1 Introduction
Discrimination of individuals based on their race, gender, ethnicity and other social identities, is a pervasive problem. While the problem is as old as humanity, it has taken on a new meaning and form at digital marketplaces and social media where our social and economic interactions increasingly take place. Evidence suggests that discrimination is prevalent in popular online platforms such as airbnb (Edelman et al. (2017) and Cui et al. (2019)), freelancing worksites (Hannák et al. (2017)), ridesharing platforms (Ge et al. (2016)), and math Stack exchange (Bohren et al. (2019)).
At first glance, online platforms are an unlikely place for discrimination to occur, given the widelyused ratings system that facilitates social learning among users. Platforms for ridesharing, housesharing, freelancing, credit, and insurance collect information about drivers, customers, guests, workers, and loan and insurance applicants based on performers’ past records and user experiences. Platforms then aggregate the information into simple ratings, and make recommendations based on these ratings. It is now routine that machine algorithms prescreen résumés of job applicants, evaluate loan or insurance applicants and freelance workers, recommend their promotion or firing, and rates recidivism of parolees.
By limiting subjective human judgment and replacing it with accurate information and objective recommendation, one would think that datadriven social learning should limit the scope for statistical discrimination. Intuitively, there should be simply less room for statistical inference on an individual based on his/her group identity—and therefore discrimination based on it—if one is guided by more accurate information about his/her individual characteristics. The logic of this reasoning is at first glance plausible, and indeed, if full information were available, discrimination, except based on tastes, should disappear.^{1}^{1}1 There is some empirical evidence that reputation/ratings ameliorate discrimination in some contexts (see Cui et al. (2019) and Bohren et al. (2019)). However, it is not at all clear that more information and social learning should necessarily lead to less discrimination.
Most importantly, it is not clear that social learning mechanisms at the heart of these platforms work fairly and unbiasedly to mitigate discrimination. At a high level, social learning involves a feedback of two processes: (1) the sampling of data (or experience) and (2) the informing (or recommending) of user decisions. The latter process is fair or unbiased, or one can at least guarantee it to be so—in keeping with the recent call for algorithmic fairness. However, the former process is neither random nor unbiased. Data is sampled when transaction occurs, and this process is dictated by the economic interests of parties: users seek reliable, trustworthy, highvalue partners with favorable ratings, not random or representative ones—a far cry from the idealized notion of statistical sampling. Without a deeper understanding of this feedback process, particularly the selective nature of sampling, one cannot truly understand the fairness of social learning and its implications for discrimination.
The purpose of the current paper is to build a model of social learning that accounts for this feedback process and investigate its implications for statistical discrimination. Specifically, we study the possibility of, and the extent to, which social learning through ratings can ameliorate or exacerbate statistical discrimination of social groups.
Our model features directed search/matching between two sides, (masses of longlived) buyers and sellers, guided by usercontributed ratings. Each seller is indexed by her social group identity and her productive type, (“high”) or
(“low”): a seller’s group identity is unchanging, but her productive type changes over time, according to a continuous time Markov chain. The group identity is payoff irrelevant since it has no effect on sellers’ productive types. Buyers seek to match and trade with sellers; the surplus generated from trade is higher if the seller is of high type rather than a low type.
The searchmatching process is frictional, and is guided by imperfect information about sellers’ types, called ratings. The rating is binary, either or , and is updated after each trade. Although it is impossible for the ratings to perfectly reveal the sellers’ types due to the ever changing nature of seller type (except in the limit), we parameterize the effectiveness of social learning, or the extent to which the ratings “track” sellers’ true types, by
—the probability that an incorrect rating gets corrected (e.g., type
seller having rated) after each transaction. Buyers direct their search attention to sellers based on their ratings and possibly on their group identities: a seller of a given rating and group matches with probability that depends on the number of buyers directing attention to the sellers with . We study the steady state of this system.Absent any ex ante bias in belief updating, discrimination may still arise from the selective nature of data sampling mentioned earlier. In our model, data on a seller is sampled whenever the seller gets matched, and matching (thus sampling) arises from buyers’ search for sellers with favorable beliefs. A positive feedback loop can then ensue since sellers with a favorable posterior belief will get sampled more, and those believed to have been sampled more often but maintained rating would enjoy even more favorable belief than those with the same rating but believed to have been sampled less often. We identify a positive feedback loop that could lead to a systematic statistical discrimination of social groups.
