor 1/(1 − η) is less than 1, and differences are understated.

i Fact 5. Group i misestimates g unless they share the same average abilities: µ˜g 6= µg if µi 6= µg. i Group i more severely misestimates group g when their abilities are further apart: |µ˜g −µg| increases with |µi − µg|.

Turning now to the beliefs of individual groups, it follows from Proposition1 that an observer correctly estimate any group that shares the same mean trait as her own group and misestimates all other groups. This is because agents’ subjective models are calibrated with respect to their own group’s experiences, and – as noted in Fact2 – two groups with the same mean trait will face the same population beliefs in equilibrium. As the gap between two groups’ actual abilities widens, so does the gap between population beliefs about the groups. As the groups accordingly face more disparite treatment, they more severely misestimate each others’ traits.

i i j j Fact 6. Observer groups agree in relative terms: µ˜g > µ˜h if and only if µ˜g > µ˜h. Relative beliefs i i are also correct: µ˜g > µ˜h if and only if µg > µh.

Comparing beliefs across observer groups, we see that they always agree in relative terms: all observers agree which group has the highest ability, next-highest, and so on. The reason is that, although different observer groups maintain different benchmarks by which they compare student scores, their opinions are derived from the same data. Groups with higher scores are always understood to have higher14 ability, regardless of what an observer thinks is the universal ability- to-score relationship. Although the exact values of equilibrium beliefs are incorrect, the ranking of groups according to population beliefs always matches the objective ranking. Intuitively, the

14This is assuming that, in the interaction game, higher ability does translate into higher scores. 14 reason for this kernel of truth amid broader inaccuracy is that students of all groups are assessed according to a common benchmark. The gap between an observer’s beliefs about two groups is

i i 1 1 µ˜g − µ˜h = 1−η (µg − µh), the same as for population beliefs. As 1−η is positive for all η < 1, it follows that in equilibrium all relative beliefs are correct.15

Fact 7. When η > 0, groups with higher abilities maintain lower beliefs about all groups:

i j µ˜g < µ˜g if and only if µi > µj.

When η < 0, groups with higher abilities maintain higher beliefs about all groups:

i j µ˜g > µ˜g if and only if µi > µj.

Because each observer group’s beliefs are determined with respect to own-group trait values, observer groups with higher trait values must necessarily infer lower absolute beliefs abut all groups when η > 0, and higher absolute beliefs when η < 0, in order to explain the same data as observer groups with lower abilities. For example, if η > 0 and male students are deemed to have higher ability than female students, then teachers will invest more in male students, giving male observers the impression that ability translates readily into test scores. Hence, given the same test scores, male observers will intuit lower abilities for all groups than will female observers.

3.2. Divergent Belief Sequences

Extreme values of η lead to divergent belief sequences. Instead of mitigating initial misesti- mations, succesive updating serves to magnify bias. For η < −1, beliefs also flip signs with each iteration of updating.16 For η > 1, however, the belief sequences tend in a single direction.

i[t] Proposition 2. If η > 1, then each misattribution sequence (˜µg )i,g∈G;t=0,1,... diverges to infin- [t] [t] ity: µ˜g −→ +∞ if µˆg > µ¯, and µ˜g −→ −∞ if µˆg < µ¯, where ! [0] X [0] µˆg = µg + (η − 1) µ˜g − µ˜h . h∈G

Proof in the appendix.

15Section4 demonstrates how this result can fail when, with more general social networks, different groups are assessed according to different benchmarks. For example, a high ability group assessed according to an even higher benchmark may be judged lower than a low ability group assessed according to an even lower benchmark. 16Because of this instability, I omit analysis of the η < −1 case. 15 As specified in Proposition2, the direction of divergence for population beliefs about each group depends critically on true ability levels as well as initial beliefs. A group is misestimated upwards to +∞ if its true trait plus a ‘boost’ term exceeds the mean ability in the population. The boost [0] P [0] term is (η − 1) timesµ ˜g − h∈G µ˜h , the gap between the initial population belief about group g and the average initial population belief. In this way, group g may be misestimated upwards either if it has a high trait relative to the rest of the population, if it is deemed to have a relatively high trait according to initial beliefs, or some combination of the two.

Fact 8. For each belief sequence, at least one group is estimated upwards, µ˜g −→ +∞, and at least one group is estimated downwards, µ˜g −→ −∞. For each group, there exist initial beliefs such that the group is estimated upwards and initial beliefs such that the group is estimated downwards.

As with equilibrium beliefs, at least one group must be overestimated, here in the sense ofµ ˜g −→ +∞, while one group is underestimated. What distinguishes the η > 1 case from equilibrium is that the determination of which groups are over- and which groups are under-estimated is not fixed by actual abilities alone. Indeed, initial beliefs can overwhelm all actual relationships between groups. Any assignment of groups to over- and under-estimation such that at least one is overestimated and at least one is underestimated can be implemented by starting with sufficiently extreme initial beliefs.

4. Model Extensions

This section relaxes two key assumptions of the main model: that agents are matched together uniformly, and that observers use their own groups as the basis for evaluating others. The extensions serve as robustness checks and as a basis for policy measures discussed in the next section. They are also more realistic depictions of many real-life situations. For example, the demographic makeup of teachers does not conform to population proportions, and there is often assortative matching along gender and racial lines between students and teachers. Similarly, using one’s own group as the sole basis for evaluating others may be less appropriate when members of different groups live together than when groups are dispersed geographically.

Model Extensions. The probability with which a group g student is matched with a group i teacher is now denoted wgi ∈ [0, 1] and describes the influence that group i has on group g. P i Accordingly, the population-wide belief about g is nowµ ˜g = i∈G wgiµ˜g. The weights wgi are 16 P non-negative and, for any student group g, the sum across all teacher groups is one: i∈G wgi = 1. This accommodates the possibility that some groups comprise an outsized share of the population or that the matching between students and teachers is asymmetric.

Instead of using only own-group experiences to calibrate their subjective models, as in Section 2, observers now aggregate the experiences of students of different groups according to empathy weights. The extent to which an observer of group i incorporates the experiences of students of group h is denoted zih ∈ [0, 1]. That is, a group i teacher implicitly assumes that all students face P an average level of teacher belief equal to h∈G zihµ˜h. As with the influence weights, it is assumed P that any teacher group’s empathy weights sum to one: h∈G zih = 1. The empathy weights model social interactions, such as among friends and family, in constrast to the professional encounters which generate the student’s exam scores.

Allowing for arbitrary influence and empathy weights, and following the original derivation of the belief updating Equations (1) and (2), group i’s updated belief about group g is now ! i0 X µ˜g = µg + η µ˜g − zihµ˜h , (3) h∈G and the population-wide revised belief about g is ! 0 X µ˜g = µg + η µ˜g − yghµ˜h , (4) h∈G where the benchmark weight X ygh = wgizih i∈G computes the degree to which the actual experiences of group h are used in forming beliefs about group g. The network is said to be fully-connected when ygh > 0 for all groups g and h. Equilibrium, when it exists, must satisfy ! X µ˜g = µg + η µ˜g − yghµ˜h (5) h∈G for all g ∈ G.

