Matchings Under Biased and Correlated Evaluations

Thank you for your thoughtful and detailed review. While our model builds on earlier formulations, we believe the structural classification into correlation regimes, the closed-form computation of thresholds, and the proof techniques for monotonicity provide nontrivial analytical advances that are central to enabling the intervention strategies in Section 4.

Below, we respond to your comments and questions.

... put main and general result in the main article not in appendix. … the general case of $\beta$.

Thank you for the suggestion. We agree that the $p < 1$ and $\beta < 1 - c$ cases merit more visibility. As the analysis in both settings is more intricate than in the $p = 1$ and $\beta \geq 1 - c$ case, it currently appears in the appendix. We will include a concise summary in the main body to highlight the broader applicability of our framework.

In the main article the Institution 1 selects only from group $G_1$...

Thank you for the comment. We believe there may be a misunderstanding: in the main body, Institution 1 does select candidates from both groups whenever $\beta\geq 1 - c$, which is the regime analyzed in detail. Line 65 mentions that candidates from $G_2$ are "not automatically excluded" refers to the case $\beta < 1 - c$, where no $G_2$ candidate can meet the threshold. We will revise this line for clarity.

The definition of fairness criteria is defined only the fraction of selected candidates from group $G_j$. Since the main point of your model is preference of institution 1 to 2. Why this point is not consider in the definition of fairness criteria.

Thank you for this suggestion. In the $p = 1$ case (main body), where all candidates prefer Institution 1, group-level selection rate remains a meaningful fairness metric. However, your point is well-taken for the general $p$ case (in the Appendix), where preferences vary. We will expand the discussion of fairness notions for $p < 1$ in the final version.

… estimate $\beta_0,\gamma_0$ …

Thanks for the question. While we treat $\beta$ and $\gamma$ as system-level parameters, they can be estimated from data. If Institution 1 uses threshold $ŝ_1^\star$ for Group 1, we can invert the formula $s_1^\star = \frac{2 - c}{1 + 1/\beta}$ to estimate: $\beta_0 \approx \frac{ŝ_1^\star}{2 - c - ŝ_1^\star}$.

Similarly, if the empirical representation ratio $\hat{\mathcal{R}}$ and Institution 2’s threshold $ŝ_2^\star$ are known, one can numerically invert $\mathcal{R}(\beta_0, \gamma)$ to estimate $\gamma_0$. We will add a brief discussion of this in the final version.

1. Would your selection procedure change fundamentally under additive bias…

Thank you for this question. Our model and analysis extend naturally to settings involving additive bias. For instance, suppose the observed utility $û_{ij}$ of a candidate $i \in G_2$ for institution $j$ is given by $u_{ij} - \beta$, where $u_{ij}$ is the true utility and $\beta \in [0,1]$ represents the additive bias penalty.

Under a natural assumption $\beta \leq c$ (analogous to the multiplicative case where we assume $\beta \geq 1 - c$) which ensures that Institution 1 admits candidates from group $G_2$, the threshold for Institution 1 becomes:

$s_1^\star = 1 - \frac{c + \beta}{2}.$

The existence of $\gamma$-thresholds continues to hold: we show an analogue of Theorem 3.1:

Theorem. Fix $\beta < c$. Then there exist thresholds $\gamma_1, ..., \gamma_4 \in [0,1]$ such that:

1. $s_2^\star + \beta \leq 1 - \gamma$ iff $\gamma \leq \gamma_1$
2. $s_2^\star \leq 1 - \gamma$ iff $\gamma \leq \gamma_2$
3. $s_2^\star \geq s_1^\star \gamma$ iff $\gamma \leq \gamma_3$
4. $s_2^\star + \beta \geq (s_1^\star + \beta)\gamma$ iff $\gamma \leq \gamma_4$.

Using these threshold conditions, along with the probability expressions developed in Section 3 for $\text{Pr}[\gamma v_{i1} + (1 - \gamma) v_{i2} \geq s_2 \,\wedge\, v_{i1} < s_1],$ we can establish an analog of Theorem 3.2 to compute $s_2^\star$ under additive bias. For example, in the regime $\gamma < \gamma_1$, the threshold equation becomes:

$s_1^\star - \frac{2s_1^\star s_2^\star - \gamma (s_1^\star)^2}{2(1 - \gamma)} + (s_1^\star + \beta) - \frac{2(s_1^\star + \beta)(s_2^\star + \beta) - \gamma (s_1^\star + \beta)^2}{2(1 - \gamma)} = c.$

We will include a brief discussion of this generalization in the final version of the paper.