To begin, the payoff irrelevance of group identity means that there always exists a “nondiscriminatory” equilibrium in which sellers of the two groups are treated identically. In this equilibrium, a seller of any given rating enjoys the same amount of attention regardless of her group identity; a rated seller enjoys a higher match/employment rate than a rated seller, due to the more favorable signal contained in the former, but identically across the group identities. Identical treatment by buyers across the social groups means that there is no bias in the sampling process between the groups, and this in turn leads to the identical updating of beliefs based on the group identity. Hence, nondiscrimination survives and perpetuates in steady state.
However, a nondiscriminatory equilibrium need not be the unique or even stable steady state.^{2}^{2}2 As will be seen, stability is defined in the usual manner, by the robustness of an equilibrium to small perturbations. There could be another steady state that is discriminatory in the following sense. Suppose buyers direct search attention toward sellers (rated sellers in group 1) away from sellers (rated sellers in group 2). This leads to sellers being sampled more often than in the steady state and to a more intensive weeding out of type sellers from sellers than from sellers. This in turn leads to sellers enjoying more favorable posterior beliefs than sellers and validates the more intensive attention sellers are receiving. Hence, the feedback loop is now complete, supporting a discriminatory equilibrium in which buyers favor sellers over sellers. Interestingly, the same feedback loop does not exist between and sellers, the rated sellers in groups 1 and 2, respectively. Say sellers receive more attention from buyers than ones. This makes the ratings of former sellers more accurate, which, however, leads to a less favorable belief for those sellers. Hence, the initial shift of the attention toward gets selfcorrected. In the steady state, therefore, and sellers are treated identically.^{3}^{3}3 Note that this is observationally equivalent to firms paying more attention to the workers in one group than those in the other, resulting in differential rewarding across the groups for (perceived) high quality. Interestingly, this behavior is consistent with the empirical findings of Bertrand and Mullainathan (2004) and Bartoš et al. (2016): Bertrand and Mullainathan (2004) performed field experiments by sending out fictitious résumés to helpwanted ads under whitesounding and blacksounding names. They find that not only résumés by whitesounding names receive more callbacks for interview s than those by blacksounding names, but the callback rates gap between highquality and lowquality résumés is significantly higher for the former group than for the latter. Similar field experiments were performed by Bartoš et al. (2016) on the prescreening behavior in job application and apartment application contexts; they find that the advantaged group receives more scrutiny in the former (“cherrypicking”) context whereas a disadvantaged group receives more scrutiny in the latter (“lemondropping”) context.
In sum, group 1 sellers are favored than group 2 sellers in this discriminatory equilibrium, despite there being no payoff relevance of the group identity and no bias in either the algorithmic rating/recommendation and belief updating. A discriminatory equilibrium need not always exist, but interestingly, when it does, the nondiscriminatory steady state may become unstable; a small perturbation in terms of buyers shifting their attention toward one group may break nondiscriminatory equilibrium and trigger a shift that leads to a discriminatory equilibrium. Our analysis shows that a discriminatory equilibrium exists if the matching friction is small and social learning friction is of intermediate value. This suggests that the advance of the online marketplace, as measured by the reduction of these frictions, may have a nonmonotonic effect on discrimination. The economy could very well begin with high enough frictions on both accounts that support only the nondiscriminatory equilibrium. With an advance in matching and social learning, frictions may diminish and a discriminatory equilibrium may emerge.