4.1. Examples

In addition to the common-influence, own-group-empathy case of Section2, several noteworthy special cases of network structures lead to particularly simple expressions for equilibrium beliefs. 17 Uniform benchmark weights. When ygh = y∗h for all groups g, h ∈ G, the benchmark used in evaluating any one group g is the same as for any other group g0. One example is the common- influence, own-group-empathy specification used in Section2. The advantage of common bench- mark weight networks is that equilibrium beliefs still admit a simple closed-form expression, which is not possible for general networks (see Lemma2 in the Appendix for details). Taking a y∗h−weighted average of both sides of Equation (4) and rearranging yields: ! η X µ˜ = µ + µ − y µ . g g 1 − η g ∗h h h∈G

As in the main model, the equilibrium population beliefµ ˜g is equal to the true value µg plus an offset term consisting of the η/(1 − η)-weighted difference between µg and a common benchmark P value h∈G y∗hµh. For η > 0, all groups with true mean traits above the objective benchmark are overestimated and all groups below are underestimated; the reverse is true if η < 0. Beliefs of individual observer groups admit a similarly straightforward expression: ! η X µ˜i = µ + µ − z µ . g g 1 − η g ih h h∈G

P i Instead of comparing group g to the objective benchmark h∈G y∗hµh, group i’s beliefµ ˜g compares P g to the corresponding value h∈G zihµh that averages other groups according to group i’s own empathy weights instead of the common benchmark weights.

Common empathy; common influence. Instead of calibrating their models according to own- group influence, each observer group weights the experiences of students in the same proportions, and the common empathy weights align with common influence weights, that is zih = z∗h = wih = w∗h, then ygh = w∗h as above. Thus population equilibrium beliefs are identical to those in the own-group empathy, common influence case: ! η X µ˜ = µ + µ − w µ . g g 1 − η g ∗h h h∈G

However, when empathy weights are common, then all observer groups have identical equilibrium g beliefs, and those coincide with the population:µ ˜h =µ ˜h for all g, h ∈ G.

Focal group. When one group d is dominant in the population, either because its members are used by all groups to calibrate their models, zgd = 1 for all g ∈ G, or because it is the only group

18 that has positive influence and has own-group empathy, wgd = 1 for all g and zdd = 1, then that group’s true mean trait becomes the benchmark against which all others are evaluated. Hence, group d is correctly estimated,µ ˜d = µd, but all other groups g for whom µg 6= µd, who do not share the same true mean level, are misestimated:

η µ˜i =µ ˜ = µ + (µ − µ ). g g g 1 − η g d

When η > 0 (η < 0) those groups with true mean traits above the dominant group d’s are over- estimated (under-estimated) and those groups with true values lower than d’s are under-estimated (over-estimated).

Own-group empathy; own-group influence. As a final example, consider a special case network which does not fit into the common benchmark paradigm: all groups have complete own-group empathy and own-group influence: zgg = wgg = 1 for all g ∈ G. This models a society in which multiple groups coexist without contact. As each group is separated from others, it is used as its own benchmark, which means there is no common set of benchmark weights. Instead, each group acts as its own single-group model and is correctly estimated at the population level:µ ˜g = µg. As in the own-group empathy, common influence case, individual observer groups have benchmark g values equal to their own true mean traits. Therefore, in general it followsµ ˜h 6= µh for g 6= h; however, because the groups are separated, out-group beliefs have no influence on outcomes.

4.2. General Results

Equilibrium existence follows from solving for a fixed point of the belief updating equation and checking stability; examining the revision equation in matrix form shows that existence as well as uniqueness are guaranteed when |η| is sufficiently small. As Lemma1 below quantifies, for any fixed network structure, the cutoff value of η guaranteeing equilibrium existence depends on

min y∗∗ ≡ ming∈G ygg, the smallest of all groups’ own-group benchmark weights. Intuitively, both η and the ygh values govern the extent to which application of the belief revision equation expands min or contracts the space of potential beliefs. As y∗∗ approaches 1, i.e. as each group becomes the benchmark by which its own members are compared, equilibrium obtains even for arbitrarily extreme values of η.

19 Lemma 1. For a given profile of true mean traits {µg}g∈G and a given network structure {ygh}g,h∈G, there exists 1/2 η ≥ min 1 − y∗∗ such that equilibrium exists whenever |η| < η. Proof in the appendix.

There are several ways to understand how misattribution equilibrium beliefs compare in general to their objective counterparts. The first perspective measures errors in absolute terms by analyzing the differences |µ˜g − µg|. The second examines relative beliefs, the ranking of groups by mean trait as implied by observers’ absolute beliefs. Finally, the third perspective analyzes how the beliefs of individual observer groups compare both with the true mean traits and with the population-level equilibrium beliefs.

As Proposition3 below establishes, misattribution equilibrium beliefs are generically wrong. Instead of producing a general over- or under-estimate of all groups, it simultaneously skews beliefs about some groups upwards and others downwards. Recall from Section2 that one group is guaranteed to be overestimated and the other underestimated. This pattern holds generally. Per

Equation (5), the equilibrium population belief about group g,µ ˜g, consists of the true value µg P distorted by a comparison betweenµ ˜g and its equilibrium benchmark h∈G yghµ˜h. With common P benchmark weights, the value of h∈G yghµ˜h for all groups g is a simple average of true traits, P P h∈G y∗hµh. As detailed by Lemma2 in Appendix A.1, in general the value of h∈G yghµ˜h depends on the average of true traits computed not only with with respect to {ygh}g,h∈G but according to a sequence of ‘higher order’ benchmark weights. When the true mean trait of a group g is equal to the weighted average according to all of its higher order benchmark weights, it is termed medial. General network structures need not admit any medial groups at all, and, as Proposition 3 establishes, a non-medial group is misestimated for almost all values of η. Proposition3 also establishes bounds on how incorrect misattribution equilibrium beliefs can be by first bounding the degree to which any one application of the revision equation can distort beliefs and then computing the limit of the compounded distortions.

Proposition 3. (a) Suppose the network is fully-connected (ygh > 0 for all g, h). In equilibrium,

(i) All non-medial groups g are misestimated, µ˜g 6= µg, for all values of η outside a set of Lebesgue measure zero. 20 (ii) All medial groups g are correctly estimated, µ˜g = µg.

(iii) At least one group is overestimated and at least one group is underestimated, µ˜g+ > µg+ + − and µ˜g− < µg− for some g , g , provided η 6= 0 and maxh1,h2∈G |µh1 − µh2 | > 0

min (b) Whenever |η| < (1/2)/(1 − y∗∗ ), beliefs are bounded from their true values by

min |η|(1 − y∗∗ ) |µ˜g − µg| ≤ min · max |µh1 − µh2 | 1 − 2|η|(1 − y∗∗ ) h1,h2∈G for all g ∈ G. With common benchmark weights and |η| < 1,

η |µ˜g − µg| ≤ · max |µh1 − µh2 | 1 − η h1,h2∈G for all g ∈ G. Proof in the appendix.

