On Statistical Discrimination as a Failure of Social Learning: A Multiarmed Bandit Approach

1. Introduction

Statistical discrimination refers to discrimination against minority people, taken by fully rational and nonprejudiced agents. Previous studies have shown that, even in the absence of prejudice, discrimination can occur persistently because of various reasons, including the discouragement of human capital investment (Arrow 1973, Foster and Vohra 1992, Coate and Loury 1993, Moro and Norman 2004), information friction (Phelps 1972, Cornell and Welch 1996, Bardhi et al. 2020), and search friction (Mailath et al. 2000, Che et al. 2019). The literature has proposed various affirmative-action policies to solve statistical discrimination, with many having been implemented in practice.

This paper demonstrates that statistical discrimination may appear as a failure of social learning. We endogenize the evolution of biased beliefs and analyze their consequences. Our model assumes that (i) all firms (decision makers) are fully rational and nonprejudiced (i.e., attempt to hire the most productive worker), and (ii) all workers are ex ante symmetric. In such an environment, an unbiased decision policy—hiring workers with superior skills—satisfies numerous fairness notions articulated in scholarly literature, including equalized odds and demographic parity. It also achieves efficiency by maximizing each firm’s payoff. However, the long-term persistence of biased beliefs could still occur. This paper demonstrates that temporary affirmative actions can effectively enhance both welfare and equality.

Although our model applies more broadly, we use the terminology of hiring markets to describe our model. We develop a multiarmed bandit model of social learning, in which many myopic and short-lived firms sequentially make hiring decisions. In each round, a firm hires one worker from a set of candidates. Each firm’s utility is determined by the hired worker’s skill, which cannot be observed directly until employment. However, as in the standard statistical discrimination model, each worker also has observable characteristics associated with their unobservable skills. Firms learn the statistical association between characteristics and skills using data pertaining to past hiring cases (shared through, e.g., private communication, social media, and recommendation letters) and use the estimators to predict the skills of candidates.

Each worker belongs to a group that represents, for example, their gender, race, and ethnicity. We assume that the characteristics of workers who belong to different groups should be interpreted differently. This assumption is realistic. First, previous studies have revealed that underrepresented groups receive unfairly low evaluations.¹ When these evaluations are used as the observable characteristics, firms should be aware of the potential bias. Second, evaluations may reflect differences in cultures, living environments, and social systems (Precht 1998, Al-Ali 2004). For instance, firms need to be conversant with the norms of drafting recommendation letters to interpret them accurately. Therefore, observable characteristics, such as curriculum vitae, exam scores, grading reports, recommendation letters, and so forth, might convey starkly different implications despite their similar presentations. If firms are unbiased and cognizant of these potential biases, they should adapt their interpretation methods for these characteristics, applying varied statistical models to different groups.

When firms learn the statistical association from data, with some probability, the minority group is underestimated because of a large estimation error raised by insufficient data. Once the minority group is underestimated, it is difficult for a minority worker to appear to be the best candidate—even if he has the greatest skill among the candidates, the firm often dismisses this fact and tends to hire a majority worker. As long as firms hire only majority workers, society cannot learn about the minority group; thus, the imbalance persists even in the long run. We call this phenomenon perpetual underestimation.

We use a linear contextual bandit model to analyze the consequences of social learning. To gauge policy performance, we utilize regret, a widely adopted measure in machine-learning literature that assesses welfare loss relative to the optimal decision rule. Regret arises if proficient minority workers are overlooked because of biased estimates by firms; hence, regret not only signifies efficiency but also encapsulates fairness. We discuss the relationship between regret and fairness criteria in Section 7.

We focus on how regret grows as the total number of firms (denoted by N) increases. When regret is sublinear in N, firms make fair and efficient decisions in the long run. We first analyze the equilibrium consequence of laissez-faire (LF) (no policy intervention). When the groups are ex ante symmetric and the population ratio is equal, laissez-faire results in $\tilde{O}(\sqrt{N})$ regret.² However, when the population ratio is unbalanced, this no longer holds, and the expected regret is $\tilde{\mathrm{\Omega }}(N)$.

Toward fair and efficient social learning, we propose a subsidy rule based on the idea of the upper confidence bound (UCB) method. UCB is an effective solution for balancing exploration and exploitation (Lai and Robbins 1985, Auer et al. 2002). By incentivizing firms to take actions that are consistent with the recommendations of the UCB, social learning can promote sublinear regret in the long run. We demonstrate that the UCB mechanism has the expected regret of $\tilde{O}(\sqrt{N})$. The subsidy required to implement the UCB mechanism is also $\tilde{O}(\sqrt{N})$.

Improving the UCB mechanism, this paper proposes a hybrid mechanism, which lifts affirmative actions upon the collection of a sufficiently rich data set. The hybrid mechanism takes advantage of spontaneous exploration: Once firms obtain a certain amount of data, the diversity of workers’ characteristics naturally promotes learning about the minority group. The hybrid mechanism achieves $\tilde{O}(\sqrt{N})$ regret with $\tilde{O}(1)$ subsidy.

We also analyze the Rooney Rule, which requires each firm to interview at least one minority candidate as a finalist for each job opening. We demonstrate that although the Rooney Rule resolves perpetual underestimation by increasing the probability of hiring minority candidates, it sometimes unjustly deprives productive majority workers of employment opportunities, suggesting reverse discrimination.