The current paper joins the longline of research on discrimination. In particular, our research follows the literature of statistical discrimination originated by Phelps (1972) and Arrow (1973).^{4}^{4}4 The earlier theories focus on tastebased discrimination. See Becker (1957). Unlike the tastedbased theories of discrimination (see Becker (1957)), this literature explains group inequality and stereotype as resulting from rational statistical inferences on groups’ characteristics. In Phelps (1972) and its modern incarnations, discrimination originates from exogenous differences in group characteristics,^{5}^{5}5 Cornell and Welch (1996) explains discrimination and its intergenerational persistence from groupspecific evaluational familiarity. Bohren et al. (2019) and Monachou and Ashlagi (2019) focus on the “misspecified” prior beliefs as a source of discrimination. whereas Arrow (1973) derives average group differences, and the associated differential treatment, as an endogenous equilibrium behavior. In the same spirit, Coate and Loury (1993b) and the subsequent literature focus on workers’ skill acquisition as a source of discriminatory group stereotyping:^{6}^{6}6 See Coate and Loury (1993a), Mailath et al. (2000), Moro and Norman (2003), Norman (2003), and a survey by Fang and Moro (2011). if employers view a certain group as less skilled and thus become more selective against them for assigning higherpaying positions, the affected group will indeed lose incentives for acquiring skills, thus fulfilling the employers’ adverse beliefs on that group.
Our paper is also related to the burgeoning literature in computer science on ethical algorithm (see Dwork et al. (2012), CorbettDavies et al. (2017), Kearns et al. (2018), and Kleinberg et al. (2018), among others). This literature explores ways to ensure that decision/recommendation algorithms satisfy a variety of fairness standards. The current paper qualifies the effectiveness of this approach, by identifying the possibility that algorithmic fairness alone may not be enough to accomplish a fairness goal. In our model, a discriminatory equilibrium may arise even when the rating algorithm treats both groups identically, as long as agents (buyers in our model) can interpret the ratings in a way that can lead to discriminatory sampling. The debate on algorithmic fairness must keep this aspect of social learning into consideration, so that either the interpretational scope is totally eliminated to guarantee outcome fairness, an approach that appears to be in line with the prescription of Kleinberg et al. (2018), or in case that is impossible, the ratings system may be designed to counteract the interpretational response by the users.
2 Model
We consider a frictional search market in which buyers (or firms) search for sellers (or workers) of unknown types.
Players.
There is a unit mass of sellers in the market. The sellers are indexed by two characteristics, type and group
. The type of a seller represents any payoffrelevant information, such as the productivity or quality of the seller. At a given moment, a seller is either of high type (
) or low type (). Each seller’s type, however, changes according to a continuoustime Markov process. Specifically, each type turns into the other type at rate . The group of a seller describes her payoffirrelevant identity, such as her gender, ethnic or racial identity. Each seller belongs to either group or , with being used as the generic index. Unlike her type, a seller’s group does not change over time. For strong symmetry between the two groups, we assume that both groups have the same total size (i.e., each group has mass ).On the other side of the market, there is mass of buyers. They search for sellers based on public information about sellers. Specifically, they condition their search on sellers’ two observable characteristics, rating and group identity .
The sellers who share the same observable characteristics, , and the buyers that search for them constitute a “submarket.” Clearly, each submarket can be indexed by . Sellers are assigned to those submarkets according to their (perfectly persistent) group identity and (evolving) ratings, while buyers choose which submarket to enter.
Matching.
We adopt the canonical searchandmatching framework to model an interaction between sellers and buyers. We assume that matching technology is common across all submarkets and exhibits constant returns to scale. The latter assumption implies that all agents’ matching rates in each submarket depend only on the ratio of buyers to seller in the submarket. We let denote a seller’s matching rate and denote a buyer’s matching rate. Note that consistency requires that for all : matching is onetoone, and thus the number of matched sellers should be identical to that of matched buyers at each point in time.
For expositional clarity, we focus on the parametric case where for some . This corresponds to the constantreturnstoscale CobbDouglas matching function and, therefore, satisfies various natural and desirable properties.^{7}^{7}7 Formally, let denote the measure of matched formed at each instant when there are mass of buyers and mass of sellers. If , then the rate at which each individual seller is matched with a buyer is given by
Trade.