An immediate corollary is that equilibrium beliefs are guaranteed to approach their true counter- parts if |η| goes to zero, if the gap between different groups’ true mean traits maxh1,h2∈G |µh1 − µh2 | goes to zero, or if own-group benchmark weights ygg all approach 1.

Unlike in the case of measuring absolute belief errors, relative beliefs are either entirely correct, in thatµ ˜g ≥ µ˜h if and only if µg ≥ µh, or they are entirely incorrect. The following simple example illustrates how equilibrium relative beliefs can be incorrect. Suppose there are four groups 1, 2, 3, 4, that µ1 > µ2 > µ3 > µ4, and that y11 = y21 = y34 = y44 = 1. The highest and lowest groups, 1 and 4, have own benchmark weights equal to 1 and therefore are correctly estimated in equilibrium,

µ˜1 = µ1 andµ ˜4 = µ4. The intermediate groups are ranked µ2 > µ3, but group 2 is evaluated according to a benchmark of µ1, higher than group 3’s benchmark µ4. Given the simple network structure of the example, equilibrium can be computed directly:

η η µ˜ = µ + (µ − µ );µ ˜ = µ + (µ − µ ). 2 2 1 − η 2 1 3 3 1 − η 3 4

As group 3 is compared favorably to its benchmark, µ3 > µ4, and group 2 is compared unfavorably,

µ2 < µ1, sufficiently large values of η yieldµ ˜2 < µ˜3 despite µ2 > µ3.

Proposition4 provides conditions that preclude such order reversals. The example above illustrates several critical factors for incorrect relative beliefs. First, it is critical that two groups may be compared to different benchmarks. With common benchmark weights, all groups share the same equilibrium benchmark, and order reversals are therefore impossible. Furthermore, it is necessary for η to be sufficiently large, because otherwise the correctness of absolute beliefs would 21 guarantee equilibrium relative beliefs to match the objective ranking. Finally, a switch in ranking requires at least two groups (like 2 and 3 in the example above) to have benchmarks that lead observers to evaluate them in the opposite order relative to their true traits. For high values of η, misattribution can lead to divergent belief paths in which the population belief about each group tends either to +∞ or −∞, with the direction solely dependent on the initial profile of beliefs. Intuitively, iterative application of misattribution only serves to amplify initial outcome gaps across groups, leading to greater and greater disparities in observers’ beliefs. Proposition4 establishes such divergent belief paths can always be found when benchmark weights are common and η is sufficiently high.

Proposition 4. (a) In equilibrium, relative beliefs are correct under the following conditions.

(i) The network has common benchmark weights. (ii) |η| is sufficiently small:

min |η|(1 − y∗∗ ) 1 min · max |µh1 − µh2 | < min |µh1 − µh2 |. 1 − 2|η|(1 − y∗∗ ) h1,h2∈G 2 h16=h2∈G

(iii) η > 0 and higher trait groups have lower benchmark weights:

X (yg1h − yg2h) ≥ 0 h∈G: µh≤K

for all K and all g1, g2 ∈ G such that µg1 ≥ µg2 .

(b) When the network has common benchmark weights and η > 1, then for any surjective assignment

(0) (n) s : G −→ {+∞, −∞} there exists an initial belief profile µ˜ such that µ˜g −→ s(g) for all g ∈ G. Proof in the appendix.

At the level of individual observer groups, equilibrium beliefs share some but not all features of their aggregate counterparts. First, all observer groups agree in relative terms about the trait ordering of different groups. Although different groups may have different empathy weights and reach different inferences, the fact that group i ranks group g higher than h reflects that xg ≥ xh, and this is the condition which determines whether any other observer group j would also rank g above h. Next, observer groups are consistent in having high beliefs or low beliefs about all other groups. Whenever an observer group i estimates a subject group g’s mean trait higher than another observer group j does, then group i is guaranteed to consistently estimate traits higher than group j. 22 P This follows from observer groups effectively using a single value h∈G zihµ˜h with which to evaluate P others; observers with low (high) values of h∈G zihµ˜h arrive at high (low) absolute beliefs when η > 0 (η < 0). Moreover, in the case of common benchmark weights, when observer groups place sufficiently high empathy on their own group, then higher (lower) ranked groups evaluate others under the implicit assumption that all groups face high (low) levels of teacher beliefs and therefore adopt lower (higher) absolute beliefs.17 This link between a group’s rank and the absolute level of its beliefs also holds in networks where all groups have complete own-group empathy: zii = 1 for all i ∈ G.

These results are collected in Proposition5 below along with a bound on the correctness of individual observer group beliefs. This bound largely reflects population level bounds, although it also shows how group i’s correctness about group g depends critically on the empathy weight zig. As the empathy weight approaches 1, group i becomes arbitrarily correct about group g even though the population aggregate belief about g may remain incorrect. Thus, in networks in which each group only empathizes with its own members, all groups are correct about their own group’s mean trait; in networks with a dominant group in which all groups place complete empathy, all observers are correct about the dominant group’s mean trait.

Proposition 5. Let µ˜ be a misattribution equilibrium. Then:

(a) Groups agree in relative terms:

i i j j µ˜g ≥ µ˜h if and only if µ˜g ≥ µ˜h

for all i, j, g, h ∈ G. (b) Groups disagree consistently in absolute terms:

i j i j µ˜g ≥ µ˜g if and only if µ˜h ≥ µ˜h

for all i, j, g, h ∈ G. (c) If benchmark weights are common, all groups have different true mean traits, and own-

group empathy is sufficiently high (mini∈G zii ≥ z¯ for some threshold z¯ < 1), then observers’

17This is predicated on η > 0. When η < 0, higher ranked groups also have higher absoute beliefs. 23 differences in absolute beliefs depend on population beliefs about the observers’ own groups:

i j µ˜g ≥ µ˜g if and only if η(˜µi − µ˜j) ≤ 0.

min (d) Whenever |η| < (1/2)/(1 − y∗∗ ), group beliefs are bounded by

i |η| |µ˜g − µg| ≤ (1 − zig) · min · max |µh1 − µh2 | 1 − 2|η|(1 − y∗∗ ) h1,h2∈G

for all i, g ∈ G. With common benchmark weights and |η| < 1,

i η |µ˜g − µg| ≤ (1 − zig) · · max |µh1 − µh2 | 1 − η h1,h2∈G

for all i, g ∈ G.

Proof in the appendix.

5. Policy Implications

This section considers the potential of three different kinds of policy measures to de-bias in- correct attributive beliefs. While the policies might not be realistically implemented in full, a comparison between them is nonetheless informative about their relative prospects for success. First, a governing body may seek to address disparities in group treatment by implementing a transfer policy. There are two ways to implement such a policy. Teachers may be compelled to treat students according to preferential or disadvantaging standards which differ from otherwise

i optimal (according to teacher belief) actions. That is, teachers of group i with beliefµ ˜g about i group g are required to take the action corresponding to the beliefµ ˜g + Dg. Alternatively, student −1 18 scores are directly adjusted such that group g is transfered (dx/dµ˜) Dg of the game product in P each match. Any profile {Dg}g∈G such that g∈G wgDg = 0 is called a transfer scheme and defines a particular transfer policy.

