\section{Introduction}
Algorithms are increasingly being used to assist or entirely replace human predictions on important outcomes. If designed carelessly, algorithms are not susceptible to only efficiency-related concerns but also unfairness, based on unacceptable characteristics like race, sex, or religion. For illustrative purposes, we focus on the scenario where a firm uses a predictive algorithm to hire or reject applicants belonging to a disadvantaged group. The generality of our framework allows for extensions to similar environments like loan applications, police stops, or parole decisions. In this paper, we abstract away from the estimation problem and instead discuss the preferences of the firm. In particular, given the firm's level of prejudice, we evaluate what type of algorithm the firm prefers: one that ``bakes in'' or one that ``reverses'' the prejudice embedded in the training data set.

Our definition of discrimination is driven by classical ideas in labor economics. We suppose the firm is both a \textit{statistical} and \textit{taste-based} discriminator. With respect to the former, we allow the firm to categorize applicants by race and vary the algorithm they use for each group. That is, the firm uses the race of the applicants to improve algorithmic prediction. With respect to the latter, we assume the firm receives a positive payoff from rejecting applicants belonging to a certain group. Throughout this paper, we consider two general algorithms specified by \cite{rr}. One algorithm carries over the prejudice contained in the training data into predictions. Following the literature, we say such an algorithm exhibits the characteristic ``bias in, bias out.'' The second algorithm treats disadvantaged applicants more favorably. Such an algorithm exhibits ``bias reversal.'' 

While the statistical behavior of these two algorithms is generally well-understood, the firm's preferences are unclear. In particular, does the extent of a firm's prejudice influence which of the two algorithms it prefers? We consider a wide range of possible levels of prejudice the firm may take: on one extreme end, the firm hires all disadvantaged applicants, and on the other end, the firm rejects all disadvantaged applicants. We find that a sufficiently prejudiced firms, on average, prefers the algorithm that transmits bias as time goes on. Conversely, a sufficiently partial firm prefers the algorithm that reverses bias. A casual restatement of our asymptotic main result is as follows.

\setcounter{subsection}{1}

\begin{theorem}[Theorem \ref{main}, Corollary \ref{gregret}]
There exists a unique level of prejudice where the firm is indifferent between the two algorithms on average. For any level of lesser prejudice, the firm prefers ``bias reversal.'' For any level of greater prejudice, the firm prefers ``bias in, bias out.''
\end{theorem}

In order to describe the firm's preferences, we must first define the preference ordering. To that end, we define a version of \textit{regret loss}, which we oftentimes refer to as \textit{regret} for brevity. A firm experiences a certain level of regret with each applicant, which varies between the two algorithms. In addition to discussing \textit{individual} regret, we also analyze \textit{average} regret, which summarizes the firm's overall payoff across all applicants. While the traditional, decision-theoretic definition of regret loss is the difference between the realized outcome and the optimal outcome, in our scenario, the firm may not observe the optimal outcome. That is, a firm never observes a rejected applicant's true potential. Thus, we define regret loss as the difference between \textit{ex ante} and \textit{ex post} outcome. We say that the firm prefers an algorithm over another when it generates less average regret. We then identify two types of environments: regular and irregular. The former is associated with arguably more common labor markets where there is a low proportion of qualified applicants. In irregular environments, the firm prefers ``bias in, bias out'' for any level of human prejudice. Importantly, the firm is able to determine which environment they are in.

A brief summary of our framework is as follows. We consider hiring decisions in two stages: in the first stage, the firm uses a human decision-maker to hire or reject disadvantaged applicants. Upon hiring an applicant, the firm observes his or her true level of qualification. At the conclusion of the first stage, the firm compiles all hiring data produced by the human decision-maker. Importantly, this data set suffers from \textit{selective labels} because the true ability of rejected applicants is unobserved. In the second stage, the firm uses this data set to construct one of the two aforementioned algorithms. Using this algorithm, the firm scores and accepts or rejects future applicants, upon which regret loss is calculated.

This paper is most closely related to the intersection of the study of discrimination and algorithmic bias, often referred to as \textit{algorithmic fairness}. In particular, we implement the theories of taste-based and statistical discrimination in the setting of algorithmic predictions. \cite{beck} defines taste-based discrimination as the act of harboring a ``taste for discrimination'' against a certain group. Following the work of Becker, \cite{phelps} and \cite{arrow} formulate the first models of statistical discrimination. \cite{fang} generally refers to statistical discrimination as the act of a human decision-maker using observed characteristics of an individual in place of unobserved yet outcome-relevant characteristics. The seminal works of Phelps and Arrow led to influential papers involving models of signal extraction in the labor market. For example, \cite{lund} incorporates the notion of skill investments in Phelps' model. \cite{cl} exploits this idea along with Arrow's model to show that two \textit{ex ante} identical groups may end up in different Pareto ranked equilibria. Adjacent areas of research investigate discrimination arising from inter-group interactions (\cite{moro}; \cite{mailath}; \cite{fang2001}). These models can be used to inform policy decisions. For instance, \cite{fang2006}, \cite{chung}, \cite{chan}, and others apply the models of discrimination to evaluate the effects of affirmative action, where the government institutes policies in favor of disadvantaged groups.

This paper aims to fit a traditional model of discrimination into the context of algorithmic bias. An empirical study conducted by ProPublica (\cite{propub}) reveals that a risk assessment algorithm used by the court system in Broward County, Florida is more likely to falsely flag black defendants as future criminals. \cite{ito}, inspired by the ProPublica report, identifies three common definitions of fairness and show that an algorithm that satisfies all three must either feature perfect predictions or equal base rates. \cite{hardt}, \cite{fpdi}, and \cite{corb} show analogous results. These findings suggest that common definitions of algorithmic fairness are challenging to achieve concurrently, if possible at all. Given this difficulty, one is naturally interested in examining the behavior of algorithms in the presence of unfairness in order to devise empirical strategies to reduce bias as much as possible. In an effort to pin down the source of such algorithmic prejudice, an influx of studies spearheaded by econometricians and computer scientists propose a number of explanations. The notion of ``bias in, bias out'' is a suspect of generating algorithmic bias. \cite{sol} and \cite{bibo} refer to ``bias in, bias out'' as the phenomenon where if an algorithm is trained on a data set generated by prejudiced human decision-makers, the algorithm would reflect these biases. Another candidate behind sowing bias into algorithmic predictions is known as the selective labels problem: a type of missing data problem with non-randomly selected samples, where a human decision-maker determines which outcomes of observations are observed.\footnote{\cite{slp}, \cite{charfair}, \cite{rr} for discussions.} Since the outcomes of observations that are not admitted are not observed by the researcher, causal inference becomes challenging when restricted to the usual econometric tools. A particular driver of this paper, \cite{rr} presents a surprising result where an algorithm reverses the prejudice of the human decision-maker under selectively labeled data, a phenomenon they refer to as ``bias reversal.'' That is, the algorithm treats the minority group more favorably, contrary to the notion of ``bias in, bias out.'' \cite{ramba} asserts that examining the question of how algorithms magnify or reduce bias through an economic perspective can lead to fruitful discussion. 

The paper proceeds as follows. In Section \ref{sec:3}, we formulate our framework and evaluate the interactions between the two algorithms. In Section \ref{sec:4}, we introduce regret loss and evaluate firm preferences. In Section \ref{sec:5}, we conclude.

\section{A Two-Stage Model}\label{sec:3}

In this section, we fit a mathematical framework on the scenario of a prejudiced firm hiring applicants in two stages. The first stage involves a human decision-maker scoring applicants. In the second stage, the firm builds a predictive algorithm by using the data generated in the first stage. We suppose the firm is born without access to an existing algorithm. Moreover, we assume that we live in a static world where the joint distribution of the population does not change over time. For this reason, we need not make any assumptions about the time duration of each stage. The distinguishing factor between the first and second stages is solely the mechanism on how applicants a scored.

\subsection{Environment and Information Structure}
In this section, we detail the environment. Consider a collection of applicants indexed by the set $I:=\{1,\hdots, n\}$. Each applicant is described by a random variable $Q$ where $Q\in\{0,1\}$ denotes qualification. We let $Q=1$ correspond to qualified applicants and $Q=0$ to unqualified applicants. We write $\P(Q=1) = \pi$. We limit $I$ to only include disadvantaged applicants because we assume that the firm is a statistical discriminator. While the firm in reality observes two sets of applicants distinguished by potentially contentious variables, e.g., race, we allow the firm to use entirely different algorithms for the two groups. In this sense, in general it may be misleading to consider which algorithms a firm prefers based on the performance of algorithmic predictions on a pooled group of applicants. We further justify and discuss our baseline environment in Section \ref{sec:4}.

While potentially valuable as an extension to this paper, we do not consider competitive interactions between firms. Therefore, suppose there exists a single firm that chooses to either accept or reject job applications from $I$. In other words, this one firm receives every application from the set $I$. The firm's payoff from an arbitrary applicant $i\in I$ depends on $Q_i$. The firm receives a positive payoff $x_q>0$ from hiring a qualified applicant. Conversely, the firm faces a loss $-x_u<0$ if the firm hires an unqualified applicant. If the firm does not hire a particular applicant, they receive zero payoff regardless of whether or not the applicant was qualified.

\begin{table}[h]
\centering
\begin{tabular}{l|l|c|c|c}
\multicolumn{2}{c}{}&\multicolumn{2}{c}{Hiring Decision}&\\
\cline{3-4}
\multicolumn{2}{c|}{}&Hire&Reject&\multicolumn{1}{c}{}\\
\cline{2-4}
    \multirow{2}{*}{Applicant}& Qualified & $x_q>0$ & $0$\\
\cline{2-4}
& Unqualified & $-x_u<0$ & $0$\\
\cline{2-4}
\end{tabular}
\caption{The firm's payoff from hiring a qualified and unqualified worker}
\end{table}

The firm and data scientist is naturally unable to observe each $Q_i$ directly. Instead, the firm associates each applicant with a realization of the pair of random variables $(\Theta, \Gamma)$. On the other hand, we restrict the data scientist's knowledge to only $\Theta$. Intuitively, $\Theta$ represents the concrete data that can be documented (e.g., highest level of education, GPA, years of experience). It follows that both the firm and data scientist may observe this information. We let $\Gamma$ represent the undocumented characteristics of an applicant that the firm extracts. For example, the applicant's demeanor, manner of speaking, and interpersonal abilities are observed by the firm oftentimes through an interview. However, $\Gamma$ does not arrive in the form of a rectangular data set, thus unobserved to the data scientist. Indeed, the comparative statics results described in \cite{rr} exploit the unobserved nature of $\Gamma$.