2. You state: "In the finite setting, stable matchings are characterised by threshold-based rules." Could you cite the specific sources supporting this statement?

Thank you for pointing this out. This result appears in Appendix C (Prop. C.1). We provide a brief sketch of the argument here and will add a short summary to the main text in the final version.

Let $M$ be any stable assignment, and let $m_1$ and $m_2$ denote the minimum observed utility among candidates assigned to the two institutions respectively. Then, the stability of the assignment implies that Institution 1 must select all candidates $i$ with $û_{i1} \geq m_1$. A similar argument applies for Institution 2. Thus, the assignment $M$ is fully characterized by the thresholds $s_1^* = m_1$ and $s_2^* = m_2$.

3. ... In practice, when $n$ is not large, how can these theoretical thresholds guide the design of effective intervention methods?

Thank you for this question. We agree that understanding how theoretical thresholds translate into practical guidance is essential. To address this, we build on a result by Azevedo and Leshno [7] which shows that the thresholds governing stable assignments converge to deterministic values $s_1^\star$ and $s_2^\star$ defined by a continuum limit. We can quantify this convergence by showing that deviations are at most $O(\sqrt{\log n / n})$ with high probability. Since $\mathcal{R}$ is a smooth function of the thresholds, it inherits their concentration properties—justifying the use of continuum analysis. (See our response to reviewer #4qyh for full details.)

To corroborate this theoretical result and illustrate its practical utility, we implemented our model for finite values of $n$ and ran 30 independent trials over randomly generated candidate profiles for each value of $\gamma$. We report results for $n = 100$ below. The empirical means closely match the continuum predictions. For $n = 250$, we observe a further reduction in standard deviation, indicating the stability and robustness of our theoretical thresholds.

$n = 100$, $c = 0.2$, $\beta = 0.9$, $|G_1| = |G_2| = 100$

These results demonstrate that even in moderately sized systems, the continuum thresholds closely approximate the empirical representation ratios. This suggests that our theoretical framework can meaningfully inform policy decisions in realistic settings. We believe this also responds to the reviewer’s suggestion for a simulation-based validation, similar in spirit to the empirical component of Celis et al. (2024). We will include detailed plots and empirical results in the final version.

4. How sensitive are the $\gamma$-thresholds and the monotonicity of $\mathcal{R}$ to non-uniform or correlated talent distributions?

Thank you for this important question. While our main analysis assumes i.i.d. uniform attributes for analytical tractability, our framework extends naturally to more general settings—including non-uniform distributions and correlated attributes.

Non-uniform distributions: For independent attributes drawn from a smooth distribution with CDF $\Phi$, the equation for the threshold $s_1^\star$ becomes:

The computation of representation probabilities also adapts using $\Phi$ and its density $\phi$. For example, in Case I when $\gamma \leq \gamma_1$:

While closed-form expressions may not always be available, the thresholds and representation metrics remain numerically tractable, and the threshold regime structure (Theorem 3.1) continues to hold. We did simulations using truncated Gaussians on $[0,1]$ to confirm the monotonicity of $\mathcal{R}(\beta, \gamma)$ across a wide range of variances.

Correlated attributes: When $v_{i1}$ and $v_{i2}$ are correlated, the sensitivity of $\mathcal{R}$ to $\gamma$ depends on the strength of the correlation. In the extreme case where $v_{i2} = \lambda v_{i1}$, the utility $u_{i2}$ becomes a scaled version of $v_{i1}$, rendering the representation ratio invariant to $\gamma$.

More generally, correlated attributes can often be expressed as functions of independent variables (e.g., via whitening in the Gaussian case). The observed utilities $û_{ij}$ then inherit this structure, allowing our framework to remain applicable. However, establishing monotonicity of $\mathcal{R}$ analytically in such settings remains challenging and is a promising direction for future work. We will include a discussion of these generalizations in the final version.