A governing body may alternatively adopt a diversity policy which either shifts the com- position of observers’ empathy weights or alters the population’s influence weights. Specifically, empathy weights are shifted towards target distribution {z∗g}g∈G, such that observer group i places weight (1−α)zig +αz∗g on group g, or influence weights are shifted towards a target profile {w∗i}i∈G,

18The factor (dx/dµ˜)−1 serves to express the transfer in terms of teacher beliefs and facilitate comparison with the preferential/disadvantaging treatment implementation. 24 such that students of group g are matched with teachers of group i at rate (1−α)wgi+αw∗i. A policy may also target both distributions simultaneously. Finally, a policy of mandated equal treatment requires that each teacher chooses a single action and applies it to all students with which she interacts. Unlike with transfer or diversity polices, equal treatment is a single specification without any parameters of variation.

The model produces starkly divergent results concerning the de-biasing potential of the dif- ferent kind of policies. Any population with incorrect beliefs can be successfully de-biased by an appropriately chosen transfer scheme, while no population can be de-biased through any diversity policy, and an equal treatment policy only de-biases for a single special case of network structure.

How Transfers Can De-Bias Beliefs. Suppose that a transfer policy is implemented by i mandating teachers to act according toµ ˜g +Dg. The belief-revision equation is accordingly modified to ! i X µˆg = µg + η µ˜g + Dg − zih(˜µh + Dh) , h∈G including D terms which appear both in the treatment of the subject group g and the comparison groups indexed by h. A transfer scheme that de-biases beliefs must yield

X µg + Dg − zih(µh + Dh) = 0 h∈G simultaneously for all groups g ∈ G. An intuitive way of accomplishing this – in fact, the only way, as Proposition6 establishes – chooses Dg such that µg + Dg is a constant. Thus, a de-biasing transfer scheme advantages or disadvantages groups according to how the group’s true mean trait differs from a fixed level. That level is uniquely determined by the constraint that transfers net to zero across the population.

Proposition 6. For any population with incorrect attributive beliefs, there exists a unique transfer policy that de-biases beliefs. The transfers are equal to

∗ Dg ≡ − (µg − µ¯) for all g ∈ G. Proof in the appendix.

How Greater Diversity Does Not De-Bias Beliefs. Under a diversity policy, the governing body chooses target profiles of {w∗i}i∈G or {z∗h}h∈G, yielding a target profile of benchmark weights 25 {y∗i}i∈G. The belief-revision equation is modified to !! X X µ˜g = µg + η µ˜g − (1 − α) yghµ˜h + α y∗hµ˜h . h∈G h∈G

Intuitively, the policy shifts equilibrium beliefs towards comparing groups against the modified P benchmark value h∈G y∗hµh, which does not de-bias beliefs but instead alters the nature of mis- estimation. As the policy intensity level α approaches 1, the population is shifted to full weight on the common benchmark, guaranteeing that at least the objectively highest or lowest mean trait group is misestimated. When the population itself has common benchmark weights, the modified equilibrium beliefs are ! η X µ˜ = µ + µ − [(1 − α)y + αy ]µ , g g 1 − η g h ∗ h h∈G demonstrating that not only is a diversity policy unable to de-bias beliefs, but that at least one group must be more severely misestimated under the policy than without it. As demonstrated in the proof of Proposition7, even in general networks it is impossible to simultaneously de-bias all beliefs through any choice of diversity policy.

Proposition 7. For any population with incorrect attributive beliefs, there is no diversity policy that de-biases beliefs. Proof in the appendix.

How Equal Treatment Typically Does Not De-Bias Beliefs. An equal treatment policy impacts equilibrium beliefs in a different manner than do transfer and diversity policies. A teacher of group i who is mandated to set a fixed action to apply to all students would optimally adopt i i i P i a∗ = aS(˜µ∗), whereµ ˜∗ = g∈G wgi · µ˜g, where wgi is the probability with which the teacher is matched to a group g student. The resulting aggregate effective belief that determines group g’s treatment is no longer the wgi−weighted average of the teacher groups’ beliefs. Instead group i g faces the wgi−weighted average of observer groups’ values ofµ ˜∗. The modified belief-revision equation can therefore be expressed in two steps as  ! X µˆi = µ + η µ˜ − z µ˜  g g g ih h  h∈G

 X X i µˆg = wgi wihµˆ .  h i∈G h∈G 26 As a special case, when all groups empathize with others to the extent that they influence them, that is wih = zih for all i, h ∈ G, equilibrium beliefs can be expressed simply as ! X X µ˜g = wgiwih µh. h∈G i∈G

Each group g is thus associated with a ‘local average’ of true mean traits determined by which students are also affected by the teachers who affect group g. If all of the local averages are the same as, for example, is the case when influence weights are common (wgi = w∗e for all g, i ∈ G), then observers are not skewed by mistakenly comparing groups against incorrect benchmarks, and equilibrium beliefs are correct. As Proposition8 below shows, the equal local averages condition is both necessary and sufficient for de-biasing regardless of the network structure. Note that even when groups are correctly estimated by teachers, however, they still face the local average level of treatment and not the level of treatment corresponding to their true mean values.

Proposition 8. An equal-treatment policy de-biases beliefs if and only if

X X X X wg1i wihµh = wg2i wihµh i∈G h∈G i∈G h∈G for all g1, g2 ∈ G. Proof in the appendix.

Literal De-Biasing. As a final point of comparison, it is instructive to consider the consequences of a policy which managed to directly mitigate observers’ FAE bias. Were it the case that, when group i evaluated group g, it placed α weight on the actual experiences of group g members, then the revision equation would be " #! i X µˆg = µg + η µ˜g − (1 − α) zihµ˜h + αµ˜g . h∈G

Thus, equilibrium beliefs would be the same as in the original specification but with η replaced by η0 = (1 − α)η. For α approaching 1, the modified equilibrium would continuously approach correct beliefs. 27 6. Discussion

This paper has introduced a model of discrimination in which members of a population are associated with outcomes through repeatedly interacting with others. Outcomes reflect both indi- viduals’ own traits and the beliefs that others have about those traits. However, observers biased by the Fundamental Attribution Error neglect the role of others’ beliefs and instead infer traits under the assumption that there is a single trait-to-outcome function which is valid for all members of the population. Observers’ inferred beliefs about different groups in turn influence group outcomes. A profile of beliefs that produces outcomes which are inferred as the original beliefs constitutes an misattribution equilibrium.

For moderate parameters of the interaction game, a misattribution equilibrium exists and is unique. Almost all groups are misunderstood by the population, with at least one group over- estimated and at least one underestimated. Members of different observer groups always agree about the relative ranking of groups by mean trait. However, they consistently disagree in absolute terms, with some observer groups having high beliefs and others low beliefs. Data from the General Social Survey, discussed below, demonstrate that these general patterns are indeed found in as- sessments about the innate qualities of different racial/ethnic groups. In terms of welfare, students benefit from being either over- or under-estimated by teachers, while teachers always suffer from any incorrectness. Policies intended to de-bias the population have mixed effects. When teachers are required to give students group-dependent preferential treatment, or transfers implement such preference, beliefs can be completely corrected. However, policies that achieve greater diversity in terms of influence or empathy cannot de-bias beliefs about all groups simultaneously.