Next, the joint law $(\Theta, \Gamma\mid Q)$ corresponds to  the joint density functions $h_q(\t, \gamma)$ when $Q=1$ and $h_u(\t, \gamma)$ when $Q=0$. Thus, we allow for the joint law of the signal to depend on the true qualification level. Denote $f_x(\t)$ and $g_x(\gamma)$ the marginal density functions where $x\in \{q,u\}$. Throughout this paper, we write $f_x(\t\mid \Gamma)$ and $g_x(\gamma\mid \Theta)$ to mean conditional density functions. We present an important assumption that characterizes the qualifications of applicants through their signal values.

\begin{assumption}[Monotone Likelihood Ratio Property]\label{as1}
The joint densities satisfy the \textit{strict monotone likelihood ratio property}, where 
\[
l(\t, \gamma) := \frac{h_q(\t,\gamma)}{h_u(\t, \gamma)}
\]
is jointly increasing and continuous in $\theta$ and $\gamma$.
\end{assumption}

\begin{remark}
In the univariate case, the space is totally ordered. In our case, the partial order is naturally given by $(\t_1,\gamma_1) < (\t_2,\gamma_2)$ if and only if $\t_1 < \t_2$ and $\gamma_1 < \gamma_2$. It should be noted that the claim need not hold for $\t_1 < \t_2$ but $\gamma_1 > \gamma_2$ for example.
\end{remark}

We opt for the strict version of the property rather than the traditional weak version, since the latter suggests that higher signals may not correspond to being a better signal. Assumption \ref{as1} intuitively suggests that qualified applicants are more likely to receive higher signals over unqualified applicants.

\subsection{First Stage: Human Prediction}
At this point, we have yet to discuss how the firm incorporates the applicants' signals in a decision rule. To do so, we consider the first stage of applications, where the firm accepts or rejects applicants by means of a human decision-maker. This setting corresponds to traditional job applications where a senior employee or human resources manager at a firm interviews applicants and decides which ones to hire.

We assume the human decision-maker is Bayesian, in the sense that they update their prior beliefs on the level of qualification of each applicant by using his or her signals $\t$ and $\gamma$. The posterior probability $\P(Q=1 \mid \Theta = \t, \Gamma = \gamma)$ that the applicant is qualified denoted by $\kappa(\t,\gamma)$ follows from Bayes' rule

\begin{equation}\label{eq:1}
    \kappa(\t,\gamma) := \frac{\pi h_q(\t, \gamma)}{\pi h_q(\t, \gamma) + (1-\pi)h_u(\t, \gamma)}
\end{equation}

Naturally, the human decision-maker assigns higher posterior probabilities to applicants with greater signals. The following powerful lemma formalizes this idea.

\begin{lemma}\label{milg}
$\k(\t,\gamma)$ is jointly strictly increasing in $\t$ and $\gamma$.
\end{lemma}

\begin{proof}
Rewrite \eqref{eq:1} as follows:
\[
\k(\t,\gamma) = \frac{1}{1 + \frac{1-\pi}{\pi}\frac{h_u(\t, \gamma)}{h_q(\t,\gamma)}}.
\]
By Assumption \ref{as1}, the denominator is strictly decreasing in both $\t$ and $\gamma$. Thus, $\k(\t,\gamma)$ is jointly strictly increasing.
\end{proof}

Next, let $\tau\in \mathbb{R}$ denote the decision-maker's prejudice against applicants. Then, the decision-maker accepts the each applicant if and only if the expected payoff exceeds the cutoff:

\begin{equation}\label{eq:2}
\kappa(\t,\gamma)x_q - [1- \kappa(\t, \gamma)]x_u > \tau.
\end{equation}

In other words, the human decision-maker is a taste-based discriminator. We may interpret $\tau<0$ as a human decision-maker who favors disadvantaged applicants. When $\tau \geq x_q$, the human decision-maker's taste for discrimination overrides the firm's payoffs leading to every applicant being rejected regardless of qualification. Conversely, if $\tau \leq -x_u$, the firm favors applicants to the point where everyone is accepted. To avoid the trivial, we focus on the interval $\tau \in (-x_u, x_q)$. Inserting the expression for posterior probability \eqref{eq:1} into \eqref{eq:2}, we define the indicator $A(\tau)$ which equals 1 if the applicant is accepted and 0 otherwise:

\begin{equation}\label{eq:3}
    A(\tau) := \mathbbm{1}\left\{ \frac{h_q(\t,\gamma)}{h_u(\t,\gamma)}  > \frac{(1-\pi)(x_u+\tau)}{\pi(x_q - \tau)}\right\}.
\end{equation}

We frequently denote $A(\tau)$ by simply $A$ for convenience, but it should be understood that $A$ critically depends on the parameter $\tau$. Once the human decision-maker scores and hires or rejects applicants according to the rule \eqref{eq:3}, the hired applicants' true types $Q$ are revealed. The firm then collects the signal and outcome for all applicants into a data set $\{D_i\}_{i\in I}$:

\begin{equation*}
    D_i:= (\t_i, A_i, Q_i).
\end{equation*}

Importantly, $D_i$ does not contain $\gamma_i$. $Q_i$ is only observed when $A_i=1$---otherwise, we treat it as a missing value. Here, we can see that the firm faces selective labels. An extension may discuss a scenario where the true types of rejected applicants are observed. For example, the productivity of rejected applicants may be observed at competing firms.

\subsection{Second Stage: Algorithmic Prediction}
With the conclusion of the first stage of hires, the firm is now able to construct algorithms using the data set $\{D_i\}_{i\in I}$. The joint distribution of the observed data crucially depends on the value of $\tau$. Since $Q_i$ is only observed for $A_i=1$, different values of $\tau$ lead to different applicants possessing a label.

We introduce a second set of applicants indexed by the set $J:=\{1,\hdots, m\}$. Identically to the first stage, each applicant is described by $Q$. We allow the data generating process from the first stage to be identical to the second stage: the signal distributions are unchanged. The sole difference is that instead of the firm personally scoring applicants, a data scientist who does not observe $\Gamma$ builds an algorithm. Indeed, the firm replaces human decision-makers with algorithms, not because the pool of applicants fundamentally changes, but for primarily businesses-related interests. While real-life firms have countless options on how to design predictive algorithms, we abstract away from the estimation problem and prioritize the behavior of the algorithms as $\tau$ changes. We consider two predictors under squared loss:

\begin{equation}
    s_1(\t, r, \tau) := \mathbb{E}[Q\mid \Theta = \t, A=1]
\end{equation}
\begin{equation}
    s_2(\t, r, \tau) := \mathbb{E}[A \mid \Theta = \t].
\end{equation}

We frequently denote the two by $s_1$ and $s_2$ for convenience, and omit some arguments when appropriate. The key difference between the two algorithms is the outcome of interest. Algorithm $s_1$ predicts whether applicants are qualified within the accepted pool while algorithm $s_2$ predicts whether applicants are accepted. It follows that $s_1$ is trained on selectively labeled data, and $s_2$ is trained on all of $\{D_i\}_{i\in I}$. It's important to note that $\tau$ affects both algorithms via the distribution of $A$.

We now present alternatives proofs of \cite[Theorem 3]{rr} and \cite[Theorem 1]{rr} respectively, making use of Assumption \ref{as1}. For a fixed $\t$, we show that $s_2(\tau)$ is strictly decreasing in $\tau$. That is, $s_2$ exhibits ``bias in, bias out.''

\begin{theorem}[{\cite[Theorem 3]{rr}}]\label{bin}
For any $\t$,
$s_2(\tau) = \mathbb{E}[A\mid \Theta = \t]$ is strictly decreasing in $\tau$.
\end{theorem}

\begin{proof}
We have that
\[
\mathbb{E}[A\mid \Theta = \t] = \P(A=1\mid \Theta = \t).
\]
Using (\ref{eq:3}), we can rewrite the right-hand quantity:
\begin{align*}
    \P(A=1 \mid \Theta = \t) &= \P\Bigg(\frac{h_q(\t, \gamma)}{h_u(\t, \gamma)} > \frac{(1-\pi)(x_u+\tau)}{\pi(x_q-\tau)} \Bigg| \Theta = \t\Bigg).
\end{align*}

Next, define:

\begin{equation}\label{eq:g}
\gamma_1(\tau) := \inf\left\{\gamma : \frac{h_q(\t, \gamma )}{h_u(\t, \gamma)} > \frac{(1-\pi)(x_u+\tau)}{\pi(x_q-\tau)}\right\}.
\end{equation}

The right-hand quantity within the brackets is increasing over $\tau\in (-x_u,x_q)$ which is our interval of interest. Recall that $h_q/h_u$ is continuous and strictly increasing in $\gamma$ from Assumption \ref{as1}. Thus, as the right-hand quantity increases, we have $\gamma_1(\tau)$ also strictly increases. Therefore, $A=1$ if and only if $\Gamma > \gamma_1(\tau)$ given realizations of $\Theta$. To conclude, it suffices to show that
\[
\P(\Gamma > \gamma_1(\tau) \mid \Theta = \t)
\]
is strictly decreasing in $\tau$. We can rewrite to:
\begin{gather*}
    \P(\Gamma > \gamma_1(\tau) \mid \Theta = \t) = 1 - \P(\Gamma \leq \gamma_1(\tau) \mid \Theta = \t)
   
   
   
   
   
   
\end{gather*}

The second term on the right-hand side is a conditional CDF, so it must be increasing in $\tau$ because $\gamma_1(\tau)$ is increasing in $\tau$. Thus, $\mathbb{E}[A\mid \Theta = \t]$ is decreasing in $\tau$.
\end{proof}

We have shown that if an algorithm is trained using the label of acceptance as the outcome of interest, greater levels of human prejudice leads to similarly prejudiced algorithmic predictions. The following analogously characterizes $s_1$:

\begin{theorem}[{\cite[Theorem 1]{rr}}] \label{RR}
For any $\t$, $s_1(\tau) = \mathbb{E}[Q\mid \Theta=\t, A=1]$ is strictly increasing in $\tau$.
\end{theorem}

\begin{proof}
Proceed as follows:
\begin{align*}
    \mathbb{E}[Q\mid \Theta =\t, A=1] &= \P(Q=1\mid \Theta=\t, A=1)\\
    &= \frac{\P(Q=1,A=1\mid \Theta = \t)}{\P(A=1\mid \Theta = \t)}\\
    &= \frac{\P(A=1\mid Q=1,\Theta =\t)\P(Q=1\mid \Theta=\t)}{\P(A=1\mid \Theta = \t)}.
\end{align*}
Applying the law of total probability to the denominator and shortening the condition $\Theta =\t$, we get

\[
\frac{\P(A=1\mid Q=1,\t)\P(Q=1\mid \t)}{\P(A=1\mid Q=1,\t)\P(Q=1\mid \t) + \P(A=1\mid Q=0,\t)\P(Q=0\mid \t)}.
\]

