Strategic Costs of Perceived Bias in Fair Selection
Authors: L. Elisa Celis
Lingxiao Huang
Milind Sohoni
Nisheeth K. Vishnoi
Venue: NeurIPS 2025 spotlight ()
Keywords: Meritocratic selection
Strategic effort
Game-theoretic modeling
Perceived valuation bias
Algorithmic fairness
Social feedback loops
Representation and efficiency
Nash equilibrium
Human-AI interaction
Primary area: social_and_economic_aspects_of_machine_learning
TL;DR: We show how perceived post-selection bias distorts strategic effort in merit-based selection, leading to disparities. Our model quantifies interventions to reduce inequity by adjusting selectivity and perceived valuation gaps.

Abstract: Meritocratic systems, from admissions to hiring, aim to impartially reward skill and effort. Yet persistent disparities across race, gender, and class challenge this ideal. Some attribute these gaps to structural inequality; others to individual choice. We develop a game-theoretic model in which candidates from different socioeconomic groups differ in their perceived post-selection value—shaped by social context and, increasingly, by AI-powered tools offering personalized career or salary guidance. Each candidate strategically chooses effort, balancing its cost against expected reward; effort translates into observable merit, and selection is based solely on merit. We characterize the unique Nash equilibrium in the large-agent limit and derive explicit formulas showing how valuation disparities and institutional selectivity jointly determine effort, representation, social welfare, and utility. We further propose a cost-sensitive optimization framework that quantifies how modifying selectivity or perceived value can reduce disparities without compromising institutional goals. Our analysis reveals a perception-driven bias: when perceptions of post-selection value differ across groups, these differences translate into rational differences in effort, propagating disparities backward through otherwise "fair" selection processes. While the model is static, it captures one stage of a broader feedback cycle linking perceptions, incentives, and outcomes—bridging rational-choice and structural explanations of inequality by showing how techno-social environments shape individual incentives in meritocratic systems.

Strategic Costs of Perceived Bias in Fair Selection
L. Elisa Celis
Lingxiao Huang Nanjing University
Milind Sohoni IIT Bombay
Nisheeth K. Vishnoi Yale University
Yale University
Abstract
Meritocratic systems, from admissions to hiring, aim to impartially reward skill and effort. Yet persistent disparities across race, gender, and class challenge this ideal. Some attribute these gaps to structural inequality; others individual choice. We develop game-theoretic model which candidates from different socioeconomic groups differ in their perceived post-selection salue-shaped by social context and, increasingly, by Al-powered tools offering personalized career or salary guidance. Each candidate strategically chooses effort, balancing its cost against expected reward; effort translates into observable merit, and selection is based solely on merit. We characterize the unique Nash equilibrium in the large-agent limit and derive explicit formulas showing how valuation disparities and institutional selec- tivity jointly determine effort, representation, social welfare, and utility. We further propose a cost-sensitive optimization framework that quantifies how modifying selectivity or perceived value can reduce disparities without compromising institu- tional goals. Our analysis reveals perception-drive -drivenn--rieennnnn bias: when perceptions of post-selection value differ across groups, these differences translate into rational differences in effort, propagating disparities backward through otherwise "fair" selection processes. While the model is static, captures one stage a broader feedback cycle linking perceptions, incentives, and outcomes-bridging rational- choice and structural explanations of inequality by showing how techno-social environments shape individual incentives in meritocratie systems.
Introduction
Meritocratic selection systems, used by institutions and firms for admissions, hiring, and content curation, aim allocate opportunities based on observable indicators of ability and effort rather than wealth, identity, or social status. They are widely viewed as promoting fairness and efficiency Examples include standardized tests such as the SAT and JEE (49 8) structured
interviews and assessments 669 13,, and algorithmic ratings online platforms 24
70 81
Yet, despite their formal neutrality, these systems often produce significant disparities in representation and outcomes. Women, racial minorities, and lower-income groups are consistently underrepresented in elite universities. leadership roles, and high-paying industries (6 [22). These gaps persist even when evaluation procedures are blind to group identity, suggesting that there are additional factors that drive inequality in merit-based processes.
One set of explanations points to structural barriers: unequal access to resources that enhance merit (e.g., quality education, extracurrieular actívities), implicit biases in selection processes, and limited opportunities due to privileged networks 84
70 81. Others suggest that individuals who face the same selection rules may simply make different choices, investing less effort because they perceive lower returns to success due to cultural preferences, opportunity costs, or labor market sorting [31 38 63 [2]. These disparities under-utilize talent, reducing innovation, diversity of ideas, and social progress 42 55 64). This raises a central question: how can differences in perceived opportunity translate into systematic behavioral disparities even when evaluation is
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symmetric?
39th Conference on Neural Information Processing Systems (NcurIPS 2025).