The remainder of this section compares the model’s predictions regarding beliefs held by mem- bers of different social groups with data from the General Social Survey, and compares misattribu- tion to other equilibrium concepts that incorporate incorrect learning in games.

6.1. Evidence from the General Social Survey

Administered by the National Data Program for the Social Sciences at the University of Chicago every one to two years since 1972, the General Social Survey reports the responses of one to two hour in-person interviews. Each wave of the survey also includes cohort-specific modules, and in 28 2000 the GSS queried N = 1, 322 respondents19 on their views regarding White, Black, Asian and Hispanic Americans. Respondents who self identified as one of the four groups provided numeric assessments regarding the intelligence, work ethic, and proclivity to violence of ‘typical members’ of each of the four groups. Each trait was measured on a scale from 1 to 7. The data thus provide an opportunity for testing the predictions of Proposition5 regarding the relationships between the average assessments of different groups. The figures below (1,2, and3) show the average ratings given by each respondent group about all four groups for each trait.

Comparing the ordering of these mean beliefs reveals support for the predictions of Proposition 5 and illustrates the value of further testing the model empirically in future work. Concerning relative beliefs, in 91.7% of specifications of an observer-group pair with a target-group pair, the two observer groups rank the two target groups in the same order with respect to intelligence (80.6% with respect to work ethic, 83.3% with respect to proclivity to violence). In 69.4% of such specifications, the observer group which gives a higher average assessment of one of the target groups with respect to intelligence also gave a higher average assessment of the other (50.0% with respect to work ethic, 61.1% with respect to proclivity to violence). Finally, in 79.2% of specifications of an observer-group pair with a given target group, the higher ranked (according to the most common ranking across different observer groups) observer group had a lower assessment of the target group relative to the lower ranked observer group with respect to intelligence (66.7% with respect to work ethic). With respect to proclivity to violence, 75.0% of specifications feature the higher ranked observer group giving the higher average assessment.20

19Per the GSS, respondents are a ‘sample of English-speaking persons 18 years of age or over, living in non-institutional arrangements within the United States.’ See http://gss.norc.org/ for details. 20The data are thus most consistent with η > 0 with respect to intelligence and work ethic and η < 0 for proclivity to violence, echoing the predictions in Section3.

29 Figure 1. GSS Assessments of Intelligence

Note: Solid dots mark the mean rating given by respondents of the group listed at far left concerning the intelligence (1-7; 7 most intelligent) of the group listed second to left. Light dots show means plus and minus one standard deviation.

Figure 2. GSS Assessments of Work Ethic

Note: Solid dots mark the mean rating given by respondents of the group listed at far left concerning the work ethic (1-7; 7 hardest working) of the group listed second to left. Light dots show means plus and minus one standard deviation.

30 Figure 3. GSS Assessments of Violence

Note: Solid dots mark the mean rating given by respondents of the group listed at far left concerning the proclivity to violence (1-7; 7 most violent) of the group listed second to left. Light dots show means plus and minus one standard deviation.

Additional Evidence from the Literature. In addition to the data from the General Social Survey, the model’s predictions also echo Gershenson, Holt and Papageorge(2016), who find that non-black teachers have lower expectations vis-a-vis black teachers about black students in their classes, and Antonovics and Knight(2009), who analyze the probabilities of police officers searching a suspect and show that while all officers search non-white suspects more often than white suspects, this effect is more pronounced for white officers. It is also worth noting how the model predicts that the level of absolute disagreement between observer groups should hinge on the difference in observer groups’ empathy weights, which comports with the general pattern that agreement between men and women with respect to beliefs about gender is higher than agreement between racial groups with respect to beliefs about race.

On the other hand, empirical studies often document men and women holding relatively similar beliefs about gender. For example, Eckel and Grossman(2008) find male and female lab subjects uniformly rating the risk tolerance of men above women, and Bordalo et al.(2019) find that both men and women in a laboratory experiment predict men to outperform women in male-typed trivia categories but underperform relative to women in female-typed categories. Reuben, Sapienza 31 and Zingales(2014), documenting that laboratory subjects acting as managers were less likely to choose a female over male employee, also found that female employers did not choose female employees at a significantly higher rate. Nonetheless, there is no clear consensus. Carrell, Page and West(2010), for one, finds that assignment to female professors helps improve outcomes in female students’ STEM subject performance levels. In a similar vein, Alan, Ertac and Mumcu(2018) study the effect of teacher attitudes on student performance and find that, while both male and female teachers include those who subscribe to more traditional gender stereotypes, the proportion is greater among male teachers.

6.2. Relationships With Other Solution Concepts

Relationships with Other Equilibrium Concepts. The equilibrium concept defined in this paper is closely related to Berk-Nash Equilibrium (Esponda and Pouzo, 2016). In their framework, players in a game are endowed with a set of possible subjective models; as the game generates feedback signals, players select and best-respond to the model whose implied distribution over feedback minimizes the Kullback-Leibler divergence with the observed distribution. In the present paper, players likewise form beliefs by fitting a subjective model with generated data and then best- respond accordingly. However, as they care only about mean traits and, as they are able to perfectly explain any observed data (mean group outcomes) with a unique model specification (mean group traits), players need not engage in the KL divergence minimizing exercise that Berk-Nash requires. In a more general setup, observers may find that outcome distributions do not perfectly correspond to any single distribution over traits. In this extension a natural generalization of the equilibrium concept specifies that agents’ beliefs, in conjunction with their subjective trait-to-outcome models, minimize KL divergence with the observed distribution.

A looser connection exists between this paper’s model of inference and the literature on Self Con- firming Equilibrium (Battigalli, 1987; Fudenberg and Levine, 1993; Dekel, Fudenberg and Levine, 2004). In such an equilibrium, players correctly understand the rules of the game but can maintain incorrect beliefs about other players’ actions and types because game feedback is too coarse to correct them. By contrast, observers with the FAE fundamentally misunderstand the production of outcomes. On account of the FAE, they assume that outcomes are produced directly from one’s trait, which ignores the role of teachers’ actions. Accordingly, observers have no beliefs about what actions teachers take against students. Given the model’s particular specification, however, any 32 affine trait-to-outcome function is mathematically equivalent to assuming that all teachers apply some common action to all students. In a more general model this connection need not hold. The presence of a dogmatic reliance on an incorrect model of game structure also has parallels in the simplified models of opponent play in Eyster and Rabin(2005) and Jehiel(2005), as well as naive social learning (Eyster and Rabin, 2010), in which players develop beliefs and conjectures that cannot withstand strategic introspection.

6.3. Concluding Thoughts

From a broad perspective, this paper illustrates how discrimination driven by agents with a cognitive bias differs substantially from classical statistical discrimination in which beliefs reflect actual circumstances and serve as coordination devices for multiple equilibria. Beliefs about groups are not only wrong, but systematically misestimated. Individual teachers, managers, or police officers maintain biased beliefs and therefore stand to benefit from de-biasing policies. These departures from classical theory evidence both the particular bias of the FAE and the population equilibrium framework within which it is studied. The population framework could accordingly serve as a template for future work examining how other biases, for example, the stereotyping of Bordalo et al.(2016) or the categorical thinking of Fryer and Jackson(2008), might analogously sustain systematically incorrect discriminatory beliefs.