Identical to the proof of Theorem \ref{bin}, we may write

\[
\P(A=1\mid Q=1,\Theta =\t) = 1 - \P(\Gamma \leq \gamma_1(\tau) \mid Q=1, \Theta = \t)
\]

which is strictly increasing---likewise for $\P(A=1\mid Q=0,\Theta = \t)$. It follows that both the numerator and denominator are decreasing. Moreover, the denominator is decreasing at a faster rate than the numerator in $\tau$. Thus, $s_1$ is strictly increasing in $\tau$.
\end{proof}

\begin{remark}
	While \cite{rr} defines the human decision-maker's scoring procedure to minimize squared loss, we use the posterior probability. It is clear these are equivalent in our setting.
\end{remark}

In other words, the algorithm $s_1$ exhibits ``bias reversal,'' where the score of an applicant increases with the firm's level of taste-based discrimination. Since firms are interested in the true qualification label of applicants, the purpose of $s_1$ is clear. On the other hand, the intuition of $s_2$ does not come immediately. Prior work discuss that the target variable that employers aim to predict widely varies, and often times data constraints akin to selective labels may lead firms to choose future employees based on who has been hired in the past. Moreover, machine learning engineers may simply be tasked by management to find a target variable that makes selection of employees easier, and scores similar to $s_2$ are leading candidates.\footnote{See \cite{evc} and \cite{bo} for a discussion on the choice of target variables in employment settings.} Before we continue, we give a lemma that further describes $s_1$ and $s_2$.

\begin{lemma}\label{cont}
$\mathbb{E}[Q\mid \Theta = \t, A=1]$ and $\mathbb{E}[A\mid \Theta=\t]$ are continuous in $\tau$ for any $\t$.
\end{lemma}

\begin{proof}
Recall that we may write
\begin{equation}\label{eq-dag}
\mathbb{E}[Q\mid \Theta= \t, A=1] = \frac{\P(A=1\mid Q=1,\Theta =\t)\P(Q=1\mid \Theta=\t)}{\P(A=1\mid \Theta = \t)}.
\end{equation}
We see that $\P(Q=1\mid \Theta=\t)$ is trivially continuous in $\tau$, since it is constant. From Theorem \ref{bin}, we have
\begin{equation}\label{eq-ast}
\P(A=1\mid \Theta = \t) = 1 - \P(\Gamma\leq \gamma_1(\tau)\mid \Theta =\t).
\end{equation}

It's easy to see that $\gamma_1(\tau)$ is continuous in $\tau$ since the right-hand side of the inequality in (\ref{eq:g}) is continuous in $\tau$, and the left-hand side is continuous in $\gamma$ by Assumption \ref{as1}. Then, since the right-hand term is a CDF, and $\Gamma$ is a continuous random variable, it takes values in the open interval $(0,1)$ and is continuous. In particular, (\ref{eq-ast}) is never 0 for all $\t$, and so the right-hand-side of (\ref{eq-dag}) is well-defined. Recall that (\ref{eq-ast}) is equal to $s_2$, so $s_2$ is continuous in $\tau$. We can identically state that $\P(A=1\mid Q=1, \Theta=\t)$ is continuous in $\tau$. Thus, $s_1$ is the ratio of two continuous functions with non-zero denominator, and we conclude that $s_1$ is continuous in $\tau$.
\end{proof}

In the above, we have shown that $s_1$ and $s_2$ are continuous functions that are strictly increasing and decreasing in $\tau$ respectively. We next evaluate interactions between the two algorithms and compare scores at given values of $\tau$. We give our first main result that proves the existence and uniqueness of a level of prejudice where $s_1$ and $s_2$ are equal.

\begin{proposition}[Equalizing Prejudice]\label{prop1}
For each $\t$, there exists a unique $\tau^\star\in (-x_u,x_q)$ where $s(\tau^\star):= s_1(\tau^\star)=s_2(\tau^\star)$. We call $\tau^\star$ the \textit{equalizing prejudice} and $s(\tau^\star)$ the \textit{equalized score}.
\end{proposition}

\begin{proof}
Recall that we may write $s_2$ as follows.
\[
\mathbb{E}[A\mid \Theta = \t] = 1- \P(\Gamma\leq \gamma_1(\tau)\mid \Theta = \t).
\]
Then we have:
\[
\lim_{\tau\to -x_{u}} \frac{(1-\pi)(x_u+\tau)}{\pi(x_q-\tau)} = 0.
\]
This implies that $\lim_{\tau\to -x_{u}}\gamma_1(\tau) = -\infty.$ Thus:
\[
\lim_{\tau\to -x_{u}} 1 - \P(\Gamma\leq \gamma_1(\tau) \mid \Theta =\t) = 1.
\]
Recall that $s_1(\t, \tau)$ is given by:
\[
s_1(\t, \tau) = \frac{\P(A=1\mid Q=1,\Theta =\t)\P(Q=1\mid \Theta=\t)}{\P(A=1\mid \Theta = \t)}.
\]
It's easy to see that as $\tau \to -x_{u}$, we have
\[
\P(A=1\mid Q=1,\Theta = \t) = \P(A=1\mid \Theta = \t) = 1.
\]
It follows that
\begin{align*}
\lim_{\tau\to -x_{u}} s_1(\t, \tau) &= \P(Q=1\mid \Theta = \t)
\end{align*}
Next, we evaluate the limit of $s_2$ as $\tau\to x_{q}$. 
\[
\lim_{\tau\to x_{q}} 1 - \P(\Gamma\leq \gamma_1(\tau) \mid \Theta = \t) = 0.
\]
For $s_1$, noting that $A=1\iff \Gamma > \gamma_1(\tau)$, by the law of iterated expectation we get:
\begin{align*}
s_1(\t, \tau) &= \mathbb{E}[Q\mid \Theta =\t, \Gamma > \gamma_1(\tau)]\\
&= \mathbb{E}[\mathbb{E}[Q\mid \Theta = \t, \Gamma]\mid \Gamma > \gamma_1(\tau)]\\
&= \mathbb{E}[\k(\t, \Gamma) \mid \Gamma > \gamma_1(\tau)]
\end{align*}
The condition $A=1$ is equivalent to $\k(\t, \gamma) x_q - (1-\k(\t, \gamma))x_u > \tau.$ For $\tau\to x_q$, the accepted applicants are such that:
\[
\k(\t, \gamma)x_q - (1-\k(\t, \gamma))x_u > x_q\implies \k(\t, \gamma) = 1.
\]
Therefore, we have
\[
\lim_{\tau\to x_{q}}s_1(\t, \tau) = 1.
\]
We have shown that $s_2$ strictly decreases from 1 to 0 over the interval $(-x_u,x_q)$, and $s_1$ strictly increases from $\P(Q=1\mid \Theta=\t)$ to 1 over the interval $(-x_u,x_q)$. By Lemma \ref{cont}, we know that both are continuous, which proves existence. Uniqueness trivially holds by Theorem \ref{bin} and Theorem \ref{RR}.
\end{proof}

\begin{figure}[H]
\centering
\captionsetup{justification=centering}
\resizebox{10cm}{!}{\input{Plots/scoreplot}}
\caption{Predicted scores from both algorithms $s_1$ and $s_2$ for\\an applicant across $\tau\in (-x_u,x_q)$}\label{fig:fig1}
\end{figure}

As the human decision-maker becomes more prejudiced, $s_1$ scores eventually become greater than their respective $s_2$ scores. Put differently, there must exist a unique equalizing level of prejudice $\tau^\star$ where $s_1$ and $s_2$ are equal for each applicant. For all levels of prejudice below $\tau^\star$, $s_2$ produces a greater score than $s_1$ and the opposite for all levels of prejudice above $\tau^\star$.

\begin{corollary}\label{greater}
Fix $\t$. Then, $s_1(\tau) > s_2(\tau)$ if and only if $\tau > \tau^\star$.
\end{corollary}
\begin{proof}
The proof is immediate from evaluating $s_1(\tau)$ and $s_2(\tau)$ for $\tau>\tau^\star$ or $\tau<\tau^\star$ using Theorem \ref{bin} and Theorem \ref{RR}.
\end{proof}

An equivalent statement of Corollary \ref{greater} is $s_1(\tau)<s_2(\tau)$ if and only if $\tau<\tau^\star$. Intuitively, if the human decision-maker is less (more) prejudiced than the equalizing level of prejudice, the algorithm $s_2$ produces greater (smaller) predictions than $s_1$. Figure \ref{fig:fig1} visually demonstrates the claims of Proposition \ref{prop1}, Corollary \ref{greater}, and Corollary \ref{cor1}.

\begin{corollary}\label{cor1}
For $\tau \to -x_{u}$, we have:
\begin{align*}
    s_1(\t, \tau) &= \P(Q=1 \mid \Theta = \t),\quad s_2(\t, \tau) = 1.
\end{align*}
For $\tau \to x_{q}$, we have
\begin{align*}
    s_1(\t, \tau) &= 1,\quad s_2(\t, \tau) = 0.
\end{align*}
\end{corollary}

\begin{proof}
The values are computed in the proof of Proposition \ref{prop1}.
\end{proof}

Corollary \ref{cor1} precisely states the limiting values of $s_1$ and $s_2$ as prejudice approaches its extreme values. That is, $\tau$ approaching $-x_u$ or $x_q$ indicates the extreme cases of partiality or prejudice respectively. To understand the behavior of $s_1$: intuitively, if the human decision-maker hires all applicants, the probability that an applicant is qualified converges to the population's proportion of qualified applicants for that specific $\t$. Conversely, if the human decision-maker's prejudice level approaches $x_q$, only the applicants with the best realizations of $\Gamma$ are accepted, leading probabilities of being qualified to approach 1. The behavior of $s_2$ is more straightforward because it matches the human decision-maker's hiring rule.

Next, we move on to analyze which of the two algorithms are preferred by the firm. We define \textit{regret loss} which succinctly captures how a firm may evaluate the quality of their own hiring ability. Regardless of which of the two algorithms a firm chooses, the procedure for computing regret loss does not vary. Hence, we arbitrarily denote the applicants' scores as $s$ which may either be $s_1$ or $s_2$. With a slight abuse of notation, we associate the score $s_i$ with applicant $i\in J$ when the algorithm $s$ is arbitrary. When discussing the score produced by $s_1$ for applicant $i$, we say $s_{1i}$ and similarly for $s_2$. After an applicant $i\in J$ is matched with a score $s_i$, the firm \textit{ex ante} payoff from applicant $i$ is given by

\begin{equation}\label{eq:6}
    \mathcal{N}(s_i) := s_ix_q - [1-s_i]x_u.
\end{equation}

The first term reflects the gain from making a correct hire, and the second term reflects the loss from making a bad hire. \eqref{eq:6} is precisely the second-stage analogue of the first-stage expected payoff in \eqref{eq:2}. The only difference is that the human decision-maker's posterior $\kappa(\t,\gamma)$ is now replaced by the machine-generated score $s$. Then, similarly to the first stage, we define the indicator $A'(s_i)$ denoting whether $i$ is accepted or not.

\begin{equation}\label{eq:7}
    A'(s_i) := \mathbbm{1}\left\{s_ix_q - [1-s_i]x_u > c\right\}
\end{equation}

where $c\in (-x_u,x_q)$ is some group invariant cutoff.  Indeed, unlike the first-stage, we assume that the firm is unable to discriminate based on taste.\footnote{When an algorithm is being used to make decisions, we assume the existence some oversight agency or regulation that restricts a concrete case of taste-based discrimination.} It's easy to see that $\mathcal{N}(s_i)$ and $A'(s_i)$ are strictly and weakly increasing (decreasing) if and only if $s$ is strictly and weakly increasing (decreasing), respectively. Once the second round of hires is complete, the firm observes the true types of the hires $Q_i$ and realizes \textit{ex post} payoff for each applicant $i$:

\begin{equation}\label{eq:8}
    \mathcal{P}(s_i) := A'(s_i)\cdot\left(Q_ix_q - [1-Q_i]x_u\right).
\end{equation}

$\mathcal{P}$ is precisely the realized gain from each applicant based on their true type. If an applicant was not accepted, regardless of $Q_i$ the firm does not make any gains.