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Expectations about what selection yields- yields-admission, employment, mobility- are shaped not only by but increasingly by algorithmic tools that mediate labor market signals (2]. Although meritocratic ideals suggest that pay should correlate with skills, productivity, and achievements, empirical studies reveal persistent wage disparities even after controlling for factors such as occupation, education, experience, and hours worked [38] 86 83 61,
historical inequalities 773 56 48
80) 75| 65] 40]. For
example, in the U.S., women earned just 83, 1 1% of what men earned in 2021. I, despite outnumbering men in the college-educated labor force [80] 26). Similar wage gaps persist across racial, class, caste, and ethnic lines, with Black, Hispanic, and Indigenous workers earning less than White and Asian peers in comparable roles [65]. Large language models (LLMs) may further exacerbate these disparities. Recent studies show that when asked for job or salary recommendations, LLMs return systematically different responses across demographic groups, even when qualifications are held constant [2]- Such signals can distort perceived opportunity and disincentivize effort long before any selection decision occurs. Taken together, these findings suggest a perception-driven bias: social and algorithmic cues about post-selection value shape pre-selection investment, reinforeing group-level disparities even under ostensibly meritocratic systems.
Our contributions. We introduce a game-theoretic model of meritocratic selection in which candi- dates from two groups differ in their perceived value of being selected. This model integrates contest theory with models of structural bias from algorithmic fairness and captures how valuation disparities influence effort, merit, and selection outcomes. While the model is static, it represents one stage of broader feedback process linking perceptions, incentives, and outcomes. Our main contributions are:
1. Modeling. We formulate a two-group contest in which n rational agents, divided into groups Gi and G2 (with proportion a E (0, 1)), compete for e n positions. Group-specific valuations follow distributions pl and p2, with p2 modeled as a p-biased version of pi for p E (0, 1] [44) representing structural disparities. Each candidate chooses effort based on their valuation to maximize their expected payoff, which is then converted into observable merit used for selection.
2. Equilibrium characterization. We prove the existence and uniqueness of symmetric Nash equilibrium in the large-agent limit, and express the equilibrium thresholds for both groups in terms of (c, a, P1 P2) (Theorem 3.1. D. We further show that the equilibrium in the finite-n setting converges to this solution at rate O V(Vlog/// n/n).
3. Micro to macro analysis of metrics. Using the equilibrium solution, derive closed-for formed-form expressions for key performance metrics metrics-group-vyee group-wise representation ratio rx. social welfare ratio rs, and institutional revenue in the case where P1 is uniform and P2 is p-biased (Proposition 4.1). These expressions reveal how small changes in p. C cor a can produce non-linear shifts in outcomes.
Fairness-aware interventions. We formulate a constrained optimization problem (Problem ( that allows institutions to trade off between increasing selectivity (c) and reducing valuation bias (p) under fairness constraints (e.g., 80%-rule). We solve this problem in closed form for linear cost functions and characterize when each intervention is most cost-effective (Figure 4
Taken together, our framework provides quantitative lens on how structural or algorithmic biases in perceived value can rationally produce effort and outcome disparities in meritocratic systems, and offers tools to design interventions that enhance both representation and efficiency.
Related work. Our work connects three areas: economic theories of meritocracy, game- theoretic models of contests, and algorithmic fairness. Social scientists have long examined the tension between meritocratic ideals and persistent disparities in outcomes, including gender and racial pay gaps [56]
38 15]. While prior models of statistical discrimination explain disparities through institutional uncertainty or group-dependent beliefs-offee despite equal agent quality 5
73
3
1666
21
23
46 our model assumes perfect institutional information and symmetric evaluation.
Instead, we show how valuation asymmetries alone can induce disparities in effort and selection outcomes. From a game-theoretic perspective, our setting builds on all-pay auctions and Tullock contests, which model competitive effort under asymmetry 776 29 27]. However, these models rarely consider group-level valuation differences. To our knowledge, we are the first to study equilibrium behavior in large-agen contests with asymmetric valuation distributions across groups. Finally, our bias model extends work in algorithmic faimess that explores valuation gaps and signal noise 45 28 [8 but previous strategic classification models typically lack inter-agent competition or valuation- based disparities [1] 29 Thus, our work offers novel integration of asymmetric group valuations into competitive contest frameworks, with implications for equilibrium behavior, fairness, and institutional design. See Section A for further discussion and detailed comparisons.

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2 Model and metrics
We consider population of n agents competing for k = en indistinguishable spots, where c C (0, 1) denotes the selection fraction. Agents are partitioned into two groups: an advantaged group G1 of size (1 - a' and disadvantaged group G2 of size an, where a E (0, 1). Each agent i Ge
(f E (1, 2)) has a valuation ~ Pr supported on S2 C R20- We model systemic disadvantage via a scaling of valuations: if P1 is the valuation distribution for G1, then G2 has valuations drawn from p2(u) = p. (²), where p E (0, 1] captures the degree of bias, implying Eu-pp [v] = pl. [e]. For instance, if p1 uniform on [o, 1). then p2 is uniform on [0, p). Such a bias model has been widely studied in the fairness literature [45] 16] 19] and serves as benchmark for understanding systemic disparities. Section D.T] D.I discusses extensions where the valuation distributions P1 and P2 are truncated Gaussians, and the bias parameter p may also be drawn from distribution, introducing stochastic heterogeneity across candidates. These structural disparities across groups may stem from unequal access to opportunity, differences in marginal returns, labor market discrimination, or broader societal narratives about value; see also Remark 2.1 for practical scenarios.
Each agent also has an initial ability a; ~ Pa supported on 2, S RR2., drawn independently. We assume that Po. is identical across groups. Agents choose policies A, S2 x S2 - R220 that map their type 0g = (v) a;) to an exerted effort = (O:).. The agent's ' score is 8; e + a, A strictly increasing merit function m IR20 + 2go maps scores merit. The institution selects the agents with the highest merit values. Each agent's payoff is
fe a₁ eyj $1,..,, sn) = [s,)ss;) among top k) -u, -- -s: (s--a;) = 1(s; among top k) u; - (s, - -a,,,
where the second equality uses the strict monotonicity of m. Agents know n. k. P1. P2, Pas their group identity, and their own type 8 = (v; a,), but not others' types. Let A = (A,...., An) denote the joint policy profile. The probability that agent is selected after exerting effort is
P.(e; as, A-) = P (e + a, is among the top k scores of (A,(0,) + ay}s U fe+ ai)), where 03 = az) are drawn i.i.d. from the respective group distributions. The expected payoff
T Ti (U. ag e; A_.) Es, (vs, as, 81, sn)] = Pe(e; a,, A-i) 14 -.
A policy profile A is a Nash equilibrium (NE) if, for all i, U, a, and e,
Te a, e; A-;) S n. (v,a, A, a); A-).
(1)
We implicitly assume that agents act rationally and strategically to maximize their expected payoff', using their knowledge of the contest structure to compute the NE policy A (74). For simplicity, we sometimes assume that Pa is point mass at 0, that policies depend only on valuations.
A special case of our model generalizes the classical undifferentiated contest (where P1 = =22), which has been extensively studied 776] 52 [2) To the best of our knowledge, our work is the first to study strategic asymmetries arising from valuation differences in settings where group sizes are known and fixed, structure commonly seen in admissions and hiring. We provide detailed comparisons with prior works 2 (27 in Section I. Remark [E.7 discusses extensions to multi-group settings and to heterogeneous effort-to- me-ier mappings, where each individual may convert effort into merit at different (non-linear) rate.
Metrics. We study three metrics to evaluate faimesss efficiency, and institutional outcomes un- der a given policy A. Define (A) as the (random) fraction of agents selected from group
Ra(A)) Ri(A) f H a metric commonly used in the fairness literature [22 17 FR (A) E [0, 1]. and low values indicate underrepresen- tation of one group. Building on standard notions of allocative efficiency (67), define group- wise social welfare as (A) := [c Eiec, (I[i selected) u ee). The social welfare ratio
Ge. The representation ratio is rv (A) == min (A) R2(A)
is rA()) == E min {
(2(A)) SI(A) }]. This metric measures disparities in average payoffs between groups. Define the average revenue as RV(A, E
(() (A)
[ E selected m(:))) capturing the average
merit of selected agents and aligns with institutional objectives [3 34
We consider the min operator since randomness can occasionally lead to equal or even higher representation for the disadvantaged group G2. This becomes vanishingly rare n
3