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38 Appendix A. Proofs

A.1. Formulas for Equilibrium Beliefs and Beliefs Sequences in General Networks

The equilibrium belief formula for uniform benchmark weight networks has a counterpart for general networks, but the structure is more complicated. Group g is compared not only against its P own objective benchmark h∈G yghµh, but also against the ygh-weighted average of other groups’ objective benchmarks and all successive ‘higher order’ averages. Formally, the k’th order benchmark (k) weights ygh are defined by

(k) X ygh = ygj1 · yj1j2 . . . yjk−2jk−1 · yjk−1h. j1,...,jk−1∈G

As Lemma2 below relates, an alternating sum of the true group means as weighted by these coefficients defines each group’s ultimate level of comparison. What makes common benchmark weights simpler is that, for such networks, all higher order benchmark weights reduce to the original (k) values: ygh = y∗h for all g, h ∈ G and k ≥ 0.

Lemma 2. In any misattribution equilibrium, ! η X µ˜ = µ + µ − y¯ µ , g g 1 − η g gh h h∈G where ∞ k−1 X (−η) (k) y¯ = y . gh (1 − η)k gh k=1 P h∈G y¯gh = 1 for all g ∈ G. For general networks, y¯gh > 0 for all g, h ∈ G is guaranteed when

min 1 y∗∗ |η| < · min . 1 − ming,h∈G ygh 1 − 2y∗∗

When benchmark weights are common, y¯gh = yh for all g, h ∈ G.

Proof of Lemma2. As a preliminary matter, note that the Taylor expansion of g(η) = (1 − η)−(k+1) around η = 0 is

∞ ∞ X ηl X ηl (k + l)! g(η) = g(l)(0) = (1 − 0)−(k+l+1) l! l! k! l=0 l=0 ∞ ∞ X ηn−k n! X n = · = ηn−k , (n − k)! k! k n=k n=k 39 where the last line substitutes n ≡ l + k. From this we obtain

∞ X n ηk ηn = . k (1 − η)k+1 n=k

Now, in matrix form, equilibrium beliefs are given by

∞ X µ˜ = ηn(I − Y)nµ. n=0

Leveraging the Binomial identity and the equivalence shown above, this can be re-arranged as follows:

∞ ∞ ! X X n   µ˜ = µ + (−1)k+1 ηn I − Yk µ k k=1 n=k ∞ X ηk   = µ + (−1)k+1 I − Yk µ (1 − η)k+1 k=1 ∞ ! η X (−η)k−1 = µ + I − Yk µ 1 − η (1 − η)k k=1 η = µ + I − Y
µ. 1 − η

Further rearranging Y in terms of Y yields

Y = Y (I − η(I − Y))−1 , so for any µ, Yµ = Yµ˜(µ), whereµ ˜(µ) is the misattribution equilibrium corresponding to µ. Decomposingµ ˜(µ) = µ + ε(µ) yields (Y − Y)µ = Yε(µ). Now leveraging the bound on population beliefs in Proposition3, by P setting µ = e = (1,..., 1) we guarantee ε(µ) = 0, which then implies Ye = Ye, so h∈G y¯gh = P 0 h∈G ygh = 1 for all g ∈ G. Finally, let µ = eg = (0,..., 1,..., 0) . From the bound in Proposition 3, min |η|(1 − y∗∗ ) y¯gh > min ygh − min . g,h∈G 1 − 2|η|(1 − y∗∗ )

Rearranging this expression showsy ¯gh > 0 whenever |η| satisfies the bound in the Lemma state- ment. ◦

40 Next, the lemma below explicitly characterizes of belief sequences as a function of initial beliefs.

(n) Lemma 3. If {µ˜ }n∈N is an attributive belief sequence, then

h   i µ˜(n) = µ + η (η(I − Y) − I)−1 [η(I − Y)]n−1 δ + (η(I − Y) − I)δ˜(0) − δ , and if the network has common benchmark weights,

η h   i µ˜(n) = µ + ηn−1 δ + (η − 1)δ˜(0) − δ , η − 1 where δ = (I − Y)µ and δ˜(0) = (I − Y)˜µ(0).

Long-run behavior of a belief sequence is influenced by the key term δ + (η(I − Y) − I)δ˜(0) or, in (0) the common benchmark weight case, δg + (η − 1)δ˜g , a combination of groups’ true differences with respect to their objective benchmarks and the differences in terms of initial population beliefs. Proof of Lemma3. Iteratively applying the revision equation to an initial belief profileµ ˜(0),

µ˜(n) = µ + [η(I − Y)]˜µ(n−1)

= µ + [η(I − Y)](µ + [η(I − Y)]˜µ(n−2))

n−2 ! X = µ + [η(I − Y)] [η(I − Y)]kµ + [η(I − Y)]n−1µ˜(0) k=0   = µ + [η(I − Y)]([η(I − Y)] − I)−1 ([η(I − Y)]n−1 − I)µ + [η(I − Y)]n−1([η(I − Y)] − I)˜µ(0)

= µ + [η(I − Y)]([η(I − Y)] − I)−1[η(I − Y)]−1     [η(I − Y)]n−1 [η(I − Y)]µ + ([η(I − Y)] − I)[η(I − Y)]˜µ(0) − [η(I − Y)]µ h   i = µ + η (η(I − Y) − I)−1 [η(I − Y)]n−1 δ + (η(I − Y) − I)δ˜(0) − δ .

In the case of common benchmark weights, the fact that Y2 = Y implies (I − Y)k = I − Y for all k ≥ 1. Returning to the third line of the derivation above,

  µ˜(n) = µ + η (1 − ηn−1)(1 − η)−1(I − Y)µ + ηn−1(I − Y)˜µ(0) η h i = µ + (1 − ηn−1)δ + ηn−1(1 − η)δ˜(0) 1 − η η h   i = µ + ηn−1 δ + (η − 1)δ˜(0) − δ . η − 1

◦

41 A.2. Proofs for Results in Main Text Proof of Proposition1. Belief convergence follows from the Contraction Mapping Theorem.

Consider two profiles of population beliefs (˜µg)g∈G and (ˆµg)g∈G. The distance between them, according to the sup norm, is maxg∈G |µ˜g − µˆg|. After one round of updating, the belief profiles are

0 µ˜g = µg + η(˜µg − µ¯) and

µˆg = µg + η(˜µg − µ¯).

The distance between these is η · maxg∈G |µ˜g − µˆg|. As |η| < 1, the belief updating mapping is a contraction, and iteration thus limits to a unique fixed point. Beliefs of individual groups similarly converge to their own fixed point. ◦

Proof of Proposition2. Iteratively substituting and then simplifying yields:

[1] [0] X [0] µ˜g = µg + η(˜µg − µ˜h ) h∈G

[2] [0] X [0] µg˜ = µg + η(µg − µ¯ + η(˜µg − µ˜h )) h∈G . .