\section{Algorithmic Preferences}\label{sec:4}

After making hiring decisions using the rule in \eqref{eq:7}, the firm observes the difference between expected (\textit{ex ante}) payoff \eqref{eq:6} and realized (\textit{ex post}) payoff \eqref{eq:8}. In the previous section, we did not make any claims about the firm's preferences. In particular, it is unclear whether the firm prefers an algorithm that transmits bias or one that reverses bias. To that end, we define the measure \textit{regret loss} which summarizes a firm's net payoffs upon hiring from a group of disadvantaged applicants indexed by $J=\{1,\hdots, m\}$. Figure \ref{fig:fig2} visually illustrates the proceedings of our two-staged model. One may question the validity of regret loss to measure firm preferences. A major difference between the traditional definition of regret loss and this paper's interpretation is what is being minimized. That is, this paper assumes that the firm does not aim to minimize regret but rather the squared loss from the definitions of $s_1$ and $s_2$. Thus, the firm is rational in the typical sense because it aims to minimize sum of squares---this paper's version of regret loss is merely an artifact of the behavior of the two algorithms. Put differently, regret in this paper summarizes the firm's attitude towards the final outcome rather than firm's focus before and during the hiring procedure.

A succinct illustration of the value of regret loss in the context of this paper is as follows. Suppose the an algorithm $s$ performs better than another algorithm $s'$ in terms of sum of squared errors. However, suppose $s$ makes more mistakes when accepting applicants than rejecting applicant: that is, $s$ hires unqualified applicants more frequently than $s'$. On the other hand, $s'$ may make mistakes in the opposite fashion: qualified applicants are rejected more frequently. In this sense, $s'$ generates less regret loss. When all hires are complete, the firm may observe that $s$ does a better job in terms of prediction accuracy but will ultimately be happier with $s'$. Thus, the purpose of our regret loss to describe firm preferences is clear.

\subsection{Individual Regret}

\begin{defn}[Individual Regret]\label{reg}
The \textit{individual regret} generated by an algorithm $s$ for an applicant $i\in J$ is given by the difference between \textit{ex ante} and \textit{ex post} payoff

\begin{equation*}
    u(s_i) := \mathcal{N}(s_i) - \mathcal{P}(s_i).
\end{equation*}

We suppress the index $i$ when we are discussing a fixed applicant $\t$.
\end{defn}

If $u(s_i)>0$, we say that the firm experiences positive regret from using algorithm $s$ on applicant $i$, where their expectations were overly optimistic compared to their actual payoffs. If $u(s_i) < 0$, we say that the firm experiences negative regret, where their expectations are exceeded, and \textit{ex post} profit is higher than \textit{ex ante} profit. When $u(s_i)=0$, the firm experiences no regret, where reality exactly matches their expectations.

We may now present comparative statics of $u(s_1)$ and $u(s_2)$ over $\tau$ for a fixed $\t$. In particular, can we expect regret loss of an algorithm $s$ to be increasing (decreasing) in $\tau$ if $s$ is increasing (decreasing) in $\tau$? Before we introduce our main result, we give a few statements to guide our proof.

\begin{figure}[t]
\centering
\captionsetup{justification=centering}
\input{Plots/flow}
\caption{An illustration of how the first and second stage produce\\ regret loss for both algorithms}\label{fig:fig2}
\end{figure}

\begin{defn}[Locally Increasing]\label{li}
Let $X\subset \mathbb{R}$. A function $f:X\to \mathbb{R}$ is said to be \textit{locally increasing} at a point $a$ if there exists a $\delta>0$ such that $f(s) < f(a) < f(t)$ whenever $a-\delta < s < a < t < a+\delta$. \textit{Locally decreasing} is defined analogously.
\end{defn}

\begin{defn}[Critical Prejudice]
We define \textit{critical prejudice} of $s$ for an applicant $\t$ as $\tau^d$ where
\[
\tau^d := \inf\{\tau: s(\tau)x_q - (1-s(\tau))x_u > c\}.
\]
We write $\tau_1^d$ and $\tau_2^d$ the critical prejudices for $s_1$ and $s_2$. Similarly, we can define the \textit{critical score} of $s$ as $s(\tau^d)=\b$ where
\[
s(\tau^d) := \frac{x_u+c}{x_q + x_u} = \b.
\]
Here, we have that $s_1(\tau^d_1) = s_2(\tau^d_2) = \b$.
\end{defn}

Intuitively, the critical prejudice of $s_1$ is the level of $\tau$ where the expected payoff from hiring an applicant is exactly $c$, so the firm does not hire. But at any $\tau+\epsilon$, the applicant is hired---and analogously for $s_2$ but with $\tau-\epsilon$. 

\begin{lemma}\label{sli}
Fix $\t$ and $\P(Q=1 \mid \Theta = \t) < \b$. Then $u(s(\tau))$ is discontinuous at $\tau=\tau^d$ for both $s=s_1,s_2$. Moreover, $u(s(\tau))$ is locally increasing if $s=s_1$ and locally decreasing if $s=s_2$ at every $\tau\neq \tau^d$.
\end{lemma}

\begin{proof}
Consider $s=s_1$. The proof for $s_2$ follows similarly. Since $s_1(\tau)$ is strictly increasing and continuous in $\tau$, it must be that $\mathcal{N}(s_1(\tau))$ is strictly increasing and continuous in $\tau$:
\[
\mathcal{N}(s(\tau)) = s(\tau)x_q - (1-s(\tau))x_u = s(\tau)(x_q + x_u) - x_u.
\]

At $\tau^d$, we have that $\mathcal{N}(s(\tau^d)) = \mathcal{N}(\b)$ and $\mathcal{N}(s(\tau^-))<\mathcal{N}(\b)$ for all $\tau^-<\tau^d$. In other words, $A'(s(\tau^+))= 1$ for all $\tau^+ > \tau^d$ and $A'(s(\tau^-))=0$ for all $\tau^- \leq \tau^d$. Using this, we compute $\mathcal{P}$ for $Q=1$ and $Q=0$:
\begin{align*}
    Q=1 &\implies \mathcal{P}(s(\tau^+)) =x_q\\
    Q=0 &\implies \mathcal{P}(s(\tau^+))= -x_u.
\end{align*}
It's easy to see that $\mathcal{P}(s(\tau^-))=0$ for all $\tau^- \leq \tau^d$. Since $x_q,x_u\neq 0$, we have that $\mathcal{P}(s(\tau))$ is discontinuous at $\tau^d$. It follows that that $u(s(\tau)) = \mathcal{N}(s(\tau)) - \mathcal{P}(s(\tau))$ is discontinuous at $\tau^d$: if $Q=0$, then we have
\[
\lim_{\tau\to \tau^{d}_-} u(s(\tau)) = s(\tau^d)\cdot(x_q+x_u) - x_u,\quad \lim_{\tau\to \tau^{d}_+} u(s(\tau)) = s(\tau^d)\cdot (x_q+x_u).
\]

If $Q=1$, then we have
\[
\lim_{\tau\to \tau^{d}_-} u(s(\tau)) = s(\tau^d)(x_q+x_u) - x_u,\quad \lim_{\tau\to \tau^{d}_+} u(s(\tau)) = s(\tau^d)\cdot(x_q+x_u) - x_u - x_q.
\]

In either case, $u(s(\tau))$ is strictly increasing over $\tau\in (-x_u,\tau^d)$ and $\tau \in (\tau^d,x_q)$. Note that for $Q=1$, $u(s(\tau))$ is not strictly increasing over all of $\tau$ since there exists an $\epsilon>0$ small enough such that $u(s(\tau^d-\epsilon)) > u(s(\tau^d + \epsilon))$, i.e., there is a downward jump discontinuity.

Next, fix $\tau\in (-x_u,x_q)\setminus \{\tau^d\}$. Either we have $\tau\in (-x_u,\tau^d)$ or $(\tau^d,x_q)$. Suppose the former since the other interval is shown similarly. Thus, we may pick $\epsilon>0$ small enough such that $(\tau-\epsilon, \tau+\epsilon)\in (-x_u,\tau^d)$. Since $u(s(\tau))$ is strictly increasing over the interval $(-x_u,\tau^d)$, it must be strictly increasing over the interval $(\tau-\epsilon, \tau+\epsilon)$. We conclude that $u(s(\tau))$ is locally increasing everywhere except $\tau^d$ whether $Q=0$ or $Q=1$.
\end{proof}

\begin{remark}
The point $\tau^d$ at which $u(s(\tau))$ is discontinuous need not be the same for $s=s_1$ and $s=s_2$. That is, $\tau_1^d\neq \tau_2^d$ in general. See Lemma \ref{shift} and Lemma \ref{uneq}.
\end{remark}

The physical interpretation for Lemma $\ref{sli}$ and $s_1$ can be understood as follows. If a score for a given applicant is increasing in $\tau$, greater prejudice leads to higher expectations for positive payoffs, i.e., $\mathcal{N}(s_1)$ is strictly increasing in $\tau$. At some level of prejudice $\tau^d$, the predicted score for the applicant becomes too good for the firm to pass up. After hiring, if the applicant is revealed to be truly qualified, the firm does not regret their hiring choice and enjoys a drop in regret loss. Conversely, if the applicant is actually unqualified, the firm experiences a spike in regret in addition to the elevated expectations resultant from higher levels of prejudice. For levels of prejudice beyond $\tau^d$, regret loss increases because the firm is naturally more pleased with a qualified hire with a lower predicted score than a qualified hire with a very high score. Figure \ref{fig:fig3} illustrates an instance of individual regret loss for a qualified and unqualified applicant. Before we continue, we identify a trivial case.

\begin{lemma}\label{trivial}
Fix $\t$. Then, $u(s_1(\tau))$ is discontinuous if and only if $\P(Q=1\mid \Theta = \t) < \b$. On the other hand, $u(s_2(\tau))$ is always discontinuous regardless of $\t$.
\end{lemma}

\begin{proof}
By Corollary \ref{cor1}, the left-hand limit of $s_1(\tau)$ is $\P(Q=1\mid \Theta = \t)$ which we denote by $\phi(\t)$ for convenience. Since $s_1$ is strictly increasing in $\tau$, we have that $s_1(\tau) > \phi(\t)$ for all $\tau$. Recall that $s_1(\tau_1^d) = \b$ and for all $s_1(\tau) > s_1(\tau_1^d)$, we have that $A'(s_1(\tau)) = 1$. Thus, we have $s_1(\tau) > \phi(\t) > \b$ for all $\tau$ and $u(s_1(\tau)) = \mathcal{N}(s_1(\tau)) - x$ where $x\in \{x_q, -x_u\}$ depending on qualification for all $\tau$, so it is continuous. Conversely, by Corollary \ref{cor1}, the image of $s_2(\cdot)$ is (0,1), so $u(s_2(\tau))$ is always discontinuous. 
\end{proof}

In some sense, the trivial case is when $\P(Q=1\mid \Theta = \t) \geq \b$. If the base rate is already bigger than $\b$ we can intuitively believe to be quite large, this must suggest that the applicant is receiving a relatively high score from $s_1$ for any $\tau$. In the following, we avoid this trivial case.