Once a buyer and a seller meet, they transact instantaneously and go back to the market.^{8}^{8}8 The assumption on instantaneous transaction makes the model more naturally applicable for oneshot relationship such as freelance independent work and insurance or loan purchase. It is conceptually straightforward, but technically complicated, to extend our model to capture the canonical persistent employment relationship. The transaction yields surplus if the seller’s type is and if the seller’s type is , where . If a buyer transacts with a seller, the buyer pays to the seller. In order to exclude trivial cases, we assume (so that there are gains from trade when a seller is of type ), but consider both the case when and the case when .
Ratings.
Market accumulates information about sellers through simple summary indices, called “ratings.” There are two possible ratings: (as in “good”) and (as in “bad”). When searching for sellers, buyers can only observe the current rating of the sellers; no information about sellers’ underlying types or their past ratings history is available to them. After each transaction, the seller’s rating may be updated to reveal her type. Specifically, with probability , a rated seller with type receives rating, and a rated seller with type receives rating. A seller with correct rating keeps the same rating after a transaction. With remaining probability , the seller’s rating remains unchanged. Note that due to the changing environment (or changing type), a correct rating may turn inaccurate.
Buyers’ beliefs over a seller’s type will depend on the rating, and the (equilibrium) behavior of all players in the system. In particular, the belief may depend on the group identity. If the two groups of agents are treated differently, the inference a buyer makes on a seller with a given rating depend nontrivially on her group identity. This will be made clear in our analysis.
Solution Concept.
We consider a steady state of the economy in terms of the distribution of sellers of different types, ratings and group identity, and the beliefs that the buyers hold for each submarket. Specifically, an equilibrium is a tuple in the stationary distribution, where is the mass of sellers of type with rating and group , is the ratio of buyers to sellers in submarket , and is the public belief on the sellers in submarket , i.e., the probability that they are of type . Note that the masses of buyers who participate in alternative submarkets are determined by the first components.
3 NonDiscriminatory Equilibrium
We begin by studying equilibria in which buyers search for sellers only based on their ratings and do not distinguish between the two groups. As in other statisticaldiscrimination models, this equilibrium always exists and provides a benchmark for discriminatory equilibria studied in Section 4. Several technical results in this section will also be useful in Section 4.
3.1 SteadyState Distribution
In nondiscriminatory equilibrium, buyers condition their search strategies only on sellers’ ratings. Therefore, effectively, there are only 2 submarkets indexed by rating . For each , let denote the measure of buyers that join submarket and denote the measure of type sellers with rating . Then, the ratio of buyers to sellers (“queue length”) in submarket is given as follows:
Our matching technology implies that a rated seller is hired by a buyer at rate , while a buyer in submarket successfully finds a seller at rate .
In steady state, ’s must satisfy the following system of equations:
In each equation, the lefthand side represents the outflow of sellers from status , while the righthand side quantifies the corresponding inflow. For example, consider . Due to our rating technology (with no false negative), a type seller with rating changes his status only when his type changes to , which occurs at rate (thus, ). On the other hand, a type seller with rating becomes type at rate (thus ). In addition, a type seller with rating improves his rating to once he meets a buyer and receives a good rating (thus ). In steady state, net flow must be equal to , which yields the first equation.
The following result is straightforward from the above equations.
Lemma 1 (Steadystate Distribution).
In steady state, the measure of sellers with type and rating is given as follows:
Letting for each ,
See the appendix. ∎
Notice that for each , the fraction of type sellers in submarket depends only on . This is due to the fact that has the same proportional effect on and , and thus is independent of
. More concretely, our rating technology involves no typeI errors, and thus a seller’s rating improves from
to only when her type is , while her rating falls from to only when her type is . This implies that in each submarket, the mass of sellers with wrong rating ( in submarket , and in submarket ) is fully determined by the mass of sellers with correct rating ( in submarket , and in submarket ) and the matching rate of the submarket (see the second and the third equations in the above system of equations). This drives the convenient independence property of .In addition, increases in , while decreases in . This is intuitive: an increase in or can be interpreted as better/faster screening of sellers. Therefore, a seller with rating becomes more likely to be type as increases. Similarly, a seller with rating becomes less likely to be type (more likely to be type ) as increases.
