Matchings Under Biased and Correlated Evaluations
Authors: Amit Kumar
Nisheeth K. Vishnoi
Venue: NeurIPS 2025 poster ()
Keywords: Stable matching
Evaluation Bias
Correlated rankings
Equilibrium analysis
Representation ratio
Primary area: theory
TL;DR: We model stable matchings under group-dependent bias and correlated evaluations, characterize equilibrium thresholds, and show how evaluator alignment amplifies or mitigates fairness loss in decentralized systems.

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 $\beta \in (0,1]$, which models systematic disadvantage faced by one group, and the correlation parameter $\gamma \in [0,1]$, which captures the alignment between institutional rankings. We study the representation ratio $\mathcal{R}(\beta, \gamma)$, 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 $\gamma = 1$, this ratio scales linearly with $\beta$.  In contrast, we show that $\mathcal{R}(\beta, \gamma)$ increases nonlinearly with $\gamma$ and even modest losses in correlation can cause sharp drops in the representation ratio. Our analysis identifies critical $\gamma$-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.

Matchings Under Biased and Correlated Evaluations
Amit Kumar
Nisheeth K. Vishnoi Yale University
IIT Delhi
Abstract
We study 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, Al-based assessments with local noise. Two key parameters govern evaluations: the bias parameter 3 E (0, I]. which models systematic disadvantage faced by one group. and the correlation parameter Y [0, 1]. which captures the alignment between institutional rankings. We study the representation ratio R(B, 7). i..., the ratio of disadvantaged advantaged candidates selected by the matching process in this setting. Focusing on regime in which all candidates prefer the same institution, we characterize the targe-marker equilibrium and derive closed-form expression for the resulting representation ratio. Prior work shows that when 7 = 1, this ratio scales linearly with [3. In contrast, we show that R(3, 7) increases nonlinearly with and even modest losses in correlation can cause sharp drops in the representation ratio. Our analysis identifies critical 7-thresholds where institutional selection behavior undergoes discrete transitions, and reveals structural conditions under which evaluator alignment or bias mitigation are most effective. Finally, show how this framework and results enable interventions for fairness-aware design in decentralized selection systems.
1 Introduction
Stable matching mechanisms are a cornerstone of allocation theory, extensively studied in game theory, economics, and, more recently, machine learning [48, 49, 50, 8, 21, 41, 32, 33, 39]. These mechanisms assign agents (e.g., students, workers, users) to institutions (c.g., schools, employers, content slots) under preference and capacity constraints, ensuring that no unmatched agent-institution pair would mutually prefer each other over their assigned match. Such matching systems are widely deployed in admissions practices, labor markets, and digital platforms.
Often, preferences across candidates are determined using evaluations such as standardized tests, interviews, or aggregate reviews. However, such evaluations can exhibit group-dependent bias systematically disadvantaging candidates from certain demographic groups [47, 28, 46, 22, 40, 55, 36]. Further, institutions often rely on overlapping signals (e.g., standardized test scores or resumes), yielding inceitinstittiinnall correlation in how candidates are assessed. These structural properties of bias and correlation can jointly skew allocation outcomes, compounding access gaps across education, employment, and economic opportunity [31, 30. 56, 26, 10].
The rise of AI-driven evaluation tools has further amplified these concerns. Predictive scoring models, automated interviews, and recommender systems are now widely used to rank candidates [43]. While scalable and efficient, these systems often inherit biases from historical data and reinforce alignment through shared features or pretraining [44, 54]. As result, modern evaluations may only be biased but also correlated across institutions in opaque or unintended ways.
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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A growing body of work has studied the impact of biased evaluations on allocation. For example, [15] analyzes a centralized matching model where common evaluation multiplicatively downweights scores for one group. They show that representation degrades linearly with bias and propose fairness- piiness-- mechanisms. In contrast, many real-world systems are decentralized: institutions make decisions independently. often using correlated but locally processed signals. For instance, school choice programs use structured tie-breaking rules that induce inter-institutionaal correlation [1, 4]. Even in the absence of bias, such correlation can generate disparities across groups [12]. These insights motivate our central question: How do group-dependen! bias and inter-institutinnal correlation jointly shape group-level representation in decentralized matching markets?
Our contributions. We build on and generalize the centralized model of [15] by introducing a decentralized stable matching framework that jointly models group- dependenn bias and inter- institutional correlation. We consider a two-institution setting where candidates belong to one of two groups: G1 (advantaged) and G2 (disadvantaged). A total fraction of candidates must be matched across both institutions. Each candidate possesses two latent attributes. Institution 1 ranks candidates solely by the first attribute, while Institution 2 uses a weighted combination of both attributes. The correlation parameter 7 E [0, 1] controls the degree of alignment between the evaluations of the two institutions. As in [15], candidates from G2 face multiplicative bias 3 E (0, 1] applied to both attributes. For our main results, we assume that all candidates prefer Institution I, reflecting prestige- driven preferences in many real-world settings (e.g.. centralized college admissions [2, 42, 35, 531).
We extend our analysis to general preference distributions in Section L. Our main contributions are:
Equilibrium characterization. We first show that, in the infinite population limit, the stable matching is governed by two thresholds, si and s2, which depend on 8, 7, and c. The threshold si admits a straightforward expression; see Equation (4). In contrast, deriving si requires resolving interdependence between institutions' rankings and cross-group orderings. We introduce regime- reduction technique that collapses sixteen potential case distinctions into three interpretable y-based regions, determined by analytically derived thresholds 71, 72, 73 (Theorem 3.1). Using this structure, we obtain closed-form expression for si (Theorem 3.2) and prove that it varies unimodally with respect to y (Theorem E.1), capturing a subtle trade-off:: increasing correlation can both enhance evaluator alignment and reduce access to high-scoring candidates due to intensified competition with Institution 1.
2. Fairness metric and monotonicity. Building on the equilibrium thresholds, we derive a piecewise closed-form expression for the representation ratio R(,,7), which measures the relative selection rates between the two groups (Theorem 3.3). We introduce a normalized variant. (V33, 7). which quantifies fairness relative to the ideal setting with full evaluator alignment. Unlike R. previously studied in the centralized model of [15], the normalized ratio isolates the effect of evaluator misalignment and enables scale-free comparisons across regimes. Because both R. and A depend on regime-speciff thresholds, direct algebraic analysis is intractable; instead, we establish their monotonicity in 3 and y by examining the equilibrium-defnie -efinnng equations themselves.
3. Studying fairness interventions. Leveraging the structure of (B, 7), we identify the Pareto frontier of minimal interventions that achieve a desired fairness level т. Specifically, we character- ize the combinations of evaluator bias 3 and alignment ry that satisfy .N (B,,)) 2 and visualize these trade-offs via contour plots. This enables system designers- given current evaluation param- eters (Bo, 70) and target to determine whether fairness can be achieved by adjusting either parameter and to select cost-effective strategies accordingly; see Section 4.
Together, these results provide structural and quantitative foundation for fairness-aware design in decentralized selection systems with biased and partially aligned evaluations.
Related work. Stable matching has been extensively studied in game theory, economics, and machine learning, with foundational contributions on stability. truthfulness, and decentralized evaluation [24, 48, 49, 50, 21, 8, 32, 33, 39]. Our work builds on [15]. who studied fairness in centralized matching under group- group-biased evaluations. We generalize their setting to allow partially correlated, institution-specte specific evaluations in decentralized markets. This aligns with recent work highlighting the role of either bias or correlation, we analyze their joint effect on representation. We also connect to literature on algorithmic fairness in subset selection, ranking, and admissions [29, 16, 25]. and to empirical
correlation in shaping outcomes [12, 4. 1]. While prior models focus on
studies showing persistent disparities in algorithmic selection systems [26, 31, 30). A full discussion of related work is deferred to Appendix A.
2

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2 Model
We consider wwo-institution matching setting where Institution has capacity en for E {1. Candidates belong to an advantaged group G1 or a disadvantaged group G2, with group sizes [G,] = vnn. We focus on the demand-eexceding-supplyy regime, where c1 + 02 < 11 + V2-
Correlated evaluations. Each candidate has two independent attributes, U/1 and U/2, drawn i.i.d. from the uniform distribution on [0, 1]. Institution evaluates candidates using 2/1, while Institution 2 uses a convex combination:
M41 = 0a and 12z = 7U-1 + (1 - 7)
where y E [0, 1] controls the degree of correlation between the two institutions' evaluations. This model captures, for instance, settings where Institution relies partly on a common criterion and partly on instiftiionspeecffc judgments.
Bias in evaluations. We model bias using a parameter 3 E (0, 1], following the framework of [29, 15]. The estimated utility ui of candidate from group G2 for Institution & is downscaled relative to their true utility:
ūæ = the if E G, and ure = Buee if G2.
This captures settings where disadvantaged candidates are systematically undervalued in evaluations, for example, due to differences in interview performance standardized tests. Our analysis focuses on 3 S 1, though it extends naturally to 3 > 1 to model overestimation, to asymmetric bias across institutions; see Appendix I), or to additive bias (Appendix J). Both institutions evaluate and select candidates from both groups; the parameter 3 only affects how candidates from G2 are evaluated.
Stable matching. A matching M assigns each candidate to institution, subject institutional capacity constraints. The matching M is stable if no candidate and institution & form a blocking pair: that is, i prefers é to their assigned institution Mi), and there exists candidate j currently matched to such that ū,e > ûze, then i must already be assigned to Stable matchings can be computed via the Deferred Acceptance algorithm [24, 50]. We consider model in which each candidate prefers Institution over Institution 2 with probability p. The main body focuses on the case p = 1, where all candidates prefer Institution 1. Extensions to general p are discussed in Appendix L.
Scaling limit. We analyze the model in the targe-marke regime, where the number of candidates + D0. This is standard approach in stable matching theory for capturing micro-to-mane behavior. In the finite setting, it is known that stable matchings are characterized by threshold- based rules; Proposition B. Specifically, for any realization of the evaluation scores û, the unique stable assignment determined by two cutoffs S and S2 such that M1) = {i: û. 2 81), M (2) = {i âu < S1 and 042 S2). Taking expectations, normalizing by n, and letting n + co, we obtain the scaling limit, where the stable matching is described by deterministic thresholds 8 and 82 satisfying:
21 Pr(um 2 81] + 12 Pr(ue1 2 s1 = C1
Pr(ua < <81, 0:2 2 82) + u2 Pr(uen < 81, ú22 2 82] C2
(2)
Here, i denotes candidate from group G1 and i' from group G2. These equations admit a unique solution ($1,$}); s5); see Appendix B for formal details. We show in Appendix B.3 that the finite thresholds S2) converge their deterministic limits (st, s5) as n + co.
Metrics. To quantify fairness in selection outcomes, we use the representation ratio introduced by [15]. Let Pj denote the fraction of selected candidates from group Gj. and define: R(M) ==
max(psspp) 'P2]' This metric captures disparity in group-level representation, with R(M) = 1 indicating
min(es.p2) .pa)
perfect proportionality across groups. In the scaling limit, the representation ratio can be expressed in closed form in terms of the stable matching thresholds (si, s5) derived from equations (1)-(2):
R(5, 7) Pr[0.124]+ Pr. 422.
Prfú. Prij-Pr &
While ((3 is useful for quantifying raw disparity, it is not always scale-free: for fixed B, the alignment parameter 7. This complicates
it may attain maximum below 1 depending comparisons across systems with different baseline levels of bias. To address this, we introduce the R(3,7) normalized representation ratio N N(3, (18,7)= == max,/'jo.11 R[B,777 which measures the fraction of ideal
representation achieved at the current alignment level. Unlike R, this normalized metric allows us to isolate the impact of misalignment independently of the underlying bias and supports principled intervention analysis.
3

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3
Main results
We characterize the representation ratio R(3,7) (and thus (B, 7)), which quantifies how equitably candidates from advantages and disadvantaged groups are selected under a stable matching. Under the scaling limit introduced in Section 2, the matching is determined by two thresholds s and sp. which are solutions to Equations (1) and (2). For clarity, we focus on the symmetric case e C2 = and 11 = 1/2 providing a complete piecewise characterization of si, s2. and the resulting R(3 as functions of the bias parameter 3 and the correlation parameter exceeding-supp -upply regime 1, where the thresholds determine partial selection. Our analysis centers on the regime B > cases where bias moderate and fairness is nontrivial. Extensions to 3 < 1 - c and to more general preference models are deferred to Appendix K and Appendix I. respectively.
We focus on the demand-
where both groups are eligible for selection by Institution I 1-capturing
Step 1: Computing the threshold sj. We begin by solving equation (1), which determines the threshold sf used by Institution 1. Since Institution evaluates candidates using their first attribute Vn, and a bias factor of is applied to candidates from group G2,, assuming "1 = 122 1 and C1 C2 c, the left-hand side of (1) becomes the average admission probability across both groups:
Pr[aa 2 s1] + Pr[ue1 2 s1] = (1 - .1) + 1 [1- )) Setting this to c yields:
o - 141/B
This expression shows that st increases with 3: as bias decreases and evaluations of group G2 improve, the institution can raise its threshold while maintaining its target capacity. For this threshold to admit candidates from both groups, it must satisfy s S 3. ensuring that the effective threshold for group G2 does not exceed their maximum possible score. Moreover, when B > 1 c, one can show that 85 S 8i (Proposition C.2). This yields the condition 3 2 1 1 c, which defines our main regime of interest. When 3 < c, Institution 1 selects only from group G1, resulting in degenerate matching; we defer this case to Appendix K. Finally, note that s depends only on 3 and c, and is independent of y, which influences only the threshold s used by Institution 2.
Step 2: Computing the threshold s.. The threshold si satisfies the second matching equation (2), which captures the fraction of candidates not admitted by Institution I but accepted by Institution 2. That is, s defines the acceptance region for Institution 2 over candidates rejected by Institution 1, based on potentially different evaluation criterion. As warm-up, consider the case y 1, where both institutions evaluate candidates using the same biased attribute U/1- This corresponds to a centralized setting previously studied in [15]. In this case, Institution 2 selects candidates whose (biased) utility falls in the interval [sh. si), so the selection probabilities for each group are: [$2, S1]] =1=32, Using equation (2), the capacity constraint
Pr[u;1 E [$2, s1]] $1
$2, Pr(u.11
becomes: [s) 82) +
S1 82
= c. Solving yields the closed-form expression: s} = $1 14178
B
This expression serves
benchmark and reveals useful structure: when both institutions are aligned in evaluation, s is simply a shifted version of sT.
In the general case 7 E (0, 1), Institution 2 evaluates convex combination of the two attributes: 44z = yUs1 + (1 7)2g. A candidate is selected if they are rejected by Institution 1 (ū. < 81) and
accepted by Institution 2 (ū.2 2 s2). The key term in (2) is the joint probability Prfuz2 2 $2 A ún
$1), which corresponds to the area above the line L(y, $2) == yx + (1-7)y- 82 inside the rectangle
[0, [1] x [0, 1]. Depending on how this line intersects the rectangle, we distinguish four cases:
Case I: 82 S min(sr7,1 - 7). The line intersects the y-axis at
$2
and the x-axis at 22 S 81,
7
yielding Pr(ua < 81, 0/2 2 82] 81
Case I1: 82 max(s17, (17,7 1 7). The line intersects the y-axis so Pr(ua1 < $1. 022 2 s2] = (1-3-10t7e1)
#2 1 and the ar-ais 22 81
2y(1->)
Case III: 817 82 < 7. The line intersects the y-axis at
52 1-7
< and the line x 81 at 82-781
1-7
leading to Pr [1 < S1, 0.2 2 22] = -
1-2
262-251
Case IV: 1- 7 s2 < s17. The line intersects the y-axis - and the x-axis at + < 51.
giving Pr(u, < $1 22 2 s2] = 31
22
1-2 27

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y
y
y
y
5: (1 1
8 (1
(0,1)
(0,1)
(0,1)
(0,1)
.92
77 82)
1 1-7
L
L
7 82)
1-y
L( 82)
1-7
81
0.9
0.9
......)
> I
> I
I
..... x
(0.0)
(0,0)
81 0.9
(0,0)
0.9
(0,0)
Figure 1: (Left to Right) Cases I, II, III, IV that arise in the probability computation.
We now repeat the analysis for group G2, whose evaluations are rescaled by 3. That is, the decision boundary becomes (3 52) := yx + (1-?)y = s2/8, and the relevant domain is [0, / x [o, 1]. The same four cases apply, denoted I' I'-IV", depending on how the line intersects this rectangle. A brute-force approach would guess the correct pair of cases for groups 11 and G2 (16 combinations total), and solve (2) under that assumption. However, this yields little structural insight into the behavior of si or the resulting representation ratio R(3,7) 7). In the next step, we characterize this structure piecewise by regime.
Step 3: Identifying structural regimes via 7-thrrehhods. Remarkably, our next result shows that only four of the sixteen case pairs (I-IV x --V')) arise in equilibrium, and that each occurs over a contiguous sub-interval of [0, 1] characterized by three values 0 S 71 72 S 73 S 1.
there exist unique values 71 S T/2 S 73 E [o, such that: (i) s² (1)/B S q if and only if S ni (ii) st(?) < 7 if and only if
Theorem 3.1 (y-thhesholdss) For any fixed B Z 1
S Y22 (iii) sg(7) 2 7s4 if and only if I 73- Moreover, these thresholds are given by: (i)
n = (87(811/3) 4(1 (1)))((()))((11))7 (ii) 72 a+1 where x the unique root in (m1:713) of
28 (8+11/))
(I
(
x #-) + (1- 0: (iii) 733 = I40'
At a high level, the reason why only four out of the possible sixteen cases arise is as follows. When 7 y= 0, L(),32) is horizontal and hence, we are in Case III (see Figure 1). We show that s²(7)/1 ? and s77//7 are monotonically increasing and decreasing functions of 7 respectively. Thus, as we raise y. the y-intercept of 827,82) increases and its x-intercepts decreases. Therefore, we transition from case III to case II and eventually to case IV (when = 1, L(7, 82) is vertical) - note that case I never happens because in this case, the relevant area above L(>, 52) is more than 1/2, but the capacity at Institution 2 is at most 1/2. The line Lg 82) also goes through these transitions as we raise y. Since these two lines remain parallel (and Lg lies above L) happen in a coordinated manner. Theorem 3.1 shows that the pair of cases corresponding to these two lines goes through the following transitions: (III, III'), (III, II',, (II, II'), (IV, IV"), and these regimes are separated by three critical thresholds 72, and 73-
change ", their transitions
These thresholds also mark qualitative changes in how the institutions evaluate candidates and balance the two attributes. For instance, when 7 < 71, disadvantaged candidates can qualify based solely on their second attribute U22, but for 71. they must also meet performance U,1. making admission more stringent. Likewise, above y/s both groups may qualify based on Us alone. Between these values, the selection regime reflects blended dependence on both attributes, with differential effeett across groups due to bias.
The proof of the first part of Theorem 3.1 relies on monotonicity of the scaled thresholds si(r) and
s3(1) that arises due to the structure of the equilibrium condition and the convex combination of T-
attributes. The values of the 7-thresholds are derived follows: for each E 2,, 3). the definition of y also yields an expression for (7;). For example, si(11) = B(1 - -.)). Since the threshold si(r) also satisfies the capacity constraint (2), we get equations for solving %
Note that, while 73 = I remains fixed for all B, we show in Proposition D.1 that the threshold functions 71 (5) and 72 (3) are strictly increasing and decreasing in 8, respectively, and both converge to a common limit as 8 + 1. In particular, lim 71 (3) lim (3) 1 < 73-
This implies that even in the absence of bias (3 1), structural phase transitions the equilibrium threshold s (and hence the representation ratio R) still occur as evaluator alignment y increases. Our
5

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framework thus provides insights not only into the compounding effects of bias and misalignment, but also into the residual impact of evaluator disagreement in the fully unbiased regime.
Finally, when 3 < 1 - C. the equilibrium admits a richer structure: there is an additional threshold 74 where 7/4 s (14)/ B. Further, there is critical value Be E [O, c] such that if B3 < Be, the relative ordering of these thresholds is 71, Ys, 72, 74; otherwise it is 71, 72, 73 74. At this critical value of 3. 72 equals 73. i.e. both the groups never satisfy the conditions in case II for given value of y. We defer a full treatment of that regime to Appendix K. We now use the threshold structure above to derive explicit expressions for s5 in each regime.
Step 4: Solving for (7(7) in each regime. The thresholds n. 72, 7a partition [0. 1] into four intervals, each corresponding to fixed pair of geometric cases for groups G1 and G2, namely (ITT, ITT"), (IIT, (IT, IT), (IV, IV") In each such regime, the expression for Pr[iuz 2 82 A 0411 < 81] is known from Step 2, and we can solve equation (2) to obtain 88(7)-
Theorem 3.2 (Equations for $ (r)).. Assume B 2 1 c and c < 1.
2- Then s2(1) satisfies:
[0, mil: si
25 -
2= >1,;
2(1-7)
(1-7)
[1,, 72]: si
2x 2
2(1-71
(1
21
c
Ta]:
(D-7-97+7913*
+
(1-7-6) o 18 /B)²
27(1
21(1-7)
Iys, 1]: s1 + 1/3) +3 3 1-7 =c
The piecewise equations above yield a unique value of $ for each 7. In particular, even when the defining equation is quadratic, the interval-specific constraints on 8 (e.g.. bounds derived from geometry or feasibility) ensure a single consistent solution. As varies, 83 transitions through qualitatively distinct behaviors-deerressimg convex, and increasigg-refleccingg subtle interplay between evaluator alignment and group-based access. We show in Theorem E. that ()(7) is unimodal (and, hence, mon-monotone w.r.t. ?) and continuous, with matching slopes at regime boundaries. Note that when 3 < 1 - c. due to the emergence of an additional threshold 7/1 Theorem K.5 shows that (1) can now have several local minima or maxima (see Appendix K.3). These characterizations of s now allow us to derive the representation ratio R(3,7) 7), which we analyze in the next step.
Step 5: Closed-form expression for representation ratio R R(3, 7). Using the expressions for st derived in Step 4, we now give piecewise closed-forn expressions for R(3,7).
Theorem 3.3 (Representation ratio). eix < 1 and $ 1-c.
Define
(3, =
Bst + ((8)) +1) 211 )]( 8)
and 0(3,7) =
7
(1 1344 (1
1-7 Then:
1+8²
1- - (3)
(0- ( ((1-0)]((+)) 1-+1-)++
if E 10.
9(1-))
if E Iul.
R(8,7)=
1 (1
1- +c + 2(1-5)
of (1 (1))0² (8.m)
if E h22
(1- +447
if E [ra, 1].
1-87 و-- 10
[1+>)(1- 11)-1193
27
The proof of this theorem uses the expression (3) for R(3, 7) in terms of s and sz. For each of the regimes defined by 71, 72, 73. we use (3.2) to eliminate the explicit dependence of this expression on s1. The monotonicity of 72) with respect to y follows from analyzing these closed-form expressions. Details are given in Appendix F. As an example, consider the regime 7 E (12, 73]- Using - and observe that the mass of the equation for s in this interval (Step 4), we define A == 1-7
group G1 candidates selected by Institution 2 is (1-7)4² This expression, together with its analog for group G2, yields an explicit formula for R(B,7) in each regime.
We present a summary of the results in this section Table 1. While our main focus is on representa- tion, the framework also supports closed-form expressions of institutional utilities (see Appendix H). In the next section, we leverage these results to examine how R(3, 7) behaves jointly with / and 7, and explore design strategics that mitigate bias while maintaining institutional efficiency.

