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Matchings Under Biased and Correlated Evaluations
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Extends a centralized biased-evaluation matching model to two decentralized institutions with correlated, group-biased evaluations, deriving closed-form large-market representation ratios and fairness interventions.
Problem
How group-dependent evaluation bias and inter-institutional correlation of candidate evaluations jointly shape group representation in decentralized two-institution stable matching markets.
Approach
Two institutions evaluate candidates with partially correlated attributes (correlation gamma), with the disadvantaged group's scores scaled by bias beta; in the large-market limit the stable matching reduces to two thresholds, giving a piecewise closed-form representation ratio with gamma-thresholds and a normalized fairness metric.
Key results
- Closed-form piecewise representation ratio R(beta,gamma) with thresholds gamma_1, gamma_2, gamma_3; R is monotonically increasing in gamma when c<1/2 and beta>=1-c.
- Example: with c=0.2, beta=0.85, gamma=0.4, R≈0.329 vs R(beta,1)≈0.510, so N≈0.644; reaching N≥0.8 requires beta≈0.911 or gamma≈0.640.
- Finite-sample simulation (n=100 and 250 per group) shows representation ratios roughly tracking the continuum predictions with decreasing variance as n grows.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Increase evaluator correlation γ | Stable matching with β ≥ 1-c and c < 1/2 · Continuum matching | representation ratio | Helped |
| Increase bias parameter β (reduce group disadvantage) | Stable matching · Symmetric continuum matching | representation ratio | Helped |
| Increase evaluator correlation γ | Stable matching · Continuum matching, c ≤ 1/2 | normalized representation ratio | Helped |
| Increase bias parameter β (reduce group disadvantage) | Stable matching · Continuum matching, c ≤ 1/2 | normalized representation ratio | Helped |
| Increase γ within the low-alignment region γ ≤ γ1 | Stable matching · Continuum matching | normalized representation ratio | No clear effect |
| Reserve seats proportional to group sizes and match within each group | Stable matching · Arbitrary β and γ | representation ratio: exactly 1 | Helped |
| Increase candidate preference probability p for Institution 1 | Stable matching with general preferences · General preference setting | representation ratio | Hurt |
Surprising findings
- Institution 2's threshold s2* is non-monotone (U-shaped/unimodal) in gamma, yet the representation ratio still increases monotonically in gamma.
- Even at beta=1 (no bias), gamma-driven phase transitions in the equilibrium threshold still occur.
Limitations
- Analysis restricted to two institutions, uniform i.i.d. attributes, and large-market (n to infinity) limit.
- Main results assume all candidates prefer Institution 1 (p=1) and beta >= 1-c; other regimes deferred to appendix.
Experimental setup
Metrics: representation ratio normalized representation ratio threshold s2*
Baselines: centralized biased matching model (gamma=1 case)
Builds on: Deferred Acceptance centralized biased-evaluation matching model (representation ratio) continuum stable matching models
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