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We analyze how the NE policy A and associated metries vary with the bias parameter p and the selection fraction c. These parameters capture systemic disparities and selection competitiveness. respectively. Even for simple instances, deriving closed-form NE strategies under asymmetric valuations is significantly more complex than in the undifferentiated case. A worked example illustrating these challenges is provided in Section B
Remark 2.1 (Practical settings with group-based valuation bias). Our model captures environ- ments where disadvantaged groups anticipate lower returns from being selected-due to structural barriers, social context, or biased algorithmic feedback (see Section 1.. For example, as discussed in Section [ persistent wage gaps across gender and race-even after accounting for qualifications as well as biased algorithmic reconmendations can diminish expectations about the benefits of selection. These lower expectations can rationally reduce pre-selection effort, even under formally fair rules, and represent the main regime study. That said, in domains such as credit, housing, educution, disadvantaged groups may instead face higher marginal returns due to limited outside options; this can be modeled by reversing which group has the compressed valuation distribution.
3 Theoretical results: Nash equilibrium and metrics for large
The first question we address is whether a Nash equilibrium (NE) policy exists for the two woougoup contest and how it can be computed. While characterizing NE policies for finite is challenging, the prge-population limit (n - 2o) reveals an interesting and tractable structure. The following result shows that in this limit, it is possible describe how the strategies of the two groups, G1 and G2, converge. However, the absence of an explicit policy formulation for finite n complicates the interpretation of convergence, which we address by adopting the notion of an approximate NE policy.
Definition 3.1 (s-Nash equilibrium (51). For an > O, policy A is said to be an e-NE policy if for any & E {1, 2), agent E G6 type (v, E S4 x Slg and effort e 2 0, the following condition is met: Te a, e; :) S (, a, A, a); A-;) t
An e-NE permits stability violations up to E. with exact NE recovered when & = 0. This notion allows us formalize the convergence of NE policies in the following theorem.
Theorem 3.1 (The two-group contest: Large limit), Let a, E E (0, 1). For (= 1, let pt be density supported on domain Qr C R2o- Let pa be a density supported on a domain St S Rzo- Let m Rzo - Rz be merit function that is strictly increasing. For { = 1,2, let Fe be cumulative density function (CDF) of the sum of valuation and initial ability such that for any 5). Suppose (S2 U (2)) + Su is connected Jan densities P1, P2. Pa are positive at any point of their own domains. Let t be the unique solution to the equation
Ç E Rz0. Fe(()= = PIyr-paarpe jv
(1 - a)))) + aF2(() = c.
Define s(v,a) = 0 + a < I. and s(v, = max (1- 0, ift
and let policy A be: each agent i E G1 uses the restriction A; = sla, xºy while each agent j E G2 uses the restriction A₁ = =l5t,xx.... Morewver, this solution is monotonically decreasing with c.
This A the unique policy such that there exists an infinite sequence A(1), A A()), ...., where A(r) is a policy for the two-group contest with n agents characterized by threshold function s(m). (22, U$22) x Q2 + B22. such that the followings hold: (1) For every integer n 2 1, agent
i E G1 uses the restriction A!m) g(1) (a,x while each agent j E G2 uses the restriction
A = (axx and lim s(n) = s; (2) Every ((n) is an En -EE policy with lim, En = O.
This theorem characterizes the policy A through a threshold function s parameterized by t, establish-
ing that there exists sequence of policies A(").... AAm).... that converge towards A, progressively approximating it. In Section E. we provide the explicit form of policies ((n) characterized by (n)) (in Theorem E.22 and a complete proof. Note that the value defines the threshold function s, and consequently, the policy A. Therefore, we focus below on analyzing t. We first remark on the uniqueness of guaranteed by Equation (2) under certain assumptions on the domains and densities (see Lemma E. which are natural in real-world contexts and satisfied by the distributions discussed in Section These assumptions ensure that each F,, being a CDF, is strictly monotonic over its
2 Here, symbol + represents the Minkowski sum of domains, where A B ba+b:ae A,D B}.