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Structural property
Result location
Closed form expeession for sj
s1
14178
2-c
(Eq. (4))
Monotonicity #j watt. B
increasing 9
(follows directly from Eq. (4)
Four regimes of
72,
(Theorem 3.1)
piecewise expression #2
via characterization Yi. "Yay "t3
(Theorem 3.2)
Unimodality of 4; w.r. Y
* is continuous with at most minimum
(Theorem E.
Monotonicity # wutt. /
increasing
(Proposirion C.I)
piecewise expression for R. Monotonicity R.
4-case formula
(Theorem 3.3)
increasing fixed
(Proposition F.I)
Table I: Summary of structural properties of thresholds and representation metrics for 3 2 c.
Extension general preferences. We also extend our analysis to the general preference setting. where candidates prefer Institution I with probability E [0, 1], as described in Appendix L. In this setting, the equilibrium thresholds si and s satisfy coupled nonlinear equations that lack closed-form solutions. In Appendix I..1, we derive these equilibrium conditions and prove the existence and uniqueness of the solution, ensuring that the model remains well-posed for all p. Using the probability expressions developed earlier, we show in Appendix L.2 that these thresholds can be computed efficiently through numerical evaluation. Building on these results, Appendix L.3 provides closed- form formulas for key outcome metrics-the the representation ratio and institutional directly in terms of si and 8g. Analytically, we establish in Appendix L.4 that both thresholds vary monotonically with p. although not necessarily strictly: (p) remains constant up to critical point p" and increases thereafter, corresponding to a regime where the selection probability Pr[u,1 < (p) A u,22 2 sp(p] is zero. We formalize this structural break in Appendix L.4.1 and confirm it numerically. Consequently, both R. and N decrease with p and exhibit a clear phase transition: they remain constant for p p* and decline sharply thereafter. Finally, numerical experiments in Appendix L.5 and Figures 8-14 reveal how thresholds, representation ratios, and utilities co-evolve as p and the correlation parameter 7 vary. The qualitative behavior of R(3, 7) and N(3, 7) remains monotone in .3 and 7 even under general preferences, while varying introduces the new structural effects described love-monntone yet piecewise regimes and distinct phase transitions that extend the continuum-limit analysis.
4 Analysis and intervention
A central goal of this work is to guide interventions that improve representation for disadvantaged groups Toward this, study how fairness metrics vary with two systemic levers: evaluator bias (3) and institutional correlation (7). While our main structural result yields closed-form expressions for the representation ratio R(3 (Theorem 3.3), its policy relevance depends on understanding how R and its normalized counterpart N evolve with these parameters. This section is organized into four parts. We begin by analyzing how R(B, 7) varies with and 7, highlighting regimes of monotonicity, sharp transitions, and diminishing returns. We then study the variation of normalized ratio (3,77) with respect to and y. Building on these insights, we formulate a framework for intervention planning under fairnsss targets and show how to use it.
Monotonicity, transitions, and concavity of the representation ratio. We begin by asking: How does R(B, 7) vary with the bias level f3 and evaluator correlation ?? For any fixed 7, the dependence bias decreases (i.e., / increases), selection thresholds si and s become
on 3 is predictable: stricter for the advantaged group, yielding a higher representation ratio (see Appendix F.2). The variation of R(3, 7) with y (for fixed 8 2 1- c), however, is more subtle. Since 7 controls how correlated the two institutions are in their evaluation criteria, one might expect that decreasing allows for more diverse evaluation and improves access for disadvantaged candidates. While s does not change with 7, $ (7) can increase or decrease with y : it follows a U-shaped trajectory as increases (see Theorem E. This raises a natural question: can R(8, 7) be non- mon-oonotoiic with y as well? The answer is no in the regime < 1/2 and 3 2 1
Corollary 4.1. Fix c < 1/2 and 3 2 1 - c. Then R(3, 7) increases monotonically in 7
In particular, R(3,7) is maximized when > = 1: R(B,7) S R(3, 1) for all 7 E (0, 1). The proof heavily relies on the piecewise formulas derived in Theorem 3.3 and is deferred to Appendix F.1. Figure illustrates the contrast: while 8/ dips and then rises with 7, the representation ratio R(9,7) increases steadily- more more sharply for larger values of B. Note, however, that this monotonicity
7

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may not hold outside the (less relevant) regime 3 221- c. In particular, when 3 < no one from G2 can be admitted Institution and this equilibrium structure can reverse the fairness trend with increasing 7 see Proposition K.8 for details.
Importantly, using the piecewise characterization from Theorem 3.3, we can generalize prior results by [14] on representation ratio. [14] considered the case Y 1 and showed that R(3, 1) varies approximately linearly with Theorem 3.3 shows that the behavior of w.r.t can be highly non-linear in the presence of We highlight this with phenomena with two examples.
Example (Quadratic decay). Let 1/2 E. On the hand, when y 1, the last case of Theorem 3.3 implies R,B 1) -(1-
B-2s which scales like 3 for small 1-29E"
enough E. On the other hand, when 7 S Y1, from the first case in Theorem 3.3, we obtain that 42e
R(B, 7) = 1+2g822 B for small enough Thus, R degrades from being linear in /3 when y = 1
quadratic in B when 7 O.
Example (Amplification in low-selectivity regime). Consider highly selective regime, where e is small and 3 ~ 1 c. For S 71. Theorem 3.3 yields: R(1 c, c1 2 c/4. In -2044e
contrast, when 7 = 1, using Theorem 3.3, we obtain: R(1 c, 1) = 2c+1c(1-ee) 3c-2c 3 for small enough Thus, while R,,,,) 7) = for all 7, a slight decrease in / (from 1 to 1 c) results in a huge gap representation ratio near the two extremes of y highly-selective setup.
To summarize, Theorem 3.3 and Corollary 4.1 imply that R(3, 7) is piccewise-defined across four structural regimes, separated by thresholds 71, 72, 73. It remains flat in the low-7 regime (7 7)), non-linearly increases through [Y), 1]. Taken together, these insights underscore that the representation ratio may not be uniformly responsive interventions changing and 7 due the compounding effect of these parameters on R.
01
011
Figure 2: Variation of (left) selection threshold s and (right) representation ratio R(3, 7) function y, for fixed 0.2.
Measuring fairness loss: Normalized representation and its response to interventions. Re- call that the normalized representation ratio is defined NV,,7) 7) max,,eejo.) R(G.5) Since R) S R(B, 1) for y E j0, 1]. this simplifies to ((3 ((R,3)/R(3, 1). Note that 0 >V(B,)) I quantifies the fraction of the maximum achievable fairness retained for a fixed 3. There are two key intervention levers: increasing 3 (reducing bias) and increasing correlation (7) increase N. However, these interventions, in the least, require that is monotonically non-decreasing in 3 and y. From Corollary 4.1, already know that R(3, 7), and hence (B, increases monotonically with 7. What is less obvious- but follows from the closed-form expressions in Theorem 3.3 is that N(8, 7) also monotonically inereasing in 33 when c S 1/2: see Theorem F.2. Figure 3 (left) illustrates this trend. The growth of N (3,3 7) is phase-dependent: flat in the low-alignment regime (7 S m) sharp in intermediate phases, and nearly linear for 7 [T3, 1]. The most severe degradation the importance of addressing both alignment and bias faimess interventions. Extending Example 1 above (the case e 1/2. When : = we have ((Br7)
R(3,y)
occurs when both 3 and are mmall-highlightinnn
by definition. But as + 0, R(B,7) = 8 while R(3,1) x B, implying N/ 7) B. In summary, even after normalization, fairness never deteriorates when bias reduced or alignment improves, and 33 and 7 jointly exert a compounded effect on M. These properties imply that the normalized representation ratio both stable and directionally reliable- that interventions targeting either parameter will monotonically improve fairness.
8

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Target-based intervention planning. We can now leverage the structure of (B,,)) to design and evaluate intervention strategies that improve representation. One may consider two classes of interventions: (1) continuous interventions that incrementally reduce bias or increase correlation, and (2) structural interventions such as capacity reservations. In the fully aligned setting (7 = 1), [15] shows that reserving seats in each institution proportional to group sizes, and applying stable matching within each group, guarantees perfect representation. A similar strategy can be shown (by extending their approach) to achieve a representation ratio of exactly for arbitrary values of B and y. This yields a non-pane parammerriic structure- intervention that ensures fairness independently of bias mitigation or evaluator alignment.
In contrast, interventions that modify the evaluation process, by increasing (5) or increasing (7). may provide additional flexibility in real-world selection systems. For instance, in Al-mediated settings, B can be increased through debiasing or anonymized scoring: 7 can increased using standardized models, calibration protocols, or shared training data. In human-mediated settings, increasing B might involve structured rubries, blind evaluation, or bias training; increasing 7 could be achieved by coordinated evaluation guidelines, common score sheets, or centralized vetting frameworks.
We now show how a system designer, given current parameters (Bo: 70) and fairness target T [0. 1]. can use our framework to plan interventions. Since ((3, 7) is increasing in both 3 and 7, any target S is attainable by increasing either parameter. We compute the Pareto frontier- the set of minimal (3, 7) pairs that just achieve ((3.7) 2 т, using Theorem 3.3. This frontier reveals all efficient tradeoff's between bias mitigation and evaluator alignment that meet the target. Figure (right) illustrates this frontier for r = 0.8, with heatmap of (13 7). The red contour shows the threshold N = т: points above meet the goal, while those below do not. The shape of the frontier reveals which direction 33 or --yiedss more efficient gains. Notably, the frontier becomes vertical when y I (3), reflecting the fact that N is flat in this region. Since "i increases with B, this insensitivity band widens bias decreases. Overall, the frontier offers practical guidance: starting from (So, 70), one can locate the nearest feasible point above the contour to meet the fairness target.
We illustrate this with a concrete example (Appendix G). Fix Bu = 0.85, To 0.40, and = 0.20. We first compute ii x 0.8276 using Equation (4), and then use Theorem 3.1 identify the applicable regime for Applying Theorem 3.3, we obtain the values R(30. 7o) = 0.329, R(S.. 1) z 0.510, and ((8; 70) 0.644. From Figure: (right), we observe that achieving a fairness target of 0.80 requires either ~ 0.640 or 3 x 0.911. This demonstrates that modest improvements in either evaluator alignment or bias reduction are sufficient to meet the desired fairness level.
Finally, note that the baseline parameters (3o, 7)) may be estimated directly from observed data. The bias parameter 3 can be recovered from the observed threshold used by Institution via 8o = 2-0-0 as follows from Equation (4). Holding / = 3o fixed, the correlation parameter y can be estimated by numerically inverting the monotone relation + R(30, 7) (or equivalently NY: tu arg min (R) R1 which admits unique solution by Theorem 3.3.
443
2-200 C-12
Ni Pal
4.00
Figure 3: (Left) NV(8, 7) and (right) Pareto frontier of (8, 7) pairs achieving NV(,7) - 0.8, for = 0.2. Starting from (3o = 0.85, 7o = 0.4). the minimum interventions required to exceed the threshold are: 8 2 0.911 (with 7o = 0.4) or y 2 0.640 (with Bgg = 0.85).

---

5 Conclusion, extensions, limitations, and future work
We introduced a mathematical framework study the structure of stable matchings in two-institution setting where candidate evaluations are partially correlated and one group experiences group-level bias. evolves with evaluator bias (3) and correlation (7). Our key technical contributions include: (i) a closed-form characterization of the equilibrium thresholds of and 6²(7) that govern the stable matching (Equation (4), Theorem 3.2);
Our focus was on how fairness measured via the representation
(ii) a piecewise expression for the representation ratio R(3, 7) (Theorem 3.3); and (iii) an analytic description of the structural 7-thresholds yi, 72, " that govern transitions in selection (Theorem 3.1).
Despite the non-monotonicity of s5()), we show that R(3 7) and its normalized variant (3,33 7) increase monotonically y (Corollary 4.1). These findings support the design of fairness interven- tions: we use (B, 7) to compute the Pareto frontier of bias-correlation combinations that achieve a target representation level т. Our main analysis focuses on the regime / preference for Institution 1. In Appendices K and L,, we extend the model to handle stronger bias (8 < c) and partial preferences, and show that several key structural properties persist.
c and full candidate
The model makes several simplifying assumptions to ensure analytical tractability. Candidate attributes are drawn independently and uniformly from 10, 1]. following standard assumptions in prior work on fairness in matching and screening [18, 29, 15]. While this enables clean derivations and actionable intervention design, it does not capture the full heterogeneity or noise present in evaluations. Some generalizations are straightforward. For example, our techniques extend naturally to settings with asymmetric capacities (c1 * c2) or candidate masses (#1 # 122). Additionally, results from [4] can be used to transfer our continuum-based analysis to finite n, with approximation error Ol)/vn); see Appendix B. The framework and techniques also extend to several richer settings beyond the symmetric, independent case. First, allowing asymmetric bias across institutions (3, * f2) shows that greater evaluator alignment (r) continues to improve representation, as less- biased institutions can "rescue" disadvantaged candidates who narrowly miss selection elsewhere (Appendix 1). Second, in the additive-bias formulation (Appendix J), where disadvantaged candidates face fixed evaluation penalty rather than a multiplicative discount, the equilibrium structure and existence of 7-thresholds persist with only quantitative changes to the cutoff equations. Third, when candidate attributes are drawn from smooth but non-uniform distribution with cumulative distribution function o, the equilibrium thresholds satisfy ((s)) + +b (3) 2(1 c1), generalizing
the uniform case (4). Although closed-form expressions are no longer available, the expressions used compute probabilities such as Pr[ui 81, 442 2 82] can be written in terms of 4 and the resulting thresholds and fairness metrics remain numerically tractable. We omit the details. That said, if candidate attributes exhibit strong dependence-for example, if all candidates have identical or highly similar profiles- the concentration results needed for the iimit-to-infinite reduction may not apply. In such cases, convergence to a deterministic limit could fail, and our monotonicity guarantees may no longer hold. Nonetheless, we expect the qualitative trend-that increasing ? improves RRtoo persist under weaker assumptions, such as mild correlations or clustered attribute structures. Extending the convergence analysis to such settings remains an open and valuable direction for future work.
Several directions remain open for future work. First, incorporating strategic behavior-such as candidates adjusting effort or signaling based on perceived bias enable a game-theoretic analysis of incentives. Second, extending the evaluation model to include multi-attribute correlation, group-dependent score distributions, or asymmetric institutional preferences could better capture practical complexity. Beyond the static analysis presented here, natural next step is to examine how the representation ratio R might evolve over time in systems that repeat or adapt across cycles-such as admissions, hiring, or promotions. In such dynamic environments, past matching outcomes could shape future candidate preferences, group participation, or institutional strategies. This feedback could lead to rich temporal effects: higher representation might enhance perceived fairness or institutional reputation, creating virtuous cycle, whereas persistent underrepresentation might discourage participation and reinforce disparities. Capturing these dynamics would require an explicitly game-theoretic or behavioral model with endogenous feedback, a direction we see as particularly promising for future work. Finally, integrating this franework with empirical data to estimate or learn (8, 7). and adapting interventions based on observed outcomes, could support adaptive faimness mechanisms in real-world deployments.
10

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Acknowledgments
We thank L. Elisa Celis for useful comments on the paper. This work was funded by NSF Award CCF-2112666 and in part by grants from Tata Sons Private Limited, Tata Consultancy Services Limited, Titan.
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Contents
Introduction
2 Model
3 Main results
4 Analysis and intervention
5 Conclusion, extensions, limitations, and future work
10
Detailed related work
17
B From finite to infinite: A continuum stable matching game
18
B.1 Proof of Proposition B.1
19
B.2 Proof of Proposition B.2
20
B.3 Thresholds in the finite- sntte-population setting
20
Monotonicity of thresholds with respect to 3
22
D Proofs of results for y-thresholds
23
Proofs results for 8
26
F Proofs of results for the representation ratio
29
F.1 Proof of Theorem 3.3 and Corollary 4.1
29
F.2 Monotonicity of representation ratios with respect to /
32
F.2.1 A lower bound V
35
Details of worked example in Section
36
H Observed utilities
36
Observed utility of Institution
37
H.2 Observed utility of Institution 2
37
H.3 Monotonicity of utilitics with respect to Ø
39
H.4 Variation of with respect to 7
40
I Extension to distinct bias parameters for two institutions
41
L.I Computing thresholds and representation ratio
41
I.2 Numerical evaluation
42
J Thresholds and representation ratio computation for the setting of additive bias
43
The setting where 3 < 1 - c
45
K.I Expression for s
46
15

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K.2 Existence of y-thheshollds
46
K.3
Variation of $2 with y
49
The general preferences case
53
L.1
Equations for the thresholds and existence/uniqueness of solution
54
L.1.1 Equations for the thresholds
54
L.1.2
Existence and uniqueness of (81 8 for general p ................................
54
L.2
Computing the thresholds
55
L,3
Expressions for metrics
................................
57
L.3.1
Representation ratio
................................
57
L.3.2
Observed utility of institutions
58
L.4
Monotonicity properties of 8 and 8 with respect
60
L.4.1
Strict monotonicity properties of the thresholds with respect to p
62
L.5
Numerical results on behavior across preference and correlation parameters
64
16

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A Detailed related work
The game-theoretcc study of assignment and selection problems, initiated by [24], has expanded into broad literature exploring stability, truthfulness, incomplete information, and machine learning approaches to preference inference [48, 49, 50, 21, 45, 34, 8, 32, 33, 39j. Comprchensive overviews provided in [51, 37].
Several empirical studies examine biases in algorithmic assignment and selection systems across real-world contexts, including centralized admissions and online labor markets [56, 26, 31, 30].
Our work builds on and extends the framework of [15], which studies a centralized multi-institution matching model where a single entity evaluates candidates. Their model assumes a uniform evaluation process across institutions and systematic bias against vastadvntteed candidates, leading unfair and inefficient outcomes. They propose fairness-awan aware selection algorithms to mitigate these issues. We generalize this setting by introducing correlated institutional evaluations and analyzing how bias and correlation jointly shape fairness and representation.
A related line of research examines the role of correlation in candidate evaluations (e.g., [6, 5, 3, 12D. [12] studies a model where institutions assign correlated scores while maintaining unchanged marginal distributions. Unlike their work, which focuses on emergent statistical discrimination due to varying correlation across demographic groups, we analyze explicit biases in evaluations and their compounded effect with correlation on stable matching
A parallel and complementary line of work studies correlated evaluations in continuum matching markets using stylized models of noisy signal generation. In particular, [57] and [58] analyze how shared or idiosyneratic noise across institutions shapes diversity, efficiency, and learning dynamics. While their models abstract away from institutional selection rules and stability constraints, they highlight how evaluation noise and correlation can lead to monoculture outcomes or suppress informative distinctions between candidates. Our work differs in modeling two-sided stable matchings with explicit score- thresholds, but shares a broader motivation: understanding how evaluation design influences representational and welfare outcomes.
Other studies explore decentralized selection systems where institutions evaluate candidates using multiple criteria. [18] examines decentralized matching but focus restiutionnsppelle fit measures and strategic behavior, whereas our model incorporates both correlated evaluations and biases. Despite these differences, both approaches highlight inefficiencies in decentralized allocation.
Our work also connects to research using a continuum model of students and institutions, which simplifies analysis by treating admissions as a uuply-and--deann problem. [17] study noisy priorities, while [7] develop a framework for characterizing stable matchings. [4] refine these models to improve approximations in finite markets. We apply similar techniques to analyze equilibrium thresholds under bias and correlation.
The role of correlation in student priorities has been examined in school choice, particularly regarding tie-breaking lotteries and welfare outcomes. [1, 4] compare settings where institutions share common lottery versus independent lotteries, corresponding to 1 (fully correlated) and = 0 (independent evaluations) our model.
Beyond school choice, other studies analyze the correlation between student preferences and priorities. [1i] examines one-to-one matching mechanisms that avoid systematic disadvantages, while [19] study how correlation affects the stability-effcciency trade-off when comparing Deferred Acceptance and Top Trading Cycles
Our work also relates to empirical studies on biases in real-world allocation systems, including school admissions and online labor markets [56, 26, 31, 30J. [12] theoretically analyzes selection under differential noise, showing that increased noise for disadvantaged groups can harm both groups by reducing the probability of securing a preferred match. In contrast, we model bias as a systematic downscaling of scores and analyze its effect on representation and fairness metrics.
and corresponding interven- tions. Several studies investigate core algorithmic problems when inputs are subject to bias, including subset selection [29, 23, 52, 13, 25, 38), ranking [16], and classification [9]. Another line of research focuses on fairness constraints in selection problems, such as ensuring proportional representation [29, 20, 27]. with the goal of mitigating bias through explicit constraints.
Finally, we discuss related work on bias in algorithmic
17

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B From finite to infinite: A continuum stable matching game
Equations for thresholds. We consider a two-institution matching setting, where the institutions have capacities e n and cyn, respectively. Candidates belong to either an advantaged group 11 or disadvantaged group G2. with group sizes |G,] = Vin and [G2] = 12n. Here, Ci2 c2, V1, P2 are constants. Recall that every candidate prefers Institution 1 to Institution 2, and that the observed utilities 0, and ررن are generated as described in Section 2. In a deterministic setting, the following result holds. The proof appears in Appendix B.1.
Proposition B.1 (Thresholds in the deterministic case). Consider a deterministic setting with two institutions and two groups such that the observed utilities ühe are all distinct. Then, there is a unique stable matching M that can be described by two thresholds 81 and 82 as follows: M- M1) = {i: ú, 2 81), and M (2) = {i -ua < 81 and ú,2 2 82)-
The key intuition behind the proof that since all candidates strictly prefer Institution any stable assignment must first allocate the top c candidates- ranked by ύ, to Institution 1. Among the remaining candidates, Institution 2 then selects the top ezn candidates based on their ú, values. The thresholds and &2 correspond to the lowest ú,1 and ü;2 values among candidates assigned to Institution and Institution 2, respectively.
In the stochastic setting, the observed utilities are random, and so are the thresholds S and S2 that determine the matching. The stability conditions in the finite setting require that exactly can candidates are assigned to Institution I and enn candidates to Institution 2. That is, conditioning on
the thresholds S1 and S2, we have
ENEG 1(aa 2 S1) + EiEG, 1(u, 2 S1 = cnn,
(5)
EieG, 1( 2 S2, in < $1 + Σieg, 1ú, 2 S2, dn11 < S1 = Cyn.
(6)
Taking expectations conditioned on S1.S2 (and then over the joint distribution of (S1, S2)), we obtain
E EiEE Pr[ua1 2 S1 S1. S2] + Eieo, Pr[u:1 2 S S1, = c,m,
(7)
E Exeo, Pr(u.22 S2, in 81 | S2] + Eiec, Pr[uz S2, in < S1 S1]] c2n.
(8)
Dividing by n, we have
E Pr 2 S1 S21l + [Pr 2 S1 S1, S2ll = c1,
(9)
EE 2 S2, an < S S1 S2]l + 122E [Pr[āş2 2 S2, 4.1 < 81 S1, Sz]] C22 where i G G2. [7] show that as n - oc the random thresholds S and S2 concentrate and converge almost surely to deterministic limits $1 and respectively. Therefore, in the large-marke market (mean-fied)) field) limit, the conditional probabilities can be replaced by the probabilities evaluated the limits, so that the above equations reduce to
(10)
1/1 Prium 2 s1] + 1/2 Pr[O.1 2 $1] = C1,
(11)
11 Pr(ú,2 2 82, dn < s1] + 12 Pr[uz2 2 $2, 4e1 < $1] = C2-
(12)
Equations (1) and (12) are the continuum or wield equations for the thresholds $ and 82. [4] extends the mean- mean-field analysis by establishing namee concentration bounds for the thresholds From his result, one can deduce that for finite n the random thresholds S1 and S₂ approximate the deterministic limits 81 and 82 with high probability, with all errors being of order 1/vnn. We now give the conditions under which these equations have unique solution.
Proposition B.2 (Existence and uniqueness of thresholds). Assume that 0i + C2 S v + V2. Then, solution (s 5 satisfying (11)-(12) exists. Moreover, the solution is unique, provided that the probability distributions of ú,1 and ú, are strictly decreasing and continuous.
The conditions of this proposition hold when the underlying distribution is uniform on [0,1], ensuring the existence of a unique solution 8i, s as functions of the model parameters. The proof appears in Appendix B.2. For a fixed 8, we also define ii,8 = min(1, s1/ and $2.9 min(1, $2/
Expressions for metrics in terms of thresholds. Given thresholds sj and z for the two institutions respectively, the ratio of the measure of candidates from G1 that are assigned to one of these
18