3.2 Buyers’ Expected Payoffs
Let denote a buyer’s flow expected payoff when he targets rated sellers (i.e., searches in submarket ). Given the steadystate queue length and the fraction of type sellers, is given by
There are the following two cases to consider.

: This case arises if and only if^{9}^{9}9 If then , and thus even if . However, this case clearly cannot be sustained in equilibrium.
This is when a buyer’s utility from a type seller falls short of the price (i.e., ) and the probability that a buyer meets a type seller is sufficiently large. In this case, clearly, no buyers search in submarket , that is, .

: Opposite to (i), this arises if and only if . In addition, it must be that a positive measure of buyers search in submarket and, therefore, : if , then a participating buyer would meet sellers infinitely frequently, each of whom gives the buyer a positive expected payoff ), and thus his expected payoff becomes unbounded.
Recall that and in steady state, is also a function only of (see Lemma 1). Therefore, also can be interpreted as a function of . Interestingly and importantly, whereas is always monotone, may not be monotone, as formally reported in the following lemma (and shown in Figure 1).
Lemma 2.
Suppose that . For both , , , and is continuous. is always strictly monotone (decreasing), while is monotone if and only if
If , then there exist and such that is strictly increasing if and only if .
See the appendix. ∎
This result is due to the fact that is decreasing in , while is increasing in (Lemma 1). Since a seller’s matching rate always decreases with relatively more sellers (i.e., higher ), the utility effect of increasing is always negative in submarket but ambiguous in submarket . Lemma 2 shows that the quantity effect (through ) always dominates the quality effect (through ) if is sufficiently small or sufficiently large: the former is because the quantity effect () is arbitrarily large when is close to , while the latter is because as tends to infinity, the quality effect vanishes faster than the quantity effect. Given these observations, it is intuitive that the quality effect can outweigh the quantity effect, and thus decreases, over an interval if and only if is sufficiently close to (so that given , is close to ).
3.3 Equilibrium Characterization
We now characterize nondiscriminatory steadystate equilibria of our model. The following result provides a necessary and sufficient condition for there to be no trade in equilibrium.^{10}^{10}10 Note that Proposition 1 argues only uniqueness of no trade outcome, not that of symmetric steadystate equilibrium. This is because in notrade equilibrium, there is no seller movement between submarkets and , and thus any distribution of sellers can be sustained as long as .
Proposition 1.
If , then it is the unique nondiscriminatory steadystate equilibrium outcome that buyers do not search for sellers, regardless of their ratings (i.e., ). Conversely, if , then buyers search for both ratings of sellers (i.e., ).
Suppose that . Since , by Lemma 1, . But then,
Therefore, (no trade in submarket ) can be an equilibrium if and only if . Since this condition is independent of rating , it is either or . ∎
This result is fairly intuitive. Since both types of sellers change their types at an identical rate, the unconditional proportion of type sellers is in steady state: recall that in Lemma 1, . If there is no trade, then there is also no market learning, that is, a seller’s rating becomes uninformative of his type (observe that if ). Therefore, no trade outcome can be sustained if and only if .
Now consider the case where , so that . In this case, searching buyers receive positive surplus (i.e., ), and thus it is necessarily the case that all buyers search. Therefore, in equilibrium, the following “market clearing” condition must hold:
Since for each , this condition can be rewritten, in terms of , as follows:
(1) 
In addition, in equilibrium, a buyer must be indifferent between submarket and submarket , that is, the following buyerindifference condition should hold:
(2) 
This equation shows a buyer’s tradeoff between ratings and . By Lemma 1, it is always the case that a seller is more likely to be type when his rating is than when his rating is (i.e., ). This makes rating attract relatively more buyers than rating , which reduces a buyer’s chance to hire a rated seller (i.e., , and thus ). In equilibrium, and are such that a buyer is indifferent between the two submarkets.
Combining the two conditions leads to the following result.
Proposition 2.
If , then there always exists a nondiscrimatory steadystate equilibrium in which .