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domain S2g + -Zg Consequently, the combined CDF ( + o must also be strictly monotonic
over the connected domain (52, U (22) S2g which guarantees the uniqueness of the solution t.
A key takeaway from Theorem 3.1 is that, in the large-n limit, each agent makes a binary decision based their combined valuation and initial ability e + either exert effort max a, 0} to ensure a score of at least t, or put in no effort at all. The threshold t, determined by Equation (2) plays central role this decision. It is chosen such that fraction 1 e ++a I f, meaning that exactly a fraction e is expected to exert effort and compete. Thus, t implicitly encodes the level of competition: higher values of reflect more intense competition, requiring higher effective scores for selection.
of the agent population has
Computing the threshold in the NE policy. Equation (2) is crucial for applying Theorem 3.1 as it enables the explicit computation of I, facilitating analysis in Section 4 Let Fa be the CDF of the initial ability Note that Fe(S) Pr Prompr,app. (M S {) Sa, Pe Pe( Fa v)du. Thus, Equation
2 is equivalent to (1 aa) ) Sn, pi(e)FalS - v)de + a Soz apz(v) Fa(S - v)dv = -1 c.
Specifically, when (2 -1 (3) for some p (0, 1]. this equation becomes becomes
(1- So, (I()) Fa(S v)dv + =/2 Soz (1)) v)dc = 1
We illustrate how to use this equation to compute the explicit form of t. Let p1 be uniform on
O = [0, 1], P2 be uniform on 222 = [0. Ø and Pa be uniform on P2g = [0. 1]. Such uniform densities often used in studies and analyses 45 16 19 serves fundamental benchmark for insights into
decision-making. allocation mechanisms, and strategic behavior. Moreover, domain (12 UU2) (0022)+(22 +22 =
[0, 2] connected for any value of p E (0, 1]. satisfying assumptions in Theorem 3.1 Since pl (v) = 1
for u E [0, 1], P2 = - for v E [0, p] and (5 v) = min (5-v)+) (Here, 24+ = max (0, x)),
Equation reduces to fo (1 - a)- min(1, - b)) )dv + Jo min(1, (S - ()) Jdu 1 c.
Consequently, the solution / is a piecewise function of parameters p. c, and o; see Proposition
R4
for its explicit form. Here, we illustrate the behavior of over a representative range where a 0.5
(equal-sized groups) and 0 < c S (high selectivity).
2- < - 2 and / = 1+3g
V Acpp11+p) p(1-p)2 if 1 2 1- 1
2VC.
(5)
1+p
1tp
Note that, while l the same for both groups, it may happen that > 1+ p (when p 2()).
implying that no agent in G2 exerts any effort. We note that for other densities, such as piecewise linear and polynomial, including Pareto, explicit forms of the solution L are achievable. For instance, consider a Pareto distribution defined by pı (v) 2 for e 2 1. p-biased density (2 p "), and Pa is a point mass at O. Here, t can be explicitly calculated: If a + - 1 > 0 and p <
V(a + c 1)/a, then t = pV/a(aa + 1) otherwise, t = v(1 - a ap²)/c.
Computing the metrics. We next ask whether the key metrics associated with the NE policy A from Section 2 can be computed in closed form. Given the simple threshold structure of A, these metrics can indeed be expressed functions of the scalar threshold f. However, for general densities p1 and P2. the expressions for the representation ratio r'R (A) and social welfare ratio (A) become more complex due the presence of the min operator and the convolution involved between pt and pa (see Theorem
F.3) For clarity, we focus on the special case where P2 is a p-biased version of pı and Pa a point mass at which admits more tractable expressions. Since t depends on the parameters P. c, and a, the resulting metrics are also functions of these parameters. The following theorem characterizes both the explicit forms and their monotonicity behavior.
Theorem 3.2 (Metries and their monotonicity). Assume p2(0) - (E) for some p (0, 1]
and Pa is a mass point at O. Let policy A be defined as in Theorem characterized by t being the unique solution of Equation (4). Then for any density Pt,
(t/p) PR(A) = I-F:(0) rs(A)
J. (x-1)))) and RV(A, m) = =m(t).
Moreover, FR (A) and (A) are monotonically increasing w.r.t p. while RV m) is monotonically increasing w.r.t. and monotonically decreasing w.r.l. e and a, for any merit function m.
'Throughout this paper, we extend the domain of a CDF F' to the entire real line R such that F is monotoni- cally non-decreasing, with F (o)oo) and F(x)) 1.
5

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teatiin
folcas enteenier
Efforte
Valuetion V
Valuation V
Valuation
(a) 72
(b) n 200
(c) n 600
Figure Evolution of group effort policies at iteration 500 for various n with p = 0.8 and c 0.2.
The proof is deferred to Section[F] F) Combined with the closed-form expression for t. this result enables direct computation of the metrics, which we use for contest analysis in Section[4 Notably, I'R (.A) and rs( increase with c, while RV(A, m) decreases, highlighting both benefits and trade-offs for the institute. The qualitative behavior of (AA) and (A) w.r.t. e and ax however, depends on the underlying densities; see Remark F2
Discussion of NE policies for finite n.. To assess the robustness and practical relevance our theoreti- cal results, we study the closeness between the finite-n NE policy and the infinite- sitiitteppplattionn threshold policy s defined in Equation (3) (see Section C). We propose a dynamic procedure (Algorithm T that initializes with
82 for groups G1 and G2, and iteratively updates them.
We simulate this dynamics under the setting pl Unif 1]. p2 Unif p = 0.8). Pa -o. with c= c= 0.2. a 0.5, and = 20, 200, 600, 1200, running 500 iterations in each case. Figure I shows representative results; full plots are in Figures
The simulations show that even moderate population sizes (n 2 600) yield policies closely tracking the infinite NE, validating its use as a practical approximation. We also observe group-level differences in convergence speed and stability (Figure 9. with smoother and faster stabilization as n increases.
Finally, using the proof of Theorem 3.1 we establish that the finite-n NE policy is Olog n/n)- close in value and yields an O( /lo n/n)) n Concretely, this means that for large the finite policy takes the form s(x) 0 for < O lll////g/// and s(x)
otherwise. For general distributions, closeness depends on the density structure and is more involved. Aligning with our empirical observations, these results reinforce the practical relevance of the large-n analysis, which captures the incentive-aligned baseline under strategic behavior. See Section C.2 for details.
Key ideas in the proof of Theorem 3.1 policy structure and verifying that it is indeed an equilibrium. We sketch the core ideas below; a full overview appears in Section E.I For clarity, we focus on the case where Pa a point mass O.
The proof involves two main steps: hypothesizing the NE
Hypothesizing the structure of the NE policy. The key challenge in characterizing the NE policy lies in the absence of its explicit form for finite Drawing intuition from the undifferentiated case with density p. where the NE policy converges to threshold function su)) = Fp (1 (1-c) c) i 2 F (1-c) and 0 otherwise, the first idea is to hypothesize that in the two-grout case, the NE policies $1, 82 also take threshold forms with group-dependent thresholds t2, tz. However, asymmetry in pl, P2 complicates the expression of winning probabilities P; and prevents straightforward computation of 4, l2. Focusing on the uniform case where is Uni 1] and p2 is the p-biased version supported on [0, p). the second idea is that in the limit n co. NE stability requires ( = l2 = L. If > (2, then some agents in G₁ benefit by reducing their effort to slightly above l2, contradicting NE; a symmetric argument holds if t < t2. Although 5 and s2 are defined on different domains, they can be viewed as restrictions of the same threshold function s characterized by t. This justifies setting $1 = $2 = s with shared threshold t, and modeling the two-group contest using an effective mixture density p (1 a a)p + op2 supported on 22 UZg. As n - oC, the contest becomes indistinguishable from the undifferentiated case with p. yielding the same threshold = F,- (1 as in Equation ( Thus, the NE policy A defined in Theorem 3.1 is a natural candidate for the limiting equilibrium. Note that this p is only used for hypothesizing NE rather than demonstrating the convergence; see discussion in Section E.4 Moreover, the above argument suggests that, in the limit, the strategic environment becomes uniform across all agents, motivating us to develop an infinite contest (Definition E.5 and provide an alternative proof; see Section E.5