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institutions to the measure of G1 is given by (here i E G1): Pr[u, 2 sil + Pr(u2 2 s2. un < <
The corresponding quantity for G2 can be expressed similarly. Thus, if E G,, G2. then representation ratio is equal to
Pr(u.) Pr R. <ei] R
(13)
Pr(an P*f]+Pr Pr/u.2 Ra1<e]]
Given the threshold si the (observed) utility derived by Institution from a candidate i Ua < si; Uh1 otherwise. Therefore, the utility of Institution is given by (here i E G1, E G2):
U1 = 11 s1(s [ss 2 sildu, + V22 So s1[s 2 silduss
(14)
where μ4 is the measure induced by the distribution of ūi for E G1 (and similarly for p!). The utility of Institution 2 can be expressed similarly.
Continuum game interpretation. We end this section by interpreting equations (11) and (12) as stability conditions in an infinite matching game, where G1 and G2 represent continuous populations rather than discrete candidates. Consider setting where candidates belong two groups, G1 and G2, with population measures in and U2, respectively, so that the total population is Vj + uo = 1. Institutions and 2 have fractional capacities e₁ and c2, satisfying the feasibility constraint: c₁ 02 S
in + 12 = 1. Admissions follow a threshold-based rule:
Candidates with á. 2 are assigned to Institution 1.
Among the remaining candidates, those with ú, 2 82 are assigned to Institution 2.
All others remain unmatched.
The thresholds (s1, 82) must satisfy Equations (11) and (12). Equation (11) ensures Institution 1 selects exactly e1 candidates, while Equation (12) ensures Institution 2 fills exactly 0g positions from candidates not admitted to Institution 1.
In finite matching models, stability requires that no candidate-institution pair forms blocking coalition, where both would prefer to be matched to each other over their current assignment. In the continuum setting, stability is implicitly enforced by the threshold structure and capacity constraints. Candidates assigned to Institution 2 or left unmatched cannot move to Institution unless 8 decreases. However, decreasing 81 would admit more than сj candidates, violating capacity constraints. Similarly, Institution cannot replace a lower-ranked candidate with a higher-ranked one without exceeding ei displacing an admitted candidate, contradicting its ranking rule. Candidates assigned to Institution 1 or left unmatched cannot move to Institution 2 unless 82 decreases, which would admit more than 22 candidates, again violating capacity constraints. Since each institution fills to capacity with the righest-ranked available candidates, and no candidate can move to more preferred institution without violating constraints, the resulting assignment is stable by construction. The equilibrium equations (11) (11)-(12) thus encode the continuum equivalent of a stable matching.
B.1 Proof of Proposition B.1
Proof. Since the number of candidates is more than the total available capacity, it is easy to check that any stable assignment must fill all the available capacity, i.e., must assign cyn candidates to Institution and czn candidates to Institution 2. Let S be the top candidates according to the utility value ūn (observe that we are using the fact that observed utilitics are distinct and hence this set is well defined). We first argue that any stable assignment M must assign S1 to Institution 1. Indeed, suppose E Sj is not assigned to Institution I by M. Then there must be a candidate i' & Si that gets assigned to Institution 1. This creates an instability: Institution prefers to, i', and prefers its assignment in M (which could be Institution or unassigned) to Institution 1. Thus, M must assign exactly S to Institution 2.
Similarly, if G denotes the set of all candidates and S₂ is the subset of \ S containing the top can candidates according to î, values, then any stable assignment must assign S2 to Institution 2. This establishes the uniqueness of stable assignment. The thresholds 81 the minimum ūn value of a candidate in S1, and similarly for $2.
19

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B.2 Proof of Proposition B.2
Proof. Step 1: Continuity and monotonicity of allocation functions.
Define the allocation
functions:
A1(s1) = 112 Pr(ua 2 $1] + 1/2 Pr[u01 2 $1),
(2 82) i Pr(uz2 2 $2, un s1] + 12 Prfüyz 2 82, 0y1 < <81].
Since Pr[ūsy 2 s] is the survival function 1 Fiy (8), where Fiy(8) is the cumulative distribution function (CDF), it follows that:
Ai (81) is continuous, strictly decreasing function of 81-
*A2($1, 52) continuous both $1 and 82.
Step 2: Boundary conditions. Consider the extreme cases:
If 81 1, then Pr[ú.: 2 1] 0, so Ai (1) 0.
If $1 = 0, then Pr[ú: 2 0] = 1, so A (0) 1/1 + 1/2
If = 1, then A2(s1, = 0.
If 32 = 0, then (21 0) reaches its maximum possible value V1 + V2.
Thus, for any feasible capacities C₁, satisfying ei + S + V2, the thresholds 81, can be adjusted ensure feasibility.
Step 3: Existence of a solution. Define the function:
F2) = (11 c1, A2(s1, S2) -2)-
Since A (si) is strictly decreasing and continuous, and A2 $2) is continuous in both variables, the
Intermediate Value Theorem (applied in IR²) guarantees the existence of a solution (si, 85) satisfying F(s1,85) 85) (0,
Step 4: Uniqueness of the solution. To prove uniqueness, assume two solutions ($11, 8/2) and (s"f sy)). Since A1 (51) is strictly decreasing, follows that:
(s)) = A (s = s =
Similarly, for a fixed si. A2(si, 82) is strictly decreasing in 82, implying that:
A2 16)) = A2 85)) + 2/2 =
Thus, (si, s5) is unique.
B.3 Thresholds in the finite- poiitepppplltion setting
In the finite model, each candidate in a group G,, where j E (1,2), has latent attributes (Mi (22) drawn independently and uniformly from [0. 112 These attributes are transformed into observed utilities ü.j. which determine candidate preferences and institutional choices. A realization w corresponds a specific draw of all (En U:2)) pairs in the population. Once is fixed, the utilities üæj are fixed as well, and the resulting stable matching is deterministic.
In this setting, Proposition B. establishes that for every realization w, the stable matching is characterized by thresholds (w) and S (w). such that Institution selects candidates with áa 2 ST (w) and Institution 2 selects those with 0:2 2 (ww).
Assuming candidate attributes are i.i.d., the fraction of candidates in each group satisfying these threshold conditions concentrates around their expectations by standard concentration inequalities (e.g., Chernoff bounds). Consequently, the empirical thresholds S (w) and S (w) converge to deterministic limits sf and s model.
oc, satisfying the equations derived in the infinite- populatoo
While our main analysis characterizes the equilibrium thresholds (s1; st) in the continuum limit, it is natural to ask how close the thresholds finite population are to their limiting values. The following result shows that these thresholds concentrate sharply around their deterministic counterparts.
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Representation Ratio (mean + std)
7
Representation Ratio (mean + std)
0.0
0.48 + 0.16
0.0
0.44 + 0.10
0.1
0.44 + 0.14
0.1
0.45 + 0.10
0.2
0.44 + 0.15
0.2
0.41 + 0.10
0.3
0.49 0.16
0.3
0.44 +: 0.06
0.4
0.33
0.4
0.53 + 0.09
0.5
0.35
0.5
0.56 = 0.10
0.6
0.61 0.19
0.6
0.58 + 0.10
0.7
0.64
0.7
0.58 + 0.12
0.8
0.64
0.8
0.63 + 0.12
0.9
0.62 0.17
0.9
0.67 + 0.16
1.0
0.73 = 0.42
1.0
0.63 + 0.13
n = 100, c = 0.2, B = 0.9, |G1 = |G2
n2 = 250, c = 0.2, 3 = 0.9, |G1] = |G2|
100.
250.
Table 2: Representation ratios for different values of and finite n. The standard deviation decreases increases, indicating closer agreement with the continuum predictions.
Proposition B.3 (Concentration of finite-populat population thresholds). Given parameters B, and Y, (8 (5) denote the unique solution to (11)-112). Consider a finite stochastic instance I with |G = G2 n, and two institations of capucities each, bias parameter 3, and correlation parameter ry. Then, with high probability, there exist thresholds 81, 82 governing the stable assignment in I such that
=O( logn
1st s71 =
for each E E (1,2).
Proof. The proof follows by standard Chernoff- argument. Given c, 3,, y Proposition B.2 ensures a unique solution ($) to (11)-(12). In the finite instance I, random thresholds $1,82 define the unique stable assignment as in Proposition B.1.
For each candidate E Gi U2, define the indicator X; = 1(ū s -J), where 0 small enough that 8f e 2 0. Let X = Σ, X₁. If E G1, then E[X.] = Pr[ua 2 sil e, and if E G2 then EE [X,]] Pr[u,1 si]. Hence,
B Z n(Pr(un 2 sil + Pr/0v1 2 (1])
nC1 ne,
where E G1, i' E G2, and the last equality follows from (11). Let & denote the event that X < ne
gives Pr[E] S n for some constant 0. If does not occur, then 81 2 sf - E: otherwise, 81 s - would violate the capacity constraint at Institution 1. A symmetric argument yields 5 s +
By the Chernoff bound, Pr[E] S =//4// Setting & = Ing
An analogous argument for 82 (using expressions for Pr(ú,2 $2 A u. $1] from Section 3) shows that [82 - s < Ce with high probability, for some constant c depending on c, 8,7. This completes the proof.
Finite-sample validation. To corroborate the theoretical results and illustrate their practical rel- evance, we implemented our model for finite sample sizes (n 100 and n 250 per group) and conducted 30 independent trials for each value of The results, summarized in Table 2, show that even for moderately sized systems (100-250 candidates per group), the continuum thresholds provide an excellent approximation to the empirical representation ratios. As expected, the standard deviation of the observed representation ratios decreases with increasing n, indicating convergence toward the continuum-limg predictions.
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C Monotonicity of thresholds with respect to B
In this section, we prove the monotonicity of s1 and s} with respect the bias parameter B. This also implies that the representation ratio is monotone with respect to 3 (Proposition F.I). We restrict the discussion to the symmetric setting VI = 1/2 = 1 and e₁ C2 = c. We further prove a useful result showing that the threshold si always remains at most 81.
Proposition C.1 (Monotonicity of sj and s w.r.t B). For any fixed value of 7 the thresholds si and 2/ are non decceeeen functions of the bias parameter B.
An easy proof of the monotonicity of si with respect to 3 considers the expression for st. Recall from 2-c if B 2 1- c, and C otherwise. This expression clearly monotone in Section that si = 1+1/3
B. However, we provide a more intuitive proof that also extends to the monotonicity of s2.
Suppose we increase 3 while keeping oं fixed. The measure of selected candidates from G2 increases (while that for G₁ remains unchanged). Since the capacity of Institution is fixed, s must also increase. The argument for the monotonicity of s is more involved version of this idea. As corollary, we find that the representation ratio is monotone in 3: increasing 3 raises both thresholds, thereby reducing the admission probability for candidates in G1.
Proof. Given value & and threshold $1, let f (3, denote Pr[un 2 s1] + Pr[u01 2 s1l. where E G1, i' E G2. Then f33, 81) ) a non-decreasing function of B and decreasing function of 81. Further, Equation (11) implies that [(3, (3))
Now consider values 31 < B3g and assume for contradiction that (11) (32). Then:
c= (( S f(92, ((())) < ((2, c,
which is contradiction. Therefore, s1 (82) 2 (3)).
We now prove the monotonicity of (B)) Rewriting Equation (12), we have:
Pr/y +1-xx++1->>>> 2 $ X < Pr|X < P
8 18x < Pr| sil c,
where X, Y are independent and uniformly distributed on [0, 1].
Let Zg be a uniform random variable on [0, s]]. Then we can rewrite the equation as:
Pr|yZ3 + (1 - >) 2 2 Pr[X- s1 + Zr + 8(1 7) 2 s Pr|BX < <s =c
Let: A(3) := - ((3) == Pr|6X < s
2 s5 (B3 = Pr172
53} p(3) = Pr[X < si].
Suppose 31 < 32 and assume s (B1) > (1²). Then using monotonicity of s (ß), we verify: (i)
A(3)) < AA(2), (ii) B(β;) < (32)). (iii) ((11) p(32), (iv) A(3) 2 B(3) for all B [0. 1).
Since A(3)p(3) + B(8) = c, the difference becomes: A(31) ((3))((3)) ( -
This difference can be rewritten as:
A(8:)(p(31) p(32)) + +1000-1311111113133---300000-100
B(32))
From Equation (11), we have p(6) + so s0 p({b) -((2) - Substituting and simplifying, we get:
A(3)) + (B2) - B(b)) <0, which contradicts the assumption. Therefore, (31 - sB²))
(A(3)) - -()))))33)--<())+)
The following result shows that when the two groups have equal size and the capacities at both institutions are equal, the threshold s; is always at most si The proof follows from the observation that if 85 > si. then a candidate G, must satisfy 2/2 2 s5 to assigned Institution 2, effectively reducing the problem to single-institution setting, which leads to contradiction.
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Proposition C.2 (s dominates 2 in the symmetric case). Suppose 1/2 = 1/ and c C2 = cv.
Then, for any fixed c, and 7, we have si S si.
Proof. Let X and Y be iid uniform random variables on [0, 1]. The threshold st satisfies: Pr[X 2 s + Pr[x 2 si,el =c
where $i ==
The threshold s satisfies:
Pr(yX (1 -)) 8q X < s1] + + (1- 2 82 X <t =
where $2.8 :- min(1 s2/8).
Assume for contradiction that x² > 81 Then:
Pr(Y > s 2 Pr(yX + (1- 2 82 X < s1).
and similarly for s2.8- Adding both terms yields:
Pr|Y > s5 + Pr|Y > s2 2c
But since sf < <5 this implies:
Pr|y > 8 + Pr|y > $1
contradicting the definition of sj. Therefore, s - si.
D Proofs of results for y-thresholds
In this section, we present the proof of Theorem 3.1 that establishes the existence of the 7-thresholds
ni, 72, and 7/ when 8 2 1 1-c c. By definition, these thresholds characterize the transitions in the evaluation probabilities Pr|yXX + (1 - >> 2 s2] and Pr[1X + (1- 2 s2 /3 as y increases from 0 to 1. Specifically, the system passes through the regime transitions: (III,III'), (III,II",, (II,I'), and (IV.TV").
We also describe how to compute the values of 7/, "/2 and Ya, and establish monotonicity properties of these thresholds with respect the bias parameter B. We restate the theorem here for convenience.
Theorem 3.1 (y-thresholds). For any fixed B 2 -1-c. c, there exist unique values S 122 S 73 E
[0, such that: (i) (T)/BB S 1- ? and only if S Ti (ii) sī(?) < 1-77 if and only if
S 72: (iii) (7 2 78 if and only i I 73- Moreover, these thresholds are given by: (i)
28 08+1 (0)))11)4(1---1 4(1-c)
71 (64(511/3) 401 +)+ ())())))(()))337
(ii) 722 . where the unique root in (7i, (3) of
(I
( r ( -
+ (1- 0; (iii) Itc'
We break the proof into two parts. In the first part, we show the existence of thresholds. Subsequently, we derive expressions for them.
Proof of existence of 7tthresholds. We show that
si() y
is decreasing function of Consider values
71 < 72 where 71, 71 E (0, Our goal is show that si(72) si(m) By definition of s²(11),
72
Pr(m X (1 Y 2 (5()), X $8] Pr/m X + (1 7 52,0(7), X < $ c,
where X and Y are independent random variables with values in [0, 1]. The above condition can be equivalently written as
82(11) X < si Pr x (1/m 1)8
( x < $ = c.
Pr X + (1/7 - 1)
71
21
Now observe that x + (1/m 1)y + (1/72 1)y for any x, E (0, 1). Thus, simple coupling argument shows that
+Pr X + (1/72 - 1)Y $2,((1)
Pr
x + (1/72 - -
< s
X < s < c.
71
71
23

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Assume for the sake of contradiction that Therefore, the above inequality implies that
((m) '
s1(2)) 72
Then,
22,0(10) 72
as well
s5(12)
22,0122
Pr X + (1/12- 1)Y
X
+ Pr X (1/72- 1)Y sी
, $1 <c.
7/2
72
Rearranging terms, we get
Pr [122 + (1 - (2 2 (2)))) X < <61 + Pr [22 + (1 -2) 2 22 (12), X < s1,e0 <
But this contradicts the definition of 82(12) Thus, we see that +2(1) is decreasing function of 7. In similar manner, we can show that =2.a(t) is a decreasing function of y, and (i(n) increasing functions of y. Since the distribution from which the attributes of candidate drawn (in this case, the uniform distribution on [0, 1) is continuous, it follows that s3 is continuous function of y.
1-7 and $2,(7) I-> are
+5((1)) 1-7 approaches 0 and when approaches 1,
(2)(7) T- approaches oc. Since
When approaches 0, +2,)())
is a monotonically increasing continuous function of y, there a unique value % call it V 1-7
such that <2,(m)) = 1. This first statement in the lemma now follows from this observation and the T-m
monotonicity of <1.(1>) Other statements in the lemma can be shown similarly. 1-7
Proof of expressions for y-thresholds. In the regime 3 S 1 - C, we know that s S 3. We also know that s S si (Proposition C.2). Thus, si, s S 8 for all y E [0, 1]. Hence, $3.5 = = for j {1, now follows from the definition of 73 and 74 that 7/3 = 74. The monotonicity of Y1 < 72. We now show that 7/2 S 73.
si I-y shows that
We first argue that "1 73. Suppose not, i.c., T3 < 7/1 For y E 7s], we are in case III and III' Therefore, $ satisfies:
2 - (s$)) s
2s - 7(7))
(15)
2(1 (1) 7)
213
By definition of 73. 82 7331 when 7 = Y3- Substituting this above, we see that
73($))²
s1 + 1/3)
2(1 73) (1+1/8²) cc
Since 73 S 71, S.1, we know by monotonicity of <i(T) that s = = S B1 - 73). Thus, we get I-7
Ast 2
3 + 23
1
c 22s
(1 + 1/B²) sj
si
where the last inequality follows from the fact that 8 < 1. But we know that 8
2 2-
T+1/3
if e < 1/2. Thus, we cannot have 7 I m- Thus 71 = min, 72, 71s) and hence, (15) holds for all
E [0, m.. Substitution s B(1 n) for 7 = in (15) yields the desired expression for 71.
Now we show that 72 S 73. Suppose not, i.e. 73 E ((1, 72). We are in cases II1 and II' in [71, 7a]. Therefore, si satisfies:
28 - 7(5))
(1-7 8 + /3) 2(1 71
s
= c.
(16)
2y(1-7)
For 7 = Y3, we know that s = 7 Substituting this above, we get:
72()))2
1 73
s 2(1 - 73) 27
Since 73 < 72, 8173 = < (1 73). Using this, we see that > sj. which is a contradiction (as argued in the previous case above). Thus, 72 S 73- The expression for evaluating 72 the statement of the lemma follows from substituting s 72 for 72 in (16).
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Finally, we show the desired expression for 73. In the interval y E [3), we are in case II and II'. Thus, s2 satisfies:
(1 (1-7-80 82 707) + (1 - -2/3 + 757 2c>(1 - 7)-
At 7 T3, $2
7381- Substituting this above, get the desired expression for 73-
Figure 4: Variation of thresholds ,...... with 3 for a fixed value of e (Left) and with e for fixed value (Right). Note that | 2 c.
We now show monotonicity of 71 and with respect to 3 observe that Ys is independent of B. Proposition D.1 (Monotonicity of y-thhrehholss w.r.t. 3). Assume с 1/2. Then (1 is monotonically increasing and (3) is monotonically decreasing function of 3 as 3 varies from
c
Proof. We first consider
?1. Using Theorem 3.1, we know that 7 is equal to
(3 + 1/8) 4(1
2s (B + 1/8) - 4(1 + (1 1/32)
a(9)
b(9)
The above can be written as +(3)+ The sign of the derivative of 71 is given by d' d(())((3) a(3)6'(3) (3), where a' (3) and b' (B) denote the derivative of a(8) and b(B) with respect to 3 respectively. is easy to verify that a(B) 2 indeed, this same as verifying 2s + 1/3) 2 2(1 c). Using I-c
si 141/3. this is same verifying
(2 cB + 1/8) 2(1 c)(1 1/3) 0.
Simplifying, the above the same as verifying
2 - 1) + c/ 200
which is true because 3 + 1 and 3 1.
Now recall that the sign of the derivative of 71 with respect to B is same as that of a a'())((3) (33) Since a(3), b(B) 2 0, suffices show that a' d()) and (3()) 0. Using the definition of sT, a' d(3 2(2 c) times
d
B + 1/3
= 32
(1 (1+3322 > 0,
d3
1 1
1/3 1+1/3)
(1 (1+
()+
because 1 2 V2 (recall that 8 2 1 1/2). Thus, d((3) > 0. We consider 6((3). Now,
B'(B) 2s dsj 2(ss))//33 Thus, 6'(3) < Of de)(0) si gs. Using sf - this inequality is /
equivalent to 1/B 1 1/BB which clearly true. Thus, (B) is an increasing function of 3.
We now show the monotonicity of (B). Suppose for the sake of contradiction that there are values 3 such that 72()) 2 72("). Now, by definition 72, we know that (32)) the following sequence inequalities:
Now, we have T-
1 *1 ()() 3))
< ()(12(3'),)
(1(()(8))88) 3')
1822(8)
1- (3"))
1 72(3")
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Here the first inequality follows from the fact that for fixed B, <9622 is increasing function of T-
and the second inequality follows from the fact that for fixed 7, s (3) is a monotonically increasing function of 3. This leads to a contradiction, and hence 7/2 (B3 is monotonically decreasing function of B.
E Proofs of results for 2
In this section, we present the proof of Theorem 3.2 and analyze the behavior of the selection threshold s as y varies. Our approach is to examine the governing equation for sž in each sub- interval of [0, 1] defined by Y1, 72, 73, where the selection dynamics transition. Using these equations, then study the non-monotonic variation of $ with yy: y increases, s2 initially decreases but eventually starts to rise again, The proof of this result relies on analyzing the equations governing si be a linear equation, analyzing closed-form solution for
from Theorem 3.2. Unless this turns s² is highly non-trivial. Our approach involves studying the equations satisfied by the first and second derivatives of s with respect to y. This enables us infer its behavior across different regions of y. We now prove Theorem 3.2. We restate it here for convenience.
Theorem 3.2 (Equations for s (r)).. Assume 2 c and c <
1. 2. Then $7(7) satisfies:
[0, si 2sjs; -187)) 2(1-7)
2 1
(3-(1-7)
[2, 8y
2s 20 2
(1-
B+g33//99
2(1-7)
2イ(1
[12, 73]: [1 (1-7-s)+7911"
(1-2-
s /B)
27(1
24(1 -)
(13, 1]: s1 + 1/3) #2
Proof. These equations follow by definition of 7i, . 74 and the corresponding cases described in Section 3. For instance, when 7 E (1), we are in case III and III', Other cases can be argued similarly.
We now show that x7(7) varies a non-monotone manner, but has only one local minimum.
Theorem E.1 (Variation of 82 with 7). Assume that the bias parameter 3 satisfies 3 2 e and 1/2. Then, 82 exhibits the following behavior as y varies from 0 1:
(i) For E [0,7]] m) s2 decreases linearly with 7
(ii) For E [Vv. 72). st is unimodal: it initially decreases and may subsequently increase.
(ii)) For E 73), st is convex.
(iv) For 7 E (1, 1). s1 is increasing.
Furthermore, the slope of s at 7/2 matches that at 7... Thus, s is unimodal.
Proof. Case 7 E [0, 71]: Here, Theorem 3.2 shows that sj varies linearly with y. In fact, s2 is equal
1
[2 + 1/8) -) 2c(1 + +1 1/83))
2( 3 B2)
The coefficient of y is equal to 2x{(1+1/32) times
2c+ (1))11+1//))22(111111) = 4c 4c-4+ ( 2 2-c
2 (1 )'
(1 82
S 4e - 4+ 42 -2- c)2/2.
1 1111/
The above is negative if 1/2. Thus, si is decreasing function of y in 71]-
26