We have shown that the regret generated by $s_1$ behaves in the opposite fashion to $s_2$ solely through \textit{ex ante} payoffs. In particular, \textit{ex ante} payoffs increases almost everywhere for $s_1$ and decreases almost everywhere for $s_2$. This implies that the algorithm of choice molds the firm's expectations as a function of prejudice. An algorithm exhibiting ``bias reversal'' encourages firms to expect more from applicants. On the other hand, an algorithm exhibiting ``bias in, bias out'' leads the firm to approach applicants more cynically as prejudice increases.

\begin{figure}[H]
\centering
\captionsetup{justification=centering}
\resizebox{15cm}{!}{\input{Plots/scoreplot_2}}
\caption{regret loss generated by algorithm $s_1$ for a\\qualified and unqualified applicant}\label{fig:fig3}
\end{figure}

The following lemmas evaluate the interplay between the critical prejudices of $s_1$ and $s_2$, and the relationship between equalized score $s(\tau^\star)$ with critical score $s(\tau^d)$. 

\begin{lemma}\label{shift}
Fix $\t$ with $\P(Q=1\mid \Theta =\t) < \b$. Then $\tau_1^d = \tau_2^d$ if and only if $s( \tau^\star) = \b$.
\end{lemma}

\begin{proof}
By Lemma \ref{trivial}, we know that $\tau_1^d$ exists. Recall that $\b=(x_u+c)/(x_q+x_u)$. From Lemma \ref{sli}, it's easy to see that for either $s_1$ and $s_2$, we have

\[
s(\tau^d)x_q - (1-s(\tau^d))x_u = c.
\]

For sufficiency, suppose $\tau_1^d = \tau_2^d$. Then

\[
s_1(\tau_1^d) = \b,\quad s_2(\tau_2^d) = \b \implies s_1(\tau_1^d) = s_2(\tau_2^d) = \b.
\]

By uniqueness of $\tau^\star$, from Proposition \ref{prop1}, we have $\tau_1^d = \tau_2^d = \tau^\star.$ Thus, $s(\tau^\star) = \b$. For necessity, we have $s( \tau^\star) = \b$. Thus
\[
s_1(\tau^\star) x_q - (1-s_1(\tau^\star))x_u = c.
\]
Since $\tau_1^d$ is unique, it must be that $\tau_1^d = \tau^\star$. An identical argument can be made for $\tau_2^d = \tau^\star$. Thus $\tau_1^d = \tau_2^d$.
\end{proof}

The above lemma states that the critical prejudices from both algorithms are equal if and only if the critical score is equal to equalized prejudice. We can further characterize this relationship when critical prejudices are not equal.

\begin{lemma}\label{uneq}
Fix $\t$ with $\P(Q=1\mid \Theta = \t) < \b$. Then $s(\tau^\star) < \b$ if and only if $\tau_1^d > \tau_2^d$.
\end{lemma}

\begin{proof}
Let $\tau_1^d > \tau_2^d$. It follows that $s_1(\tau_2^d) < s_2(\tau_2^d)$ and $s_1(\tau_1^d) > s_2(\tau_1^d)$ since $s_1(\tau_1^d)=s_2(\tau_2^d)$. Thus, $\tau_2^d < \tau^\star < \tau_1^d$ by Corollary \ref{greater}. Then, $s_2(\tau_2^d) > s_2(\tau^\star) = s_1(\tau^\star)$ by applying $s_2$ which is strictly decreasing. We have $\b > s(\tau^\star)$ as was to be shown. For necessity, suppose $s(\tau^\star) < \b$. Then, $s(\tau^\star) < s_1(\tau_1^d) = s_2(\tau_2^d)$. In other words, $s_1(\tau^\star) < s_1(\tau_1^d)$ and by Theorem \ref{RR}, we have $\tau^\star < \tau_1^d$. A similar argument can be given for $\tau_2^d < \tau^\star$.
\end{proof}

Indeed, Lemma \ref{uneq} is equivalent to saying $s( \tau^\star) \geq \b$ if and only if $\tau_1^d \leq \tau_2^d$. Combining this with Lemma \ref{shift}, we have that the inequalities are strict.

\begin{lemma}\label{between}
Fix $\t$ with $\P(Q=1\mid \Theta =\t) < \b$. Then, $\tau^\star \in (\tau_1^d, \tau_2^d)$ if $\tau_2^d > \tau_1^d$. If $\tau_2^d < \tau_1^d$, then $\tau^\star\in (\tau_2^d, \tau_1^d)$.
\end{lemma}

\begin{proof}
Let $\tau_1^d < \tau_2^d$. Recall that for all $\tau \in (\tau_1^d, \tau_2^d)$, we have that $A'(s_1(\tau)) = A'(s_2(\tau))= 1$. Thus, for $\tau \in (\tau_1^d,\tau_2^d)$:
\begin{align*}
Q = 0&\implies u(s_1(\tau)) = \mathcal{N}(s_1(\tau)) + x_u,\quad u(s_2(\tau)) = \mathcal{N}(s_2(\tau)) + x_u\\
Q = 1&\implies u(s_1(\tau)) = \mathcal{N}(s_1(\tau)) - x_q,\quad u(s_2(\tau)) = \mathcal{N}(s_2(\tau)) - x_q.
\end{align*}
In either case, by Lemma \ref{cont}, we have that $u(s_1(\tau)) - u(s_2(\tau))$ is continuous in $\tau$ in this open interval. Observe that
\[
\lim_{\tau \to \tau_{1+}^d} u(s_1(\tau)) = \b x_q - (1-\b)x_u - x = \lim_{\tau \to \tau_{2-}^d} u(s_2(\tau))
\]
where $x\in \{x_q, -x_u\}$ depending on qualification. Since $u(s_1(\tau))$ and $u(s_2(\tau))$ are strictly increasing and decreasing, respectively in $\tau \in (\tau_1^d, \tau_2^d)$, we have that
\[
\lim_{\tau \to \tau_{1+}^d} u(s_1(\tau)) - u(s_2(\tau)) < 0,\quad \lim_{\tau \to \tau_{2-}^d} u(s_1(\tau)) - u(s_2(\tau)) > 0.
\]
By choosing $\epsilon > 0$ small enough, these two inequalities hold at the boundary of the closed interval $[\tau_1^d + \epsilon, \tau_2^d - \epsilon]$, and by the intermediate value theorem, there must exist a $\tau$ such that:
\[
u(s_1(\tau)) = u(s_2(\tau)),\quad \tau \in (\tau_1^d, \tau_2^d).
\]
By Proposition \ref{prop1}, we get uniqueness of this $\tau$, and so this $\tau$ is precisely $\tau^d$. The proof for $\tau_1^d > \tau_2^d$ is identical.
\end{proof}

Lemma \ref{between} states that the equalizing prejudice is in-between the critical prejudices for each $\t$. In addition, noting the montonicity of both algorithms, we have that, between the critical prejudices, either both of the algorithms accept the applicant or both reject the applicant, depending on which critical prejudice is larger. Finally, we evaluate the preferences of the firm based on the regret loss generated by each algorithm.

\begin{defn}[Algorithmic Preference]\label{algpref}
For an individual application $\t$ and fixed $\tau$, we say that the firm \textit{prefers an algorithm} $s$ to $s'$ for $\t$ when $u(s(\t, \tau)) < u(s'(\t, \tau))$.
\end{defn}

\begin{lemma}\label{prop2}
For each $\t$ with $\P(Q=1\mid \Theta = \t) < \b$, we have that $\lim_{\tau\to \tau^{\star}_+} u(s_1(\tau)) = \lim_{\tau\to \tau^{\star}_-} u(s_2(\tau))$.
\end{lemma}

\begin{proof}
Fix $\t$. Recall that $s(\tau^d) = \b$ and $u(s)$ is discontinuous at $\b$. Consider the case $\tau^d_1\neq \tau^d_2$. By Lemma \ref{shift}, we have $s_1(\tau^\star) = s_2(\tau^\star) \neq \b$. Thus, $u$ is continuous at $s(\tau^\star)$ by Lemma \ref{cont}, and we have
\[
\lim_{\tau \to \tau_+^\star}u(s_1(\tau)) = u(s_1(\tau^\star)) = u(s_2(\tau^\star)) = \lim_{\tau \to \tau_-^\star} u(s_2(\tau)).
\]
For the other case $\tau^d_1= \tau^d_2$, and we have $s_1(\tau^\star) = s_2(\tau^\star) = \b$ by Lemma \ref{shift}. Consider the subcase where $Q=1$. It's easy to see
\[
\lim_{\tau \to \tau^\star_-} u(s_2(\tau)) + x_q = \lim_{\tau \to \tau^\star_+} u(s_2(\tau)),\quad \lim_{\tau\to \tau_-^\star}u(s_1(\tau)) = \lim_{\tau \to \tau_+^\star} u(s_1(\tau)) + x_q.
\]
Since $s_1(\tau)$ is strictly increasing in $\tau$, we have that $s_1(\tau)\to \b$ as $\tau \to \tau_-^d$. Similarly, since $s_2(\tau)$ is strictly decreasing in $\tau$, we have that $s_2(\tau) \to \b$ as $\tau \to \tau_+^d$. Setting the two equal gets our result. For $Q=0$, we see
\[
\lim_{\tau \to \tau_-^\star} u(s_2(\tau)) - x_u = \lim_{\tau \to \tau_+^\star} u(s_2(\tau)) ,\quad \lim_{\tau \to \tau_-^\star} u(s_1(\tau)) = \lim_{\tau \to \tau_+^\star} u(s_1(\tau)) - x_u.
\]
We can similarly see that $\lim_{\tau \to \tau_+^\star} u(s_2(\tau)) = \lim_{\tau \to \tau_-^\star} u(s_1(\tau))$. Again combining the two equations we get our result.
\end{proof}

\begin{remark}
Indeed, we can equivalently show that $\lim_{\tau\to \tau_-^\star} u(s_1(\tau)) = \lim_{\tau \to \tau_+^\star} u(s_2(\tau))$, by plugging the identity in Lemma \ref{prop2} into the defining equations in the proof.
\end{remark}