This paper is framed as a positive analysis elucidating how discrimination arises from social learning conducted by small, rational, and unbiased firms. Alternatively, our study could be viewed as a normative analysis showcasing an efficient hiring policy targeting the long-term average skill of workers hired by a large firm (as explored by Li et al. 2020). For this latter scenario, our results for the hybrid mechanism indicate that a firm can cease affirmative action once it has accumulated reasonably comprehensive information about minority workers.

2. Related Literature

2.1. Statistical Discrimination

Various studies have analyzed statistical discrimination both theoretically (Phelps 1972, Arrow 1973, Foster and Vohra 1992, Coate and Loury 1993, Cornell and Welch 1996, Mailath et al. 2000) and experimentally (Neumark 2018 is an excellent survey). We contribute to this literature by articulating a new channel of discrimination: endogenous data imbalance and insufficiency. Similar to previous studies, we assume otherwise ex ante identical individuals from different groups to demonstrate how discrimination evolves and persists. Meanwhile, our results provide further indication that demographic minorities suffer from discrimination as an inevitable consequence of laissez-faire.

Hu and Chen (2018) examine a dynamic reputation model in a labor market, where workers can endogenously select their skill level. As highlighted by Foster and Vohra (1992) and Coate and Loury (1993), statistical discrimination can potentially discourage minorities from enhancing their skills. Implementing a fairness constraint through affirmative action at the entry level may rectify inequalities within the entire labor market. Our study adds to this body of literature by demonstrating that short-term affirmative action successfully tackles inefficiency and inequality, even when the skill level is fixed.

Kannan et al. (2019) study how a college can design an admission and grading policy to achieve fair employment, assuming employers form a Bayesian belief about students’ skills based on the information provided by the college. We also consider a government that introduces an affirmative-action policy taking into account endogenous responses. Che et al. (2019) examine a rating-guided market, demonstrating that feedback loops can cause discriminatory inferences concerning social groups. We identify endogenously created informational disparities because of feedback loops within a distinct model inspired by a hiring market, deliberate on the underlying causes (demographic imbalance) that instigate discrimination, and propose policy solutions.

Bohren et al. (2019, 2023) and Monachou and Ashlagi (2019) have demonstrated how misspecified beliefs about groups generate discrimination. Thus far, this literature has attributed belief misspecification to bounded rationality. In contrast, we demonstrate that misspecified beliefs may evolve and persist endogenously. Through a laboratory experiment, Dianat et al. (2022) reveal that affirmative action’s impact becomes fleeting if the measure is discontinued before beliefs undergo transformation. Our hybrid mechanism offers a resolution by optimally choosing the timing to terminate the program.

2.2. Social Learning

The economics literature has extensively studied herding, information cascade, and social learning (e.g., Banerjee 1992, Bikhchandani et al. 1992, Smith and Sørensen 2000). Additionally, various papers have studied improvements to social welfare through subsidy for exploration (e.g., Frazier et al. 2014, Kannan et al. 2017) and selective information disclosure (e.g., Kremer et al. 2014, Papanastasiou et al. 2018, Immorlica et al. 2020, Mansour et al. 2020). We propose novel policy interventions to improve social learning in fairness and efficiency.

2.3. Multiarmed Bandit

A multiarmed bandit problem stems from the literature of statistics (Thompson 1933, Robbins 1952). This problem is driven by the question of how a single long-lived decision maker can maximize his payoff by balancing exploration and exploitation. More recently, the machine-learning community has proposed the contextual bandit framework, in which payoffs associated with “arms” (actions) depend not only on the hidden state but also on additional information, referred to as “contexts” (Abe and Long 1999, Langford and Zhang 2008).³

Several previous studies have considered a linear contextual bandit problem and studied the performance of a “greedy” algorithm, which makes decisions myopically in accordance with the current information. Because firms take greedy actions under laissez-faire, their results are also relevant to our model. Bastani et al. (2021) and Kannan et al. (2018) have shown that a greedy algorithm leads to sublinear regret if the contexts are diverse enough. We characterize the relationship between the diversity of contexts and the rate of learning. Moreover, we show that the population ratio is crucial to the regret rate. As an efficient intervention, Kannan et al. (2017) consider a contextually fair UCB-based subsidy rule. Although our subsidy policy also originates from the idea of UCB, we establish the hybrid mechanism that reduces budget expenditure by utilizing spontaneous exploration.

The multiarmed bandit approach has recently found applications in labor market analyses. Bardhi et al. (2020) demonstrate that a minor difference in initial beliefs about each worker’s type can ultimately yield a substantial disparity in workers’ payoffs. Johari et al. (2018) examine how a labor platform can discern workers’ skills to attain an optimal worker assignment when the platform only observes the outcomes generated by teams, not individual workers.

Li et al. (2020) portray a large firm’s hiring process as a multiarmed bandit problem and empirically compare the performance of the status quo (screening via manual work), a greedy policy, and a UCB method. They reveal that a UCB method not only screens job applicants efficiently but also preserves diversity. Their findings suggest that a UCB method is both fairer and more efficient when implemented by a large firm. Interpreting this paper as a study of an efficient hiring policy by a large firm, our results enhance Li et al. (2020) by providing theoretical foundations that outline the performances of a greedy policy (corresponding to laissez-faire) and a UCB method. Moreover, we illustrate that affirmative action can be discontinued shortly by characterizing the performance of a hybrid mechanism.