---

Case Y E 72]: We rewrite the constraint in Theorem 3.2 for this case as follows:
2y(1 (s] - c) 2 + + + (s1)² + (1 s2/B + //) - 0.
(17)
Differentiating this, we get (here (sg) denotes the derivative of 25 w.r.t. 7):
2(1 2 - -2sis 2y(
$/ + 7
si
2 - y
1
3
3
3
82
7s
(sg) 2755 + 3
1 1
+
= 0.
B
3
We first show that (sg)' < 0 at = 7. For this, we substitute 85 = 3(1 - 71 in the equation above.
Since the coefficient of (st) is negative, it suffices to show that the remaining terms not involving (sg)' is negative, i.e., we need to show:
2 <0. B
(1 1 21)ss c) -2s (011--1)) 71) + +22(5))
The l.h.s. is linear function of 71. Since 71 2 0 and 3(1 -1) 2 sini (because 71 S 7s), it suffices to check the above condition for 711 = 0 and ?1 = st+;
8 For 712 O, the above condition holds
because o1 S B. When 71- 3-37 7/1 2 1/2 because sj S B. Therefore, (1 - ) c) < O. Using this value of ", B1 - 7n = =- Thus, the above condition holds here as well. Thus, we have shown that s3 < 0 at = 71- It remains to show that s is unimodal. This follows from the following general fact:
8
Fact E.2. Let f(y) be a twice differentiable o on closed interval (21.72l such that the following condition holds:
ff + CC
where B, C. D are independent of C + 0, and A(r) is a continuous function of o and there is a positive constant E such that A(7) E. Then f(7) is unimodal on [T,.7al 72,, i.e., if f'(Ta) = ((( = 0, then ('(7) = 0 for all y E [b).
Proof. First, consider the case when D 0 (the case D < O analogous). Let u(7) denote the f' (?). Assume that there are two distinct values Ta, 7h E [2), [ Ta < 7% such that v(?a) = u(7)) O. but u(70) A 0 for some 7o E [.] Since u(7) is continuous, we can assume, by suitably selecting Ta and 7h. that u(>) is non-negative in [Ta, 76]- By the equation stated in the condition above, we see that there is constant
such that if [u()]] < E, then u'()) < 0. By continuity of u(>), there exists a 8 >> 0 such that for all points E := 17: min(7. + 6, 70)]. 0 < uu>) < E. Now, consider value 7 E L, where 7 # Va- By mean value theorem, there exists value % E ITa 7 such that u' (Tr) = u(7)=w(Tk)
7-7a > O. But this is a contradiction, because u(7)) < E and hence, u' (1e)
u(a)
7-%
must be negative.
Finally, consider the case where D = 0. Assume that C > O; the other case can be handled similarly. follows that there is a constant e such that if < u(7) S E, then u' (7) < 0. The above argument now carries over analogously.
Let us verify that the conditions stated in the above fact hold in our setting. Differentiating (17) twice and grouping terms, we get (here (s²y and (st)" denote the derivative and the second derivative of s w.r.t. y respectively):
22 B
2 82 (s))))++
4 [
+ +2877) (st)"
As
11-$81 -1-
si 01-22
O.
Here A denotes
+
781 B
Note that A > because s S s/. and hence,
3.
7
8 (1 71 B) 2 0.
A > 1
7
B
27

---

We also note that
4
4s
3 [1 (1- s5/3) 40.
Indeed, if the Lh..s is equal to O, then 3s = 1- - // 3 i.c. si + 1/3) = L. Now, B 8+14 1/3 2 2 and si > 1/2. Therefore, this cannot happen. Thus, the conditions stated in Fact E.2 hold with f(7)
denoting s5()), a(7) + 2(1 2(2))² st $)
# 2A + 2s17, b 3
4st 3 (1 si/8) and d = - c)
We now claim that (s²(r))' can be for at most one Y E [2], Indeed, suppose is at two distinct points Noa and 7t. where Yo < 78. Then, Fact E.2 shows that s5(7) is constant during the entire interval [p]: But we argue that this cannot happen. Indeed, suppose 85(7) = during this interval. Substituting this value in (35) and multiplying both sides by 2y(1 ([]]-)), we see that satisfies a non cen- quadratic polynomial with constant coefficients. But this can have at most two roots. This is a contradiction because all y E [Te: 76] satisfy this equation. Thus, we see that s7 has at most one local maximum or local minimum in [7,, 7/l. Since (sgY < 0 at 7ı, we conclude that s2 can have at most one local minimum and no local maximum in [11: 72]-
Case 7 E [12, Ya]: We rewrite the equation for this case in Theorem 3.2 as:
(1 - 7 si + ? + (1 - 7 2 / + 78/ 207 (1 - 7)-
Differentiaiing the above equation twice with respect to 7, we get:
(() 2(f + 2f(r)) (1) +
+ 2fa(t) -4c,
where f(1) denotes (1 - - s/ + 7 and fa()) denotes (1 7 $2/ 33 Since $3 S si.
f(7) 1 7 - 5/ + 784 2 7 of + 7s7 (1 (1 7) > 0.
Similarly, fa(r) 0. Finally,
f"(7) =
= /B.
It follows that (s2)" when 7 (72, 73). This shows that si is convex.
Case 7 E [73, 1]: Solving for z in the last case mentioned in Theorem 3.2, we get:
si + B) + 3(1 7) -c3y
1+
The coefficient of y is equal to I+3 times
s1 + 3) 3 cB cB=
c3 2c3.
If c < 1/2, see that this is increasing function of 7. Thus, have shown the desired behavior of si in each of the given intervals. It remains to argue that 8z is unimodal. The only case to check is if s5 has local minimum in both [2) (where it has been shown to be unimodal) and in (72, 73] (where it has been shown to be convex). We can directly check by differentiating the equations for the second and the third cases in Theorem 3.2 that (st) at 7 = 72 in both the intervals is the same. Thus. if s has a local minimum in 72]. then it is an increasing function at = 72. Thus, it will remain an increasing function in [12: 73] (because it is convex in this range).
It remains to check that (sg)' at y = 7/2 is the same according to the second and the third equations
in Theorem 3.2. We consider the second equation first. Let g(7) denote (1-7-2//847e1/0) Diffcrentiating both sides and substitution sn = - 72, we get:
27(1-7)
28 (x5) - (+7)²
2s - 72) -s1)?
+ g'(2)2) = 0.
2(1-72) 72)
2(1 - 22
After simplifying, the above becomes:
g((12) + 28 (s)) + (st)² - 25 (1 72) 0.
(18)
2(1 - )2
Similarly, differentiating both sides of the third equation in Theorem 3.2 and substituting 8 = 1 1/2-2 we get the same equation as above. This shows that (st) 7 72 according either of the two equations in Theorem 3.2 is the same. This completes the proof of the theorem.
28

---

F Proofs of results for the representation ratio
In this section, we prove Theorem 3.3 that presents explicit formulas for the representation ratio R(B,7) 7) across the four regimes determined by the threshold values ",, 72, Ya. under the assumption based the governing equations for the thresholds si
that [ 2 1 C. These expressions (Equation (4)) and s (Theorem 3.2), and are used to establish both correctness and monotonicity results. Specifically, we analyze how the formula for R(3, 7) evolves as y increases, showing that in all four regimes the representation ratio is either constant or strictly increasing in 7, These results culminate in the proofs of Theorem 3.3 and Corollary 4.1.
In Appendix F.2, we also show that for fixed 7 R(3, 7) is monotone in B, and prove that the normalized representation ratio ((8, 7) = R(3, 7) [B, 1) is non-decreasing in 8 over the interval [,
Throughout, we evaluate the representation ratio via
R(3 7) = Pr(u,1 2 s1] + Pr(u.1 $1, úv2 2 s5]
Pr[u. 2 8i] + Pr(uai < s1; 0,2 2 st
We recall Theorem 3.3 and Corollary 4.1 here.
Theorem 3.3 (Representation ratio). Fix < and -
A88 7) = Bst + (( + +1 1)- 2(1 2 1)((1-8))) -c)
Define
and 0(3,7) =
4124) (1
-B(1-81+
Then:
1+8²
1 - (d
(A- 1- 1-13
if E 10,
2(1-5)
if E 12,72l.
1- +- 2
R(3,7)=
1- 4 (1-))
if7 E (22,2))
(1- B)))
if E (ra.) 1].
1mat (1m-10"(am) 2y
Corollary 4.1. Fix < 1/2 and 3 2 c Then R(B,7) increases monotonically in n.
F1 Proof of Theorem 3.3 and Corollary 4.1
We derive closed-form expressions for the representation ratio R(B, 7) using the equations governing si (Theorem 3.2), and use them to prove Theorem 3.3 and Corollary 4.1. The expressions defined piecewise over the intervals determined by the thresholds 71, 72, 7s. and also yield the desired monotonicity y.
Representation ratio for 7 S 7/- We know that Pr|uz 2 s2 A t41 < s + Pr/uy2 2 $2 u01 s1,g
and for y S1, we are in Cases and I'. Hence,
Pr[uz 2 82 A ua < <8 = 25 - 7(8))2
2(1 77
and
Priuy2 2 82 A uvi .T s
25 (55))
2.8
For sake brevity, let & denote 20(0)- Then we get: 2(1-7)
s1 1/3)
(1 + 1/32) =c
Using si 141/8. we see that
2(1 c)
3 1+ 1/32
Therefore,
- 1/82 182 - 1 2c
R(B,r)
1 1-
B- 2c
Monotonicity. It is clear from the expression above that R(B,7) does not vary with 7.
29

---

Representation ratio for "1 Y S 72.
When 7 E (7), Theorem 3.2 shows that:
2 2s1)2 2(1 - 7)
55 + 7
c.
(19)
2y(1 77
Further, the representation ratio is given by
R(B 7) = 1 8//B + (1-7-3
2 (-
(20)
2x -=9 2(17))
Let A denote <-1ei Then, Equation (19) can be written as:
7(5))2 2(1 7) i A
1
(1-
=c
(21)
27
Let denote 1 1 A/3. Then the above can written:
1 1-Y u² + + (1 - 3)st
((8))
2(1 -7) 0.
(22)
27
The positive root of this equation is:
-Bs y (Bst)2 2(1-2)(( 3411s
201-))((381115 -0 (s7)²
(1 7)/7
Observe that the fraction of selected candidates from G2 is which can be written as:
7
1 1-st/3+
27 u²
1 - s +
7 A
2(1 7)
where
A(3,))==-P2 + (8T) + +) 9(1-))((-P) + 1)st c)
7
Monotonicity, In (22), we let v denote
1- u. Then this equation can be written as:
2 + s +
7(s))2
u + (1 B)st c 2(1 77
2
Diffcrentiating both sides with respect to 7, we see that the sign of v' (7) is same as that of
+ (s1)² - usi 1- + (s1)²
7
Using the fact that u = 1 A/8, the above expression is positive iff 3(1 - 7) - A/8) S (81)7, which using the definition of A, is same as s 2 ((1-)). This fact holds true because 7 2 71. Thus we see that v' (v) > 0. Since the fraction of selected candidates from G2 is given by o2/2+1-s5 - we see that as we raise y, this fraction increases. Hence R(3, 7) increases with increasing 7.
Representation ratio for 72 < S 73-
In this range of we have the equation:
7 $ + 7
(1 71-7- $ + /
c.
2y(1 7)
2 - 7)
The representation ratio is given by
1- 1-//// + (1-7- 1/ + 1893 2- 7)
R(y)
(1- +148) )
- 8 +
2
30

---

Let A denote 4- Then the above equation can be written as:
20y
(1 + 1
Let denote 1 - A/B. Then, the above equation becomes:
20y
2 + (1 - 3 + Bu)²
1-
The non-negative root of u is given by:
3((1 -1 + -(1 B)2
((3 7) ==
1-7
+ _
Note that the fraction of selected candidates from G2 is given by
(1 7)000 (3, 7)
1 1 -is//3- +
2-y
Monotonicity. We shall show that 0(3,7)
is an increasing function of 7. This, along with the expression for the fraction of selected candidates from G2, will show that the latter fraction an increasing function of y. Notice that 0(B,7)
v
1-2 given by 7
(1 (1+ (++3))(3,1)
/ ----- -3(1 -31---33100--01111111 3)
-B11-
3p 1 +v- +V-1--pp
1-7 / 7
2c(1 + 82).
is easy to check by inspection that the above increases as we increase 7.
Representation ratio for 2 7s- When 2 73.
82
7 2y
Pr[uz2 2 8 A un si1 ss
?
and
82
1 -
Pr[ue2 82 A uo1 < si,el =si,3
7
27
Thus, we have the equation:
(13)) s1 + 1/3)
1 1-7 c.
Using s1 = I+1/3 the above equation, we get:
2-c
$2 7(1-20 2c) +1
? 7(1 + 1/3)
Therefore,
1+7 7(1 2c
R(3,7) = 27 7/(1-1/3)
3 8-1+7(8-1
1tr 7(1-20)+1
1- 1- 43c)
27
7(1+1/8)
Monotonicity. Differentiating the expression for R(3,7) with respect ?i we that its sign is the same as the sign of
(1- - 3 + 7(1 - - - 1 + 4c)) - 4Bc),
which is the same as 4c(1 3)2 0. This shows the monotonicity of the representation ratio in this case.
This completes the proof' of Theorem 3.3 and Corollary 4.1.
31

---

F22 Monotonicity of representation ratios with respect to 3
We now analyze how the representation ratio R(3, 7) varies with the bias parameter 5, for fixed ". We show that R(3,7) 7) is non-decreasing in 9. and further establish that the normalized representation ratio ( 7)
also non-decreasing over the interval 3 [1 c, 1], under mild assumptions on The first result follows from the structural monotonicity of the selection thresholds, as shown in Appendix C. The second result relies on case analysis based on Theorem 3.3.
Proposition F.1 (Monotonicity of representation ratio w.r.t (). For any fixed y, the representation ratio RiB 7) is non-decreasing function of the bias parameter B3.
Proof. Adding (11) and (12), we obtain
Pr((X 81) V (>X (1-7)Y 2 a)) + Pr[(3X 2 sj) ( -5)]] 2c.
As shown in Proposition C.1, both ii and 22 are non-decreasing in 8. Therefore, the first term on the left-hand side is non-increasing in 3, implying that the second term must be non-decreasing. Since the representation ratio is the ratio of the second term to the first, it follows that R(3, 7) is non-decreasing function of 3. Formally, since A 7) + B(3, 7) = 2c and si 82 are non-decreasing in 3, A(B,r) y) is non-increasing and B(3, 7) nen-decrggiing in B. Hence for 31 B2.
B(32)
B(3)) A(3))(((3)) B(3)) + ((3) A(32)) 2 0,
( A(3))
A(3:)A(B))
so R(3,1) 7) is non-decreasing.
Theorem F.2 (Monotonicity of normalized representation ratio for 3 > c). For S 1/2, if 3 2 1 - G, then for every fixed ", the map 3 + ((3,7) 7) is non-decreasing.
Proof. Our goal is to show that the normalized representation ratio non-decreasing in / over the interval [1 c, 1]. Since S and - (1-,,1), A(3
R(B,
and hence ((3,a) x) S c.
The strategy as follows: We consider the logarithmic derivative 0, In 7), and aim show that it is non-negative throughout the interval B E c, 1].
Let (B, and B(8,7) denote the fractions of candidates selected from groups G1 and G2, respec- tively. By definition, the representation ratio is given by
B(3,
R(B, 7) = A(3,
To establish the desired monotonicity, it suffices to show that 0; 7) decreases with increasing
Fact F3. 1f0, 2 03 1), then O3 In 2
Proof. By definition,
N(3,7) 3(3,7) A(3,
A(3,7) B(3,1)
Observe that for any 7, A(8,7) B(3,)) = 2c, the total capacity in the two institutions. Therefore, the above can be expressed as:
N(8,7) = B(3,7) 2c B(8,1)
2c - 2c-B(33n) B(3,1)
direct calculation yields:
0g In.N(B 7) BgB(3,7)
+ BgB(3,
0g
0gB(3,1) 2c- - B(3,1)
B(B,7)
2c - B(3,7)
B(3,1)
32

---

Proposition F.1 implies that (3, 2 0 for all Since ((3, + A(8,)) = 2c and ((3, - A(3,)) r), we see that B( S c. Therefore, 20- B(3m) > 0. Assuming 0, 2 0, 1). the r.h.s. above is at least 0 B 1) times
1
B(3,7) 7)
B(3,7)
B(3,1)
B(3.1)
One can easily check that + 2c-z above quantity non-negative. Thus, we have shown that Og In (13,7) > 0.
a decreasing function of x when S x S Therefore, the
Thus, it is enough to check that 0₂ 2 0gB(3,1) for all 0 S 1. We now consider the various regimes in which y lies:
Case 4 (< 2 Ys 1/(1 + c)):
The analysis for this case in Appendix F.1 shows that
B(3,)) = 22.3
1 1-7
7(1 2c) +
1 1-2
?
27
7 33
27
Differentiating with respect to 3, we get:
0B (1 (1- +
(1+ 3)2
Clearly (3,(3,7) ?) is decreasing function of y and hence, Og,( 2 0gB((31) when y 2 79- We also note the following for rest of the proof:
2(1 cc c) B3
(23)
(1+ 8)2
Case (r S 71 (3)): The argument for case 1 in Appendix F.1 shows that
B(3 = 2(1 - c)
1 32
Therefore,
0BB(3,n) = 4,(1 c)
(1 + 32)3
Since [1 [1-c, c, 1] and hence 8 2 1/2, we verify
45 - c)
2(1 c) c)
(1+ (1 + 3)2
because multiplying by positive denominators gives 2 + 3)2 2 (1+ + 88)3, which holds for 3 2 1/2. Using (23), we see that 0g 2 B 1) when 7 S m1
Case (7 E (r1 (3), (7),(3))2(()) (2()))) Recall the abbreviations
(B = Bst + v (st)2(B2 +1) 2(1 (01 - 3)s c)
c
$3) = 1 +++1 1/8'
The analysis Appendix F.1 for case shows that
B(8,7) = s
A
8
2(1 - 7)
Recall that 8
I+1/3- Using this fact in the equation above and differentiating with respect to B
we get
2 2-c
27A(B,1) 7)
0g
+
2(1-7) 0g,(3,n)
(24)
(1+ 8)3
33

---

The argument in Appendix F.1 shows that that A(3,7) is positive multiple of 1 oj si S 3, have
5-77
Since
311-7)
(33, 7) = 82 781
3(1 2 1 -7-78 3(1-7) = si
0
(25)
and the same argument shows ((3, 7) 2 0, where 0(8,)) is defined in the case 72 7 7a- We
show in Lemma F.4 that 03 7) 0. Thus, it follows from (24) that
c
0₂B(B, 2
(1 + 3)2
Comparing with (23), we see that 0, n) 2 B 1) when (71(3),22()))
Case 3 (r E (7/((3), 7a)): this regime Theorem 3.3 shows that
B(3 =
si +
g² = 1+ B
c
+
1
(26)
B
2y
27
where
2c7(1 BB)
-3(1 8) (11 - 3)2 [3)=
1 1-7
(337)) =
+ ,2
Since ((3r))
<3-18) the argument in case 2 above shows that ((8,)) 2 0. We show 1-7
Lemma F.4 that 7y0(3,7) 2 follows from the expression in equation 26 that 0g B(3 2 2-c
(1+3)2- Comparing with (23) that 0g Bg, 7) 2 0g B(8, 1). Thus, for all the values of y lying in the range (0, B 7) 2 ((3, 1). Since (33, 7) is continuous and piecewise differentiable in
y, the inequality 0g B(3,7) 2 0g (B, extends all boundary values of y, completing the proof. Theorem F.2 now follows from Fact F.3.
L
Lemma F.4 (Monotonicity of A and f in B). Fix c E (0, and y E (0, For B E [1 - c, 1] define
2 c
(1000))++(000) ([()) +1) 2(1 (1 (111((1((33))
(B) = 1+1/8"
AB3 = : +
0(3,7) = -3(1 - 3) + (11-3)2 20 (1 + B2)
y-7
1 32
Then both A(3,)) 7) and ((3, 7) are strictly increasing on the interval c, 1).
Proof. 1. (B) is increasing. Write s(B) = si(B) (2-c)
direct derivative gives
1+3
2 > 0.
(B) = (1 (12
Using the above expressions for (3) and derivative, we note that
(3)
sT(3) 3 32
(27)
dB
2. Monotonicity of A (B3, 7). Put
v(3) =(3) +1) 2(1 7)((1 -3)s(() c)
We show V'(8) 0.
34

---

(i) The first term. Because both s(B) and 8 = 1 increase, their product s(B) + 1) has positive derivative.
(ii) The second term. Define h( == -1- 3)a(3) - c. Since s(3) is increasing but multiplied by (1 (1-39) 3), have (3) = - (1 - (3) s(3) < 0 on c, 1]. Hence - -(2 (//1->)/>)))) h(3) has
positive derivative. Combining (i) and (ii) yields (3) > 0.
Therefore VV(B) is increasing. The linear term (3) decreases (its derivative -3 Bs' < 0),
but the growth of (3) dominates because V (3) > 8² 2Bss' (Appendix F.2.1). Hence
d
(B,n)) -33) (3) + (3) > 0.
d3
2 (()
3. Monotonicity of (3, Let
(6) -1 3)2 20y(1 + (.2
T(8) = V((3).
1 1-7
Since (1 - 3)² decreases and (1 + B²) increases, FB(B) - 2(8 + 1 > so (3) > 0. Write
g(3)
0(3,7)= = + 32 -(3) == B - () + +(B). Then
- + ( 0 (because ('33) >1 1211-2231
23 on
c, 1))
Finally
( = (1+ + g 2B g(3)
> 0,
(1+
since g(3) and (3) are positive.
Thus, both A(3, 7) and (B8,7) 7) are strictly increasing for 3 E [1 - c, 1].
F.2.1 A lower bound on ('(3)
For completeness, we record the derivative computation used in Lemma F.4. Recall
s(3) (2 c)3
(B) = ( +1) 2(1 (11 -
(1--)))111111)))1 (3))
B
with c < 1. 7 E (0,1) and 8 e [1- - c, 1].
Step 1: derivative of s(B).
2 - > 0.
s((3)
(1+ (1+
Step 2: derivative of v. Split (3) = v (8) (2) with 1 = =2( 12 = 2(1-7) ((1
7
B)s c).
- 23))= +1) 2,82, V2(3) 2(1 - 7)) (s - (1-3s'3)) < O,
because the bracketed factor is positive. Hence
V v'(3) (B) - (3) > 2ss (B) + 1) + 2852.
Step 3: comparison with s2 + 2,3sss. Since (B2 + 1) 2 and s' s((3) > 0,
2ss' (32 + 1) > 25s (3),
So
V'(B) 2 2ss 23) 23s = 52 + 23ss' 1s² 2ss s²1.
Because the bracket is non-negative, the inequality follows.
35

---

G Details of worked example in Section 4
This appendix elaborates the example presented in Section 4, providing step-by-step derivation of the representation ratio and minimal interventions using our analytical framework.
Step 1: Set parameters.
Fix:
Sn = 0.85,
7a 0.40,
c 0.20.
We begin by computing the threshold for Institution 1:
2 -
1.8
1.8
0.8276.
1/B 1+ 1/0.85
2.176
Step Identify regime.
Using Theorem 3.1, we compute the relevant thresholds:
1
711 R 0.084, 722 ~ 0.2904,
2 0.833.
1 1+
1.2
Since 7/ < <0 = 0.40 < 7a. this falls into Case of Theorem 3.3.
Step 3: Compute R (Bo, yo). In Case 3, we define:
0(7) BB(1 B)
B)2
(1+
1-7
Substituting Bu = 0.85, = 0.2, and %o = 0.40, we get:
((%)) ~ 0.3096.
Then:
- 3 + R(3, 70)
8 (1-))))
0.1044
2%0
0.329.
1 87
(-(1-20)
0.3170
2%0
Step 4: Compute R (Ba, 1) and normalize.
When 7 1, use Case 4 of Theorem 3.3:
-2(1 (b) + 4c
2(0.15) + 0.8
0.5
R(9. 1) 2(1 BB) 4,0c
~ 0.510.
0.3 + 0.68
0.98
Hence, the normalized representation ratio
R(3o, 70)
0.329
N 7o) R(3,, 1)
2
~ 0.644.
0.510
Step 5:
Determine minimal interventions. To reach fairness target of 0.80, we solve for the minimal interventions (3',7)) that yield ((B'', no) 2 0.80 or (( 7') 2 0.80.
We find:
If 70 = 0.40, 8 N 0.911, If 3. = 0.85, '' ~ 0.640.
Thus, either reducing bias to 8 x 0.911 or improving evaluator alignment to y zs 0.640 suffices to meet the target. / 0.80.
H Observed utilities
In this section, we consider the observed utility Ut received by each institution E (1,2), defined the total utility from the candidates assigned to it. Let M denote the deterministic assignment function that maps each candidate to an institution É, based on whether their observed scores cross the respective selection thresholds. Then the observed utility is given by:
Us =
Σ use-
(EM-1(2)
We derive explicit expressions for 24 and U2, analyzing how they depend on the parameters / and For Institution 1, the utility can expressed closed form using the threshold st. For Institution the utility depends on the geometry of the selection region defined by the stochastic evaluation rule, and we provide piecewise expressions across different y-rreimms. Finally, we show that both 24, and Up are non-decreasing functions of 8, complementing our faimess analysis.
36