Thank you for your thoughtful and constructive comments. We appreciate your careful reading of the paper and the insightful questions raised. Below, we respond to each point in detail, clarifying our assumptions, highlighting key contributions, and discussing extensions and open directions suggested by your feedback.

The complex analysis of the equilibrium strategy of the second institution reduces the clarity of section 3 and some details of their analysis could be deferred to the appendix for the sake of clarity.

Thank you for this helpful suggestion. We agree that the derivation of the equilibrium strategy for the second institution is a technically involved part of Section 3 and may affect the overall readability. In the final version, we will move some of the more intricate case analyses and algebraic derivations to the appendix, while preserving the key insights and structural results (e.g., unimodality and threshold characterization) in the main text. This should help streamline the exposition without sacrificing the conceptual contributions.

Additionally, I was interested in more discussion on the extent to which the monotonicity of $\mathcal{R}$ with respect to $\gamma$ depends on the assumption that attribute 1 is drawn iid for each candidate. That is, should we expect correlation of evaluation strategy to only improve representation if the institutions are correlating on an attribute where all candidates have the same distribution over values and each are independently assigned a value?

We thank the reviewer for raising this insightful question about the role of the i.i.d. assumption in our analysis of the monotonicity of the representation ratio $\mathcal{R}(\beta, \gamma)$ with respect to the correlation parameter $\gamma$.

To rigorously establish monotonicity, our analysis first reduces the finite-population model to a continuum limit. This reduction leverages the result of Azevedo and Leshno [7], which shows that as the number of candidates $n$ grows, the outcome of the stochastic finite model concentrates around a deterministic infinite-population limit.

To be more precise, in the finite model, each candidate $i$ in group $G_j$ has latent attributes $v_{i1}, v_{i2}$, drawn independently from a distribution over $[0,1]^2$ (uniform in the baseline case). These attributes are transformed into observed or biased utilities $û_{ij}$, which determine candidate preferences and institutional choices. A realization $\omega$ corresponds to a specific draw of all these $(v_{i1}, v_{i2})$ values for each candidate in the population. Once $\omega$ is fixed, the utilities $û_{ij}$ are also fixed, and the stable matching is deterministic.

In this setting, Proposition C.1 (Appendix) establishes that for every realization $\omega$, the stable assignment is governed by thresholds $S_1^\star(\omega)$ and $S_2^\star(\omega)$, such that Institution 1 selects candidates with $û_{i1} \geq S_1^\star(\omega)$ and Institution 2 selects those with $û_{i2} \geq S_2^\star(\omega)$.

Now, under the assumption that candidate attributes are i.i.d., the fraction of candidates in each group satisfying these threshold conditions concentrates around their expected values due to standard probabilistic concentration (e.g., Chernoff bounds or the central limit theorem). This implies that the empirical thresholds $S_1^\star(\omega)$ and $S_2^\star(\omega)$ converge to deterministic values $s_1^\star$ and $s_2^\star$ as $n \to \infty$, which satisfy explicit equations derived in the infinite-population model.

We restate the formal convergence result below:

Proposition. Given parameters $c$, $\beta$, and $\gamma$, there exist unique values $s_1^\star$ and $s_2^\star$ such that the following holds. Consider a finite stochastic instance $\mathcal{I}$ with $|G_1| = |G_2| = n$, and two institutions with capacities $cn$ each, bias parameter $\beta$, and correlation parameter $\gamma$. Then with high probability, there exist thresholds $s_1, s_2$ such that the stable assignment in $\mathcal{I}$ is governed by these thresholds and satisfies $|s_\ell - s_\ell^\star| = O\left(\sqrt{\frac{\log n}{n}}\right)$ for each $\ell \in \{1, 2\}$.

This proposition justifies the use of the continuum model to analyze properties such as the monotonicity of $\mathcal{R}$ with respect to $\gamma$, as the finite model increasingly resembles the limit as $n$ grows.

That said, if candidate attributes exhibit strong dependence—for example, if all candidates have identical or highly similar attributes—the law of large numbers and related concentration results no longer hold. In such cases, the convergence to a deterministic limit may fail, and our monotonicity results may not apply.