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Pot of soll pip--aa e-05
(8.95 0.01
18590 C
102
00
944
06
18
07
38 08
30
344
05
3.
07
34 08 10
(a) PR (A) v.s.
(b) FR (A) V.S. C
(c) (A) v.s.
(d) rs (A) v.s. c
Figure 2: Plots of the representation ratio r (A) and the social welfare ratio rs (A4) as parameters p and e vary for Proposition 4.1, with default settings of (p, c, a) = (0.8,0.1, 0.5). A dotted line in these plots indicates the threshold at which FR (A)) = 0.8 (A) = 0.8.
Showing A is NE, Although we have solid guess for the NE policy A, a key challenge arises: the NE policy for finite n lacks an explicit form, making unclear how to define convergence to A. Towards this end, the third key idea is to provide a different proof of convergence to a threshold function for an undifferentiated case with density p. without using the explicit NE formulations. The first attempt to prove that A is an e-NE for sufficiently large fails- even in the simple case with p = Unif(o (0, 1] and c = 0.5. One can construct a valuation v = 0.2 such that the agent benefits by exerting small effort = 0.01, achieving winning probability Pa(e; A-2) z 0.5 and obtaining payoff of approximately 0.09. To bypass this, the idea is to define proxy sequence A(r) with threshold policies (()) that I) converge to as n + oo and 2) ensure that P:(e; ( + for e < t. making ((") an ENENE with En - 0 (see Section E.1.3.. To this end, we define s (v) - if 2 2 t .Vlog n/'n n and 0 otherwise, so that a (c + vlog fraction of agents put in effort t. This yields Pr(e; (T) I v1/n by concentration, ensuring the payoff from deviation is at most v //n, Finally, adapt this new proof technique to the two-group case with general pr. P2 by carefully selecting the following threshold shift (Definitions E. and
and A()) is (//lon///))
A = min (F)(t) - Vlogn/n,, F2 (F2(t) - /logn/m)3
and set (v) tiff 2 An. otherwise (see Theorem E.2
We find that limn- An t.
indicating that (m) converges to s. Crucially, such a A, ensures at least (c + n /logn/n)-ffractinn -fraction of agents, in expectation, putting in effort 1. Using this property, we establish a concentration tail bound for the winning probability P. analogous to the undifferentiated contest: Pe(e; A + 0 if
e < I. (see Lemma E.4. Furthermore, this minimal winning probability guarantees that (m) is an NEE with lim, oo En 0 (Lemma E.6 This analysis concludes Theorem 3.1
Uniqueness of the NE guaranteed by Theorem 3.1, First, if 22 U is not connected, the CDF
(1 - aiFi (v) + (v) may not be strictly monotonic over 22 U P2. For instance, if c 0.5. Pi is uniform on 12 = [0, 1] and p2 is uniform on f2g = [2, 3]. then (1 - F1 + a1/(2))11) = (1-a) + aFF2 (2) = 0.5 1 1-6 Consequently, there may exist two distinct points f = 1 and t2 = 2 in 22 U 222 leading to non-unique NEs. In contrast, when 22 U Q2 is connected and each Pe is positive on P4. the unique solution to Equation (3) ensures a unique NE policy. To this, by Corollary 3.2 of (20). symmetric agents use symmetric policies in NE, so we hypothesize a policy pair 82) for G and G2, respectively. As -> oc, both 81 and 82 manifest threshold functions, leading to 81 = 82 = 8, ensuring the uniqueness of NE, Additionally, it arises because if 81 * 82, the NE would destabilize, as agents from one group would adjust their thresholds gain higher payoffs. This proof can be easily extended to multiple groups and non-identical cost of effort; see Remark E.7
4 Analysis: metric behavior and intervention design
Variation of metries with p and c. Using Theorems 3.1 and 3.2 we analyze how the
ratio T'e (A), social welfare ratio Ts (A), and average revenue RV,A, resoond to changes in the bias parameter P and the selectivity parameter c. We study whether these effects are linear non-linear, and whether they exhibit sharp thresholds.
Setup. We adopt the setting from Section 3 pı is uniform on [0, 1], P2 is its p-biased variant, uniform on [0, p). and pa is point mass at O. This isolates the effect of asymmetric valuations while simplifying calculations. The corresponding CDFs are (v) = v and F2(v) = v/p. with both saturating to I outside their support. Unless varied explicitly, use default values p 0.8, 0.1,
7