---

H.1 Observed utility of Institution 1
Given the threshold si, the (observed) utility derived by Institution from a candidate Ua < si: uu otherwise. Therefore, the utility of Institution 1 is given by (here i E G1, E G2):
U1 So silss 2 stldus v2 Su s1 sildp's
(28)
where is the measure induced by the distribution of ūn for i E G1 (and similarly for p4)).
Theorem H.1 (Observed utility of Institution 1). Assume 12 and C c2 1. Then the
observed utility of Institution is given by
(51)
+ 1 1 (s
u1 2
Proof. From Equation (28), the utility computed as the sum of two integrals over the observed scores for candidates in G1 and G2, respectively. The first term is:
S. 1/ 2 si]dx do
[ dr 84
[x271 1 (a1)²
2 of
2
For candidates in G2, the observed utility downes by 3, and the integral becomes:
B .(( x 1 si
1 - (s)
si B
Summing these two terms yields the result.
H.2 Observed utility of Institution 2
We now turn to the observed utility 42(() derived by Institution from the candidates selects. Recall that the observed utility from a candidate is zero unless they are selected, in which case equals their score ú,2. Thus, the total utility can be written as
U2 V1 Blua2 L(ū,1 si, 02 2 { + v2 Elūsz 1(6rr si,g, 2r2 s2,e
where i G1. i' E G2, and the first term represents candidates from G1 assigned to Institution 2, and the second term represents candidates from G2 assigned Institution
To evaluate these expectations, we consider the geometry of the selection region under the stochastic evaluation rule. Given 7 define A,, the region within the rectangle with cor- ners (0, 0), (s1, 0), (sT, 1), (1, 0) that lies above the line 7r + 11--)y = =2(() (as illustrated in Figure 1). Similarly, let B., denote the corresponding region for group G2, bounded by
(0, 0), (), (1,0) and the line 7at + - 7)y = s /33
Let (x+) vz4) and (x.6.0 xf) denote the centroids of A, and By, respectively. Then, the total observed utility (7()) can be expressed as:
U2(7) Area(A,)) (, + (1-7)9;) + B Ar) (NZ + (1-7)87)
It is useful to observe the following result:
Lemma H.2. Let T be the triangle with vertices (0, 0). (0, )n)) and (5. where and
7 (0, 1). Then
[| (100
83
(1 7)9) dxr dy
3r (1 - 7)
37

---

Proof. Because y + (1 - y is an affine (degree-one) function, its integral over a triangle equals its value at the triangle's 's centroid multiplied by the triangle's 's area.
Centroid. The centroid CT = (Z, (7) triangle with vertices (), (0, --)), (5. is the average the vertices:
s T
g = 31--
7
Evaluating the integrand at the centroid gives
S(Cr) - q + - (1-7)
+ (1 - 7)
8
2s
3(1 - 7)
3 3 3
2. Area. The base of the triangle is ; and its height is 1-7,
8 8
x²
Area(7) (T) = 2 1 1-7 2 (1 - 7)
3. Integral. Multiplying the centroid value by the area,
JL, (10 + (1 7))) dr dy- = f(Cr) (T)a(T) = 3 2 (1 -1))
28 82
8
3 (1 - 7)
We now use Lemma H.2 to evaluate U2 for various cases.
Case (s) < min 1 - 7)): In this case, the line intersects y-axis at T-yy
+ < and the æ-axis
at si 2 sj. The integral of the utility over the entire rectangle sj] [0, 1] can be given follows:
7
area of the rectangle is si and its centroid is at (#) 4). Therefore, the integral of yæ + (1 7)y over the entire rectangle is
75 + 1117)
8
2
Subtracting out the integral of the utility over the triangle formed by (0,0), - (+%
see that the observed utility (for Institution 2) of the selected candidates from G2 is
s (7st + 1 11-))
(s))3
2
3- 7)
Case II (s² 2 max 1 - -)): In this case, the line intersects the y-axis at 1-y > 1 and the x-axis at 12 > si. Thus, its intersection point with the line is at (1-(1-2) and its intersection point with the line sf is at $ -5-2 The centroid of the triangle is
8/ (1 (1-7) 2ys 2 27 s 7
1 1-7
_ - ? + 2s + 7s1). Thus, the utility of Evaluated on the line yr + (1 y)y, we get the value institute from G is:
3( (1 + 2s + 7s4) (1 7 + 787)2
27(11 7)
Case III (s1r < 2 7): In this case, the line intersects y-axis at - and the line st
at si= We break the area into two parts, the upper rectangle whose centroid is 1
sss 1-7t
The area of this rectangle is s1 The area of this triangle is si 79
(1
<2 The centroid of the lower triangle is 3 (28
30 -
Thus, the total utility
1-7
si (1 1
31111177
ys
1
y
si
7s
22ys
3s Ys
1-
+
+
2
2
2
1 1-9
3
3
38

---

Case IV (1 - 7 < 2 < 8)7): In this case, the line intersects the y-axis at - 1 and the a-axis
- St. 1.
We break the area into two parts, the right rectangle whose centroid is 1 2
sf + $2 1) The area 303-(1-2,2) 2). 7
of this rectangle is 81 =2. The centroid of the left triangle is ! triangle is 1 1_2, Thus, the total utility is
The area of this
3
7
82 8 + 82
1-7 7 + 2
(1-7)_ 27
353 (1 (1-2)
3 + 2(1 7)
7)
88
2
3
The above quantities can be evaluated for G2 in an analogous manner (by replacing si and s by s and $2 respectively, and multiplying the resulting expression by B). We can now write down the total (observed) utility of Institution 2.
Theorem H.3 (Characterizing U2 (7)). Assume that 3 2 1 cand e < 1/2. Then, U2(1)) is:
yor E [0:71::s)
(1-
8/
1-7+82
+
of
7ai
279
+
303-281
1-7
2
2
1-7
3
+ (1 (1- B(1-73)
23
2/
+
1-7-3//1 R)
2
+ 23
I-7
rai
3/
+
325
3,3
Fory E (n; 72.: 81 (1
(1-
o/
28i
1
st
2yys
3e] y
+ 3
+
1-7
2
2
1-7
3
7 +222//B + 781/3) 18-
/8)
2-(1
For E 7al: } 1---+ 2s² + +78))=
(1 +-97
+ {11 7 + 282/B + 188133
+
27(1 7)
(1-9- 1-99-2//4400/11
27(1-7)
For E 1hss 11:
(si -
-2 7) (70i t-2
+ 1-7
+
(1-7) (36--(1-7)
7) + 2(1-7)
27
+ {s8
s (
(78118002/0 + 1-7
9(1-))
9(1
38
13-11-11))
+ 201 (1-2))
27
3
Proof. When y E (0, 71), we are in cases III and III', When y E (2), we are in cases III and II' When 7 (22,73 72), we are in cases II and II', Finally, for 7 E I1a, 1), we are in cases IV and IV". O
Plugging in the values of sj and s into the utility expressions, one can carry out an analysis analogous to that for st and the representation ratio. We omit the details and instead present plot in Figure 6, which shows how 22 varies with y. We observe that the behavior of 2/2 mirrors that of sj. which is expected since increasing the selection threshold typically results in higher utility.
Specifically, U2 is larger for smaller values of 7. An intuitive explanation is that the area Area(A,)-whioe) corresponds to the fraction of G2 candidates selected by Institution 2-is decreasing function of 7. Likewise, the threshold 8j increases as 7 decreases. This stands in contrast to the behavior of the representation ratio R, which increases with Thus, increasing y induces two opposing effects: it improves representational fairness (as measured by R) but decreases the total observed utility of Institution 2 (as measured by U2).
H.3 Monotonicity of utilities with respect to /
We now show that the observed utility of each institution non-decreasing function of the bias parameter B, for fixed 7; see also Figure 5.
Proposition H.4 (Monotonicity of Uz). For any fixed y, the observed utility of Institution I is non-decreasing function of the bias parameter B.
Proof. For simplicity, we assume C = C2 = c: the argument generalizes to arbitrary capacities. From Theorem H.1, the utility of Institution is given by
(si) 2
1 1- (s
u
+
2
39

---

Differentiating with respect to 3, we obtain:
d
(si,a)
ds
dsi.s d3
2 81
B
dB
d3
It suffices to show that this derivative is non-negative. From the threshold constraint (see Equa- tion (11)),
(1 (1-87 5 + - -1-58,00
and hence,
dep
dsi.6
+
0.
d3
d3
Using this, the sum of the last two terms becomes
ds
dsi
de d3
sT.3
(s -s]
dB
d3
This expression is non-negative because ii sj and dai non-decreasing in B.
de 2 by Proposition C.I. Therefore, Ui is
Proposition H.5 (Monotonicity of U2). For any fixed % the observed utility of Institution 2 is non-decreasing function of the bias parameter B.
Proof. Let U, denote the utility contribution to Institution from a candidate in group Gj. Since
the expectation of non-negative random variable Z can be written B|Z] So' Pr[Z 2 du, have:
B
Pr(U, 2 8²] du +
Pr[U, 2 u/ du
/' 03
5 Pr(U, 2 sz +
Pr(U; 2 u² du,
since Pr(U, 2 u] = Pr(U, 2 st] for all 0 82
Differentiating with respect to and applying the Leibniz rule, we obtain:
d B(U,]
d=ż dB
d
d Pr 2 u] du
dsž Pr st]-
PrUU;
2 $11 8 dB
Pr(U,
+z]+
d3
03 d3
d3
The first and last terms cancel, leaving:
d
d
B
= d3
Pr
s +
Pr(U, 2 u] du.
d3
d3
Since an increase in 3 increases the effective score of group G2 candidates and thus the probability that they are selected, both terms on the right-hand side are non-negative. Hence, the expected utility E[U,] and therefore Uz is non-decreasing function of B.
H.4 Variation of Uf2 with respect y
We investigate how the observed utilities of institutions vary with respect to the correlation parameter
y. As shown Figure 6, the utility Institution 2 exhibits non-monotonic relationship with 7, displaying a U-shaped pattern across all tested values of B. This arises due to competing effects: higher y increases alignment between institutional rankings, reducing the chance of instittttntt gains, while also increasing selectivity.
40

---

030
Figure 5: (Left) Utility of Institution as a function of / for fixed y 0.5 and c == 0.2. (Right)
Utility of Institution 2 as function of for fixed 0.2 and varying 7 {0.2, 0.5,0.880 0.8}.
Figure 6: Utility of Institution as a function of ry for various & at fixed e 0.2. showing non- monotonic behavior.
I Extension to distinct bias parameters for two institutions
Our main analysis assumes common bias parameter 3 applied uniformly across both institutions. In practice, however, evaluation standards may differ-one institution may adopt more rigorous faimess safeguards
less biased scoring process than the other. To capture this asymmetry, the model to allow distinct bias parameters (812, 822) with respect to Institutions and 2 for the candidates in group G2 respectively. Analogous
extend
c assumption, we assume that both the parameters B12, 822 are at least 1 - c. For the sake of concreteness, we also assume 322 2 812- In Section I.1, we show that the thresholds and representation ratio can be computed numerically. In Section 1.2, we demonstrate numerically that the qualitative properties of these metrics remain unchanged, and we further verify this analytically an extreme parameter regime.
the
1.1 Computing thresholds and representation ratio
The expression for si remains same as in (4) with replaced by 812:
$1
1+1/8*
2-c
(29)
82, We now outline the steps for computing s and R(y). Given values 81, 82, 7. let P(si, 82, 7) denote
Pr[you + (1 7)022 2 82 A uil < 81]. This can be evaluated using the four cases mentioned in Section
Since 312 2 1 - c, it follows that s < B12. However, unlike Proposition C.2, it may not happen that 8 B22.. We first check if s < B22. For this, evaluate P(s 322, If P( 822, 72) c, then we know that si has to be at least 322 otherwise the fraction of candidates assigned to institution would exceed its capacity. Similarly, P(s 822, 7) c, then s B22.
Case 5 > B22: In this case, all the candidates selected by institution 2 are from Gy. Therefore, si is obtained by solving the equation:
P 7) =
Further, R(7)
1-93/192
c+(1-+1)
41

---

Case $2 < B22: To evaluate s2, we need to solve for the following equation:
P(s $2, 7) + P ( 312, 5 822. 7) = c.
Note that there will be four cases for each of the terms on the lh..s. above. More precisely, while evaluating P(s), $2 7), we will have to guess one of the following cases:
(i) - min 7) 7)
(ii 82 2 max(s17, 1 7)
(iii) 817 < 82 < 1-7
(iv) 1 - 82 817.
Similarly, while evaluating P12 / (22, 122,)). 7). we have to guess one of the following four cases. Let و denote s1/B12 and 2 denote 82 322
(i)' 82,9
01 7)
(ii)' 82,3 2 max $1,07, +)
(iii)' 81,97 82,8 < 7
(iv)' 1-7 < 82,8 < 81,37.
For each combination of 16 possibilities, solve for s and then take the solution that is consistent with the corresponding guess. After solving for s3, the representation ratio is given by:
$i + P(s $2,0
s P(s),
1,2 Numerical evaluation
Numerical experiments in this setting indicate that, across a wide range asymmetric configurations, s is unimodal in while R(y) increases consistently. For example, the plots in Figure 7 show this behavior for different values of B12 and 322-
To illustrate this phenomenon analytically, we show that even an extreme asymmetric setting, the representation ratio R(y) is strictly higher when the institutions are fully aligned (7 1) than when they evaluate candidates independently (7 = 0).
Proposition I.1. Consider setting where 312
and Bo2 1. Then, R(7) when
7 and R(7) 20812 Bi2+1
2c-812-1
< when 7
Proof. When y 1, the two institutions are fully aligned in their evaluations. A candidate is selected if either (ă 2 8i or thz2 2 sÿ. We first claim that s3 S si. Suppose not. The fraction of candidates from G1 admitted to Institution is 1 si. which must be at most by the capacity constraint. If s3 > s1. then no candidate from G1 would assigned to Institution Moreover, the fraction of candidates from G2 admitted to Institution 2 is less than 1 1- o S. violating the capacity condition at Institution 2. Hence, s I sT. Consequently, any candidate with ú,2 2 s is selected.
Since 322 1, an equal fraction of candidates from both groups satisfy ú,2 si, implying that R(7) =
We now consider the case = 0. In this regime, the institutions evaluate candidates independently. Institution 1 admits candidates based on the threshold s 1+1/0.2 The fraction of candidates from G not admitted to Institution is si, while that for G2 is s /1/312. Since 322 = 1 and the attribute est i.i.d across groups, the fraction of candidates from G, admitted Institution is "/++"//1a" and the corresponding fraction for G2 is
es /810 Hence, the overall representation ratio is given by #] +///11
ext 1812
+ + -1- (1-11/312)
+ /B12
+ + =a)
Substituting the expression for sj from above and simplifying yields the desired result.
42

---

Figure 7: Variation of thresholds sj and s/ and representation ratio R with y when c 0.2,
0.9, B22 1.
J Thresholds and representation ratio computation for the setting of additive bias
We now consider the additive-bias model in which the observed utility of a candidate i G2 for institution j üay = Viy 33 where Ui the true utility and [0. 1] an additive bias parameter (observe that, unlike the multiplicative bias parameter setting, smaller values of f3 imply less bias). Under the natural assumption S c- the analogue of the multiplicative regime 3 2 e that ensures Institution 1 continues to admit some candidates from G2 the admission threshold s at Institution satisfies the equation:
s + (1 si B) = c.
Thus, we get
81 1 c+ B
(30)
2
The geometric structure of the selection regions remains unchanged up to translation by 3, and the y-threshold framework developed Section continues to apply. Thus, the expression for Pr Tr[ < 81, 7,2 2 82] in the four cases 7, ...., TV corresponding to group G1 remain unchanged. Similarly, we have the corresponding cases 1', .'.....IV' .I' for G2, where we redefine st.s = min(1, st 3) and $2 - min(1, s3 B).
Similarly, the proof of Proposition C.2 showing s2 s extends directly. We now establish the existence of the y thresholds. As in Theorem 3.1, we show that there is a constant co < 1 such that if the available capacity e co. the relative ordering of the y thresholds is fixed.
Theorem J.1. Assume co 0.36, and & There exist thresholds 0 m 72 73 74 S such that:
(1) + + 75 n
(2) s <1 1-7 = 93592
(3) s/ gst = 5 93.
(4) 5 + 32 (s + +) = 7 m4 7
Proof. We first show the existence the thresholds .....,, 74- Arguing as in the proof of Theo-
rem 3.1, we can show that <(()) is decreasing function of y and st(ol is an increasing function of T-q
y: the argument follows verbatim with si.8 and s² denoting min(1, s + (8) and min(1, s 3) respectively. Since 3 < c, < 1 Therefore, s sf + 3. Since s I sT. it follows that $2.8 = + 3 as well.
+A(2) is an increasing function of +b()+ Since I-7
1-y is also an increasing function of 7. When 7 = 0, this ratio s 8 I s + < 1; and when approaches 1, this ratio approaches infinity. Thus,
43

---

there is unique value 71 E [0, such that =(m)) 1. Further, 51(1++8 1-71
< iff < 71- The 1-7
existence of the thresholds 72..... 74 can be shown similarly.
1- < 1. Therefore, the definition of 722 implies that y S 72- Similarly, 73 < 74. It remains show that 722 S 73. The proof proceeds in two steps: (i) we first show that
At 71, 1-71
s5(m)
و
(73,74). 72 $ [74, 11, and (ii) then we show that 72 & (73, 74). This will prove that 72 S 73, and hence,
71 S 12 S 73 - 7/-
Proving 72 & Ira, 1J. Assume for the sake of contradiction that 72 E h4, 1]. Then, at 7 = -/2. <5(7) 1 Y and s5(7) S 77)(7) (because 7/2 2 74 2 -a). Thus
in case IV for G1-
Similarly, we are in case IV' for G2. Therefore, constraint (2) implies that
82
+ 1 + +
82 + 3
1- =c
+
?
2y
7
27
Substituting 8 1= above, we get
1 11?
2s - (1+ 33)
When 7 = 72, 7 =
7-== =x](7)= S 787. Thus,
1-2
S <s. Using this fact and (30), we get:
7
1 (1-ct c+ (1 Sc.
2
The l. above is decreasing function of B3 Since 3 S we get
(1 c)- Sc
But this contradicts the fact that e < co 0.36. Therefore, 7/2 & l/4 1].
Proving 72 & (7s, 74). Assume for the sake of contradiction that E (7). Then we are in case IV and II' when y 72. Thus, constraint (2) implies
85
1 1-7 +
(1 7 82 -B + (s) + (3))))²
8
27
27 7)
Substituting s = above, get:
1-7
+ ((s) 337 3)
s 27
27(1 -7)
At 7 = 72. 1-1 S s7. Therefore,
s + ((s + 317) -322
S.
2 7)
Now observe that (st + +) - (st(7) + B)y - 3 (1 7 + Keeping fixed, as we raise 3, of decreases. Thus, the l.h.s. above is a decreasing function of B. Since S c, and s1 = e when 3 = e, we get:
(111-))11-)) >
- c (7 c)²
2 27(1 -7) S
+
Simplifying the above, get:
307² (5c + 1) + 6² S.
When = 72, - < 7 S 7. This implies that , 2 1/2. It easy to verify that the l.h.s. above an increasing function of y when 7 [1/2, 1]. Indeed, the derivative w.n.t. y of the l.h.s. above is
6c7 - 5c + 1 2 3c 5c + 1 2c > 0 because e < 1/2. Therefore, substituting = 1/2 above,
we get
7c + < 0, 4
2
2
which not possible if e < 0.36. Thus, we see that ?/2 & (73, 74). This shows that 72 S3, completes the proof of the theorem.
73, and
D
44

---

Using these aegile-cchrraterizationn together with the probability computations from Section 3 for Pr|yua (1 222222 $2 U/ < $11. we obtain a piecewise set of equations that determine si
under additive bias:
Theorem J.2 (Equations for $ (7)). Assume S and < 60 = 0.36. Then (5 satisfies:
[0, 71]: s
2=10)~9(3i)
2(1-7) ss 265
+.)++9)
2(1-7)
72:72]: 81
2aj
+ (1-7- (2) +))
2(1-7)
2y(1-7)
7a]:
(1-g-0
27(1-
+
(1-)-(33+3)+
2x(1-7)
hs: 81 5/2 +
1-2
(1- (+)+)))+(((++)))
2%
2x(1-7)
Iy,, 1): 2s + +
+. 9
1-7
Proof. The proof follows along the same lines as that of Theorem 3.2. For example, when Y, Theorem J.1 shows that we are in cases III and IIII. Therefore, for candidate i G1.
si
2s 788
Priua < st. Uu22 2 s2)
2
2
1 1-7
Similarly, for a candidate E G2.
+ +
(
2(s² + 3) -7(s + +
Pr(ua < 81, 022
2
2
The equation for 82(7) now follows from (2). Other cases can handled similarly.
We can use the equations in Theorem J.2 to solve for the threshold sz. Once s and s are determined, expression (3) yields the representation ratio R. We omit the details.
These results show that the geometric and analytical structure underlying the multiplicative-bia model extends naturally the additive setting, providing unified framework for quantifying how evaluator bias influences equilibrium thresholds and representation outcomes.
K
The setting where B < 1 -
In Section 3, analyzed the setting where the bias parameter satisfies 3 This condition ensures that Institution admits at least some candidates from the disadvantaged group G2, and the analysis led to clean three-phase structure characterized by the thresholds 7n, 72: 73 with fixed ordering.
In this section, we depart from that setting and study the regime where 8 1 1 This regime qualitatively distinct and exhibits complex structural behavior in the equilibrium thresholds and selection dynamics. To analyze this, again use finiteto-iinfiite reduction framework analogous to Appendix B, deriving mean-field equations for thresholds and examining their qualitative behavior function of the correlation parameter 7
Our main findings in the 3 c regime are summarized above and contrasted with the 2 case in Table 3.
The number of 7-thresholds increases from three to four: 71, 72, Ya 74. Moreover, unlike the 3 > 1 case, the ordering of these thresholds is no longer fixed, but instead depends sensitively on the value of 3.
We establish the existence of critical value B, such that for Be, the thresholds follow the order 71 < 73 < 72 < 7/, whereas for 3 > 3.. the order is 71 < 72 < 73 < 74-
The threshold s2 (7) may no longer be unimodal. Unlike the earlier regime where s1(7) had single minimum, we show that it may have multiple local minima and maxima. This non-monotonicity also manifests the representation ratio R(3,7) 7), which need not be monotonic in 7
45

---

Structural property
Result for B <
3
Closed form expression for sj
si c (Bg. (31))
1+3/6
2-c
(Rg. (43)
Monotonicity sj wott. B
Independent B (31) (31))
Increasing in (follows from Eq. (4)
Regimes of Y
"2 < & 74 < "s <
X (Theorem 3.1)
"ta S 74 (Theorem K.I. Proposition K.3)
Piecowise expression for
Characterized via 71....174 (Theo-
Characterized rem 3.2)
via 72, (Theo-
rem K.5)
Variation of #
Continuous, possibly with multiple local
Continuous, with most one minimum (Theorem E.1)
extrema (Theurem K.6)
Monotonicity 4g W.L.
Increasing ( (Propusition C.I)
Increusing (Proposition C.I)
Monotonicity R w.r.l
Increasing in for lixed y (Proposiiinn F.1)
Increasing in 9 for lixed y (Proposition E.I)
Monotonicity of R w.r.
Non monotonic in for lixed 8 (Proposi-
Non Non-decreusing for fixed (Corol- lary 4.1)
tion K.8)
Table 3: Comparison of structural properties of thresholds and representation metrics between the regimes C and | 2 c.
This section is organized follows:
In Appendix K.2, we generalize the notion of --thresholds and formally state the structural result Theorem K.1 and Proposition K.2. We also define and characterize the critical value Be in Proposition K.3 and provide lower bounds on its value in Proposition K.4.
In Appendix K.3, we characterize the piecewise structure of s) using Theorem K.5, and show how it varies across the threshold intervals. We also prove in Theorem K.6 that s5()) exhibits multiple local extrema. Finally, we demonstrate in Proposition K.8 that the representation ratio R(7) can decrease with increasing 7, in contrast to the monotonic increase observed in the 3 2 1 regime.
K.1 Expression for i
When B <
then si 2 8. Indeed, otherwise the fraction of candidates from G₁ selected by institution would be least
s 2 1 8 2 c, which is contradiction. Thus, s1 2 8, and hence, $i.3 = 1. This implies that no candidate from G2 gets selected. Thus, we have the following
equation for institution I: 1 - s = c, which implies
=1=1-c
(31)
K.2 Existence of y-thresholds
We first show the existence of y thresholds.
Theorem K.1 (Existence of four --thresholds). Assume c 1/2 and B> 1-2c unique values 71, 72; 73, 7/4 E [o, 1] such that for any y E [0, 1). the following hold:
1-c" Then, there exist
(i) (), (7) < (1 - 7) if and only if S m.
(ii) st()) $ (1 7) if and only if T2-
(iil) s;(n) 2 i if and only if S 73-
(iv) (7) 2 7 if and only i S 74.
In case 2c < B < 1-2 the thresholds .' exist with the above mentioned properties. When 3 - 2c, the thresholds 72, 73 with the above-mentioned properties exist.
and Lh follows from the same arguments in the proof of Theorem 3.1. However, we need to show that these ratios achieve the desired values.
the oi $2 Proof. Monotonicity of I- 7
When 7 = 0, 1-y (0) = =3(0) < 1 2 1-20. As approaches 1, (0)a(0) "I-- approaches
(0)
infinity. Therefore, intermediate value theorem and monotonicity of $2. I-7 shows that this ratio equals
46