Nonetheless, we believe the qualitative trend—that increasing $\gamma$ improves $\mathcal{R}$—likely persists under weaker assumptions, such as mild correlations or clustered structures among candidate attributes. Extending the convergence analysis to such settings remains an open and valuable direction for future research.

We will add a discussion of this modeling assumption and possible extensions in the final version. We thank the reviewer again for encouraging us to clarify this aspect.

Is it possible to simplify the theorem statements in the main text (and keep full statements in the appendix) to more clearly show that and have the properties necessary for intervention as explored in Section 4?

Thank you for the helpful suggestion. We agree that clarifying the key properties of the main results could improve the flow into the analysis in Section 4. In the final version, we plan to simplify the presentation of the main theorems—focusing on the properties most relevant for intervention design—while retaining full technical statements in the appendix. This should make the analysis section more accessible without requiring substantial changes to the structure.

(Extension question) How might $\mathcal{R}$ change over time if this model is extended over time, especially if previous admission decisions influence candidate preferences in future time steps?

Thank you for this thought-provoking question. Our current model is static and analyzes a single-round matching process. That said, we agree that in settings where candidate preferences, group participation, or institutional strategies evolve over time—based on past match outcomes—the representation ratio $\mathcal{R}$ could change in nontrivial ways.

Capturing such dynamics would likely require a game-theoretic or behavioral framework with endogenous feedback. For instance, if increased representation improves perceived fairness or institutional reputation, this may in turn affect future applications—potentially creating a virtuous cycle. Conversely, persistent underrepresentation could lead to disengagement and long-run disparities. Modeling such feedback loops could illuminate when correlation amplifies or corrects inequities over time.

While these dynamics are outside the scope of our current analysis, we believe this is a compelling direction for future work—especially in persistent systems like recurring admissions or hiring cycles. We will mention this direction in the final version as a promising avenue for future work.

Small question, but what happens if $c=1/2$, i.e., is there a special case when both institutions can admit everyone?

Thank you for the question. In our model, there are two groups of size $n$, so the total candidate pool has $2n$ individuals. Each institution admits a fraction $c$ of the total population, i.e., $cn$ candidates per institution. So full admission—where every candidate is matched—occurs when $2cn = 2n$, which implies $c = 1$.

Therefore, the case $c = 1/2$ is still selective: only half the total population (i.e., $n$ candidates) is admitted across both institutions. In the regime $\beta \geq 1-c = 1/2$, we can compute the values of the $\gamma$-thresholds using Theorem 3.1, the thresholds $s_2^*$ using Theorem 3.2 and the representation ratio using Theorem 3.3.

In the case $c = 1$, all the candidates are admitted and hence, the representation ratio is $1$. We can compute the values of the thresholds $s^*_1$ and $s_2^\star$ using equations (4) and Theorem 3.2 respectively (observe that we shall always be in the regime $\beta \geq 1-c$ here).

If we have misunderstood your intent or if you're referring to a different version of the model (e.g., with institutions admitting their own groups), we would be happy to clarify further.

We thank you for your thoughtful comments and encouraging overall assessment. We're especially glad that the paper’s analysis and framing were found clear and of high quality. Below, we respond to your questions and comments.

My only complaint would be that the techniques don't seem very novel or if they are, they haven't highlighted it in the main body of the paper. But that is minor and is to be expected with these kind of problems.

We appreciate the reviewer’s comment regarding the novelty of techniques and agree that clarifying this in the main body would strengthen the paper. While our model builds on established components in matching theory and bias modeling, we believe the technical contributions are substantial and novel in several directions:

1. Regime reduction via $\gamma$-thresholds. We introduce a new classification of parameter regimes that collapses 16 case distinctions into 3 interpretable ones using carefully identified $\gamma$-thresholds. This structure emerges from non-trivial comparisons between institution-specific thresholds and cross-group rank orderings, and allows us to express key outcome metrics in unified forms across the regimes.

2. Characterization and unimodality of $s_2^\star$. We provide a closed-form description of the second institution’s optimal threshold $s_2^\star$ in terms of $\beta$, $\gamma$, and $c$, and rigorously prove that $s_2^\star$ behaves in a unimodal fashion with respect to $\gamma$. This behavior reflects a non-obvious trade-off: while higher correlation can align evaluations and boost the expected quality of selected candidates, it can also lead to reduced access to high-scoring candidates due to competition with Institution 1. Capturing this crossover requires careful comparative statics and monotonicity arguments.