Let be the implicit function defined by equation (1). It is straightforward that and . In addition, since the righthand side of equation (1) increases in both and , is monotone.^{11}^{11}11 For the result with , one can express the righthand side as follows:
Figure 2 explains the argument behind Proposition 2. The red dashed curves represent the market clearing condition (1). They are always decreasing from infinity to zero as increases, which is because the righthand side is increasing in both and . The blue solid curves capture the buyerindifference condition (2). As shown in Figure 2, they are not necessarily monotone, which is because, whereas always decreases, may increase over an interval (Lemma 2). Nevertheless, they are always continuous, start from and eventually stay away from , and thus the two curves always intersect.
As exemplified by the right panel of Figure 2, there may exist multiple equilibria. This is, again, because the buyerindifference condition may produce a nonmonotone relationship between and (i.e., the implicit function may not be monotone). Clearly, if (in which case is monotone), then there always exists a unique nondiscriminatory equilibrium. Even if , it is often the case that nondiscriminatory equilibrium is unique (see the left panel). However, there is a nonnegligible set of parameter values that yield multiple equilibria.^{12}^{12}12 If there are multiple equilibria, then they are ranked in terms of buyer surplus: since the marketclearing condition is always decreasing in , if there are two equilibria, and , and , then . In this case, buyers’ expected payoffs are necessarily higher with , because is decreasing in (see Lemma 1), and thus has not only a higher value of but also a higher value of .
4 Discriminatory Equilibrium
In this section, we investigate discriminatory equilibria in which buyers condition their search strategies on sellers’ identities as well as their ratings. We first derive a condition under which such equilibria exist and then compare them to nondiscriminatory equilibria studied in Section 3.
4.1 Notation and Assumption
We use the same notation as in Section 3 but distinguish the two groups with superscripts . For example, we denote by the measure of buyers who are targeting rated sellers in group , and the measure of sellers with type , rating , and group .
Within each group , sellers follow the same transition dynamics as in the nondiscriminatory case. Therefore, Lemma 1 applies unchanged to each group. In particular, for each , the proportion of type sellers in submarket is given as follows:
In addition, buyers’ expected payoffs are determined as follows:
Proposition 1 also applies unchanged. Specifically, by the same logic and proof as for nondiscriminatory equilibria, there is no trade in any submarket if and only if . In addition, if , then trade must take place in all four submarkets. Since the analysis is trivial for the notrade case, from now on, we maintain the following assumption:
Assumption 1.
, and thus for all and .
4.2 Existence of Discriminatory Equilibrium
We begin by presenting a necessary condition for the existence of discriminatory equilibria, namely that the function must be nonmonotone, which is the case if and only if .
Proposition 3.
If is monotone (i.e., ), then there does not exist a discriminatory equilibrium in which or .
In equilibrium, buyers must be indifferent over all 4 submarkets, that is,
Since is monotone, it is always the case that . If is also monotone, then it is also the case that . Therefore, a discriminatory equilibrium cannot exist. ∎
Let us first explain how the nonmonotonicity of can lead to the existence of discriminatory equilibria. Suppose that is nonmonotone, and fix such that for two distinct values, : in Figure 3, it suffices to choose . Given and , let be the value that supports as a nondiscriminatory equilibrium, that is,
Similarly, let be the corresponding value for , that is,
Suppose that , and consider a steady state in which , , and . By construction, buyers are indifferent over all 4 submarkets, that is,
In addition, for both , we have
Therefore, this (discriminatory) strategy profile is a steadystate equilibrium.
This discriminatory equilibrium gives higher expected payoffs to group 1 sellers than to group 2 sellers. Intuitively, this is because the two groups have the same matching rate with rating , but the former have a higher matching rate than the latter with rating (i.e., ).^{13}^{13}13 To be precise, the following composition effect needs to be taken into account: more group 2 sellers are rated than group 1 sellers (i.e., ). However, it can be shown that this composition effect never outweigh the effect due to the difference in matching rates: the unconditional matching rate is increasing in both and . Let us emphasize that this discriminatory outcome arises even if the two groups have no fundamental differences (including the size of the group) and, perhaps more importantly, despite the fact that there may exist a unique nondiscriminatory equilibrium. In other words, unlike in some related papers, the possibility of discriminatory equilibria relies neither on any intrinsic differences between the two groups nor on the equilibrium multiplicity of our underlying environment.