---

1 for a unique 7, which we denote 71- In case << 1-2e I-c (0 1, and hence the ratio above for all y E [0, 1].
the T-y
stays
Since (0) - 1, the ratio s 1-y is at most at = 0. Therefore, exists for the entire range
of B. Similarly, the ratio s approaches infinity approaches O, and is equal to s²(1) I si (using Proposition C.2) when 7 = Thus, the threshold 77 also exists for the entire range of [3.
When / 2 - c, the condition for 73 is the same as that for 74- Hence, assume / < 1 c. In this case, .1 1. If 3 1 2c, (1) I B. Thus, the condition for 74 becomes: s5(n) 2 Byy
Since - - B at 7 1 and approaches infinity at 7 0, existence of 74 follows. Observe that when 1 - 2c, the only value of satisfying (iv) is 7 1, and hence, 74 does not reveal any useful information.
7
For sake of completeness, shall set "n O when / I-c and 74a to 1 when , < 1 2c. We now show that the only possible orderings of these thresholds are (i) 91 72, Ys Ya, (ii) "/, Yas 72, Ya-
Monotonicity properties of (1)) and *5(1) show that "1 S 7/2 and 7/3 S 74- We now show that 7 1-y
the smallest and 74 is the largest among "I...., 74.
Proposition K.2 (Extremal ordering of thresholds). Assume 1/2 and B < 1 Then n S 7 for all j E (1....,4) and 7/4 2 Yg for all j E (1....,4)
Proof. Assume for the sake of contradiction that Y is not the smallest among these thresholds. Then, the only other choice is 7/3 the smallest, i.e., Y/ < Y1- follows that |3 > 2c, otherwise 71 For any 7 E 73].
ys - (n) - 3(1-7), 7),,
because 7a 71 (and hence, 35 3ラ
Substitution 'a above, get
B
73 B 3+s7 s
(32)
Now for y E 7sl. we are in case III and 11I' Therefore, s satisfies:
s (2s) 781)
2x2/B -
(33)
2(1 7)
2(1 2)1-7)
Substituting y = Yss and 8 si in the equation above, we get:
(st)²
+ 2/
s 2(1 73)
2(1 73)
Solving for y and using the fact that
$1, we get
4s
3
73 4s + (51)² 2 -
81 + B
where the second inequality follows from (32). Simplifying the above, get:
4(s1) 2s < (((s))
1). -).
Now, the r.h.s. above is negative, but the l.b.s. is positive because s = 11= 1/2. This leads to a contradiction, and hence, must be the case that 7/1 I 73-
Now we proceed to show that 74 is the largest among 71...., 7t. Suppose not. Then, 72 must be the largest value (since " S 74). Thus, the ordering of these thresholds would be 71, 73, 74, 72- In the interval 7 E 72)), we are in the case T, TV' (and s3 S 7B E (). Thus, s satisfies:
(s))²
8
1 1- = =c
+1
2y(1 7)
By
27
When 7 7= 72, s} = 72- Substituting this fact above and solving for 72, we get
12
1+ 2
47

---

Since 7a < 72. we know that 1 - )/2 < 872 i.e., 7/2 > 1
I+. Using this fact above, we see that
si < 1/2, which is contradiction because si 1 2 1/2. Thus, that "/2 S 74- This completes the proof of the desired result
remains to decide between the orderings Y1, 72, 73, 7/2 and 21 72: 74- turns out that for
fixed c, both of these orderings can emerge as we vary / from 1 2c to C, but there is a critical value of 9. denoted 9e. such that for all B Be. the latter ordering occurs; and we get the former ordering when 3 > Be.
Proposition K.3 (Existence of a critical Be where the threshold order changes). Assume < 1/2. Then there exists value Be E [0, 1 such that for 2c S 3 < So. (B) S 7a( S (2(8) S
7 (3); and for all B. < 3 I 1 71 (3) S (3) S 73(8) S 74(3).
Observe that if 8. above, the ordering remains 71 72, 73. 74 for the entire interval [0, 1 c) (and hence for the interval [o, 1)).
Proof. The proposition follows from the monotonicity of (B( and (3) as functions of B. We shall show that (B)) is non non-increasing function and (B) is non-decreasing function of (. The
result now follows from the observation and the fact that when 3 = 1 c, 7 I 72 I 73 I 74 (Theorem 3.1).
We shall treat s3 as function of both /3 and y. Now observe that the ratio (3.r) I-y is monotonically
increasing if we fix and increase y and monotonically non-decreasing if fix y and increase Now, suppose there are values 3 < 3 such that (3) < (8").. By definition of 72,
8 (2())
,3', (2)
- (3)
72 (8')
Since 72(8) < 72(8"), monotonicity of =5(8.)) 1-7
in y implies that
(3())2(3) r2('))
1 12)
1 - (8')
Since < 8', the monotonicity of $5(8.n) in 3 yields:
1-y
$ (3,,722 72())) (B))
- (B'))
which is contradiction.
We can prove monotonicity of (3(3) in similar manner (note that st 1 C, and hence, is independent of 3 when B < c).
We now show that for 3, to be strictly larger than 0, c must be at least 1/3. In other words, if c < 1/3, then the ordering will be "/, 72, 73. 74 for all B E [0, 1--, c). Proposition K.4 (Lower bound on 3a). Assume e < 1/2 and B. > 0. Then c 2 1/3.
Proof. Suppose e < 1/3 and 3. > O. Then consider the setting where 3 = 3c. Here, (2(3.) (a().) We consider the equation satisfied by s5 in the interval [a). If s5()) S B, it satisfies:
25 - 7ss)2
(1 (1-32/3)2
c.
2(1 (11-))
2y 7)
Otherwise, it satisfies:
s 28 - -
2(1 7)
In any case, we have
C c² 8 2818 - 7(8))2
2(1 - 7)
48

---

Now, = which is also equal to 'a. we have 52 = 1 - and 82 ysj. Using both these conditions in the inequality above, we get:
(1 - 7)8 81 1 - c
c st
2(1 - 7)
This shows that 2 1/3.
We make a few observations:
Unlike the >> 1 - case, can have four distinct thresholds. As we decrease B, some these thresholds may not exist. fact when 3 goes below 1-2c T-c" threshold "i does not appear. Intuitively, this means that for such small values of 3, a candidate from G2 cannot get admission in institute based on 8:2 parameter. Similarly "Ya does not appear when 3 goes below 2c.
alone, irrespective of the correlation
The relative order of the thresholds may depend on the value of J. This again different from the 3 2 1 setting, where there was no such dependence on 33.
K.3
Variation of s with y
In this section, we consider the variation of s with y. Theorem E. we had shown that s can vary in a non-monotone manner. We consider whether such non monotone behavior exists in the 3 <<B
c regime.
Since the relative order of the y-thresholds depend on whether / < Be. our results (and their analysis) split into two corresponding cases. The following proposition follows analogously Theorem 3.2.
Theorem K.5 (Characterization of 8 for low 3). Assume that B < 1 - c, B < Be and e < 1/2. Then, assuming s2(7) S B8 satisfies the following equations as 7 varies from 1.
(i) [0, ml:
2x 22 ((1122
20. 202/8-2
8
+1
(34)
21-7)
2(1
(ii) msl
2x - ,,,²²
(1=2218)2
87
2(1-))
2y(1
=
(35)
(m)) 72]:
(x))²
(1-83/0)²
8) 2
+
2/(1-?)
c.
(36)
(iv) [12, 7a]:
s1 1 + 1-x
(L=x1/0)²
(37)
2x(1
(v) []:
8
(38)
By
When 3 < 1 c and 3 2 Be, < 1/2, the equations satisfied in the intervals [0, Y1] and Ir 1]
remain above. For the remaining cases, have:
(i) [2]:
81
25 -(
(1-82/0)2 2-(1 C.
(39)
(i) (12, ra]:
1- 2 1
(1-x) /8)2
2-(1 7)
2-(1-99)
C.
(40)
(ii)) [Tss ral:
(1=a//0)2
8
+
(41)
27
27(1-9)
The above equations only consider the case when +5()) S 3. case s5(7) > B, we need to remove
the terms corresponding to G2 in the corresponding equation above. The following result shows that as we vary 7, 85 (?) can have several local minima or maxima. This in contrast to the 3 > c setting, where there was only one local minimum.
49

---

Theorem K.6 (Threshold in low 33r regime). Let E [0. 1/2). Assume that the bias parameter B
lies in the range [0, cl. When 3 S Se. 82 varies as follows as we vary Y from 0 1:
(i) [0, ml: 82 is linearly decreasing function 7.
(ii) naa:: 52 a unimodal function of has no local maxima and has at most one local minimum.
(ii) [ra, 12) 52 is unimodal function of q/z it has no local minima and at most one local maximum.
(iv) [12, Ya]: s2 a unimodal function; it has no local maximum and most one local minimum.
(v) (74,, 1]: 82 is linearly increasing function of 7
When > Brs 52 behaves as the above case for the intervals [0, 70 h4, 1]. For the remaining three intervals, 82 varies follows:
(i) (1.722:: s a unimodal function of has no local maxima and has at most one local minimum.
(ii) Tal: 82 is a convex function, and hence, has at most one local minimum and no local maximum.
(iii) (ys: s3 is unimodal function. If c 1/4. it has no local maximum and most one local minimum. If S 1/4, it has no local minimum and at most one local maximum.
Proof. First, consider the case when B < Be. We consider various cases:
Case y E [0, 71: Since 7 S 71. 82 < 3. Further, (34) shows that s is linear function of
with the slope given by 2s +2/B times 4si + (s1)² + 1. The latter quantity negative if s1 2 1/2 (which is the case when 1/2).
Case 7 E Ys]: We know that s2(1)) $ B. If s2()) 2 B, then s5(7) satisfies the following truncated version of (35):
2s s - 7
si 2(1 - 7)
This becomes linear function in s2 with slope proportional (st)² 4(s1) 2 which is positive iff e 22 In case /2 1, and if s exceeds B, it will remain above B (and increase linearly) for the rest of the interval. However if /2 1, s2 remains below / during this interval. Thus, there is value 7/8 I 73. such that s remains at most 3 during [n, 73]. and remains above B during (7/- 73]. Note that 7 may equal 73. in which case the latter interval empty. We now study the behavior of s in [7, Tal. We know that it satisfies (35).
Using $1. rewrite (35)
( 1)r(1 - 7) 28 + + + s2/B)² = 0.
Differentiating with respect to 7, we get
2(1 - (2s - 1) 32 + (x()) 2(si7 111(111111111111
- -3/33)(ss)) = -.
Differentiating again, we get
2(s;) + 1/3(1 $3 ga
4s) 8s - -(87)) - 4=
Note that 8s; - 4ss))² < 0. It follows that when [(5)'] is close to 0. (s²)" becomes positive. Thus, once the slope (sg)' becomes positive, it does not become again. Therefore, there can only be a local minimum in [71, 73].
Case y E [7s, 72]: We know that if 8 I B, it satisfies (36). In case, s > it would satisfy:
(sg)²
si
27 -7) = c.
50

---

y < 1/2, will be an increasing function. At Y3, we know that si = 1 11-10 Y3, which implies that ?/ 1//2. Thus, if (7%) 2 3, there would be an initial sub-interval hax 7/1 of haa [2] where 22 will be above B and would subsequently stay below f3.
Now, we consider those values of y where 85 S B. Multiplying both sides of (36) by 2-(1 7) and differentiating, we see that (s5) is equal to positive quantity times [sî - - 2y). Since c < 1/2, si = 1 c > c. Thus, (sg)* is an increasing function till 7 = 1/2 and then it becomes decreasing function. Combining the above observations, we see that s2 will be initially increasing, and then will become decreasing note that 7/2 > 1/2 because
7/2 s S si72 < 72.
Case Y E [2): If s2 S 5, then 83 satisfies (37). Otherwise, it would satisfy:
2 + 1-2
si
(42)
27
In this case, multiply both sides of (42) by 27 and differentiate w.r.t. 3 to get:
2(s))' 2(s] - c) - 1- 4c < 0,
because >> 1/4 (Proposition K.4). Thus, 85 would be a decreasing function till it becomes at most B. Hence we get: there is a value 3/2 E [12.7a] such that s7)) 2 3 for all y E (72, and is decreasing function of y during this interval. Note that 7/2 could be same "/2 in which case (12) S B.
Now we consider the remaining interval I/2- 74]. When s2 S B, it satisfies (37). Multiplying both 81), we get:
sides by 2y(1 - 7) and differentiating (and using
2 (1 3)/3)(8))' (1 (1-7)(82) 0.
2 c) 27) + 2s² 2(1 7)
Differentiating the above equation again, we get:
2 -2 ((22))) - 3
2
A(7)(s3)) -
(s2) + 4(s) -c) c) 2- g2
where A(7) = (1-7)
(1- -1-)=
((1-2c) -2 2- Sc < 1-74. Further, 4 -) 2 -
because e 2 1/3 (Proposition K.4). Thus, when (s²) becomes small enough, (*3)" 1 becomes positive. Therefore, s2 does not have local maximum in 172. 74]. Further, (244 A74 < <B. Therefore,
*5 S B throughout the interval I7/- 74] Combining the observations about [72, 7/2 and [1á, 7al. we
see that s initially decreases and does not have local maximum in [4].
Case 7 E IY r1]: Since s $ By < / throughout this interval, it satisfies (38) for all y E h4, 1]. It follows from (38) that
25137 + 31
8 +
Since o1 > 1/2, si is an increasing function of 7-
This completes the discussion various cases when B .e. It is also worth noting from the discussion for intervals [71, 73] and [73, 72] the set of y values where s (7) 2 3 forms an interval.
Now we consider the case when 3 > Bc. Clearly the cases [0, 71] and h4, 1] follow as above. We now consider the remaining settings:
Case y E [Y,, 72]: In this interval, used in the case 7 [na 7a] when B. apply in the same manner here.
83 satisfies (39) which is identical to (35). Thus, the arguments
Case 7 E [a] In this case, 8i satisfics (40) if assuming sz S 3. In case, s > 3, satisfies:
(1 - 7 - s5 + 7
(43)
2 7)
Multiplying both sides by 27(1 - 7) and differentiating twice, we see that (st)² > 0. When (7) S 3, satisfies (40). Again, multiplying both sides of this equation by 2-(1 - 7) and differentiating twice, we see that (s²)"= > O. Thus (st)" > 0 at all points in [72, 73. (one can check
51

---

that if s2()) = B, then the slope (si())) given by the two equation (40) and (42) are identical).
Hence, 8/ is convex in this interval.
Case Y E [7s, Yal: this case, s satisfies (41) which is identical to (37). Thus, the arguments used in the case for the case 3 [12. 7al when 3 < Be apply in similar manner long > 1/4.
We complete the argument when e < 1/4. case 85(7)) > 8, it would satisfy (42). Multiplying both sides by 27 and differentiating, we see that 82 will be non-decreasing linear function (and hence, remain above B throughout this interval).
Now consider the case when (73) - 3. As in the case 3 E h22 74 for 3 < Sc, we get the equation:
2 f2
A())(s2)) ((s)))) 3+
(52) + 4( - c) - = 0,
3
where A(7) (1 - +)+ 2 21 s3 > 1 - 74. Now, if c > 1/4, 4(s) - c) - 2>> 0, and rest of the arguments the B < Be case hold.
We have shown that (12) < B. If < 1/4, we see that 4 c) - < 0. This shows that if (s*Y close enough to then (s5)" < O. Thus, does not achieve a local minimum, though it may local maximum. This completes the discussion on all the cases.
The result above shows that when B < 1 - c, (7) can behave in a highly non non-mono non-monotone manner as vary 7- Similar observations hold for the representation ratio:
Variation representation ratio with 7. In Theorem 3.3, we showed that when 3 2-1-c c, R(y) is monotonically non-decreasing function of y. However, when 3 < 1 - it is possible that R(7) decreasing function of y sub-interval of [0, 1] and is an increasing function of y in another sub-interval. Thus, R(y) may not be monotone function of 7. We now show that unlike the c case, R(y) can decrease with y. We shall first need a useful result:
implicit constraint). Let f.g : RR2 - R be functions of two variables, and 2 [0, 1] + TR. be real valued function defined on the interval [0, 1]. Suppose the following relation holds for all 7 [a,b) for some
Fact K.7 (Monotonicity under
b 1:
((1,2(1)) + g(7, =(7)) =c
where is constant. Let Jnp f2 denote the partial derivatives of f with respect the coordinates (and define Ir Iz similarly). Suppose f2z92 > O and
fx 9x
(44)
fo 9=
for all E [0, 1). Then f(7,=(7)) is non-increasing function of E (a, b). Similarly, if, 9z < 0 and
f 9
(45)
f= 9=
for all y E (a, b). Then ((1,((1)) is non-increasing function of y E [a, b).
Proof. First assume fz, 0= 0. Differentiating
f(7,2(7)) + +g(7, =
with respect to y (and using z' to denote
ds we get
fr faz' + 9. + gz = 0.
Therefore,
J + 91 fz 9z
x'
Since fz. Iz > (44) implies that
ff + gr
fx
fa + 92. Jz
52

---

Therefore, 5 Now,
df(,,((1)) = foz + f fn S 0.
dy
This proves the desired result. The argument when fa. 9s < 0 is similar.
We now show that the representation ratio may decrease with increasing 7;
Proposition K.8 (Representation ratio decreases in mid-bias range). Assume < 1/2 and c. Then representation ratio decreases monotonically with v for E [),
1-2c 3 < T-
Note that the condition
1-2c < / is needed to ensure that 71 > 0 (Proposition K.2).
T-c
Proof. For 7 E [0, m0).)). s2 S 8 and hence, 82 satisfies Equation (34). We show that the representation ratio decreasing using the notation in (44). Here, we define 2 = [I-?] (the negative sign is needed because we want 92,92 2z to be positive). Then, the terms in the l.h.s. (34) corresponding to G1 and G2 can be expressed as:
7(s7)2 2(1 71)
[(7,)) = si +))
=
+ 2(1 - 7)
routine calculation shows that
fx f= 2(1 7)2²
91
9=
2(1 - 7)²
Since ii 2 B, the reverse of (44) holds. This shows that 2)),2), which represents the fraction of
selected candidates from G₁, is an increasing function of y. Thus R(y) is a decreasing function of y.
L The general preferences case
In this section, we extend our analysis to the general preference setting, where candidates prefer Institution 1 with probability p E (0, Unlike the when Institution 1 is always preferred, the equilibrium thresholds s and s now satisfy nonlinear equations that do not admit closed-form solutions. Nevertheless, this generality reveals rich structural behavior.
In Appendix L. we derive the equilibrium conditions that implicitly define si and sj for arbitrary p. We further prove the existence and uniqueness these thresholds.
In Appendix L.2, we show how compute these thresholds using closed-form probability expressions developed earlier, enabling efficient numerical evaluation.
In Appendix L.3, we provide formulas for computing key outcome metrics in terms of the thresholds: the representation ratio and each institution's 's observed utility.
In Appendix L.4, we prove that the thresholds vary monotonically with holding other parameters fixed. However, this variation is not necessarily strictly monotonie, illustrated in Figure 8. In particular, the threshold s remains constant for values of p up to critical point p", beyond which it increases strictly with p.
This behavior arises because, for small values of p. the threshold (p) is sufficiently low while s5 (p) is relatively high, resulting in the probability Pr[u, sp) A u22 > s (p)] being zero. Consequently, all candidates with un 2 (p) are selected by one the two institutions, regardless of the value of Since this selection condition does not depend on p. the threshold s (p) remains unchanged for p < p². We formalize this argument- along with similar statement for (p) in Appendix L.4.1.
In Appendix L.5, we numerically examine how thresholds, faimeess metrics, and utilities evolve as and the correlation parameter y vary. key contribution of this section lies in the comprehensive set of plots (Figures 8-14) that reveal monotonicity, phase transitions, and other emergent behaviors, offering insights that go beyond what closed-form analysis can capture.
53

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L.1 Equations for the thresholds and existence/uniqquenes of solution
L.1.1 Equations for the thresholds
We now derive the threshold equations for the two-group model, where each candidate independently prefers Institution 1 over Institution 2 with probability p E [0. 1), and belongs to either group G1 or G2. Candidates in group G2 face multiplicative bias (0, 1] in their evaluations. Let i E G1,
i' G2, and define the bias-adjusted thresholds:
81 min (1, 81/8), $2 minm1,
We assume uniform group sizes (v1 = 1/2 = 1) and symmetric capacities (c1 02 = c). Our approach follows a finite-to-infinite reduction analogous to that in Section B, transitioning from random threshold behavior in the finite setting to deterministic thresholds in the continuum limit. The capacity constraints reduce the following pair of equations:
((1 -1) 81) + (1-81,))) + (1 - p) (Pr(uz 2 81 A u22 < 822] + Pr(ug1 2 81,8 A u22 < (2,3)) = C,
(46)
(1 -1-27 p) (Prfu 22) Prfurz 2 s2,el) (Pr(ua < A u2 2 s2) Prfuvi $1,9 uv2 82,3)) =
(47)
L.1.2 Existence and uniqueness of (si, sz) for general p
The proof of existence and uniqueness also follows the p setting closely (see the proof of Proposition B.2). Define the residuals
(1 82) [(1 s1) (1 s1,0)] + + -1-11 p) (1o0(s1.00 22) + $10,0 s2s
F2(81, 82) 82) = (1- - Vo((2) + (822 p @oi 82) @01,5(81, 82) -c
where, for example, (1) = Pr/ui 2 "ar"vvariatss are defined analogously.
s2l. Vo(s2) = Pr/ue 2 s2). and the
Proposition L..1 (Existence and uniqueness). The system
Fi s2) 0, F2(s1, 52)
($1, 82) 10,
admits a unique solution (81,8) Moreover, the map + (st $5) is continuous on (0, 1).
Proof. 1. Monotonicity of each equation. By the uniform noise formulas, one checks that
OF
OF
0F₂
0F2
0,
>
<0
0s
dsz
0s
Os2
throughout 1) In particular, each residual is strictly monotone in each variable.
2. Existence via one-dimensional roots. For each fixed E [0, 1]. the function + Fi(si, continuous, strictly decreasing, and one verifies
Fi (0, s2) > 0,
Fi 82) < 0.
Hence there a unique root 51 A(s2) E (0, 1) with F (A(22) s2) 0. By strict monotonicity in $1, A is continuous and strictly increasing in
Likewise, for each fixed $1 E [0, 1), the function $2 + F2 82) is continuous, strictly decreasing, and satisfies
(2(51,0) 0) O, (s 1) < 0, so there is unique root $2 [(s1) E (0, 1). Again by striet monotonicity, I is continuous and strictly decreasing in $1.
3. Uniqueness by graph intersection. The solution (si, +t) must satisfy both
A(s2),
$2 l(s1).
54