2.4. Algorithmic Fairness

This algorithmic-fairness literature has implicitly assumed exogenous asymmetry in worker skills and pursued the approaches to correct between-group inequality. To this end, “discrimination-aware” constraints such as equalized odds (Hardt et al. 2016) and demographic parity (Pedreschi et al. 2008, Calders and Verwer 2010) have been proposed, with several papers applying these constraints in the context of multiarmed bandit problems (Joseph et al. 2016) or more general sequential learning (Raghavan et al. 2018, Bechavod et al. 2019, Chen et al. 2020). Although these fairness goals are conflicting in general, we analyze an environment in which many fairness goals are aligned and demonstrate how affirmative action improves them.

3. Model

3.1. Basic Setting

We develop a linear contextual bandit problem with myopic firms (agents). We consider a situation where N firms (indexed by $n=1,\dots ,N$) sequentially hire one worker for each.⁴ In each round n, a set of workers I(n) (i.e., arms) arrives. Each worker $i\in I(n)$ takes no action, and firm n hires only one worker $\iota (n)\in I(n)$. Both firms and workers are short-lived. Upon round n ending, firm n’s payoff is finalized, and all rejected workers leave the market.⁵

Each worker $i\in I$ belongs to a group $g\in G$. We assume that the population ratio is fixed: for every round n, the number of workers belonging to group g is $K_g\in \mathbb{N}$ and $K={\sum }_{g\in G}{K}_g$. Slightly abusing the notation, we denote the group worker i belongs to by g(i). Each worker i also has observable characteristics ${\mathit{x}}_i\in {\mathbb{R}}^d$, with $d\in \mathbb{N}$ as their dimension. Finally, each worker i also has a skill $y_i\in \mathbb{R}$ that is not observable until worker i is hired. The characteristics and skills are random variables.

Because each firm’s payoff is equal to the hired worker’s skill y_i (plus the subsidy assigned to worker i as affirmative action, if any), firms want to predict the skill y_i based on the characteristics ${\mathit{x}}_i$. We assume that characteristics and skills are associated as $y_i={\mathit{x}}_i^{\prime }{\mathit{\theta }}_{g(i)}+{ϵ}_i$, where ${\mathit{\theta }}_g\in {\mathbb{R}}^d$ is a coefficient parameter, and ${ϵ}_i\sim \mathcal{N}(0,{\sigma }_{ϵ}^2)$ i.i.d. is an unpredictable error term. We assume $\Vert {\mathit{\theta }}_g\Vert \le S$ for some $S\in {\mathbb{R}}_{+}$, where $\Vert ·\Vert$ is the standard L2-norm. Because ϵ_i is unpredictable, $q_i≔{\mathit{x}}_i^{\prime }{\mathit{\theta }}_{g(i)}$ is the best predictor of worker i’s skill y_i.

The coefficient parameters ${({\mathit{\theta }}_g)}_{g\in G}$ are initially unknown. Hence, unless firms share information about past hires, firms are unable to predict each worker’s skill y_i. We assume that firms share information about past hiring cases.⁶ Accordingly, when firm n makes a decision, in addition to the characteristics and groups of current workers ${({\mathit{x}}_i,g(i))}_{i\in I(n)}$, firm n observes the characteristics, groups, and skills of previously hired workers ${(x_{\iota (n\prime )},g(\iota (n\prime )),y_{\iota (n\prime )})}_{n\prime =1}^{n-1}$. We refer to all realizations of these variables as the history in round n and denote it by h(n). Formally, h(n) is given by

Note that h(n) does not include information about (i) the worker hired by firm n, or (ii) that worker’s actual skill. This is because the notation h(n) represents the information set firm n faces when it makes a hiring decision. We denote the set of all the possible histories in round n by H(n). The firm’s decision rule for hiring and the government’s subsidy rule are defined as a function that maps a history to a hiring decision and the subsidy amount (described later). For notational convenience, we often omit h(n).

3.2. Prediction

We assume that firms are not Bayesian but frequentists. Hence, firms do not have a prior belief about the parameter $\mathit{\theta }$ but estimate it only using the available data set.⁷

We assume that each firm predicts skill using ridge regression (L2-regularized least square).⁸ Let $N_g(n)$ be the number of rounds at which group-g workers are hired before round n. Let ${\mathit{X}}_g(n)\in {\mathbb{R}}^{N_g(n)\times d}$ be a matrix that lists the characteristics of group-g workers hired by round n: each row of ${\mathit{X}}_g(n)$ corresponds to ${\{{\mathit{x}}_{\iota (n\prime )}:\iota (n\prime )=g\}}_{n\prime =1}^{n-1}$. Likewise, let ${\mathit{Y}}_g(n)\in {\mathbb{R}}^{N_g(n)}$ be a vector that lists the skills of group-g workers hired by round n: each element of ${\mathit{Y}}_g(n)$ corresponds to ${\{y_{\iota (n\prime )}:\iota (n\prime )=g\}}_{n\prime =1}^{n-1}$. We define ${\mathit{V}}_g(n)≔({\mathit{X}}_g(n))\prime {\mathit{X}}_g(n)$. For a parameter $\lambda >0$, we define ${\overline{\mathit{V}}}_g(n)={\mathit{V}}_g(n)+\lambda {\mathit{I}}_d$, where ${\mathit{I}}_d$ denotes the d × d identity matrix. Firm n estimates the parameter as follows:

Firm n predicts worker i’s skill q_i, while substituting the true predicted skill ${\mathit{\theta }}_g$ with estimated skill ${\widehat{\mathit{\theta }}}_g(n)$: ${\widehat{q}}_i(n)≔{\mathit{x}}_i^{\prime }{\widehat{\mathit{\theta }}}_{g(i)}(n)$. Hence, ${\widehat{q}}_i(n)$ and ${\widehat{\mathit{\theta }}}_g(n)$ depend on the history h(n). The ordinary least squares (OLS) estimator corresponds to the ridge estimator with λ = 0. We use the ridge estimator instead of the OLS estimator to stabilize the small-sample inference. For example, for some history, ${\mathit{V}}_g(n)$ may not have full rank, and the OLS estimator may not be well defined. Even for such histories, the ridge estimator is always well defined.

For analytical tractability, we assume that for the first $N^{(0)}$ rounds, each firm n must hire from a prespecified group, g_n. We refer to the first $N^{(0)}$ rounds as the initial sampling phase. We assume $N^{(0)}$ to be small and deal $N^{(0)}$ as a constant.⁹ Let $N_g^{(0)}≔{\sum }_{n=1}^{N^{(0)}}\mathrm{𝟙}[g_n=g]$ as the data size of initial sampling for group g, where $\mathrm{𝟙}[\mathcal{A}]=1$ if event $\mathcal{A}$ holds or zero otherwise. The initial sampling phase is exogenous. That is, we ignore the incentives and payoffs of firms and assume that the characteristics x of the hired candidate constitute an i.i.d. sample of the corresponding group. We analyze mechanism, social welfare, and budget after round $n>N^{(0)}$. The initial sampling phase can be interpreted as data that have already been produced in history. The welfare cost is already sunk, and the government can no longer make policy interventions for the event that has already occurred in the past.

3.2.1. Mechanism.

In addition to worker skills, firms are also concerned about subsidies. We assume that firm preferences are risk neutral and quasilinear. Hence, if firm n hires worker i, its payoff (von Neumann-Morgenstern utility) is given by $y_i+s_i$, where $s_i\in {\mathbb{R}}_{+}$ denotes the amount of the subsidy assigned to worker i.

In the beginning of the game, the government commits to a subsidy rule $s_i(n,·):H(n)\to {\mathbb{R}}_{+}$, which maps a history to a subsidy amount. Hence, once a history h(n) is specified, firm n can identify the subsidy assigned to each worker $i\in I(n)$. Firm n attempts to maximize

Firm n’s decision rule $\iota (n,·):H(n)\to I(n)$ specifies the worker that firm n hires after history h(n). We say that a decision rule ι is implemented by a subsidy rule s_i if for all n and h(n), we have

Throughout this paper, any ties are broken arbitrarily. We call a pair of a decision rule and subsidy rule a mechanism. We often drop h(n) from the input of decision rule ι when it does not cause confusion.

3.3. Regret

Regret is a standard measure for evaluating the performance of algorithms in multiarmed bandit models:

Because ϵ_i is unpredictable, it is natural to evaluate the performance of the algorithm (or the equilibrium consequence of the policy intervention) by comparing it with q_i. If the parameter ${({\mathit{\theta }}_g)}_{g\in G}$ were known, each firm could easily calculate q_i for each worker i and hire the best worker, $i^{*}(n)≔{\text{arg max}}_{i\in I(n)}{q}_i$. In this case, regret would be zero. The goal of the policy design is to establish a mechanism that minimizes the expected regret $\mathbb{E}[\text{Reg}(N)]$, where the expectation is taken on a random draw of workers. This aim is equivalent to maximizing the sum of the skills of workers hired.

Following the literature, we often evaluate the performance by the limiting behavior (order) of expected regrets. A decision rule ι is said to have sublinear regret if $\mathbb{E}[\text{Reg}(N)]=O(N^a)$ for some a < 1. Small regret implies not only efficiency but also fairness. Regret measures the disparate impact that is not justified by skill disparity, and sublinear regret is achieved if firms hire the most skillful workers without regard to the group of workers. The relationship between regret and fairness is discussed further in Section 7.

3.4. Budget

Some of the policies we study incentivize exploration through subsidies. The total budget required by a subsidy rule is also an important policy concern. The total amount of the subsidy is given by $\text{Sub}(N)≔{\sum }_{n=N^{(0)}+1}^N{s}_{\iota (n)}(n)$.

4. Laissez-Faire

This section analyzes the equilibrium under laissez-faire, that is, the consequence of social learning in the absence of policy intervention.

4.1. Symmetry and Diverse Characteristics

To illustrate a failure of social learning, we make three assumptions. First, as a minimal environment to analyze discrimination, we focus on the two-group case.

When we consider asymmetric equilibria, we refer to group 1 as the majority (unprotected) group and group 2 as the minority (protected) group. The two-group assumption enables the elucidation of how the minority group is discriminated against.

Second, we assume that groups are symmetric.