[t] t−1 t [0] X [0] µg˜ = µg + (η + ··· + η )(µg − µ¯) + η (˜µg − µ˜h ) h∈G  −t  −t ! η − η 1 − η η − 1 X [0] = µ + ηt (µ − µ¯) + (˜µ[0] − µ˜ ) . g η − 1 η − η−t g η − η−t g h h∈G

For η > 1, η − η−t  ηt −→ +∞, η − 1

[t] [0] soµg ˜ approaches +∞ when the right-bracketed expression is positive, i.e. when µg +(η−1)(˜µg − [0] [0] [0] µ˜i ) > µi, and approaches −∞ when the expression is negative, i.e. when µg +(η −1)(˜µg −µ˜i ) <

µi. ◦

Proof of Lemma1. The equilibrium characterizing Equation (5) in matrix form isµ ˜ = µ + η(I − Y)˜µ, so whenever I − η(I − Y) is invertible, it must follow that

µ˜ = (I − η(I − Y))−1µ.

42 Invertibility of I −η(I −Y) fails only when the matrix admits 0 as an eigenvalue, which is equivalent to Y having an eigenvalue of 1 − 1/η. Since Y can have at most |G| distinct eigenvalues, there are at most |G| values of η such that the model lacks a unique equilibrium. Furthermore, since Y is row-stochastic, all of its eigenvalues have modulus less than or equal to 1. As 1 − 1/η > 1 for η < 0 and 1 − 1/η < −1 for η < 1/2, any populations without equilibrium existence must have η ≥ 1/2.

To show the existence of a cutoff value of η that determines stability, fix a population network, and suppose the fixed point is stable for some value of η. By Lemma3, it follows that

n−1 (n) n n (0) X k k −1 µ˜η = η (I − Y) µ˜ + η (I − Y) µ −→ (I − η(I − Y)) µ k=0 for allµ ˜(0). In particular this holds forµ ˜(0) = 0, so

n−1 X lim µ˜(n) = lim ηk(I − Y)kµ =⇒ lim ηn(I − Y)nµ˜(0) = 0. n−→∞ η n−→∞ n−→∞ k=0

Now suppose |ηˆ| < |η|. As |ηˆn(I − Y)n| < |ηn(I − Y)n| , it follows thatη ˆn(I − Y)n −→ 0. For all g ∈ G, and as a function of η,

n−1 X h i ηˆk (I − Y)kµ g k=0 is a power series centered around 0; on account of the series converging forη ˆ = η, it followsη ˆ is in (n) the radius of convergence. This showsµ ˜ηˆ is convergent. Now define

n (n) (0)o η ≡ sup η :µ ˜η convergent ∀ µ˜ .

(n) (0) If |η| > η, then by the definition of η the sequenceµ ˜η is not convergent for someµ ˜ and therefore the equilibrium is not stable. If |η| < η, then by the definition of η there existsη ˜ such

(n) (0) that |η| < |η˜| ≤ η andµ ˜η˜ is convergent for allµ ˜ . By the first part of the proof, it follows from |η| < |η˜| that the equilibrium is stable for η. To establish a general bound on η, note that by the Contraction Mapping Theorem, the fixed point of the belief updating equation is stable whenever the revision equation, here denoted f, is a contraction. Using the metric given by the supremum norm, it suffices to show there exists K < 1

43 such that (1) (2) (1) (2) (1) (2) kf(˜µ ) − f(˜µ )k∞ = kη(I − Y)(˜µ − µ˜ )k∞ < Kkµ˜ − µ˜ k∞ for allµ ˜(1), µ˜(2). To that end, note that for any v ∈ RN ,

X k(I − Y)vk∞ = max (1 − ygg)vg − yghvh g∈G h6=g

min 1 X ≤ (1 − y∗∗ ) · max vg − P yghvh g∈G ygh h6=g h6=g

min ≤ (1 − y∗∗ ) · 2kvk∞.

min Thus, f is a contraction and the equilibrium is stable whenever |η| · 2(1 − y∗∗ ) < 1. In the case of (n) common benchmark weights, let {µ˜ }n∈N by any attributive belief sequence. As, for all n ≥ 1,

X (n) X y∗gµ˜g = y∗gµg, g∈G g∈G so, for n ≥ 2, " !# " !# X (n−1) η X µ˜(n) − µ˜ = µ + η µ˜(n−1) − y µ˜ − µ + µ − y µ g g g g ∗h h g 1 − η g ∗h h h∈G h∈G  !! 1 X (n−1) 1 = η µ(n−1) − µ − y µ˜ − µ g 1 − η g ∗h h 1 − η h h∈G ! (n−1) X (n−1) = η µ˜g − µ˜g − y∗h(˜µh − µh) h∈G

(n−1) = η(˜µg − µ˜g).

From this expression it is clear thatµ ˜(n) −→ µ˜ for allµ ˜(0) if and only if |η| < 1.

◦

44 Proof of Proposition3. (a-i,ii) By Lemma2, the equilibrium population misestimation of g as a function of η is

2 ∞  k ! η X η X (k) − µ − y µ . 1 − η 1 − η g gh h k=1 h∈G

If group g is medial, then all of the difference terms are zero and the function is identically zero, proving claim (ii). Otherwise, let ( ) X (k) k = min k : µg 6= y µh , k≥1 gh h∈G and note that the population misestimation is zero if and only if ρ(ϕ) = 0 where ϕ = η/(1 − η) and

∞ ! X k X (k) ρ(ϕ) = ϕ µg − ygh µh . k=1 h∈G

As an analytic function, ρ is either identically zero or admits only isolated zeros. Let k be the P (k) smallest value of k such µg 6= h∈G ygh µh; then the k’th derivative at ϕ = 0 is ! (k) X (k) ρ (0) = k! µg − ygh µh 6= 0, h∈G so ρ is not identically zero. Hence the set of ϕ, and equivalently η, such that population misesti- mation is zero has Lebesgue measure zero, proving claim (i).

(a-iii) Suppose η > 0 and denoteµ ˜max = maxg∈G µ˜g andµ ˜min = ming∈G µ˜g. It must be that µ˜ > µ˜ ; otherwise max min  µ˜g ≤ µg + η(˜µmax − µ˜min) = µg

µ˜g ≥ µg + η(˜µmin − µ˜max) = µg, which impliesµ ˜g = µg for all g ∈ G, which would then imply

µ˜max − µ˜min = max µg − min µg > 0, g∈G g∈G a contradiction. Thus  µ˜max > µmax + η (˜µmax − µ˜max) = µmax

 µ˜min < µmin + η (˜µmin − µ˜min) = µmin.