In other words, Lemma \ref{prop2} states that there exists at least one level of prejudice where the regret induced by both algorithms is equal when $\tau^d_1\neq \tau^d_2$. Otherwise, we can be sure there exists a point where the limit of the regret induced by both algorithms are equal as $\tau$ approaches this point. This result is clearly driven by Proposition \ref{prop1} where this point is exactly the equalizing prejudice. However, it is less clear if an analogue of Corollary \ref{greater} holds for individual regret. The following statements characterize the connection between individual regret and algorithmic predictions.

\begin{proposition}\label{analogue}
Fix $\t$ with $\P(Q=1\mid \Theta = \t) < \b$. If $\tau \in (\tau_1^d,\tau_2^d)$ or $(\tau_2^d, \tau_1^d)$ then $s_1(\tau) > s_2(\tau)$ if and only if the firm prefers $s_2$ to $s_1$. 
\end{proposition}

\begin{proof}
We show the claim assuming $\tau_1^d < \tau_2^d$. The other inequality is shown identically. As an immediate consequence of Lemma \ref{between}, we have that $\tau^\star\in (\tau_1^d, \tau_2^d)$ and $A'(s_1(\tau)) = 1 = A'(s_2(\tau))$. Thus, for all $\tau\in (\tau_1^d, \tau^\star)$, we have that
\[
u(s_1(\tau)) < u(s_2(\tau)) \iff \mathcal{N}(s_1(\tau))-x < \mathcal{N}(s_2(\tau)) -x \iff s_1(\tau) < s_2(\tau)
\]
where $x\in \{x_q, -x_u\}$ depending on qualification. When $\tau_1^d > \tau_2^d$, everything proceeds identically except with $x=0$. If $\tau_1^d = \tau_2^d$, the claim follows from Lemma \ref{prop2}.
\end{proof}

Proposition \ref{analogue} demonstrates that the two individual regret curves emulate the behavior of the score curves as shown in Figure \ref{fig:fig1} in the interval $(\tau_1^d, \tau_2^d)$ or $(\tau_2^d, \tau_1^d)$ which contains the equalizing prejudice $\tau^\star$. Importantly, Proposition \ref{analogue} together with Lemma \ref{between} imply that when both algorithms give scores that both lead to acceptance or rejection, then analogous claims of Proposition \ref{prop1} hold.

Casually speaking, this result suggests that if both algorithms accept/reject an applicant, the firm should opt to believe the algorithm that gives the more conservative score in order to minimize regret. However, for $\tau$ outside of this interval, it is unclear which of the algorithms the firm should believe. The following corollary shows a correspondence between Proposition \ref{prop1} to the preference case for unqualified applicants.

\begin{corollary}\label{better}
For each applicant $\t$ with $Q=0$ and $\P(Q=1\mid \Theta = \t) < \b$, we have that $s_1(\tau) > s_2(\tau)$ if and only if the firm prefers $s_2$ to $s_1$.
\end{corollary}

\begin{proof}
Fix $\t$. By Corollary \ref{greater}, we have that $\tau < \tau^\star$ if and only if $s_1(\tau) < s_2(\tau)$. Observe that
\begin{align*}
    s_1(\tau) < s_2(\tau) &\iff \mathcal{N}(s_1(\tau)) < \mathcal{N}(s_2(\tau))\\
    &\iff \mathcal{N}(s_1(\tau)) + x_u < \mathcal{N}(s_2(\tau)) + x_u.
\end{align*}
We consider three cases: $\tau_1^d < \tau_2^d$, $\tau_1^d>\tau_2^d$, and $\tau_1^d = \tau_2^d$. In the first case, consider $\tau$ in four possible intervals: $(-x_u, \tau_1^d)$, $(\tau_1^d, \tau^\star)$, $(\tau^\star, \tau_2^d)$, and $(\tau_2^d, x_q)$. For $\tau \in (\tau_1^d,\tau_2^d)$, Proposition \ref{analogue} holds. For $\tau \in (-x_u,\tau_1^d)$, we have that $u(s_1(\tau)) = \mathcal{N}(s_1(\tau))$ and $u(s_2(\tau)) = \mathcal{N}(s_2(\tau)) + x_u$, and so the claim holds. The second case shown analogously. Lastly, in the third case, we have two intervals $(-x_u, \tau^\star)$ and $(\tau^\star, x_q)$ by Lemma \ref{shift}. For $\tau$ in the first interval, we have $u(s_1(\tau)) = \mathcal{N}(s_1(\tau))$ and $u(s_2(\tau)) = \mathcal{N}(s_2(\tau)) + x_u$, so the claim holds. For $\tau$ in the second interval, we have $u(s_1(\tau)) = \mathcal{N}(s_1(\tau)) + x_u$ and $u(s_2(\tau)) = \mathcal{N}(s_2(\tau))$ and so the claim holds.
\end{proof}

\begin{remark}
Equivalently, Corollary \ref{better} suggests that $s_1(\tau) < s_2(\tau)$ if and only if the firm prefers $s_1$ to $s_2$. Moreover, if $s_1(\tau) = s_2(\tau)$, the firm is indifferent between the two algorithms.
\end{remark}

Corollary \ref{better} precisely draws a connection between higher scores and algorithmic preference for unqualified applicants. Indeed, for unqualified applicants, the firm would naturally prefer the algorithm that returns the lowest scores possible. The following corollary states that a similarly nice claim does not hold for applicants with $Q=1$. In particular, there need not be just one point where $u(s_1(\tau))$ and $u(s_2(\tau))$ are equal.

\begin{corollary}\label{qualc}
Fix $\t$ with $Q=1$. Suppose $\P(Q=1\mid \Theta = \t) < \b$ and $\tau_1^d \neq \tau_2^d$.
\begin{enumerate}
	\item If $x_q-c > x_u$ and $\mathcal{N}(\P(Q=1\mid \Theta = \t)) <0$, there exist exactly three $\tau\in (-x_u, x_q)$ such that $u(s_1(\tau)) = u(s_2(\tau))$.
	\item If $x_q-c > x_u$ and $\mathcal{N}(\P(Q=1\mid \Theta = \t)) \geq 0$, there exists exactly two such $\tau \in (-x_u,x_q)$.
	\item If $x_q - c \leq x_u$, there exists at least one such $\tau$ and at most three.
\end{enumerate}
\end{corollary}

\begin{proof}
For convenience, again denote $\P(Q=1\mid \Theta = \t) = \phi$. We assume $\tau_1^d > \tau_2^d$, since the converse is identically shown. We compute
\[
\lim_{\tau \to -x_{u}} u(s_1(\tau)) = \mathcal{N}(\phi) = \phi x_q - (1-\phi)x_u,\quad \lim_{\tau \to -x_{u}} u(s_2(\tau)) = 0.
\]
Since $\mathcal{N}(\b) = c$, it follows that $\mathcal{N}(\phi) < c$. The minimum of $\mathcal{N}(\phi)$ occurs at $\phi=0$ where $\mathcal{N}(0)=-x_u$. Thus, we have
\[
\lim_{\tau \to \tau_{2-}^d}\mathcal{N}(\phi) > -x_u,\quad \lim_{\tau \to \tau_{2-}^d} u(s_2(\tau)) = \mathcal{N}(\b) - x_q = c-x_q.
\]
In the interval $(-x_u, \tau_{2}^d)$, we have that $u(s_1(\tau)) - u(s_2(\tau))$ is continuous since $\tau_2^d < \tau_1^d$. Since we assume $\mathcal{N}(\phi) < 0$, this difference is negative at the far left. Since $c-x_q > x_u$, the difference is positive approaching $\tau_2^d$ from the left. Moreover, the difference is strictly increasing since $u(s_1(\tau))$ and $-u(s_2(\tau))$ are strictly increasing. Thus, there exists a unique $\tau\in (-x_u,\tau_2^d)$ such that $u(s_1(\tau)) = u(s_2(\tau))$. We can identically show that there exists a unique $\tau\in (\tau_1^d, x_q)$ upon recognizing that
\[
\lim_{\tau \to x_{q_-}}u(s_1(\tau)) = 0,\quad \lim_{\tau \to x_{q_-}}u(s_2(\tau)) = -x_u,
\]
and an identical argument holds. By Lemma \ref{prop2}, there must exist $\tau^\star\in (\tau_2^d,\tau_1^d)$. Thus, in this case, there are exactly three $\tau$. If we assume $\mathcal{N}(\phi)\geq 0$, there no longer exists $\tau \in (-x_u,\tau_2^d)$ since $u(s^1(\tau)) > u(s^2(\tau))$ for all $\tau$ in this interval. However, the argument for $\tau$ in the other two intervals still holds. In the last case, we cannot guarantee the existence of $\tau\in (-x_u,\tau_2^d)\cup (\tau_1^d,x_q)$, but there must exist at least one by Lemma \ref{prop2}.
\end{proof}

Corollary \ref{qualc} precisely states that there can be at most three $\tau$ where the regret from $s_1$ and $s_2$ are equal for $Q=1$. Note that when $\tau_1^d=\tau_2^d$, by Lemma \ref{prop2} and Proposition \ref{analogue}, there exists exactly zero $\tau$. But, for all $\tau<\tau^\star$, the firm prefers $s_1$ and the opposite for $\tau > \tau^\star$. Figure \ref{fig:fig4} visually demonstrates the main claims of this section where $c>0$ indicates a more conservative firm. It can be visually seen that $x_q-c > x_u$, which leads to three unique $\tau$ in the left-hand plot. As was shown, it is generally unclear for a qualified applicant which algorithm should be preferred at a $\tau$, since there are several points where regret is equal. Thus, it may be more meaningful to evaluate an aggregate measure of regret.

\begin{figure}[H]
\centering
\captionsetup{justification=centering}
\resizebox{15cm}{!}{\input{Plots/regretplots}}
\caption{regret loss generated by $s_1$ and $s_2$ for a fixed $\t$}\label{fig:fig4}
\end{figure}