Note that although we assume that groups are symmetric, firms do not know the true parameters and therefore apply different statistical models to different groups. That is, even though the true coefficients are identical (${\mathit{\theta }}_g={\mathit{\theta }}_g^{\prime }$ for all $g,g\prime \in G$), firms estimate them separately; thus, the values of the estimated coefficients are typically different (${\widehat{\mathit{\theta }}}_g(n)\ne {\widehat{\mathit{\theta }}}_{g\prime }(n)$ for $g\ne g\prime$).

Although Assumption 2 is unrealistic (because the characteristics should evidently be interpreted differently), it is useful for elucidating how laissez-faire nourishes statistical discrimination. Under Assumption 2, agents are ex ante identical (as assumed in Arrow 1973, Foster and Vohra 1992, Coate and Loury 1993, Moro and Norman 2004), and therefore the differences we observe in the equilibrium are entirely attributed to social learning.

Furthermore, when groups are symmetric, disparate impact is unambiguously unfair. It is well known that popular fairness notions aim at different goals and are compatible with each other only in highly constrained special cases (see, e.g., Kleinberg et al. 2017). The symmetric environment specified by Assumption 2 is one such exception: In this environment, sublinear regret implies many fairness goals conflicting with each other in general (see Section 7). Because this paper’s focus is not to debate which of the various types of fairness notions should be respected, we will concentrate only on the symmetric environment.¹⁰

Third, we assume that characteristics are normally distributed, and therefore, the distribution is nondegenerate. This assumption captures the diversity of workers.

We consider essentially the same results to hold more generally as long as the characteristics are sufficiently diverse. Note that when we have both Assumptions 2 and 3, then there exist ${\mathit{\mu }}_x,{\sigma }_x$ such that ${\mathit{\mu }}_{xg}={\mathit{\mu }}_x$ and ${\sigma }_{xg}={\sigma }_x$ for all $g\in G$. Hence, ${\mathit{x}}_i\sim \mathcal{N}({\mathit{\mu }}_x,{\sigma }_x^2{\mathit{I}}_d)$ for all i.

4.2. Perpetual Underestimation

To determine whether social learning incurs linear expected regret, it is useful to check whether it results in perpetual underestimation with a significant probability.

4.3. Sublinear Regret with Balanced Population

This section analyzes the case of only one candidate arriving from each group during each period. The contextual variation implicitly urges firms to explore all the groups with some frequency. Consequently, laissez-faire has sublinear regret, implying that statistical discrimination is eventually resolved.

Let ${\mu }_x=\Vert {\mathit{\mu }}_x\Vert$ and $\mathrm{\Phi }$ be the cumulative distribution function of the standard normal distribution. The constant on the top of $\mathbb{E}[{\text{Reg}}^{\text{LF}}(N)]$ is inverse proportional to $1-\mathrm{\Phi }({\mu }_x/{\sigma }_x)$, which approximately scales as $\exp (-{({\mu }_x/{\sigma }_x)}^2/2)$.

All proofs are presented in Online Appendix E.

To prove Theorem 1, we characterize the condition with which underestimation is spontaneously resolved. Let indices i₁ and i₂ denote the majority candidate and the minority candidate. With a constant (i.e., independent of N) probability, the minority group is underestimated (i.e., ${\widehat{\mathit{\theta }}}_2(n)$ is misestimated in such that ${\mathit{x}}_{i_2}^{\prime }{\widehat{\mathit{\theta }}}_2(n)\ll {\mathit{x}}_{i_2}^{\prime }{\mathit{\theta }}_2$ often occurs) in early rounds because of a bad realization of the error term. Even in such a case, there is some probability of the minority candidate being hired. Because characteristics are diverse (i.e., ${\sigma }_x>0$), with some probability, the majority candidate i₁ is not very good (i.e., ${\mathit{x}}_{i_1}^{\prime }{\widehat{\mathit{\theta }}}_1(n)\approx {\mathit{x}}_{i_1}^{\prime }{\mathit{\theta }}_1$ is small). In such a round, ${\mathit{x}}_{i_1}^{\prime }{\widehat{\mathit{\theta }}}_1(n)<{\mathit{x}}_{i_2}^{\prime }{\widehat{\mathit{\theta }}}_2(n)$ holds despite group 2 being underestimated, and the minority candidate i₂ is hired. In such a case, firms update their belief about the minority, leading to a resolution of underestimation. Such events occur more frequently when workers have more diverse characteristics, that is, ${\mu }_x/{\sigma }_x$ is small.

As anticipated by the theory of least squares, the standard deviation of ${\widehat{\mathit{\theta }}}_g(n)$ is proportional to ${({\overline{\mathit{V}}}_g(n))}^{-1/2}$, and we demonstrate that its diameter ${({\lambda }_{\text{min}}({\overline{\mathit{V}}}_g(n)))}^{-1/2}$ shrinks as $\tilde{O}(1/\sqrt{n})$, where ${\lambda }_{\text{min}}$ is the minimum eigenvalue of a matrix. The regret per error is defined by this quantity, with the total regret being $\tilde{O}({\sum }_{n\le N}(1/\sqrt{n}))=\tilde{O}(\sqrt{N})$.