The mechanism behind our discriminatory equilibria is the novel feedback loop between ratings and trade. More buyers search for group 1 sellers because their ratings are more informative (i.e., ). Conversely, group 1 ratings are more accurate because they are hired and reviewed more frequently (i.e., ).
As shown by Proposition 3, this feedback effect never results in the existence of discriminatory equilibria if is monotone. It may not work even if is not monotone, depending on . Nevertheless, the following result shows that whenever is not monotone, there is a positive measure of ’s that give rise to discriminatory equilibria.
Theorem 1.
If is not monotone (i.e., ), then there exist and such that a discriminatory equilibrium exists if and only if .
See the appendix. ∎
In order to see why a discriminatory equilibrium does not exist if or , note that in our model, buyers use group identities only to improve informational content of ratings. If is sufficiently small, then for all , and thus ratings do not contain much information (i.e., for all ). To the contrary, if is sufficiently large, then each is large, and thus ratings convey precise information about sellers’ types (i.e., for both , , while ). In both cases, group identities do not add much more information to ratings, and thus it is unlikely that buyers condition their search strategies on them.
4.3 Stability of Ratingsbased Discrimination
The analysis so far has demonstrated that nondiscriminatory and discriminatory equilibria can coexist. The possibility of discriminatory equilibria is intriguing by itself, but they would be unlikely to matter in practice if they could not be sustained in any robust manner. We now examine whether (and when) discriminatory equilibria exhibit desirable stability properties.
Since discriminatory equilibria can exist only when is not monotone, we mainly focus on the case where throughout this subsection. In addition, so as to streamline the analysis, we also restrict attention to the parameter space that yields a unique nondiscriminatory equilibrium.
Assumption 2.
For all , there exists a unique nondiscriminatory equilibrium.
Since the market clearing condition is monotone in both and , this assumption holds whenever the function does not increase too sharply, which is guaranteed if is close to .
In order to provide a relevant stability concept, let denote buyers’ equilibrium expected payoffs in the nondiscriminatory equilibrium when the total measure of buyers is given by . In other words, if is a nondiscriminatory equilibrium with measure of buyers, then . Proposition 2 and Assumption 2 ensure that is welldefined for all .
Both nondiscriminatory and discriminatory equilibria can be summarized by a pair such that (market clearing) and (buyer indifference among all 4 submarkets). The only difference between them is that an equilibrium is nondiscriminatory if , while it is discriminatory if . Based on this observation, we make use of the following tractable notion of stability.
Definition 1.
An equilibrium with is stable if and unstable otherwise.
In order to understand this definition, fix a steadystate equilibrium with , and suppose that a small measure of buyers move from group 1 to group 2, so that measure of buyers search for group 1 sellers and measure of buyers search for group 2 sellers. After the change, if buyers targeting group 1 receive a higher expected payoff than those targeting group 2, then those buyers who left group 1 will move back to group 1, restoring the original equilibrium with . If it is the opposite, then even more buyers would move to group 2, making the economy drift further away from the equilibrium . Clearly, the (original) equilibrium is stable in the former case and unstable in the latter case. Our stability definition captures this idea in a particularly simple fashion. For the simple condition in the definition, observe that, since ,
which reduces to in the limit as tends to .
We first apply our stability notion to nondiscriminatory equilibria.
Proposition 4.
Fix a nondiscriminatory equilibrium in which and for both . The equilibrium is stable if and only if is decreasing at .
In a nondiscriminatory equilibrium, . Therefore, it is stable if and only if . Since the market clearing condition (1) always expands as increases, this is equivalent to the equilibrium value of increasing in (so that decreases), which in turn holds if and only if is decreasing at .
∎
Recall that if , then there exists a unique steadystate equilibrium, which is nondiscriminatory (Propositions 2 and 3). Proposition 4 suggests that the unique equilibrium is stable, which is desirable. It further suggests that even if a nondiscriminatory equilibrium coexists with discriminatory equilibria, the former may be stable, but not always. If an equilibrium lies on a decreasing region of (either or ), then it is stable. Otherwise (i.e., ), the equilibrium is unstable.