\subsection{Average Regret}

Now that the behavior of regret loss generated by an individual applicant is well understood, we wish to evaluate the regret generated across an entire set of applicants $J$. For the purposes of this section, we rephrase our framework as follows. Denote $|J| = m$ and let $\{Q_i, \Theta_i\}_{i\in J}$ be pairs of independent and identically distributed random variables with the data generating process specified as per Section \ref{sec:3}. In this section, we let $m$ increase as more applicants apply to the firm. A succinct measure of total regret then is \textit{average regret}, which is easily defined:

\begin{defn}[Average Regret]
A firm's \textit{average regret} generated by an algorithm $s(\tau)$ and set of applicants $J$ is denoted by $U(s(\tau))= \frac{1}{m}\sum_{i \in J}u(s_i(\tau))$.
\end{defn}

We discuss $U$ as a function of $\tau\in (-x_u, x_q)$ as usual and examine the comparative statics as $\tau$ increases. An immediate claim follows from the definition of $U(s)$.

\begin{lemma}\label{every}
Average regret $U(s(\tau))$ is locally increasing (decreasing) almost everywhere for $s=s_1$ ($s=s_2$).
\end{lemma}
\begin{proof}
Since $U(s(\tau))$ is defined as a sum of locally increasing functions with either zero or one discontinuity in each, we have that $U(s(\tau))$ is discontinuous at, at most, $m$ unique points.
\end{proof}

To guide our main results, we define two possible states of the environment, which the firm is able to recognize.

\begin{defn}[Regular]
	We say that the environment is \textit{regular} when:
	\begin{equation}\label{cond}
		\frac{x_u}{x_q} > \frac{\frac{\pi}{1-\pi}\P(\mathbb{E}[Q\mid \Theta_1] \leq \b\mid Q=1) - 1}{\P(\mathbb{E}[Q\mid \Theta_1] \leq \b\mid Q=0) + 1}
	\end{equation}
Otherwise, we say the environment is \textit{irregular}.
\end{defn}

\begin{remark}
	A regular environment is defined with a strict inequality in \eqref{cond} to ensure that the average regret loss generated by $s_1$ is strictly less than that of $s_2$ for $\tau \to -x_u$. This implies that there is a $\tau\in (-x_u,x_q)$ where the average regret are equal. Conversely, for an irregular environment, the weak inequality implies that as $\tau\to -x_u$ we allow $U(s_1(\tau)) = U(s_2(\tau))$, so there does not exist a $\tau\in (-x_u,x_q)$ where the average regret are equal. This arrangement is made clear in Corollary \ref{last}.
\end{remark}

It is easily seen from \eqref{cond} that if the environment is irregular, then $\pi> 0.5$ because the numerator of the right-hand-side of \eqref{cond} must be greater than 0. In other words, an irregular environment has a high baseline probability $\pi$ of applicants being qualified. Conversely, if $\pi \leq 0.5$, the environment is regular. Indeed, the environment is characterized as regular when there is a reasonably low proportion of qualified applicants, which is arguably common in most competitive labor markets. We note that even if $\pi > 0.5$, the environment may still be regular under specific circumstances. For instance, when $c$ is negative and large in magnitude, which suggests that the firm is strongly resistant to risk. In such a scenario, we would expect that the firm has quite a low $x_u$. A prime example of such a scenario would be a job that does not require high levels of skill: this would intuitively explain a large $\pi$, a low $c$, and a low $x_u$. While it could be argued that this specific scenario is not ``regular'' in the colloquial sense, we choose this terminology to simplify discussions. In any case, it is assumed that the firm has information on the joint distribution of $Q$ and $\Theta$, meaning that the firm is capable of determining whether the hiring environment is regular or irregular. In both cases, we give theoretical guarantees for algorithmic preferences.

\begin{proposition}\label{prop3}
As $\tau\to -x_{u}$ , the average regret generated by $s_2$ is greater than that of $s_1$ if and only if the environment is regular. That is:
\[
\lim_{\tau \to -x_{u}} U(s_2(\tau)) - U(s_1(\tau)) \to \mathbb{E}\left[\lim_{\tau \to -x_{u}} u(s_2(\tau)) - u(s_1(\tau))\right] > 0
\]
almost surely as $m\to \infty$.
\end{proposition}

\begin{proof}
Recall that $u(s_2(\tau)) \to 0$ as $\tau\to -x_{u}$ for any $\t$ with $Q=1$ and $u(s_2(\tau)) \to x_q + x_u$ as $\tau\to -x_{u}$ for any $\t$ with $Q=0$. We have that
\[
\lim_{\tau \to -x_{u}} U(s_2(\tau)) = \sum_{i=1}^{m} \mathbbm{1}\{Q_i = 0\}(x_q+x_u) / m.
\]
Define the sequence of random variables $Z_i = \mathbbm{1}\{Q_i = 0\}(x_q+x_u)$. We know that $Q_i$ are independent and identically distributed, so $Z_i$ are as well. Let $S_m = \sum_{i=1}^{m} Z_i$. Compute $\mathbb{E} Z_i = \P(Q_i = 0)(x_q+x_u) = (1-\pi)(x_q+ x_u)$ which is obviously finite. By the strong law of large numbers, we have
\[
\lim_{\tau \to -x_{u}} U(s_2(\tau)) \to (1-\pi)(x_q+x_u)
\]
almost surely. This is trivially equal to the expectation of $u(s_2(\tau))$ as $\tau \to -x_{u}$. Next, recall that $\lim_{\tau \to -x_{u}} u(s_{1}(\tau)) = \mathcal{N}(\phi(\Theta)) - \mathcal{P}(\phi(\Theta))$. Plugging this in, we get
\[
\lim_{\tau \to -x_{u}} U(s_{1}(\tau)) = \frac{1}{m}\left(\sum_{i=1}^m \phi(\Theta_i)x_q - (1-\phi(\Theta_i))x_u - \mathbbm{1}\{\phi(\Theta_i) \geq \b\}(Q_ix_q - (1-Q_i)x_u)\right).
\]
Let $W_i$ denote the summand above. Again applying the strong law of large numbers, we easily see that $W_i$ has finite mean and so:
\[
\lim_{\tau \to -x_u} U(s_1(\tau)) \to \mathbb{E}\left[\lim_{\tau \to -x_u}u(s_1(\tau))\right]
\]
almost surely. And so the first claim has been shown. For the equivalence, we first compute the expectation using the law of iterated expectation:
\begin{align*}
	\mathbb{E}\left[\lim_{\tau \to -x_u}u(s_1(\tau))\right] &= x_q\pi - x_u(1-\pi) - x_q\P(\phi(\Theta_1) > \b, Q=1) + x_u\P(\phi(\Theta_1) > \b, Q=0)\\
	&= x_q\pi\P(\phi(\Theta_1) \leq \b\mid Q=1) - x_u(1-\pi)\P(\phi(\Theta_1) \leq \b \mid Q=0)
\end{align*}
We wish to show that this is less than $(1-\pi)(x_q+x_u)$. Denoting $\a_q := \P(\phi(\Theta_1) \leq \b\mid Q=1)$ and $\a_u:=\P(\phi(\Theta_1) \leq \b\mid Q=1)$, we proceed as follows:
\begin{align*}
	x_q\pi\a_q - x_u(1-\pi)\a_u < (1-\pi)(x_q+x_u) &\iff \frac{\pi}{1-\pi}x_q\a_q - x_u\a_u < x_q+x_u\\
	&\iff \frac{\pi}{1-\pi}x_q\a_q - x_q < x_u\a_u + x_u\\
	&\iff \frac{x_u}{x_q} > \frac{\frac{\pi}{1-\pi}\a_q- 1}{\a_u + 1}
\end{align*}
where the last line is true by regularity.
\end{proof}

Proposition \ref{prop3} tells us that in the extreme case of partiality, the firm on average will prefer $s_1(\tau)$ to $s_2(\tau)$ as time goes on. In particular, since the average regret for $s_1(\tau)$ is negative and $s_2(\tau)$ is positive, this further suggests that the firm will be happy with $s_2$ and unhappy with $s_1$.

\begin{proposition}\label{prop4}
As $\tau\to x_{q}$, the average regret generated by $s_1$ is greater than that of $s_2$. That is:
\[
\lim_{\tau \to x_{q}} U(s_2(\tau)) - U(s_1(\tau)) \to \mathbb{E}\left[\lim_{\tau \to x_{q}} u(s_2(\tau)) - u(s_1(\tau))\right] < 0
\]
almost surely as $m\to \infty$.
\end{proposition}

\begin{proof}
Note that $\lim_{\tau \to x_{q}}u(s_2(\tau)) = -x_u$ for all $\Theta_i$, deterministically. Thus:
\[
\lim_{\tau \to x_{q}} U(s_2(\tau)) = -x_u,\quad \mathbb{E}\left[\lim_{\tau\to x_{q}} u(s_2(\tau))\right] = -x_u.
\]
For the other term, we recognize that $\lim u(s_1(\tau))=x_q+x_u$ with probability $\P(Q=0)$ and $0$ with $\P(Q=1)$. We again use the strong law to see that
\[
\lim_{\tau \to x_{q}} U(s_1(\tau)) = \mathbb{E}\left[\lim_{\tau \to x_{q}} u(s_1(\tau))\right] = (1-\pi)(x_q+x_u).
\]
Since this is positive and $-x_u$ is obviously negative, their difference is negative.
\end{proof}

In contrast to Proposition \ref{prop3}, Proposition \ref{prop4} states that in the extreme case of prejudice, the firm prefers $s_2$ to $s_1$, with average regret generated being negative and positive respectively as time goes on. Finally, we may compare average regret as $m\to \infty$, with the help of a few remarks.

\begin{lemma}\label{atoms2}
$\tau^d_2(\Theta)$ is a non-atomic random variable for any $\tau\in (-x_u,x_q)$.
\end{lemma}

\begin{proof}
Recall that
\[
s_2(\t, \tau) = \P(A=1 \mid \Theta = \t) = \P(\Gamma > \gamma_1(\t) \mid \Theta = \t).
\]
As $\t$ increases, we see that $\gamma_1(\t)$ continuously decreases, and since $\Gamma$ is a continuous random variable, we have that $s_2(\t, \tau)$ continuously increases in $\t$. Recall the definition of $\tau_2^d(\t)$:
\[
s_2(\t, \tau^d_2(\t)) = \b.
\]
Thus, for $\t+\epsilon > \t$, we have that $s_2(\t+\epsilon, \tau^d_2(\t)) > \b$ which implies that $\tau^d_2(\t+\epsilon) > \tau^d_2(\t)$ since $s_2(\tau)$ is strictly decreasing in $\tau$. Thus, $\tau_2^d(\t)$ is strictly increasing in $\t$, and since $\Theta$ is a continuous random variable, we conclude that $\tau_2^d(\Theta)$ is also a continuous random variable, i.e., non-atomic.  
\end{proof}

While $\tau_2^d(\Theta)$ is trivially non-atomic, it is not the case for $\tau_1^d(\Theta)$ based on our model. Recall that we can write $s_1(\t) = \P(Q=1\mid \Theta = \t, \Gamma > \gamma_1(\t))$. As $\t$ increases, we see that $\gamma_1(\t)$ decreases, leading to possibly smaller realizations of $\Gamma$. Thus, we give an important, realistic assumption which leads to our final claim.