Theorem 1 indicates that statistical discrimination is resolved spontaneously when candidate variation is large. At a glance, this appears to contradict widely known results that state laissez-faire (greedy) may lead to suboptimal results in bandit problems because of underexploration. However, the variation in characteristics naturally incentivizes selfish agents to explore the underestimated group, and therefore, with some additional conditions, the probability of perpetual underestimation is bounded.

4.4. Large Regret with Unbalanced Population

Although Theorem 1 implies that statistical discrimination is spontaneously resolved in the long run, it crucially relies on one unrealistic assumption—the balanced population ratio. In many real-world problems, the population ratio is unbalanced, and the protected group is often a demographic minority in the relevant market. We indeed find that the population ratio crucially impacts the equilibrium consequence under laissez-faire.

In the proof of Theorem 2, we evaluate the probability that the following two events occur: (i) ${\widehat{\theta }}_2$ is underestimated, and (ii) the characteristics and skills of the hired majority workers are not very bad throughout rounds (i.e., ${\max }_{i:g(i)=1}{x}_i{\widehat{\theta }}_1\ge c{\mu }_x\theta$ for some constant c > 0). The probability of (i) is polylogarithmic to N (i.e., $\tilde{\mathrm{\Theta }}(1)$), and the probability that (ii) consistently holds for all the rounds $n=N^{(0)}+1,\dots ,N$ is polylogarithmic if $K_1>{\log }_2 N$. When both (i) and (ii) occur, we always have ${\max }_{i\in I(n)\setminus \{i_2(n)\}}{x}_i{\widehat{\theta }}_1>x_{i_2(n)}{\widehat{\theta }}_2$ (where $i_2(n)$ is the unique minority candidate of round n); thus, the minority worker is never hired. Note that the majority group does not suffer from perpetual underestimation (with a significant probability) because the event that all the minority workers are bad occasionally occurs.

Theorem 2 indicates that we should not be too optimistic about the consequences of laissez-faire. A small imbalance in the population ratio (the ratio of majority to minority is just ${\log }_2 N$ to 1) could lead to a substantially unfair job allocation. Once the minority group is underestimated and the majority candidate pool is reasonably large, then the minority group is afforded no hiring opportunity, perpetuating underestimation. This insight applies to many real-world problems because unbalanced populations are commonplace.

We conjecture a substantial probability under a broader environment than the premise of Theorem 2. Specifically, the assumptions of d = 1 and $K_1>{\log }_{2}N$ are made only for analytical tractability, and (approximately) linear regret should be obtained under a weaker set of assumptions. Theorem 2 (i) focuses on perpetual underestimation, which is an extreme form of statistical discrimination, and (ii) evaluates the probability of perpetual estimation occurring loosely. In Section 9, we demonstrate that perpetual underestimation occurs with a significant probability even under the assumptions of d = 5 and $(K_1,K_2)=(10,2)$, where the premise of Theorem 2 does not hold.

5. The Upper Confidence Bound Mechanism

Section 4 has discussed the equilibrium consequences of laissez-faire. We observed that an unbalanced population ratio leads to a substantial probability of underestimation being perpetuated. Policy intervention is demanded to improve social welfare and the fairness of the hiring market.

This section proposes a subsidy rule to resolve underestimation. We employ the idea of the UCB algorithm (Lai and Robbins 1985, Auer et al. 2002), which has widely been used in the literature on the bandit problem. The UCB algorithm balances exploration and exploitation by developing a confidence interval for the true reward and evaluating each arm’s performance according to its upper confidence bound to achieve this balance. Firms are generally unwilling to follow the UCB decision rule voluntarily; therefore, the government needs to provide a subsidy to incentivize firms to hire a candidate with the greatest UCB index. This section establishes a UCB-based subsidy rule and evaluates its performance.

The adaptive selection of candidates based on history can induce some bias, meaning the standard confidence bound no longer applies. To overcome this issue, we use martingale inequalities (Peña et al. 2008, Rusmevichientong and Tsitsiklis 2010, Abbasi-Yadkori et al. 2011). We here introduce the confidence interval for the true coefficient parameter, ${({\mathit{\theta }}_g)}_{g\in G}$.

There are three remarks. First, $\tilde{O}(\sqrt{N})$ regret is the optimal rate for these sequential optimization problems under partial feedback (Chu et al. 2011). Hence, Theorem 3 states that the UCB decision rule effectively prevents perpetual underestimation and is asymptotically efficient. Second, Theorem 3 relies only on Assumption 3, and therefore, the regret under UCB is sublinear even when groups have a fundamental disparity besides their group sizes. Third, differing from the case of laissez-faire, where the factor depends on the variation of the context (Theorem 1), Theorem 3 provides a reasonably small regret bound even when σ_x is very small.

To implement the UCB decision rule, we need to satisfy the firms’ obedience condition (1) in conjunction with the UCB decision rule (2). In the following, we propose one of the most straightforward subsidy rules.

6. The Hybrid Mechanism

The UCB mechanism has one drawback: it continues subsidies in perpetuity. Even for a large n, there remains a gap between estimated skill ${\widehat{q}}_i(n)$ and the UCB index ${\tilde{q}}_i(n)$. This is undesirable for several reasons. First, introducing a permanent policy is often more politically difficult than introducing a temporary policy. Second, a long-term distribution of subsidies tends to increase the required budget. Third, the permanent allocation of the subsidy features (unmodeled) administrative costs.