Can a discriminatory equilibrium be stable? The following result argues that (there is a sense in which) discriminatory equilibria are more likely to be stable than nondiscriminatory equilibria.
Theorem 2.
Whenever there exists a discriminatory equilibrium, there exists a stable discriminatory equilibrium.
Suppose that a discriminatory equilibrium exists. By Theorem 1, this is the case if and only if is nonmonotone (i.e., ) and . Now consider the function such that
Since is finite, while , if is sufficiently close to . In addition, the existence of discriminatory equilibrium implies that there exists such that (i.e., ). Combining these with continuity of , it follows that there exists such that
The first condition implies that is a discriminatory equilibrium, while the second condition implies that the equilibrium is stable. ∎
Note that Theorem 2 does not claim that all discriminatory equilibria are stable. There may exist an unstable discriminatory equilibrium. Theorem 2 implies that if such an unstable equilibrium exists, then there exists another discriminatory equilibrium that is stable. Of course, it also implies that if there is a unique discriminatory equilibrium, then it is necessarily stable.
4.4 Rating Quality and Discrimination
As ratings have become increasingly more prevalent, the associated technology also has improved, extracting more (accurate) information from more users (buyers). Will such a technological advance bring more fairness by weakening the role of prejudice in decision making, or can it actually worsen discrimination? The following result shows that the effect is nonmonotone in general: an improvement in rating quality, measured by , may create discrimination initially.^{14}^{14}14 Note that and always enter the steadystate system together (see Lemma 1), and thus they cannot be separately identified. Intuitively, measures how fast ratings get corrected, while measures how fast ratings become obsolete. Therefore, their ratio, , is the proper measure of rating quality. However, if the technology becomes sufficiently effective, then discrimination becomes unsustainable.
Proposition 5.
Fix and . There exist and such that a discriminatory equilibrium exists if and only if .
Let . Then, the two equilibrium equations, (1) and (2), can be written as follows:
and
Notice that appears only on the lefthand side of the first equation, together with . This implies that if there is an equilibrium (whether discriminatory or nondiscriminatory) with and , then effectively the same equilibrium exists with and as well, and vice versa. Combining this observation with Theorem 1, it follows that for a fixed value of , a discriminatory equilibrium exists if and only if , where and . ∎
For an intuition, recall that discrimination arises in our model because a seller’s group identity may provide extra information about her productive type. Such extra information is of little value when ratings are sufficiently uninformative (i.e., is close to ) or sufficiently informative (i.e., is large). It can make a large enough difference that can sustain a discriminatory equilibrium only when rating quality belongs to an intermediate range.
5 Comparison to Coate and Loury (1993a)
Just as in our paper, Coate and Loury (1993a) (CL, hereafter) demonstrated that discriminatory outcomes can arise despite no exogenous differences between two groups. They considered a labor market model in which workers invest in their human capital and employers assign a job to each worker based on the worker’s group identity and a noisy signal about the worker’s human capital. They showed that discrimination can result from a “selffulfilling prophecy”: employers believe that group 2 workers are less likely to invest in their human capital than group 1 workers and, therefore, assign less group 2 workers to a better job than group 1 workers. Expecting lower returns, group 2 workers indeed invest in their human capital less than group 1 workers.
The underlying mechanism behind statistical discrimination in our model differs from that of CL in several important ways. First, the discrimination in our model is based on rational inference on sellers’ transaction histories (“ratings”), and thus it does not require endogenous human capital acquisition as in CL. The perspectives on the cause of discrimination also differ. In the latter, “future” anticipated discrimination discourages the candidates in the discriminated group from acquiring skills, thus validating the discrimination in equilibrium. In our theory, “past” discrimination disadvantages the discriminated sellers/workers in the inference formed by prospective buyers/employers about their good ratings, thus perpetuating discrimination.
Second, unlike CL, the existence of discriminatory equilibrium in our model does not rely on multiplicity of nondiscriminatory equilibria. In CL, discrimination arises when (and because) different groups coordinate on different equilibria, making multiplicity of nondiscriminatory equilibria a necessary and
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