3. Implicit monotonicity of $\mathcal{R}(\gamma)$. Although $\mathcal{R}$ can be written in terms of the parameters $\beta$ and $\gamma$, its piecewise definition and dependence on regime-specific thresholds make direct analysis challenging. We establish its monotonicity not by direct computation, but by examining the properties of the equilibrium-defining equations that $\mathcal{R}$ satisfies in each regime. This strategy leverages structure in the stable matching mechanism and can generalize to other fairness metrics where explicit formulas are difficult to analyze.

We will revise the final version to highlight these contributions more clearly in the main text.

I was a bit surprised that the representation ratio was positively correlated with the correlation factor gamma, given that the tone of the introduction seemed to be making a case against positive correlation. I don't know if this was intentional or not. In hindsight, this positive correlation makes sense since the two dimensions of the utilities are independent and the bias factor applies equally to both of them. Would this correlation flip if the \beta for u_2 was slightly higher (ie less biased) than that for u_1

Thank you for this thoughtful observation. We understand the reviewer’s surprise and agree that the positive correlation between the representation ratio $\mathcal{R}$ and the correlation parameter $\gamma$ might initially appear to contradict the tone of the introduction. In the introduction, our concern with correlation stems from its potential to entrench shared biases across institutions. However, as our results show, the effect of increasing $\gamma$ on $\mathcal{R}$ is more nuanced and depends critically on the bias parameter $\beta$.

In particular, we prove that $\mathcal{R}$ is monotonically increasing in $\gamma$ only when $\beta \geq 1 - c$. Below this regime, monotonicity can fail, and the representation ratio may behave non-monotonically; see Table 2 in Section I for a comparison of these cases. We agree that this distinction could be made more explicit in the paper and appreciate the opportunity to clarify it.

Regarding asymmetric bias across institutions (i.e., $\beta_1 < \beta_2$), our findings suggest that increasing $\gamma$ can still improve representation, because higher correlation allows the less-biased institution to better “rescue” disadvantaged candidates who narrowly miss the threshold at the more-biased institution. This effect persists even when the bias levels differ.

To illustrate, consider the following extreme case:

Propsition. Let Institution 1 apply bias with parameter $\beta_1$, where $1 - c < \beta_1 < 1$, and let Institution 2 be unbiased, i.e., $\beta_2 = 1$. Then:

• When $\gamma = 1$, $\mathcal{R}(\gamma) = 1$,
• When $\gamma = 0$, $\mathcal{R}(\gamma) = \frac{2c + \beta_1 - 1}{2c\beta_1 - \beta_1 + 1} < 1$.

Proof idea. When $\gamma = 1$, the institutions are fully aligned, and any candidate with $u_{i1} \geq s_2^\star$ is selected by at least one institution, regardless of group. In this case, the threshold $s_2^\star$ becomes $1 - c$, and each group contributes a fraction $c$ to the total admits—yielding a representation ratio of 1 despite the asymmetry in bias. In contrast, when $\gamma = 0$, the first institution selects candidates based on a threshold $s_1^\star = \frac{2-c}{1+1/\beta_1}$. For the remaining candidates, the distribution of $v_{i2}$ is i.i.d. and hence, the fraction of candidates from each group that get selected to Institution 2 depends on their group-dependent acceptance probabilities.

Additional simulations. To further support this point, we have extended our implementation to allow different bias parameters $\beta_1$ and $\beta_2$ for Institutions 1 and 2, respectively. Our numerical experiments indicate that the monotonicity of $\mathcal{R}(\beta_1, \beta_2, \gamma)$ with respect to $\gamma$ continues to hold across a range of asymmetric settings. For instance, we have verified this property for $c = 0.3$, $\beta_1 = 0.8$, and $\beta_2$ varying from $0.8$ to $1$ (where $\beta_1$ and $\beta_2$ are the bias parameters for the two institutions, respectively). Similar trends hold for other values of $c$ and for $\beta_1 \geq 1 - c$.

Due to constraints of the rebuttal process, we are unable to attach plots, but we will include this discussion—along with illustrations—in the final version.