To overcome these limitations, we propose the hybrid mechanism, which initially uses the UCB mechanism but switches to laissez-faire by terminating the subsidy at some point. We abandon the UCB phase upon receiving sufficient minority-group data to induce spontaneous exploration. Similar to the UCB mechanism, our hybrid mechanism has $\tilde{O}(\sqrt{N})$ regret. Furthermore, its expected total subsidy amount is only $\tilde{O}(1)$, whereas the UCB mechanism needs $\tilde{O}(\sqrt{N})$ subsidy.

The construction of the hybrid mechanism is as follows. Let $s_i^{\text{U-I}}(n)={\tilde{q}}_i(n)-{\widehat{q}}_i(n)$ be the size of the confidence bound. Note that $s_i^{\text{U-I}}(n)$ corresponds to the amount of the subsidy allocated by the UCB index subsidy rule (Definition 5). The hybrid index ${\tilde{q}}_i^{\text{H}}$ is defined as

The hybrid index is literally a “hybrid” of estimated skill ${\widehat{q}}_i(n)$ and the UCB index ${\tilde{q}}_i(n)$. If the difference between the UCB index and estimated skill surpasses the threshold (i.e., $s_i^{\text{U-I}}(n)>a\Vert {\widehat{\mathit{\theta }}}_{g(i)}(n)\Vert$), then the hybrid index is equal to the UCB index ${\tilde{q}}_i(n)$. The confidence bound $|{\tilde{q}}_i(n)-{\widehat{q}}_i(n)|$ is large when society has insufficient knowledge about group g(i), which is typically the case during early stages of the game. Once this gap falls below the threshold (i.e., $s_i^{\text{U-I}}(n)\le a\Vert {\widehat{\mathit{\theta }}}_{g(i)}(n)\Vert$), then the hybrid index switches to the estimated skill ${\widehat{q}}_i(n)$.

The hybrid decision rule hires the worker who has the greatest hybrid index.

The order of the regret under the hybrid decision rule is the same as the original UCB, and the subsidy amount is reduced to $\tilde{O}(1)$ (with respect to N). This is a substantial improvement from the UCB mechanism, which requires the $\tilde{O}(\sqrt{N})$ subsidy.

The threshold for switching from the UCB mechanism to laissez-faire is crucial for guaranteeing the performance of the hybrid mechanism. Our threshold, $a\Vert \widehat{\mathit{\theta }}(n)\Vert$, is determined such that the hybrid decision rule ${\iota }^{\text{H}}$ satisfies proportionality, a new concept that this paper establishes. We prove that the amount of exploration exerted by the hybrid decision rule is proportional to the UCB decision rule. This property guarantees that the hybrid rule resolves underestimation and secures the expected regret of $\tilde{O}(\sqrt{N})$. The formal statement appears in Lemma 28 in Online Appendix E.5.

7. Regret and Fairness Criteria

This section analyzes the connection between sublinear regret social learning and equalized odds and demographic parity, fairness notions prevalent in fair machine-learning literature (see, e.g., a survey by Makhlouf et al. 2021). For clarity, we invoke Assumption 1, specifying that group 1 is the majority (unprotected) group and group 2 is the minority (protected) group.

7.1. Sublinear Regret

Thus far, we have mainly used regret as a fairness notion. Recall that a decision rule ι is said to have sublinear regret if $\mathbb{E}[\text{Reg}(N)]=O(N^a)$ for some a < 1. This implies that the decision rule eventually makes an unbiased decision in the sense that each firm hires based on accurately estimated skill predictors, q_i, uninfluenced by the workers’ group affiliation. Consequently, sublinear regret implies asymptotic unbiasedness in firms’ decision-making processes.

7.2. Equalized Odds

Equalized odds, defined below, is the fairness notion most directly related to sublinear regret.

The intuition is as follows. The violation of equalized odds leads to persistent biased decisions by firms. Consequently, society experiences enduring, nondiminishing regret, which signifies a failure in sublinear-regret learning. Therefore, the realization of sublinear regret inherently necessitates the fulfillment of equalized odds.

7.3. Demographic Parity

Demographic parity is defined as follows.

Theorem 7 elucidates that in instances of group symmetry, equalized odds and demographic parity are compatible, eliminating the contention over the choice of fairness notions. Moreover, these two properties are guaranteed by sublinear regret.

Although our discussion has centered on equalized odds and demographic parity, the machine-learning literature has developed numerous other fairness notions. It is anticipated that many of them align within symmetric groups, but typically exhibit conflicts with sublinear regret and equalized odds in more general environments.

8. Rooney Rule

Subsidy rules can address statistical discrimination, but practical implementation challenges persist. In Online Appendix B, we examine the Rooney Rule, which mandates that firms interview at least one candidate from every group. This rule sidesteps the complexities of subsidies and hiring quotas, making it straightforward to apply. We summarize our key findings below.

Our two-stage model facilitates the analysis of the Rooney Rule. In this model, a firm only invites a limited number of candidates for interviews and receives additional signals on their skills. This rule compels firms to interview at least one representative from all groups. The outcomes are mixed. The Rooney Rule reduces the underexploration of minority groups by boosting their hiring probability, thus averting persistent underestimation. Yet, it occasionally and undesirably excludes qualified majority candidates in the first stage. Simulations suggest that phasing out the Rooney Rule can mitigate this drawback. For a comprehensive discussion, refer to Online Appendix B.