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and a = -0.5, representing moderate bias, high selectivity, and balanced group sizes. We refer to Section SectionD.22oor analogous analysis under truncated Gaussian distributions.
Closed-form metrics. Fixing the density setup above, we apply Theorems 3. and 3.2 to derive
closed-form expressions for rR (A)),, and (A)).. These summarized below (proof in Section F.
Proposition 4.1 (Metrics for uniform densities). Let pı be uniform on [0, 11, P2 uniform on [0. pl.
and Pa a point mass at Let A be the NE policy as T - oC. Let Pe == __
I-a' Then:
1 p(1 - c)
1= T-a
p Pe:
if 2 Pe
0 -~ 0p + a
(R(A) = - ap app+00 + a
(x(A)= =0
if < Pa:
if p 2 Pc:
a - appptoo
if < Po: Ts(A) plp ap a + c- 1)2
rs(A)= =0
p z Pc.
aa- (a- ap + cp)2
= m(() for any merit function ml.); FR (A) and (A( are monotonically increasing functions of parumeters p; a, a.
Moreover, RV)
As in Equation 5]. Proposition 4.1 reveals sharp threshold at p= 1
1-a When P <11 1-
I lies above the maximum valuation in G2, implying that no agents from that group participate. Consequently, FR (A) and (A) are zero and independent of p. When p crosses this threshold, these metrics become positive and increase monotonically with p. reaching 1 at 1, the symmetric case.
Metric behavior: Figure S plots the representation ratio T' (A) and the social welfare ratio (A))) as functions of the bias parameter / and selectivity parameter c. The corresponding threshold values are shown in Appendix Figure [3(b] Both (A) and r. rs(A) (4) exhibit non-linear growth with increasing and c, and drop sharply-super super-linearly- when when these parameters decrease. For instance, with e 0.1, a 0.5, and p S 0.85, we observe that FR (A) < 0.2, indicating notably low representation for group G2. This highlights a key practical insight: in highly selective environments, such as contests with 1-in-10 selection rate, strategic behavior amplifies disparities. These trends echo empirical findings on under-representation in competitive domains [77] 61]. Moreover, reductions in e (i.c., increased selectivity) lead to pronounced declines in both representation and social welfare.
These trends of metrics offer designers of meritocratic selection processes critical insights into strategies for countering under-representatiin and elevating F' (A) (or mitigating disparities in average payoffs and elevating rs(A), ). We recall the two main criteria for identifying representation bias: 1) Ensuring the selection of at least one agent from every group, and 2) adhering to the 80% rule, which serves as a guideline for identifying potential adverse impact the hiring rate for G2 falls below 80% of that for G1, i.e., FR(A) 0.8. Given the fixed nature of a within the population structure, the main avenues for interventions aimed at improving / (A) focus on adjusting the parameters p or c. Below, we explore potential interventions for both approaches:
(I) Increasing p effectively means increasing the valuation of agents in G2. Various strategies have been proposed and implemented to achieve this goal. For instance. [38 highlights several approaches to narrow the pay gap, including enhancing workplace flexibility, decreasing the cost associated with temporal flexibility, and improving the availability of high-quality, affordable childcare. These interventions aim to increase job valuation for women, analogous to increasing p. Figure 2(a) quantifies the required increase in p. to ensure at least one agent from G2 is selected, P must exceed 0.8; to adhere to the 80%-rule, it should be at least 0.976.
(2) As shown in Figure 2(b) raising e above 0.1 satisfies the criterion for selecting at least one agent from Gg, while elevating it to 0.5 meets the 80%-rule requirement. Increasing e represents a more straightforward approach than boosting P and might be more feasible for institutions. This could involve pre-selecting a larger subset of candidates and applying a distinct selection process to this subset, based on institutional priorities and the likelihood of successful candidates following the expected trajectories.
Finally, regarding the average revenue RV,A, m) m(t), it immediately follows from Proposition 4.1 that RV,A,, m) significantly decreases as competition within the entire population intensifies- either through a decrease in p or an increase in Given that average revenue is indicative of the benefit of the institute, this trend underscores the critical need for contest designers to mitigate systemic biases in valuations. The decline in average revenue with increasing bias compromises not
8

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only the faireess of the contest, but also the overall quality of the outcomes it produces. This aligns with research that has explored the losses attributed systemic biases 42 55 64
Optimizing interventions under cost and fairness constraints. Having identified two policy revers-reducing bias (p) and increasing selectivity (c)-a natural question arises: how should institutions choose between these interventions to improve outcomes such as the representation ratio or social welfare ratio? To address this, we formulate a constrained optimization problem for cost-effective intervention design. We allow two interventions: increasing by Ap E [0, 1 - pl, and increasing e by A. [0, 1 cl. Let r'R (Ap.. p. Ae) denote the representation ratio under the NE policy with updated parameters + Ap. e + A.. The goal is to ensure r& (Ap, Ae) 2 T while minimizing intervention cost. We define two components of the cost function:
(1) Resource cost of increasing Let ʃ [0, -] - R2o be monotonic, modeling the institutional cost of boosting valuation. A simple form is linear: f(Ap) aDpp. justified by first-order Taylor approximation when A, is small. Other variants include f f(Ap) a for B representing the increase in the marginal cost of continuously improving bias.
(2) Cost via revenue loss. Increasing reduces average revenue RV(A, m) m((), as it lowers the score threshold I. Let A p. A.) = =m((p, c)) m(((p + Ap; e + Ac)); represent the revenue decline. Since the institution seeks to maximize value, this loss contributes to total intervention cost.
Optimization problem. We formalize the intervention design as:
a -.) f(Ap) + 9(Ap, Ac) s.t.
TR TR Ae) 2
(6)
This framework also applies to reducing welfare disparities by replacing with r' in the constraint.
Empirical calibration. To demonstrate real-world applicability, calibrate the model using gender- disaggregated data from JEE Advanced 2024, highly competitive entrance exam for India's IITs. Of 180,200 candidates, 139,180 were male and 41,020 female; 40,284 males and 7,964 females qualified. This yields admit rates of 28.9 98.9% for males and 19.4% for females, giving an observed representation ratio of Tobs x 0.671. The overall selection rate is 0.268, and the female applicant fraction is z 0.228. These values anchor our analysis of strategic disparities and potential interventions. In Section G.T under the uniform density setup in Proposition 4.1 compute p R 0.882 using the explicit form of Tohs = (A)
a-aptep (1c)(1-p)
Explicit solution under uniform densities. We now solve the optimization problem under this uniform density setup with (p.c. a) (0.882, 0.268, 0. 0.228), setting m(e) e, f(Ap) 54 and g(Ap: A.) = t(p.c) - t(p + Ap.c c + Ac). This yields the objective: 5A 1.1 [t(p + Ap, + Ae) - t(p, c)). subject to the condition Fx ( Ae) 2 and feasibility constraints.
Insights. Figure shows the optimal intervention as function of the target threshold For 7 < 0.92, increasing e (lowering selectivity) is more cost-effective. For r > 0.92, increasing p (mitigating bias) becomes preferable. This suggests that expanding access is more impactful under high disparity, while improving group valuation better when gaps are narrower.
We conducted additional simulations by varying and beyond the default values (see Section G.2). The results remain consistent with our main findings, confirming the robustness of the above key insights. In Section G.3] we also offer a concrete example to illustrate how our model supports interpretable predictions and can inform data- -uta-ggounded interventions while also noting what is required to operationalize it in practice.
Optinal
min Ap010,1 P.. 5A (p + 21)(1 - De)
(1 a) + A,) +a
s.f. (1 a)(p A,) + + 4.
alp 4,) te+ A= Ao) 2
A.
p ,+ 1-a
Figure 3: Explicit form of Problem 6
Figure 4: Plot of optimal interventions (Ap, Ae) for various E (0.671, 1).