\begin{assumption}
$s_1(\Theta)$ is a non-atomic random variable for any $\tau\in (-x_u,x_q)$.
\end{assumption}

In other words, we suppose that for a fixed $\tau$, changes in $\t$ must lead to changes in $s_1(\t)$. Indeed, the economic intuition behind this assumption states that applicants with stronger or weaker profiles should receive different scores from an algorithm that predicts qualification. This assumption is benign because $s_1(\Theta)$ is atomic only under very restrictive mathematical circumstances: if and only if there is some interval where $s_1(\t)$ is constant in $\t$. To see why this does not occur in general, recall that $s_1$ is given by:
\[
s_1(\t) = \mathbb{E}[\k(\t, \Gamma) \mid \Gamma > \gamma_1(\t)],\quad \gamma_1(\t) \equiv \inf\left\{\gamma : \frac{h_q(\t, \gamma )}{h_u(\t, \gamma)} > \frac{(1-\pi)(x_u+\tau)}{\pi(x_q-\tau)}\right\}.
\]
By Assumption \ref{as1}, we know that the likelihood ratio is strictly increasing in $\t$ and $\gamma$ which means that $\gamma_1(\t)$ is strictly decreasing in $\t$. Moreover, we know that $\k(\t,\Gamma)$ is strictly increasing in $\t$ and $\gamma$. Thus, $s_1(\t)$ being constant in $\t$ corresponds to the expectation of the posterior distribution remaining constant against competing forces of $\t$ increasing and the condition $\Gamma > \gamma_1(\t)$ admitting smaller observations of $\Gamma$. It is clear such a scenario would only occur under extreme assumptions imposed on the joint distribution of $\Theta$ and $\Gamma$, and we can be convinced such an event is remarkably uncommon in reality. Then, we can give an analogous statement to Lemma \ref{atoms2}.

\begin{lemma}\label{atoms1}
$\tau_1^d(\Theta)$ is a non-atomic random variable for any $\tau\in (-x_u,x_q)$.
\end{lemma}

\begin{proof}
Since $s_1(\t, \tau_1^d(\t)) = \b$, it follows that for all $\t+\epsilon$ with $\epsilon\neq 0$, we have $s_1(\t+\epsilon, \tau_1^d(\t)) \neq \b$. Thus, $\tau_1^d(\t) \neq \tau_1^d(\t+\epsilon)$ for all $\t$ since $s_1$ is strictly changing in $\tau$.
\end{proof}

\begin{theorem}\label{main}
For a regular environment, as $m\to \infty$, there exists a unique $\Bar{\tau} \in (-x_u,x_q)$ almost surely such that $U(s_1(\Bar{\tau})) = U(s_2(\Bar{\tau}))$.
\end{theorem}
\begin{proof}
Let $\Tilde{\tau}$ be a point such that $U(s(\Tilde{\tau}))$ is discontinuous. Define the random variable
\[
T(\Tilde{\tau}) = \lim_{\tau\to \Tilde{\tau}_-} U(s_1(\tau)) -  \lim_{\tau\to \Tilde{\tau}_+} U(s_1(\tau)).
\]
The value of $T(\Tilde{\tau})$ depends on whether for each $\Theta_i$ with $s_1(\tau_1^d(\Theta_i))=\b$, we have $Q_i=1$ or $0$. Clearly, we have
\[
T(\Tilde{\tau}) = \sum_{\{i : \tau_1^d(\Theta_i) = \Tilde{\tau}\}} Q_i x_q - (1-Q_i)x_u.
\]
From Lemma \ref{atoms1}, we have that
\[
\P(|\{i : \tau_1^d(\Theta_i) = \Tilde{\tau}\}| > 1) = 0.
\]
In other words, there is only one $\Theta_i$ almost surely such that $\tau_1^d(\Theta_i) = \Tilde{\tau}$. Rewriting, and dividing by $m$, we have
\begin{align*}
    T(\Tilde{\tau}) / m &= \left(\sum_{\{i : \tau_1^d(\Theta_i) = \Tilde{\tau}\}} Q_ix_q - (1-Q_i)x_u\right) / m\\
    &\overset{a.s.}{=} Q_ix_q - (1-Q_i)x_u / m \to 0
\end{align*}
as $m\to \infty$. But note that $T(\Tilde{\tau})$ is defined as the height of the jump discontinuity at $\Tilde{\tau}$ of $m\cdot U(s_1(\tau))$. In other words, the height of the jump discontinuities of $U(s_1(\tau))$ approach 0, and $U(s_1(\tau))$ is continuous almost surely as $m\to \infty$. Moreover, by Lemma \ref{every}, since $U(s_1(\tau))$ is continuous at the limit and locally increasing almost everywhere, indeed $U(s_1(\tau))$ is strictly increasing at the limit. An analogous proof can be given for $U(s_2(\tau))$.

By Theorem \ref{prop3}, Theorem \ref{prop4}, and the intermediate value theorem, as $m\to \infty$ there exists $\Bar{\tau}\in (-x_u,x_q)$ almost surely such that
\[
\mathbb{E}\left[u(s_2(\Bar{\tau})) - u(s_1(\Bar{\tau}))\right] = 0
\]
and uniqueness follows from the fact that $U(s_1(\tau))$ and $U(s_2(\tau))$ are strictly increasing and decreasing respectively in $\tau$ as $m\to \infty$.
\end{proof}

Theorem \ref{main} bridges the claim of Proposition \ref{prop1} to the case where firm preferences are of interest. In the same way there exists an equalizing prejudice, there exists $\Bar{\tau}$ where average regret generated by $s_1$ and $s_2$ are equal as $m\to \infty$. Then, we give a similar remark to Corollary \ref{greater}.

\begin{corollary}\label{gregret}
For a regular environment, as $m\to \infty$, $U(s_1(\tau)) < U(s_2(\tau))$ almost surely if and only if $\tau < \Bar{\tau}$. 
\end{corollary}
\begin{proof}
As $m\to \infty$, Theorem \ref{main} shows that $U(s_1(\tau))$ and $U(s_2(\tau))$ are strictly increasing and decreasing respectively and continuous almost surely. Thus, for $\tau < \Bar{\tau}$, we have as $m\to \infty$
\[
U(s_1(\tau)) < U(s_1(\Bar{\tau})) = U(s_2(\Bar{\tau})) < U(s_1(\tau)).
\]
\end{proof}

Equivalent to Corollary \ref{gregret}, as $m\to \infty$, we have that $U(s_1(\tau)) > U(s_2(\tau))$ if and only if $\tau > \Bar{\tau}$ almost surely. Theorem \ref{main} and Corollary \ref{gregret} exactly show that sufficiently prejudiced firms with $\tau > \Bar{\tau}$ will be less happy with $s_1(\tau)$ than $s_2(\tau)$ on average. Conversely, firms that sufficiently favor disadvantaged applicants with $\tau < \Bar{\tau}$ generally prefer $s_1(\tau)$ over $s_2(\tau)$ on average.

Because Theorem \ref{main} only holds for regular environments, we consider the irregular environment and find that the firm prefers $s_2$ for all $\tau$ as $m\to \infty$.

\begin{corollary}\label{last}
	The firm almost surely prefers $s_2$ over $s_1$ on average for all $\tau$ as $m\to\infty$ if and only if the environment is irregular.
\end{corollary}

\begin{proof}
	In the proof of Proposition \ref{prop3}, we see that regularity is equivalent to:
	\[
	\mathbb{E}\left[\lim_{\tau \to -x_u} u(s_1(\tau))\right] <\mathbb{E}\left[\lim_{\tau \to -x_u} u(s_2(\tau))\right].
	\]
	This implies irregularity gives the reverse inequality. As $m\to \infty$, the left-hand-side is strictly increasing and continuous almost surely in $\tau$ and conversely for the right-hand-side. Thus, the left-hand-side is greater than the right-hand-side for all $\tau$. 
\end{proof}

Corollary \ref{last} can be understood by the nature of an irregular environment. In particular, irregular environments should be intuitively linked to labor markets with a high proportion of qualified applicants yet high firm-payoffs for qualified labor. The algorithm $s_2$ would be preferable in this arguably uncommon setting because it scores all applicants too generously, leading to inflated expectations. Indeed, our definition of regret loss rewards more conservative algorithms.

It should be noted that the above discussion is limited to disadvantaged applicants. For advantaged applicants, the algorithms $s_1$ and $s_2$ do not change in $\tau$. Since we are interested in the comparative statics of firm preferences across human prejudice, we isolate the outcomes of the firm to be strictly related to disadvantaged applicants. On the other hand, since the scores for advantaged applicants do not change in $\tau$, the comparative statics of regret loss discussed in this paper are not lost. That is, the average regret across all applicants generated by $s_1$ strictly increases in $\tau$ as $m\to \infty$ and conversely for $s_2$. Despite this, when considering all applicants, it may be the case that $s_1$ is preferred over $s_2$ on average for all $\tau$ or vice versa---this is an artifact of the joint distribution of the population rather than human prejudice. In this sense, our discussions about the effects of human prejudice on the firm preferences are preserved.

\section{Closing Remarks}\label{sec:5}

In this paper, we constructed a rich two-stage model that incorporates a statistical and taste-based discriminating firm hiring applicants through the use of a human decision-maker and an algorithm. We first provided alternative proofs to comparative statics results presented in \cite{rr}. Upon comparing two algorithms $s_1(\tau)=\mathbb{E}[Q\mid \Theta, A=1]$ and $s_2(\tau)=\mathbb{E}[A\mid \Theta]$, we found that there exists a unique level of prejudice $\tau^\star$ where $s_1(\tau^\star) = s_2(\tau^\star)$. Next, we defined a version of regret loss generated by an algorithm, which allows us to compare the firm's preferences over $s_1$ and $s_2$. We define a regular environment and prove the almost sure existence of a unique point where average regret from $s_1$ and $s_2$ are equal as time goes on. Given the locally increasing and decreasing nature of $U(s_1(\tau))$ and $U(s_2(\tau))$, it immediately followed that less prejudiced firms will prefer $s_1$ over $s_2$ on average, while more prejudiced firms will prefer $s_2$ over $s_1$. Intuitively, firms partial to disadvantaged applicants prefer their algorithmic predictions to follow suit. Thus, the algorithm that reverses prejudice in favor of disadvantaged class is naturally preferred. Conversely, strongly prejudiced firms wish for their prejudice to be reflected in their predictions. For irregular environments, the firm prefers $s_2$ over $s_1$ for all $\tau$ as time goes on. In this paper, we focused on individual firms, but an extension may consider competitive interactions in the context of screening problems. Moreover, in special scenarios, outcomes of rejected applicants may be observed in local communities, which may lead to a more refined interpretation of regret loss.

\nocite{*}
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