9. Simulation

This section presents the outcomes of our simulations.¹⁴ Unless specified, model parameters are set as $d=5,\mathit{\theta }=(1,1,1,1,1),{\mathit{\mu }}_x=(1.5,\dots ,1.5),{\sigma }_x=1,\hspace{0.17em}{\sigma }_{ϵ}=0.5$, λ = 1, and N = 1,000. Group sizes are set to be $(K_1,K_2)=(10,2)$. The initial sample size is $N^{(0)}=K_1+K_2$, and the sample size for each group is equal to its population ratio: $N_1^{(0)}=K_1,N_2^{(0)}=K_2$. We draw 4,000 paths independently for each simulation scenario. The value of δ in the confidence bound is set to 0.1.

9.1. The Effects of Population Ratio

We test how the population ratio impacts the frequency of perpetual underestimation. The decision rule is fixed to LF. We fix the number of minority candidates in each round to two (i.e., $K_2=2$) and vary the number of majority candidates ($K_1=2,10,30,100$).

Figure 1 exhibits the simulation result. Consistent with our theoretical analyses, we observe that (i) as indicated by Theorem 1, laissez-faire rarely produces perpetual underestimation if the population is balanced (i.e., K₁ is close to $K_2=2$), and (ii) as indicated by Theorem 2, the larger the population of majority workers (i.e., K₁ increases), the more frequently perpetual underestimation occurs. With $K_1=10$, perpetual underestimation occurs in more than 2% of runs, which is large enough to ensure that laissez-faire produces (approximately) linear regret.

9.2. Laissez-Faire vs. the UCB Mechanism

Figure 2 compares the regret associated with the LF decision rule and the UCB decision rule. As indicated by Theorem 2, our simulation shows that laissez-faire has a significant probability of underestimating the minority group. Consequently, laissez-faire sometimes causes perpetual underestimation, and regret grows (approximately) linearly to n. Furthermore, because of the possibility of perpetual underestimation, the confidence intervals of the sample paths (denoted by the red area) are very large, indicating the highly uncertain performance of laissez-faire. In contrast, consistent with Theorem 3, the UCB decision rule performs much more stably. Because the UCB rule avoids underexploration, it does not cause perpetual underestimation.

9.3. The UCB Mechanism vs. the Hybrid Mechanism

Next, we compare the performance of the UCB and hybrid mechanisms. The parameter of the hybrid mechanism is set to be a = 0.5. Figure 3 shows the associated regret. As Theorems 3 and 5 anticipated, the regrets associated with the two decision rules are similar (these two decision rules have the same order: $\tilde{O}(\sqrt{N})$). Figure 4 compares the subsidy rules. As Theorems 4 and 5 predicted, the subsidy required for the UCB index rule grows at the rate of $\tilde{O}(\sqrt{N})$, whereas the hybrid index subsidy rule only requires only a constant subsidy, implying that the policy intervention can be terminated at some point. Furthermore, the hybrid index subsidy rule requires a much smaller budget than the UCB index subsidy rule. To summarize, the hybrid mechanism produces similar regret as the UCB mechanism with a much smaller budget.¹⁵

In Online Appendix G.1, we demonstrate that a nonindex subsidy rule implements the UCB decision rule with a smaller total budget; its total subsidy cannot be bounded by a constant.

9.4. Fairness Metrics

Figure 5 illustrates the disparity in equalized odds, representing the empirical analog of the sum of the terms on the left-hand side of Equation (3) and Equation (4). Similarly, Figure 6 displays the demographic disparity, corresponding to the empirical analog of the term on the left-hand side of Equation (9).¹⁶ For every n, both the UCB and hybrid decision rules outperform laissez-faire concerning the two fairness metrics. Notably, with our simulation setting, the differences in performance related to equalized odds are especially pronounced.

10. Conclusion

We have studied statistical discrimination using a contextual multiarmed bandit model. Our dynamic model articulates how a failure of social learning produces statistical discrimination. In our model, the insufficiency of data about minority groups is endogenously generated. This data shortage prevents firms from accurately estimating the skill of minority candidates. Consequently, firms tend to prefer hiring majority candidates, leading the data insufficiency to persist. We have demonstrated that an unbalanced population ratio leads laissez-faire to tend toward perpetual underestimation, an unfair and inefficient consequence.

We analyzed two possible policy interventions. One is subsidy rules that incentivize firms to hire minority candidates. Our hybrid mechanism achieves $\tilde{O}(\sqrt{N})$ regret with $\tilde{O}(1)$ subsidy. Another intervention is the Rooney Rule, which requires firms to interview at least one minority candidate. Our result indicates that terminating the Rooney Rule at an appropriate point would resolve statistical discrimination while maintaining the social welfare level. These results contrast with some of the previous studies (e.g., Foster and Vohra 1992, Coate and Loury 1993, Moro and Norman 2004) demonstrating the possible counterproductivity of affirmative-action policies.

Our analyses of the two interventions provide a consistent policy implication: Affirmative actions effectively resolve statistical discrimination caused by data insufficiency, but such actions should be lifted upon acquiring sufficient information. Accordingly, a temporary affirmative action constitutes the best approach to resolving statistical discrimination as a social learning failure.