45 An analogous proof holds when η < 0. (b) Note that ∞ X µ˜ − µ = [η(I − Y)]k(η(I − Y)µ). k=0 Combining the bound in Lemma1 with the relation

min k(I − Y)µk∞ ≤ (1 − y∗∗ ) · max |µh1 − µh2 |, h1,h2∈G it follows ∞ X k min kµ˜ − µk∞ ≤ [η(I − Y)] (1 − y∗∗ ) · |η| max |µh1 − µh2 | h1,h2∈G k=0 ∞ ∞ X min k ≤ (2|η|(1 − y∗∗ )) · |η| max |µh1 − µh2 | h1,h2∈G k=0 min |η|(1 − y∗∗ ) = min · max |µh1 − µh2 |. 1 − 2|η|(1 − y∗∗ ) h1,h2∈G

The bound on absolute errors in the case of common benchmark weights follows by inspection from the formula for equilibrium beliefs. ◦

Proof of Proposition4. (a-i) When the network has common benchmark weights,

(˜µg − µ˜h) = (1 + η)(µg − µh), and as stability requires η > −1, it follows that relative beliefs are correct. (a-ii) By definition, relative beliefs must be correct when

1 max |µ˜g − µg| < min |µh1 − µh2 |, g∈G 2 h16=h2∈G i.e. absolute beliefs are so close to their objective counterparts that their potential regions are non-overlapping. Applying the bound in Proposition3 yields condition (a-ii).

(0) (0) (0) (a-iii) Suppose the condition holds, and letµ ˜ be any initial belief profile such thatµ ˜g1 ≥ µ˜g2 for all g1, g2 ∈ G such that µg1 ≥ µg2 . Then by the revision equation, ! (1) (1) (0) (0) X (0) µ˜g1 − µ˜g2 = (µg1 − µg2 ) + η(˜µg1 − µ˜g2 ) − η (yg1h − yg2h)˜µh . h∈G

46 (0) The first two terms are clearly positive. Concerning the third, note that since the ordering ofµ ˜h P (0) (1) (1) follows the order of µh, condition (a-iii) guarantees that h∈G(yg1h − yg2h)˜µh ≤ 0, soµ ˜g1 ≥ µ˜g2 . (n) (n) (n) Iteratively applying the revision equation,µ ˜g −→ µ˜g for all g ∈ G andµ ˜g1 − µ˜g2 for all n ≥ 1.

Thusµ ˜g1 ≥ µ˜g2 . (n) (0) (b) By Lemma3, ˜µg diverges in the direction of δg + (η − 1)δ˜g , so it suffices to find values of µ˜(0) such that 1 δ˜(0) > − δ˜(0) g η − 1 g P if and only if s(g) = +∞. Set mg = 1(s(g)=+∞) − 1(s(g)=−∞), so mg − h∈G y∗hmh > 0 if and only if s(g) = +∞. Thenµ ˜(0) = K · m satisfies the above inequality for all g when K > 0 is sufficiently large. ◦

Proof of Proposition5. (a) For any i, j, g, h ∈ G,

i i j j 2 (˜µg − µ˜h)(˜µg − µ˜h) = [(µg − µh) + η(˜µg − µ˜h)] ≥ 0.

(b) Similarly, !2 i j i j 2 X (˜µg − µ˜g)(˜µh − µ˜h) = η (zih − zjh)˜µh ≥ 0. h∈G If benchmark weights are common, ! i X µ˜g = µg + η µ˜g − zihµ˜h h∈G       η X X η X = µ + η µ + µ − y µ − z µ + µ − y µ g  g 1 − η  g ∗j j ih  h 1 − η  h ∗j j j∈G h∈G j∈G ! η(1 + η) X = µ + µ − z µ . g 1 − η g ih h h∈G

For anyz ¯ ≤ min{zii, zjj}, letz ˆii = (zii − z¯)/(1 − z¯),z ˆjj = (zjj − z¯)/(1 − z¯), andz ˆih = zih/(1 − z¯) P P for all h 6= i andz ˆjh = zjh/(1 − z¯) for all h 6= j. Note that h∈G zˆih = h∈G zˆjh = 1. Then ! η(1 + η) X X µ˜i − µ˜j = − z µ − z µ g g 1 − η ih h jh h h∈G h∈G !! η(1 + η) X X = − z¯(µ − µ ) + (1 − z¯) zˆ µ − zˆ µ , 1 − η i j ih h jh h h∈G h∈G

47 so if max |µ − µ | z¯ > h1,h2∈G h1 h2 minh16=h2∈G |µh1 − µh2 | + maxh1,h2∈G |µh1 − µh2 | it follows i j µ˜g − µ˜g ≥ 0 ⇐⇒ η(µi − µg) ≤ 0 ⇐⇒ η(˜µi − µ˜j) ≤ 0.

(c) By the revision equation, ! i X µ˜ − µg = η µ˜g − zihµ˜h h∈G   1 X = η(1 − zig) µ˜g − P zihµ˜h zih h6=g h6=g ! 1 X 1 X = η(1 − zig) µ˜g − P zihµ˜h + (˜µg − µg) − P zih(˜µh − µh) . zih zih h6=g h6=g h6=g h6=g

Applying the bound from Proposition3, it follows

 min  i |η|(1 − y∗∗ ) |µ˜ − µg| ≤ |η|(1 − zig) · 1 + 2 min · max |µh1 − µh2 |, 1 − 2|η|(1 − y∗∗ ) h1,h2∈G which simplifies to the expression in the proposition statement. ◦

Proof of Proposition6. In the case of transfer policies, any transfer values {Dg}g∈G which de-bias beliefs must satisfy ! X µg = µg + η µg + Dg − ygh(µh + Dh) , h∈G P implying µg + Dg = h∈G ygh(µh + Dh) =µ ¯, for all g ∈ G. As a transfer policy it must be that P g∈G wgDg = 0. Thus ! ∗ X Dg = − µg − whµh . h∈G ◦

Proof of Proposition7. To show that no diversity policy can de-bias beliefs, suppose for the sake of contradiction that one does, and denote y∗h to be the benchmark weights implied by either

48 the target weights wit∗ or zi∗. It must be that

X X µg = α y∗hµh +(1 − α) yghµh, h∈G h∈G | {z } | {z } µ¯ cg where cg denotes the benchmark comparison (in true value terms) for group g. Let µmin, µmax refer to two groups with the lowest and highest true mean traits, respectively. We know µmax > µmin, as otherwise, by Proposition3, equilibrium beliefs would be correct. Since the above equation is satisfied simultaneously for all groups, the value of α must be

(c − c ) − (µ − µ ) α = max min max min . cmax − cmin

For this value to lie in the range [0, 1], it first must be that cmax − cmin > 0, as otherwise α ≤ 1 would demand µmax − µmin ≤ 0, a contradiction. Second, it therefore follows that

µmax − µmin ≤ cmax − cmin ≤ µmax − µmin, implying α = 0. This proves the proposition statement. ◦

Proof of Proposition8. In the case of the equal treatment policy, it is straightforward to verify that correct beliefs satisfy the equilibrium conditions under the modified belief-revision equation when true mean traits satisfy the condition in the proposition statement. To show sufficiency, note that the policy de-biases beliefs only if

X µ˜g − zihµ˜h = 0 h∈G

for all g ∈ G, which holds only ifµ ˜g1 =µ ˜g2 for all g1, g2 ∈ G, which holds only if

X X i X X i wg1i wihµ˜h = wg2i wihµ˜h i∈G h∈G i∈G h∈G

i for all g1, g2 ∈ G. Now, if the policy de-biases beliefs,µ ˜h = µh for all i, h ∈ G and the above equality simplifies to the condition in the proposition statement. ◦

49