Matchings Under Biased and Correlated Evaluations

TL;DR

The paper studies a decentralized two-institution stable matching with group-biased and partially correlated evaluations. It derives a continuum equilibrium characterized by thresholds and presents closed-form, regime-dependent expressions for the representation ratio $\mathcal{R}(\beta,\gamma)$ and its normalized form $\mathcal{N}(\beta,\gamma)$, revealing how bias and evaluator alignment jointly shape representation. It identifies three critical $\gamma$-thresholds that partition regimes and demonstrates monotonicity of fairness metrics in $\beta$ and $\gamma$ under a key parameter regime, enabling Pareto-frontier interventions to achieve targeted fairness. The framework supports practical fairness design in decentralized selection, including planning interventions (bias mitigation vs. alignment) and extending to general preferences, additive bias, and nonuniform candidate attributes. Overall, the work provides structural insights and actionable guidelines for fairness-aware design in stable matching with biased and correlated evaluations.

Abstract

We study a two-institution stable matching model in which candidates from two distinct groups are evaluated using partially correlated signals that are group-biased. This extends prior work (which assumes institutions evaluate candidates in an identical manner) to a more realistic setting in which institutions rely on overlapping, but independently processed, criteria. These evaluations could consist of a variety of informative tools such as standardized tests, shared recommendation systems, or AI-based assessments with local noise. Two key parameters govern evaluations: the bias parameter $β\in (0,1]$, which models systematic disadvantage faced by one group, and the correlation parameter $γ\in [0,1]$, which captures the alignment between institutional rankings. We study the representation ratio, i.e., the ratio of disadvantaged to advantaged candidates selected by the matching process in this setting. Focusing on a regime in which all candidates prefer the same institution, we characterize the large-market equilibrium and derive a closed-form expression for the resulting representation ratio. Prior work shows that when $γ= 1$, this ratio scales linearly with $β$. In contrast, we show that the representation ratio increases nonlinearly with $γ$ and even modest losses in correlation can cause sharp drops in the representation ratio. Our analysis identifies critical $γ$-thresholds where institutional selection behavior undergoes discrete transitions, and reveals structural conditions under which evaluator alignment or bias mitigation are most effective. Finally, we show how this framework and results enable interventions for fairness-aware design in decentralized selection systems.

Figures (15)

• Figure 1: (Left to Right) Cases I, II, III, IV that arise in the probability computation.
• Figure 2: Variation of (left) selection threshold $s_2^\star$ and (right) representation ratio $\mathcal{R}(\beta, \gamma)$ as a function of $\gamma$, for fixed $c = 0.2$.
• Figure 3: (Left) $\mathcal{N}(\beta, \gamma)$ and (right) Pareto frontier of $(\beta, \gamma)$ pairs achieving $\mathcal{N}(\beta, \gamma) \geq \tau = 0.8$, for $c = 0.2$. Starting from $(\beta_0 = 0.85, \gamma_0 = 0.4)$, the minimum interventions required to exceed the threshold are: $\beta \geq 0.911$ (with $\gamma_0 = 0.4$) or $\gamma \geq 0.640$ (with $\beta_0 = 0.85$).
• Figure 4: Variation of thresholds $\gamma_1, \ldots, \gamma_3$ with $\beta$ for a fixed value of $c$ (Left) and with $c$ for a fixed value of $\beta$ (Right). Note that $\beta \geq 1-c$.
• Figure 5: (Left) Utility of Institution 1 as a function of $\beta$ for fixed $\gamma = 0.5$ and $c = 0.2$. (Right) Utility of Institution 2 as a function of $\beta$ for fixed $c = 0.2$ and varying $\gamma \in \{0.2, 0.5, 0.8\}$.

Theorems & Definitions (81)

• Theorem 4.1: (γ)-thresholds
• Theorem 4.2: Equations for (s_2^⋆(γ))
• Theorem 4.3: Representation ratio
• Corollary 5.0
• Proposition 6.0: Monotonicity of $s_1^\star$ and $s_2^\star$ w.r.t. $\beta$
• proof
• Proposition 6.0: $s_1^\star$ dominates $s_2^\star$ in the symmetric case
• proof
• Theorem 7.1: (γ)-thresholds
• proof : Proof of \ref{['thm:gammathreshold']}