---

Alternative potential interventions. Having discussed interventions based on adjusting p and we next consider alternative approaches that modify the contest structure itself. These interventions can further reduce disparities in representation or social welfare ratios, though they depart from the baseline two-group contest formulation. A detailed analysis of these extensions provided in Section G.4
Introducing preference heterogeneity. One approach is apply group-specific merit mappings of the form m²(s) = Iys + ye for group Gr (L = 1, 2), with parameters Iz, Me 2 0. Here, Iy acts as a scaling factor or "handicap." and ye as an offset or "head start" (see also Appendix A With this intervention, we can still compute the Nash equilibrium for infinite (Theorem G.I.. which implies:
(AA and (A) increase with p. c, and a, as well as with Z2. y/, and decrease with 1 y1:
Choosing merit parameters with 22 > 21 and y/ > y1 can sustain high representation and welfare ratios (e.g., TR((A, rs (A) 2 0.8) even in highly selective settings;
Increasing T2 or y/2 can thus serve as an effective disparity-reducing intervention.
Incorporating outside options. Another possibility is assign each agent in group Ge reservation payoff X; 2 if not selected. Because this payoff is earned only upon losing, a higher X, lowers the marginal benefit of effort, acting opposite the merit parameters ry and ye- Hence, increasing A (the outside option for the advantaged group) reduces their effort incentives and can help narrow representation and welfare gaps.
Setting group-specific selection rates. Finally, the institution can set separate capacity constraints for each group- for instance, selecting c-fraction of agents from G1 and G2 independently. This decomposes the overall model into two within- within-group contests, fixing T' (A) = under equal selection rates. Compared to the combined contest, agents in G2 now face lower bar for selection and exert more effort on average.
5 Conclusion, limitations, and future work
This work highlights a central tension in modern meritocratic systems: even when selection mech- anisms are formally unbiased, systemic disparities in how groups perceive value can lead rational agents to behave in ways that perpetuate inequality. Our model captures this dynamic through a strategic contest framework that extends all-pay auctions to multi-group group settings. By analyzing Nash equilibria in the large population limit, we characterize how group-level biases (p) and selectivity (c) affect fairness and institutional metrics such as representation, social welfare, and revenue. A central contribution is Theorem 3.1, which provides an explicit form for equilibrium strategies under broad conditions. Our framework enables interpretable predictions and supports Lata-grounder policy interventions.
Our model makes simplifying assumptions to enable analytical tractability. Most notably, it assumes agents are fully rational and that merit is captured by single-dimensional notion of effort. In practice, decision-makinn is shaped by uncertainty, cultural context, and multifaceted criteria for merit, Extending the framework to incorporate bounded rationality, noisy information, or multidimensional effort remains an important direction for future work. Several application-driven extensions are also promising. One involves modeling university admissions systems with external incentives (e.g., brand-based free-riding), which may result in over-representation of certain groups (p > 1). Another is to study how affirmative action or group-dependent costs reshape equilibrium behavior. These variants would help bridge theory with institutional design. Beyond these extensions, an important avenue is to embed this static framework within dynamic feedback environments where perceptions evolve over time in response outcomes and institutional signals. Such models could capture how bias propagates or attenuates across repeated selection cycles. Finally, while our model isolates tractable facet of systemic inequality, real-world disparities-especiial especially in Al-mediated evaluations- demand broader integration with social and historical context. As algorithmic tools shape hiring, admissions, and promotion, framework helps explain how group-level differences in perceived value can interact with selection to amplify or mitigate bias. More broadly, we view this work step toward unifying rational-choice and structural perspectives inequality through formal, data-driven modeling. We hope this work informs the design of more equitable, data-driven decision systems.
10