---

Equivalently, the continuous curve + A(s2) (in the $1 s1-direction) and the graph of the inverse r meet. Since A strictly increasing and I strictly decreasing, there is exactly one intersection in 10. This proves uniqueness.
4. Continuity in p. One checks that the Jacobian det De .2) (Fi, F2) is strictly negative on (0, 1)2 is 0z, Fi 0.z 22 0s F2 O), so by the Implicit Function Theorem, the unique solution (si, $i) depends -mmsmoohyly on p. In particular, it is continuous.
I.2 Computing the thresholds
In this section, we describe our approach for computing the thresholds s and sz. Specifically, we show how the probability expressions in equations (46) and (47) can be evaluated using the analytical results developed in Section 3. Unlike the special case when
1, it is now possible that s2 > si.
which necessitates careful treatment of events such as {uzz 2 #2 A u41 < 81). In particular, when
82 2 817 + 7)
this event has zero probability due to the structure of the support, i.e., Pr[uz 2 82 A un < <s1]= = 0. similar observation applies when evaluating the corresponding bias- aaaassddsteed probabilities for candidates in group G2.
Our procedure consists of the following steps:
Rewriting probability expressions. We express the probability terms in (46) and (47) using quantities of the form
Pr(ma < 81 A u22 2 22] - Pr|uz2 2 82].
and Pr[uz2 2 s2] for candidates i G1, and analogously for 1' E Gz. These forms simplify the evaluation using the results from Section 3.
Case identification. The expression Pr[u.1 < 81,3 a A u22 2 82.0] depends on the relative positioning of 82 with respect to 81 and 1 y. Section 3 identifies four distinct cases under the assumption that s I si, but this assumption may not hold in the general setting. For instance, if 8 2 s1Y+ (1 -)), then the probability is zero. Geometrically, this corresponds to the line L(y, $2) not intersecting the rectangle [0, $1] x [0, 1].
Similar logic applies to Pr(up 2 $2]. Overall, we identify seven mutually exclusive cases for computing Pr[u: 81 A u22 82] Prfu22 2 s2] and Pr[uz2 2 $2). For candidates in G1, only seven cases are feasible. For candidates in G2, we must consider twenty-eigh total configurations, accounting for the seven cases above along with independent binary assumptions on whether S 3 and whether 82 < B.
Solving and verifying. For each guess of the applicable case(s), we substitute the corre- sponding closed- form expressions into equations (46) and (47), resulting in a system of two equations in the unknowns 8 and $2. We solve this system and verify whether the solution satisfies all assumptions made in the guess. Since the solution to the threshold equations is unique, exactly one of the constant number of guesses will yield a consistent and valid threshold pair (s1, sz).
We now elaborate on each of the steps above:
Rewriting probability expressions. We first consider (46). We apply the identity:
Pr[un 2 $1 A 422 < s2] (1 (1) s1) Pr|uz 2 s2] + Prium $1 u2 2 s2l,
and similarly for i', This yields:
(1 -1-88 81) + (1 s1 + (1 - (1-p) (Pr(un < 81 A u2 2 82) Pr[uz2 2 82]
(48)
+ Pr[up1 < $1.3 A UF2 2 s2,2l - Pr[uz2 2 82,91) c.
Note that Pr|um1 2 82] can be expressed as Pr[un 2 82 ^ un S 1].
The second equation (47) remains unchanged the probability terms already expressed in terms of the desired events.
Case identification. Both the equations (48) and (47) involve Pr[ua < Au,2 2 x2] -Pr(u,2 2 s2] and Pr[um 2 82] for a candidate i E G, and similarly for a candidate i' E G2. We now show the possibilities for a candidate i G1:
55

---

(i S max(1 - -7, 7s1s) we are in Case I while computing both Pr(u, < A u2 2 sy) and Pr[u22 2 82). Thus,
Pr|ua < <1 A u22 2 82) - Prruu2 2 82] = 81 - 1,
and
s2
Pr[u2 2 52] = 2-((1 7)
(ii) 1 1- < S max - 71,781) we are Case IV for the both the probability events.
Hence,
Pr/ua <1 A U12 2 82] - Pr[uz 2 82] = = 11
and
Pr(uz 2 s2] = =1 82
1 7-7
? + 24
(iii) 22 [s1, 1}, 22-2es [0, 1] Observe that 7/81 S 82 S - ? and 82 S min(7; 1 - 7).
Thus, we are in Case III for Pr[ua < 81 A 142 2 82). but in Case I for Pr[uz 2 82). Therefore.
8
81 (282 781)
Pr(ua 81 A u2 2 82] Pr[uz 2 82] = 2y (1 - -7)
2(1 7)
and
52
Pr(u22 2 s2] = 2y ?)
(iv) E
[s1, 1]. -/ > 1: Observe that $2 2 781 + (1 - 7) and S 82 S y. Thus,
1-7
Pr[ua 81 u2 82] 0, but we in Case IV for Pr[uz 2 82). Therefore,
82 1 1-7
Pr(ua < 81 A ux2 2 82]
Pr[uz 2 82] +
7
27
and
82 1 1-Y
Pr[uz2 2 82] =
7 27
(v) 1, 89-781 1-7
1 Observe that 2 781 + (1 7) and max(y, 1-
7)). Thus,
Pr[um
81 u2 0, but in Case II for Pr[u- 2 82). Therefore,
(1- 82)
Pr/ua < 81 Au 2 s2) - Pr[u-2 2 s2] 2 7)
and
(1 82)2
Pr[uz2 2 s2] 2 - 7)
(vi) 1, #2-701 E (0,1],82 82 < y In this Case, 817 82 < - and y 82 1 7-
1-y
Thus, we Case III for both probability events. Hence,
81 (282 781)
282 7
Pr[ua < 81 A u2 2 82] Pr(uz 2 s2] =1
2(1 - 7)
1 1-7
and
282 -
Pr[u:2 2 32] = 2(1 -7)
(vii) 82 1, ب Thus, we are in Case for both the probability expressions. Thus,
[0, [0.1], 2 1 Here $2 max(y, - 7) and 82 2 max(?, 8
1 1 7 82 781)
(1 (1-82)²
Pr|um A u2 2 s2l - Pr/uz s2]
21(1-))
2 - 177)
and
(1 - 82)
Pr|u2 2 sz]
2 (1 - 7)
56

---

Similarly, we can express Pr(ue1 S1, A uv2 2 s2) - Pr[ur2 2 s2,8).
Formulation of system of equations for sj and s.. We now illustrate example of the system of equations that arises from specific guesses made in the previous step. Suppose, we consider Case (v) for G1 and Case (vi) for G2. Further, also assume that s S 3 but sp 2 8. Under these assumptions, we have
(1 82)
Pr(ua < 81 U22 2 82] Pr(u12 2 82]
2y((1 - 7)
for a candidate i E G1 and
$1 - -781/B)
2 - 1-7
Pr[uei < 81,8 A us2 2 82.8] Pr(uz2 2 82,88
-1
2/(11 ?)
Thus, Equation (48) can be written as:
(1 - 82)2
- 781133)
2 8-7
(1-81) p(s1/B 1) (1 p)
(1 p)
+
27 - 7)
2 -7)
1
The second equation obtained from (47)
(1 32 2- 7 + 27
(1 82 2
s1 91(2-781/2)
- 7
+
2(1
2y 7)
B
28 7)
Together, these yield system of two equations in the unknowns 81 and 82, whose solution gives a candidate threshold pair s5). We then verify whether this solution satisfies the following consistency conditions:
s5).
1. s S B and 2 > 8,
2. Case (v) applies to (s s5), i... s > 7 and si-rsi > 1, I-7
3. Case (vi) applies to (si/3, 1), i.e., - E (0, 1].
If all conditions are met, the thresholds (sf. 25)) are valid. Otherwise, we proceed to test one of the remaining possibilities.
Unlike the p case, where the equations admit closed-fome solution for sf and piecewise formula for $2, the general E (0, 1) case yields a coupled nonlinear system that typically does not admit closed-form solution. Even in the symmetric case p = 1/2, the equations remain analytically intractable due to the interaction between the thresholds and regime-specifie probability expressions. Numerical solutions for various parameter settings are provided in Appendix L.5.
L.3 Expressions for metrics
In this section, we give expressions for computing the representation ratio and the observed utility of each of the two institutions.
L.3.1 Representation ratio
The representation ratio is defined the ratio A/B, where the numerator A corresponds to the fraction of candidates from the disadvantaged group G2 who are selected, and the denominator B is the corresponding quantity for the advantaged group G1.
The numerator can be written as:
A = (1 $1,9) + (1 p) (Pr(ur1 < $1,8 A uP2 2 32,0] Pr[ure 2 s2,el)
(1 - p) Pr[uz2 82,9] + Pr|uei < 81,8 A uy2 2 82,9]
(1 $1,9) + Pr[ue1 < $1.9 A u/2 2 22,2l.
where the final simplification follows by cancellation of terms.
Similarly, the denominator is:
B = (1 - $1) + Pr[um1 < A u2 2 s2).
Hence, the representation ratio is given by:
(1 - -1.8) Pr Pruur 8188 A uy2 2 82
R(p,8, B,
(1 81 + Pr[ua < 81 A 02 2 82]
57

---

Special case: 1. We now analyze a simplified setting where and 2 1/2 (the case p 1/2 symmetric). Since more candidates prefer Institution 1, it follows that (p) (p)). Under full correlation (? = 1), the observed utilities are deterministic functions of the true values, so:
A candidate is selected if their utility to either institution exceeds the corresponding thresh- old.
In this case, both groups face the same selection criterion: being above sj (p).
Thus, the capacity constraint becomes:
(1 sg()) + (1 sT(P) 2c,
B
which yields:
2(1 c)
sz(D) 1+ 1/8
Note that (p) is independent of p.
Now consider the assignment to Institution 1. A candidate from G1 is assigned to Institution 1 if:
1. They prefer Institution (which happens with probability p). and
2. Their utility exceeds (p). Otherwise, they go to Institution 2 (if their utility exceeds sp)))).
The capacity constraint for Institution becomes:
p(1 sp)) + p - $i()) = c,
which simplifies to:
(1) 2-c/1
1/B
Since selection depends on whether the candidate's 's value exceeds (p), the representation ratio becomes:
1 sr)\33
1 - (p)
This reflects the differential admission likelihood due to the biased utility scaling between the two groups.
L.3.2
Observed utility of institutions
In the general preference setting, the observed utility of Institution (assuming #1 = D2 = 1) is given by:
u1 p Eg [u. (0 2 sill + (1 p) EG 0.a 1 1 2 s7 A 022 < <s,]
+ EG, [uyn [0v1 2 r.,]]] + (1 p) R.,, [Tv [T 2 si A Typz < s
(49)
where Git denotes the subset of candidates in group G, who prefer Institution
To simplify the utility calculation, we define the following integral:
I ( $1, 82) = S f x- 1/y + (1-7) 82 A 2 s1]di dy.
Proposition L.2 (Observed utility of Institution 1 in the general preference setting). The utility of institution 1, U2 is given by:
3
811
s7,0)
+ +11 - (4177, $1, s2) + 11,(11,,,110 $1,9, s2.9))
2
2
58

---

Proof. For a candidate i E G11.
[" sds (1 - sî)
Blua Ju 2 81]]
2
For a candidate i E G12, For candidate G12. the definition of the integral 1T shows that Blu 1 [ū,] 2 si A U22 < 85]] is equal to (7, Arguing similarly for the two sub-groups of G2 and using (49) implies the desired result.
remains to evaluate [1 $2) This integral can be evaluated based on the following cases:
Case (82 7811): In this case, (7 $1, s2) = O. For given y. varies from 81 to s3-(1-7hy
82 81. Thus, the range of xr is empty for any E [0, 1].
Case 11 ( E [s1, 1], 72 E [0, 1)): Note that æ varies from $1 to 1. For a given x, 3 1-7
1-7
varies from 0 to min(1, "=²). Since 81, 9-
52
I
However, 1-y 2
only if x S 82/7 E [81, 1]. Thus, the integral is
82/7
82 72
vdl
1
82
$25
751
1-7
7
62
2
3
Case III ( ( E [81, 11, #2-2#1 > 1): The argument here is similar to case above, #2-22
1-7
T-7
may not remain at most for all æ E [81, 1J. In fact, when æ S #2-(1-7) min(1, 20 ===2=)
1, and then varies from 0 to 1. Thus, the integral is
(1-4)
52//
32 (1 (1--1)-1) ) 7
$2 72
vdr +
vdx
(22-1-99
1 1-7
82
82 (1 7)
+
1
62
Case IV ( > 1,
72
[0, 1)): The argument is again similar to case II, except for
1-7
the fact that 82 z remains non-negative for x E [s1, 1]. Therefore, the desired integral is
7 1-7 xdx
82 (1 s 3/4 st)
Case V ( > 1, 82-281 > 1): As x varies from 81 to 1, -2-241 remains non-negative
I-y
because 82 2 but this expression exceeds 1 when x exceeds 52 (1-)) Thus, the desired integral is
82 TCE rdr
82 1--(111 (1-7)) 7)
rdr +
(2-(1-9)
1 1- ?
7
82 ?
(1 7) 2
1-7
7
2
3
We now give the expression for the expected utility of Institution 2. Define the integral:
12 $1, 82) :- f/ (12 + (1 (1)-))))eer + +(1 (1- -) 2 s2 Ax sı]drdy.
Arguing as in the proof of Proposition L.2, we get
Proposition L.3 (Observed utility of Institution 2 in the general preference setting). The utility of Institution 2, Ug given by:
(1 p) (12(7, 52) + (12(1,1,2)) + (12(7, 82) + B12 $1,8, $2,0))
59

---

Finally, observe that the integral I2(y, $1, s2) is exactly the integral computed in the four cases stated in Appendix H.2. We also need to take care of fifth possibility when s exceeds siy + (1 - 7) in this case, (2 81, 82) = 0. Thus, we have:
Case I (s) < min(st7,1 7) ): In this case, the integral is
si + - 7) (s2)²
2
3-(1 -7)
. Case II -1
s 2 $2 2 max (8 1 - 7)); The integral is
1
(1 - 7 53 + 7
7 + 778))
3
27 (7)
. Case IIII (517 < 8/2
1 - >): The integral is
si 1 (1-1- 1
82
78|
1 1
82
81 78
1y88 +
382 781
+
2
2
2
3
Case IV (1 - 7 <
82 < si>): The integral
si 82
7s1 + 8/
1-7
(1-7)
[352 (303
(1 7)
2(1 - 7)
+
2
2
27
3
. Case V (s) > 8 + (1 7))); In this case, the integral is 0.
L.4 Monotonicity properties of s and 5 with respect to p
In this section, we show that for any fixed values of the parameters c, 3,3, and y, the thresholds si and s are monotone functions of the parameter More formally, we show:
Theorem L.4 (Monotonicity of thresholds w.rt. p). For any fixed values of c, c,B,7, the thresholds $ (p) and (p) are decreasngg and non-increasing functions of p respectively.
Proof. The equations for solving (p) and (p) can be written as follows:
p(Pr/un 2 l1] + Pr[ur1 2 s1,el) + (1 (Prfun 2 A u2 < s2]
Pr[ur1 2 $1 uy2 < $2,8l) C1
(50)
p(Pr[un < $1 A u2 2 s2] + Prfuur < $1,3 A u22 2 s2,8l)
+ (1- (Prluz 2 32] Pr[upe 2 s2,el) = C2.
(51)
Since the set of selected candidates from G₁ is exactly those for which Us1 2 81 or u22 2 82 and similarly for G2, we get
Pr[un 2 $1 V 42 2 sz] Pr[uvi 2 $1,3 V uy2 2 $2 = c + C2
(52)
Note that the lh..s. of the above equation does not depend on p explicitly. formal proof of the above statement can be given by adding (50) and (51). Indeed, the sum of the terms involving candidate i in the two equations is:
p Pr[um 2 s1] + (1 p) Pr[um 2 A u2 < s2] + + < A u2 2 82] + (1 - p) Pr(ua 2 82)
= =Pr[ua 2 s1] (1 p) Pr[ua 2 81 A u2 2 s2] + Pr[u, 2 52] - p Pr[uc 2 81 A u2 Z 82]
Pr(un 2 s1] + Pr(u,22 2 s2] Pr[um 2 A u2 2 s2] Pr[ma 2 V u2 2 s2).
Arguing similarly for i', we get (52). Given values 81, $2, let [(p 51, s2) and g(s), s2) denote the
lh..s of (50) and (52) respectively, i.e., fp, 81, 82)
p(Pr(un 2 sr]+Pr|um 2 s1 +(1 (1- (1-1) (Pr(un 2 ss Aug < priuur1 2 s1,0Aur2 < s2,9l),
and
g(s1, == Pr(un 2 51 V U,2 D sz] + Pr[uei 2 $1,8 V uv2 2 s2,2].
We now show monotonicity properties of these functions.
60

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Fact L.5. Let 81, $2. 10, 1] such that S s and $2 S 8/2 Then, g($1, S 9(6), 82).
Further, g(s1. 92) g(1 82) if si s and $2 < 52
Proof. The first statement follows from the fact that Pr(ua 2 81 V u42 2 82] and Pr[uei 2 $1,8 uy2 2 $2,8] are non-increasing functions 81 and 82.
For the second statement, assume < $1 and < $22 Observe that
Pr[um 2 $1 V u2 2 $2] Pr|ua 2 5f V u2 2 [q2
= Pr(($ E 141 S $ A 422 S x/1 + Pr(un S 8 A S 422 S s22]
Now, if Pr/ua S <& 82 I u22 I ز = 0, then it must be the case that u,2 I 82 whenever un S 81.
ie., 82 2 817 + 7))- But then, 0, 2/2 whenever u41 S 81. Thus,
Pr[si S uil S 1 u2 و Pr[si S un 51 >
This shows that ( s2) < 9 s))).
Now we show the monotonicity properties of the function f-
Fact L.6. Let $1, $2, $2 P.p' E [0, 1) such that 81 2 $1, 82 S $2, S p'. Then, f(p, $1, 82) S
fp', i $2). Further,
(i) f(p,81,82) < f(p', ifp
(ii) f(p, < f(p' s1, ifst 8/.
Proof. We first show monotonicity with respect to p. Note that
p Pr|uut 2 s1] + (1 - p) (Pr jua 2 s1]
Priua 2 81, 422 2 s2))
Pr|ua $1] (1 Pr[u,1 81, 422 2 s2]
Similarly,
p Priuri 2 s1 + (1- p) p) (Prfupi 2 $1 Pr|uei 2 $1,0. uv2 2 s2,el)
= Pr|ur1 2 $1 - ((1-p) Pr|up1 2 $1,8, 2 82,91
Thus,
f(p', 81, 82) - f(p, $1, 82) 2 (p' - Pr 2 s1 A u2 2 32) >0.
(53)
Now we show monotonicity with respect 81: since Pr[u, > $1 Pr[u, 2 81, U/2 $2]
Pr[um 2 $1,6), Pr[u,1 $1,6, uy2 < $2,2l, are non-increasing functions of s1 (and the first one is strictly increasing), we see (using the definition of f) that
f(p' $2, 2 ((p,,1,,50
(54)
with the inequality being strict if s < $1. The monotonicity with respect s2 follows similarly: Pr|um 2 $1, u2 < s2], Pr[uc1 2 $1,0, e uv2 $2,e] are non-decreasing functions of s2, and hence.
f(p' $/ 25) 2 f(p', $1,
(55)
The desired statements now follow by combining (53), (54) and (55).
We are now ready to show the monotonicity property of sj. Let 0 S p < p' S 1, and assume for the sake of contradiction that (p) < (p'). Two cases arise:
(p) < (p"): follows by Fact L.55 that (x) 83())) < g( (z/), (p)),) which is contradiction because both of these quantities must be equal Ci + 02 (using (52)).
61

---

iii (p( 2 (p'): Using Fact L.6 (and part (ii) of result),
f(p, (p). (p)) f(p', si sp()),
which is again a contradiction because both of the above expressions should be equal to ci (using (50)).
Thus, we see that neither of the two cases above can happen and hence, si()) monotonicity property of (p) can be shown similarly by analyzing the properties of the function h(p, 81, 82) that denotes the l.h.s. of (51).
sf/)). The
We note useful corollary of this result: Corollary L.7. Given parameters c, B,7 and p.p' [0, 1], it cannot happen that st(p) spp') and (p) = (p()).
Proof. Assume w.l.o.g. that p p'. Assume for the sake of contradiction that si spp) and (p) = sy()). Then, Fact L.6 shows that
f'y', s1()),,)))))))))))))) s5(()) flp, s1() (1()))))3())))) - s1))
which is a contradiction because both f(p, sp(). sp() and f(y sT()), $ sz(')) are equal to cj-
L.4.1 Strict monotonicity properties of the thresholds with respect to
Theorem L.4 showed that (p) and (p) are non non-decreasing and pon-increasing functions of respectively. In this section, we show strict monotonicity properties of these thresholds
Proposition L.8 (Strict monotonicity of 8 w.rt. p). For any fixed values of the parameters c, B, % there exists a value p" E [0, such that (p(p) is strictly increasing for p E [p", 1] and does not vary with p for E DD)")). Assuming 3 2 (e1 + (2), (p) = 2-6
1+1/3 for all p E [0,p") and sp(p)
is strictly decreasing in (0.p"). Further, p" 0 iff (0) > (0) + (1-7).
We first prove an auxiliary result.
Fact L.9. For any fixed values of the parameters 7,81,82 E (0, 1), Priun Si A u2 2 82) = isf 82 781 + (1 7).
Proof. Suppose Pr[un I $1 A U22 s2] 0. Assume for the sake of contradiction that 82 ys1 + (1 - For the sake of concreteness, let := $1 + (1 -7) Recall that u;1 0/1 and
U22 = y0n (1 7)us2 where Usi and 0/2 are independent and distributed uniformly in [0, 1). Now E, 1], then
if ti E [0, - e), s1] and 0/2
u2 2 ( <) + (1 - (1 -) = 181 + (1 7) - = 82.
Since 81 > O, we see that Pr(u, < $1 A u,2 2 s2 = 0, a contradiction.
For the converse, suppose 82 2 $17 7)) If u4 $1, then U2 = yyu (1 7)2z2 can exceed 82 only when U:2 2 1, which is a zero probability event.
We now complete the proof of Proposition L.8. Define
I = {p E [] s((p) 2 (s)(p) + 1-7).
Theorem L.4 shows that if / is non-empty, then it is a closed interval of [0, 1]. Assuming / is non-empty, let it be of the form (0. p*] (in case I is empty, p* = 0). We now show that (p) does not vary with p when < p"
Fact 10. Assuming p > 0, (p) does not vary with for p E (0, If 2 1 (a + c2), s (p) 2(1cc) 1+1/3 for all p [0, p²). Further, st( is strictly decreasing function of p [0.p*].
62

---

Proof. Suppose < p². Then Fact L.9 shows that Pr(u;1 $1 A u:2 2 s2) = 0. We show that
(p) 2 , + -
as well. Indeed, if (p) 1, this follows trivially. Hence, assume (
si(p)
Then
B
(1) z 7s7
7 2 si + n
3
follows by Fact L.9 that Pr[uei S 81,3 uv2 2 $2,81 = 0 also. Thus, (52) becomes:
Pr[um 2 [1) Pr|uy1 2 $ (p)] = cı + c2.
In other words,
(1- (p)) + - (1~si,6 (p)) c1 C2. First assume that sp 2 B. Then the above equation has the solution:
(p)
(cr + c2).
Thus s1 (2) does not vary with p. 3 2 (cl + 22). this case cannot occur. Thus, the above equation becomes:
(1=))(p)) + - x1(p) c + c2,
which has the solution:
(p( 2 (c1 +22
1 + 1/B
Thus, (p) does not vary with in [0. The fact that sp(p) strictly decreases in this interval follows from Corollary L.7.
Thus have shown the statements in Proposition L.8 when p [0,p") We now show that sj (p) is a strictly increasing function of for > p". We first need the following extension of Fact L.9: Fact L.11. Consider 81 E 10, and 82, s/2 [0 such that 82 < 8/2 S 817 (1 7)- Then,
Pr|ua < 81 A S u,2 < s > 0.
Proof. First, consider the case when s/2 S 1 - y. Let > 0 be small enough positive constant
(whose value will be fixed later). Suppose un E [0, {] and 0/2 E [-=7 - Here e is such
$1, 2 1-7 2 s2. It is easy to verify that u.2 yuu + (1 7)022 [32, s2). Thus,
that
Pr|u,1
$1 A un 2/1 0.
Now consider the case when s/2 > 1 1 7. Express & as ary + (1 (1-7) for some a S $1. A similar argument above applies with Un [a E, al and U/ 1,)
Fact L.12. The threshold (p) is a strictly increasing function of p for p E "p",1].
Proof. Consider values p.p' E [p", 1] and assume p < p'. Assume for the sake of contradiction that (p) = (T/).. Let s denote this common value. Constraint (52) can be expressed as:
Pr|u, 2 s1] + Pr[ua < $1 u2 2 s2] + Pr[uy1 2 s1 + Pr[upr < $1,9 A urz 2 $2,0l = C1 + +2- Substituting (si, (p)) and st(")) and equating the two equations, we get (let $i.g denote min(1,
Pr[um < A u2 2 sp)] + + < si,g A up2 2 [2,)
= Pr[uu s7 A U22 s3(r') Pr[um < $t,s A uvz 2 $2,)
We know from Theorem L.4 and Corollary L.7 that sz(p) > (p'). Therefore, the above can be written as:
Pr|uei < A sz(l/) < u:2 < <z((p)] + Pr|umi < و 22 (p') S uv2 < 22 (p) = 0. This implies that Pr(ua < s1 A xz(p') S u2 < st(p)] = 0, but this is a contradiction because of Fact L. Thus, (p) (p').
63

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We now show the last statement in Proposition L.8: Fact L.13. The quantity p' is strictly positive iff s5 > 7s( + (1 (1-7).
Proof. Suppose p* > 0. Then 0 and hence, s3(0) 2 (0)
Assume for the sake of
contradiction that 82 (0) = 7s (1)+(1->). Letp E (0.p'). We know from Theorem L.4 that sÿ (p) S
(5) and s (p) 2 (0), and at least of these inequalities must be strict (using Corollary I..7).
But then (p) < (p)(p) + (1 7), contradiction. Therefore, (0) > (s) + (1 -7).
For the converse, assume (0) (0) (1 7). Then continuity of (p) and (p) shows that p² 0.
This completes the proof of Proposition L.8.
Calculating p" Assume that (5) > 70] + (1 - 7). Then monotonicity of s (p) and sz())
(using Theorem L.4) shows that p² satisfies:
55()) = 708 + +(1 - -7)-
Now, Proposition L.8 shows that
sp(p") 7/2 c2)
(1 -
(56)
1 1/3
Since Pr[u,2 2 (p') A 4:1 < {p()) = 0 and similarly for i', we can express Equation (51) as: (1- p" (Pr [u 2 s (2 + Pr(uze 2 $2 (p=)(p")]] = c2.
Equation (56) shows that s (p*) > 1 - and hence, (p)" > 1 - 7. Now, three cases arise:
Here, si(x") 7. Then 8 (x") 2 y as well. Then, px is given by
2027 77 (1*))² + 2 (p*))²
1-p"
Here sā(p") < y but s2,)(p") 7. Then, pd is given by
1 1-p" (1=
(p*) 1-7
2y(1-7)
27
Here $2,8 < and hence s (p*) <> 7 as well. Then, p¹ is given by
C2
p
2 22 (1=)))))s(pp" + 1-7
So far, have discussed the strict monotonicity properties of (p). One can analogously show the following lemma regarding the strict monotonicity of sp()). We omit the proof.
Proposition L.14 (Strict monotonicity 8 w.rt. p). For any fixed values of the parameters c, B, 7 there exists a value q² E [0, 1) such that s (p) is strictly decreasing for p E [0,q*] and does not vary with p for p E (q", 1). Assuming qº 1, sp() p E (q*, .1. is given by the following equation:
Pr|uz 2 s2] + Pr(upz2 $2.0] + 02 Further, q < 1 iff (1) > (1).
L.5 Numerical results on behavior across preference and correlation parameters
This section numerically examines how equilibrium quantities respond changes in the candidate preference parameter p and the evaluator correlation parameter 7. Figures 8-12 present four key quantities across varying parameter settings: (i) the equilibrium thresholds of and 85. (ii) the representation ratio R. (iii) the normalized representation ratio N, and (iv) the utilities Ui and Uz achieved by the two institutions. addition, Figure 14 explores how fairness metries vary with evaluator correlation 7 for fixed preference weight p and varying levels of group bias 3.
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Effect of preference alignment (p). Figures 8-11 11 fix 0.2 and vary p for different values of B
and 7
In Figure 8 (3 0.8. 0.9). increased preference alignment with Institution leads to widening disparity: sf rises, s falls (consistent with Theorem L.4, Proposi- tion Propoottnn L.14), and both R and N drop significantly monotonically, Institu- tion I's utility rises with p, while Institution 2's utility declines (both monotonically).
In Figure 9 (3 = 1.0, = 0.9), evaluations are unbiased. Representation ratios remain flat at and utilities vary symmetrically with p. highlighting faimess preservation under bias-free conditions.
Figure 10 (3 = 1.0. = 0.0) shows a similar fairness pattern: no correlation and no bias yield uniform thresholds and balanced utilities, with representation unaffected by preferences.
Figure 11 (3 0.8, y 0.8) exhibits gradual threshold divergence and monotonic decline in R and N, indicating compounding effects of preference, bias, and correlation.
Effect selectivity increase. Figure 12 repeats the previous setting with c: 0.5, revealing sharper phase transitions: R, fall rapidly beyond a critical value of and the utility gap between institutions grows more pronouneed. This demonstrates how institutional capacity amplifies the effect of bias under correlated evaluations.
Effect of correlation (7). Figure (for 3 0.8, p= = 0.6) highlights the non-monotonic impact of increasing As the correlation grows:
The selection threshold s5))) initially decreases and then increases, resulting in a convex- shaped utility curve for Institution 2, consistent with the behavior described in Theorem E.
Both the representation ratio R(y) and the normalized selection share M(r) increase monotonically, as established in Corollary 4.1. In addition, Figure 14 shows that for any fixed and p. the representation ratio R(y)) and the normalized selection share N(7) increase monotonically with B this pattern closely mirrors the trend observed Figure
First-choice ratio parameter. When p < 1, we can further define faimess metric called the first-choice ratio, denoted by F, which measures relative likelihood that candidates from the two groups receive their first-choice institution. Formally, for given parameters p, B, 7, let ʃj denote the fraction of candidates from group Gj who are assigned their first-choice institution under the stable assignment, i.e., given the stable matching thresholds (sī, s2).
fj = Pr(ai,1 2 si] + (1 -1-11 pp Pr Pr[us,2 2 st],
where i; denotes candidate from group Gj. Then, F = f2/fi This metric was also studied in the centralized selection setting by [15].
Figure 15 illustrates how F varies with for different values of 8 and 7 when == 0.3. We find that is typically unimodal, attaining its maximum when p is near 1/2. At this point, the top-ranking candidates from G₁ are evenly distributed across both institutions, allowing a larger fraction of G2 candidates secure their most preferred institution. As expected, increasing 5 reduces the bias against G2 and consequently increases F.
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Normalized representation ratio. A Threstrods of the two institttions
0.80
0.385
0.78
Resresentation ratio. R
0.76
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Observed utilitees of the two Institutions
1.00
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Figure 8: Variation thresholds, representation ratio, normalized representation ratio and utilities with p for fixed value of 0.2, 8 = 0.8, and ? 0.9.
Thresholds of the two instituiion
0.90
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Representation
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formalized representation ratio.. A
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P
Figure 9: Variation of thresholds, representation ratio, normalized reference ratio, and utilities with for fixed value of 0.2, 3 and 7 = 0.9.
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0.900
1.04
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Representation
0.894
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Normalized representation
0.1898
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Figure 10: Variation thresholds, representation ratio, normalized representation ratio, and utilities with for fixed value of 0.2, B = 1, and 0.
Normalized represeniaa ratio Threshelds of the two instittions
0.80
0.78
0.37
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Representaaion
0.36
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0.2
Figure 11: Variation of thresholds, representation ratio, normalized representation ratio, and utilities with p for fixed value of 0.2, 8 = 0.8, and ? 0.8.
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Normalized representation rabio Thresholds the two irstituions
0.80
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Repressntation
0.78
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Observed utilities of the two insitutions
1.00
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Figure 12: Variation thresholds, representation ratio, normalized representation ratio, and utilities with for fixed value of = 0.5, B = 0.8, and 0.8.
0.76
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Representation
0.20
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the Institutions
Normalized representation AV
0.175
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Figure 13: Variation of thresholds, representation ratio, normalized representation ratio, and utilities with y for fixed value of 0.2, 0 0.8, and p 0.6.
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0.6
Norma stdd nepo copresentaiion ratn.
0.0
0.
0-0.b
9-0
0=0.00
Rt: 9:
P-10
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Figure 14: Variation of representation ratio R and normalized representation ratio N with for different values of 3 E {0.8, 0.85, 0.9, 0.95, 1.0} at fixed e = 0.2 and p = 0.6.
0.79
0m0
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68
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07
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A=c.
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p=0.
02
Figure 15: Variation of first choice ratio F with for varying values of 3 and 7. The left panel uses 0.2 and 0.9. The right panel uses 7 0.8 and 0.3.
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Reviews:
[{"review_id": "zxVjs3gUWq", "reviewer": "Reviewer_hvVj", "summary": "The authors study a matching problem where two sets of candidates apply to two institutions. One set of candidates is advantaged, and the other is disadvantaged. Both candidates prefer one institution over the other. Their priorities according to the institutions are randomly drawn but the priority of a candidate at one institution is correlated to their priority at the other. The disadvantaged candidates have their priorities further scaled down by a bias factor. The authors study the behavior of the representation of these students at the institutions as a function of the bias factor and the correlation factor between the priorities at different institutes. They fully characterize this behavior in a simple setting where the number of candidates is large (tending to infinity). They show that in a particular regime, the representation of the disadvantaged students increase with decreasing bias and increasing correlation.", "questions": "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. \n\nWould this correlation flip if the \\beta for u_2 was slightly higher (ie less biased) than that for u_1?", "limitations": "yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "The paper is very well-written and studies an interesting problem. Their problem, even though it uses a simple setting, seems to require a tedious analysis but they do a good job of highlighting the more interesting parts. Their interpretation and discussion of their analytical results is also very thorough and they also discuss useful simplified examples where appropriate.\n\nMy 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.", "was_revised": true, "final_justification": "The authors addressed my comment on the novelty of their technical contribution", "rebuttal": "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.\n\n> 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.\n\nWe 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:\n\n1. **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. \n\n2. **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.\n\n3. **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.\n\nWe will revise the final version to highlight these contributions more clearly in the main text.\n\n\n> 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\n\n\nThank 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 $\\beta$.\n\nIn 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.\n\nRegarding 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.\n\nTo illustrate, consider the following extreme case:\n\n**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:*\n- *When $\\gamma = 1$, $\\mathcal{R}(\\gamma) = 1$,*\n- *When $\\gamma = 0$, $\\mathcal{R}(\\gamma) = \\frac{2c + \\beta_1 - 1}{2c\\beta_1 - \\beta_1 + 1} < 1$.*\n\n**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. \n\n**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 $\\beta_1 \\geq 1 - c$. \n\nDue 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."}, {"review_id": "0RKxq9WZEe", "reviewer": "Reviewer_4qyh", "summary": "In this work, authors study how institutional bias and inter-institutional correlations in evaluation affect the ratio of advantaged and disadvantaged populations under matching of candidates and institutions. Here, disadvantaged means that the quality may be underestimated multiplicatively by a bias factor. The main claim is that both reducing bias and increasing correlation between evaluations of institutions can improve the representation ratio of disadvantaged groups. The authors support this claim with detailed analytical results of how the bias and correlation parameters affect the representation ratio at equilibrium. Finally, their framework can extend to many natural generalizations which makes this framework nice for evaluating decentralized selection decisions on the demographics of selected candidates.", "questions": "- Is it possible to simplify the theorem statements in the main text (and keep full statements in the appendix) to more clearly show that $s_2^*$ and $\\mathcal{R}$ have the properties necessary for intervention as explored in Section 4? \n- (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? \n- Small question, but what happens if $c= \\frac{1}{2}$, i.e., is there a special case when both institutions can admit everyone?", "limitations": "Yes.", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 4, "originality": 4, "strengths_and_weaknesses": "$\\textit{Strengths}$: The authors present a well-motivated problem and well-motivated model parameters that correspond to key intervention points of evaluating decentralized admission procedures. Additionally, their model is very clearly explained, assumptions are clearly stated, and the authors highlight possible generalizations whenever they state simplifying assumptions (such as uniform candidate preference for one institution over another and sufficiently unbiased institutions). The organization of Section 3 and the supporting figure and table create nice intuition which makes it easier to follow along and understand their main conclusions, even though their analysis admits somewhat messy closed form solutions. Finally, Section 4 has a nice discussion of example interventions for both $\\beta$ and $\\gamma$ and the numerical experiments support the extent to which the interventions can improve $\\mathcal{R}$ (and $\\mathcal{N}$).\n\n$\\textit{Weaknesses}$: 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. 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?", "was_revised": true, "final_justification": "My confidence has increased since the authors addressed my questions of model assumptions and extensions in detail and I have become more convinced of the strength of the results they present in this work.", "rebuttal": "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.\n\n> 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. \n\nThank 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.\n\n> Additionally, I was interested in more discussion on the extent to which the monotonicity of $\\mathcal{R}$\n 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?\n\nWe 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 $\\gamma$.\n\nTo 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.\n\nTo 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.\n\nIn 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 $û_{i2} \\geq S_2^\\star(\\omega)$.\n\nNow, 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.\n\nWe restate the formal convergence result below:\n\n**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 $\\ell \\in \\{1, 2\\}$.*\n\nThis 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.\n\nThat 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.\n\nNonetheless, 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.\n\nWe 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.\n\n> 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?\n\nThank 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.\n\n\n> (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?\n\nThank 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.\n\nCapturing 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.\n\nWhile 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.\n\n\n> Small question, but what happens if $c=1/2$, i.e., is there a special case when both institutions can admit everyone?\n\nThank 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 $c = 1$.\n\nTherefore, 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. \n\nIn 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). \n\nIf 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."}, {"review_id": "mruw55KLdR", "reviewer": "Reviewer_H4Bb", "summary": "The authors develop a large‐market stable‐matching model with two evaluating institutions whose signals are both group-biased (disadvantaged candidates’ scores are down-scaled by a factor $\\beta\\le 1$) and imperfectly aligned (correlation parameter $\\gamma\\in[0,1]$). They derive closed-form formulas for the pair of admission cut-offs that uniquely characterize every stable matching, reveal three critical $\\gamma$ thresholds that partition the parameter space into four analytic “phases,” and obtain a piecewise expression for the disadvantaged group’s representation ratio $R(\\beta,\\gamma)$. Although each institution’s threshold can behave non-monotonically, $R(\\beta,\\gamma)$ is proven to increase with either less bias (higher $\\beta$) or tighter signal alignment (higher $\\gamma$). A normalized version of $R$ cleanly separates these two effects, enabling the authors to trace Pareto frontiers of bias-reduction versus alignment-improvement interventions that achieve any target fairness level—thus offering actionable guidance on when debiasing, standardization, or coordination yields the greatest gains in representation.", "questions": "1. Would your selection procedure change fundamentally under additive bias, or could your proofs adapt directly?\n\n2. You state: “In the finite setting, stable matchings are characterised by threshold-based rules.” Could you cite the specific sources supporting this statement?\n\n3. The main results in the article focus on the closed‐form solutions across different parameter phases. In practice, when $n$ is not large, how can these theoretical treshholds  guide the design of effective intervention methods?\n\n4. How sensitive are the $\\gamma$ thresholds and the monotonicity of $\\mathcal{R}$ to non-uniform or correlated talent distributions?", "limitations": "Yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 4, "clarity": 3, "significance": 3, "originality": 4, "strengths_and_weaknesses": "## Strengths\n\n* The paper presents a complete large-market analysis, deriving closed-form formulas for both admission cut-offs *and* the resulting representation metrics; proofs are laid out carefully in the main text and extensive appendices, with clear intermediate lemmas (e.g., Theorem 3.1–3.3 and Table 1’s structural summary) that make the logical flow easy to audit .\n  \n* By turning the representation ratio into an explicit function $R(\\beta,\\gamma)$ and normalised variant $N(\\beta,\\gamma)$, the authors translate their math into concrete design tools (e.g., the Pareto-frontier plots in Fig. 3) that a practitioner could use to pick a cost-effective debiasing or standardisation strategy .\n  \n* Prior studies either assume common rankings with bias or study correlation without group-specific bias (Brilliantova & Hosseini, 2022). This paper is, to my knowledge, the first to analyse the *interaction* and to prove monotonicity of a fairness metric despite non-monotone intermediate thresholds .\n  \n* The writing is well-structured: Section 2 walks through the model step-by-step, and “Step 1 / Step 2” derivations of $s_1^\\star$ and $s_2^\\star$ (p. 6-7) are particularly reader-friendly .\n  \n* Summaries and visual aids (Table 1, Figures 2–3) make the complex piece-wise behaviour intuitively accessible .\n\n## Weaknesses\n\n* Uniform score distributions, two institutions of equal capacity, and a single prestige-dominant preference profile (all candidates prefer institution 1) abstract away many features of real markets. The authors acknowledge some of these limitations themselves , but they remain a barrier to external validity.\n  \n* All results live in an asymptotic continuum. Even a small simulation study or a synthetic-to-real case (as done by Celis et al., 2024 for the one-institution model) would have strengthened the quality argument.\n  \n* The four-case formula for $s^\\star(\\gamma)$ and the ensuing phases are relegated to dense algebra; newcomers to matching theory may struggle to interpret why the geometric regimes matter.\n  \n* Some notation choices (e.g., switching between raw, biased, and scaled utilities) require multiple passes to parse. A running toy example early on could improve approachability.\n  \n* Real clearinghouses (college admissions, labour platforms) routinely involve dozens of institutions and many protected attributes, so scaling of results is unclear.\n  \n* In the main article the probability of each candidate prefers Institution 1 over Institution 2 is equal to 1 and the general case put on the appendic. In my opinion as far as possible it is better put main and general result in the main article not in appendix. We have the same situation fo for consider the general case of $\\beta$.\n  \n* In the main article the Institution 1 selects only from group $G_1$ but it claims general case is consider in the appendix.\n  \n* 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\n  \n* As my understand to run intervention planning we need to estimate $(\\beta_0,\\gamma_0)$, but in this work do not explain how can estimate it by emperical data.\n  \n* Without empirical or simulated finite-market evidence, practitioners can’t judge how sharp the analytic thresholds are for real-sized cohorts.\n  \n\nThe paper is mathematically rigorous, clearly written, and addresses a problem of growing practical importance. In my view, it has substantial impact and surpasses several previously published studies on the topic. Its main drawback, however, is the overall structure. Owing to conference page limits, the manuscript exceeds 60 pages and relegates many key discussions and facts to the appendix, which diminishes the work’s immediate value. Although I am not a specialist in this field, I believe that submitting a journal version—where the principal results are included in the main text rather than the appendix—would give the paper a strong chance of acceptance and make it better suited for a journal than for a conference.", "was_revised": true, "final_justification": "**Final Justification for Recommended Score**\n\nAfter considering the rebuttal, discussions with the authors, and exchanges with other reviewers and the AC, my evaluation is as follows:\n\n**Resolved Issues**\n\n* **Clarity of Section 3:** Authors agreed to move detailed derivations and technical case analyses to the appendix while keeping key structural insights in the main text, improving readability.\n* **i.i.d. Assumption and Monotonicity:** Authors provided a rigorous explanation of why monotonicity results rely on the i.i.d. assumption, clarified the role of the continuum limit, and acknowledged limitations under strong dependence. They will add a discussion of possible relaxations and extensions.\n* **Presentation of Theorems:** Authors committed to simplifying theorem statements in the main text to highlight intervention-relevant properties, retaining full proofs in the appendix.\n* **Special-case Clarification:** Authors clarified the case of full admission and its implications for the representation ratio.\n\n\n**Conclusion**\nGiven the strength of the core results, the clear motivation, the responsiveness to feedback, and the improvements promised for the final version, I increase my original score.", "rebuttal": "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.\n\nBelow, we respond to your comments and questions.\n\n>...  put main and general result in the main article not in appendix. … the general case of $\\beta$.\n    \nThank 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.\n\n> In the main article the Institution 1 selects only from group $G_1$...\n\nThank 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.\n\n> 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.\n\nThank 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.\n\n\n> … estimate $\\beta_0,\\gamma_0$ …\n \nThanks 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: $\\beta_0 \\approx \\frac{ŝ_1^\\star}{2 - c - ŝ_1^\\star}$.\n\nSimilarly, 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.\n\n\n> 1. Would your selection procedure change fundamentally under additive bias…\n\nThank 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. \n\nUnder 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:\n\n$$ s_1^\\star = 1 - \\frac{c + \\beta}{2}.$$\n\nThe existence of $\\gamma$-thresholds continues to hold: we show an analogue of Theorem 3.1:\n\n**Theorem.** *Fix $\\beta < c$. Then there exist thresholds $\\gamma_1, ..., \\gamma_4 \\in [0,1]$ such that:*\n\n1. $s_2^\\star + \\beta \\leq 1 - \\gamma$ iff $\\gamma \\leq \\gamma_1$  \n2. $s_2^\\star \\leq 1 - \\gamma$ iff $\\gamma \\leq \\gamma_2$  \n3. $s_2^\\star \\geq s_1^\\star \\gamma$ iff $\\gamma \\leq \\gamma_3$  \n4. $s_2^\\star + \\beta \\geq (s_1^\\star + \\beta)\\gamma$ iff $\\gamma \\leq \\gamma_4$.\n\nUsing 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:\n\n$$ s_1^\\star - \\frac{2s_1^\\star s_2^\\star - \\gamma (s_1^\\star)^2}{2(1 - \\gamma)} + (s_1^\\star + \\beta) - \\frac{2(s_1^\\star + \\beta)(s_2^\\star + \\beta) - \\gamma (s_1^\\star + \\beta)^2}{2(1 - \\gamma)} = c. $$ \n\nWe will include a brief discussion of this generalization in the final version of the paper.\n\n> 2. You state: \"In the finite setting, stable matchings are characterised by threshold-based rules.\" Could you cite the specific sources supporting this statement?\n\nThank 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.\n\nLet $M$ 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 $s_2^* = m_2$.\n\n> 3. ... In practice, when $n$ is not large, how can these theoretical thresholds guide the design of effective intervention methods?\n\nThank 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.)\n\nTo 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\n---\n**$n = 100$, $c = 0.2$, $\\beta = 0.9$, $|G_1| = |G_2| = 100$**\n| $\\gamma$ | Representation Ratio (mean ± std) |\n| :---: | :----------------------------------: |\n|  0.0  |          0.48 ± 0.16           |\n|  0.1  |          0.44 ± 0.14           |\n|  0.2  |          0.44 ± 0.15           |\n|  0.3  |          0.49 ± 0.16           |\n|  0.4  |          0.53 ± 0.25           |\n|  0.5  |          0.55 ± 0.19           |\n|  0.6  |          0.61 ± 0.19           |\n|  0.7  |          0.64 ± 0.28           |\n|  0.8  |          0.64 ± 0.16           |\n|  0.9  |          0.62 ± 0.17           |\n|  1.0  |          0.73 ± 0.42           |\n\n---\nThese 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.\n\n> 4. How sensitive are the $\\gamma$-thresholds and the monotonicity of $\\mathcal{R}$ to non-uniform or correlated talent distributions?\n\nThank 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.\n\n**Non-uniform distributions:** For independent attributes drawn from a smooth distribution with CDF $\\Phi$, the equation for the threshold $s_1^\\star$ becomes:\n\n$$\n\\Phi(s_1^\\star) + \\Phi\\left( \\frac{s_1^\\star}{\\beta} \\right) = 2(1 - c_1).\n$$\n\nThe computation of representation probabilities also adapts using $\\Phi$ and its density $\\phi$. For example, in Case I when $\\gamma \\leq \\gamma_1$:\n\n$$\n\\Pr[u_{i1} < s_1, u_{i2} \\geq s_2] = \\Phi(s_1) - \\int_{0}^{s_2/\\gamma} \\Phi\\left(\\frac{s_2 - \\gamma x}{1 - \\gamma}\\right) \\phi(x) \\, dx.\n$$\n\nWhile 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.\n\n**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 $\\gamma$.\n\nMore 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."}]

Author rebuttal:

Meta-review: This paper studies stable matching with group-dependent bias and correlated evaluations across institutions, deriving closed-form expressions for representation ratios and characterizing equilibrium thresholds that enable fairness-aware interventions in decentralized selection systems. All reviewers provided positive assessments regarding the paper's technical rigor, clarity, and practical relevance for fairness-aware system design. Reviewers appreciated the well-motivated problem formulation, thorough analytical results, and the framework's potential for real-world interventions. The authors provided comprehensive rebuttals addressing reviewer concerns and commitment to revisions including making improvements to Section 3 clarity, theorem presentation, discussion of model assumptions and extensions, etc. This paper merits acceptance and authors are encouraged to incorporate all the reviewer suggestions in the final version.
Decision: Accept (poster)