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Acknowledgments
LH acknowledges support from the State Key Laboratory Novel Software Technology, the New Cornerstone Science Foundation, and NSFC Grant No. 625707396. LEC was supported in part by NSF Award IIS-2045951. NKV was supported in part by NSF Grant CCF-21 12665 and by grants from Tata Sons Private Limited, Tata Consultancy Services Limited, and Titan.
References
[1] Erwin Amann and Wolfgang Leininger. Asymmetric all-pay auctions with incomplete informa- tion: The two-player case. Games and Economic Behavior, 14:1-18, 1996.
[2] Jiafu An, Difang Huang, Chen Lin, and Mingzhu Tai. Measuring gender and racial biases in large language models: Intersectional evidence from automated resume evaluation. PNAS Nexus, 4(3):pga8089, 03 2025.
[3] Kenneth Arrow. The theory of discrimination. Technical report, Princeton University, Depart- ment of Economics, Industrial Relations Section., 1971.
[4] Kenneth J. Arrow. What has economics to say about racial discrimination? Journal of Economic Perspectives, 12(2):91-100, June 1998.
[5] Jackie Back and Ali Makhdoumi. The feedback loop of statistical discrimination. SSRN Electronic Journal, 2023.
[6] Solon Barocas, Moritz Hardt, and Arvind Narayanan. Fairness and Machine Learning. fairml- book.org, 2019. httpp:/www.. faw fairmlbook org
[7] Yasar Barut and Dan Kovenock. The symmetric multiple prize all-pay auction with complete information. European Journal of Political Economy, 14(4):627-644, 1998.
[8] Surender Baswana, Partha P Chakrabarti, V Kamakoti, Yash Kanoria, Ashok Kumar, Utkarsh Patange, and Sharat Chandran. Joint seat allocation: An algorithmic perspective, 2015.
[9] Yahav Bechavod, Katrina Ligett, Aaron Roth, Bo Waggoner, and Zhiwei Steven Wu. Equal Opportunity in Online Classification with Partial Feedback. Curran Associates Inc., Red Hook, NY, USA, 2019.
[10] Gary S. Becker. The Economics of Discrimination. Economic Research Studies. University of Chicago Press, 2010.
[11] Francine D. Blau, Marianne A. Ferber, and Anne E. Winkler. The Economics of Women, Men, and Work. Pearson, Boston, MA, 7 edition, 2014.
[12] Francine D. Blau and Lawrence M. Kahn. The gender wage gap: Extent, trends, and explana- tions. Journal of Economic Literature, 55(3):789 865, 2016.
[13] Iris Bohnet. What Works: Gender Equality by Design. Harvard University Press, 2016.
[14] Yang Cai, Constantinos Daskalakis, and Christos H. Papadimitriou. Optimum statistical estimation with strategic data sources. In COLT, volume 40 of JMLR Workshop and Conference Proceedings, pages 280-296. JMLR.org, 2015.
[15) Emilio J. Castilla and Stephen Benard. The paradox of meritocracy in organizations. Adminis- trative Science Quarterly, December 2010.
[16] I. Elisa Celis, Chris Hays, Anay Mehrotra, and Nisheeth K. Vishnoi. The Effect of the Rooney Rule Implicit Bias in the Long Term. In Proceedings of the 2021 ACM Conference on Fairness, Accountability, and Transparency, FAccT '21, page 678-689, New York, NY, USA, 2021. Association for Computing Machinery.
[17] I. Elisa Celis, Lingxiao Huang, and Nisheeth K. Vishnoi. Multiwinner voting with fairness constraints. In IJCAI, pages 144-151. ijcai.org, 2018.

---

[18] L. Elisa Celis, Amit Kumar, Anay Mehrotra, and Nisheeth K. Vishnoi. Bias in evaluation processes: an optimization- model. In Proceedings of the 37th International Conference on Neural Information Processing Systems, NIPS '23, Red Hook, NY, USA, 2023. Curran Associates Inc.
[19] L. Elisa Celis, Anay Mehrotra, and Nisheeth K. Vishnoi. Interventions for ranking in the presence of implicit bias. In FAT*, pages 369-380. ACM, 2020.
[20] Shuchi Chawla and Jason Hartline. Auctions with unique equilibria. In EC, pages 181-196. ACM, 2013.
[21] Stephen Coate and Glenn C Loury. Will affirmative-action policies eliminate negative stereo- types? The American Economic Review, pages 1220-1240, 1993.
[22] Joanne McGrath Cohoon, James P. Cohoon, Seth Reichelson, and Selwyn Lawrence. Effective recruiting for diversity. IEEE, FIE 2013, pages 1124-1124, 2013.
[23] Ashley C Craig, Roland Fryer, et al. Complementary bias: A model two-sided statistical discrimination. Technical report, National Bureau of Economic Research, 2017.
[24] Jeffrey Dastin. Amazon scraps secret AI recruiting tool that showed bias against women, October 2018. hhtps://reut: .st rs/2N1dzRJ
[25] Shanglyu Deng, Qiang Fu, and Zenan Wu. Optimally biased tullock contests. Journal of Mathematical Economics, 92: 10:21,,11,200 2021.
[26] Drew DeSilver. A majority of u.s. colleges admit most students who apply. Pew Research Center, April 2019. http:://ww.perreeercc..oo www.eaarcccoo/ffa.ttteteo hataaat/tttaan//////////////// S
http:///0/////majjo/////// -o- -osttt-t-tttttttteett--------- who- apply/
[27] Edith Elkind, Abheek Ghosh, and Paul W. Goldberg. Contests to incentivize target group, In IJCAI, pages 279-285. ijcai.org, 2022.
[28] Vitalii Emelianov, Nicolas Gast, Krishna P. Gummadi, and Patrick Loiseau. On fair selection in the presence of implicit variance. In EC, pages 649-675. ACM, 2020.
[29] Vitalii Emelianov, Nicolas Gast, and Patrick Loiseau. Fairness in selection problems with strategic candidates. In EC, pages 375-403, ACM, 2022.
[30] Gil Epstein, Yosef Mealem, and Shmuel Nitzan. Lotteries vs. all-pay auctions in fair and biased contests. Economics & Politics, 25(1):48--666 60, 2013.
[31] Warren Farrell. Why Men Earn More: The Startling Truth Behind the Pay Gap and What Women Can Do About It. AMACOM, New York, NY, 2005.
[32] Jörg Franke, Christian Kanzow, Wolfgang Leininger, and Alexandra Schwartz. Effort max- imization in asymmetric contest games with heterogeneous contestants. Economic Theory, 52:589-6300 630, 2013.
[33] Jörg Franke, Wolfgang Leininger, and Cédric Wasser. Revenue maximizing head starts in contests. Ruhr Economic Puper, (524), 2014.
[34] Jörg Franke, Wolfgang Leininger, and Cédric Wasser. Optimal favoritism in all-pay auctions and lottery contests. European Economic Review, 104:22-37, 2018.
[35] Jörg Franke, Christian Kanzow, Wolfgang Leininger, and Alexandra Schwartz. Lottery versus all-pay auction contests: A revenue dominance theorem. Games and Economic Behavior, 83:116-126, 2014.
[36] Qiang Fu and Zenan Wu. On the optimal design of all-pay auctions. Technical report, Working Paper, 2021.
[37] Malcolm Gladwell. Outliers: The Story of Success. Little, Brown and Company, New York, 2008.
12

---

[38] Claudia Goldin. A grand gender convergence: Its last chapter. American Economic Review, 104(4):1091-1119, April 2014.
[39] Anthony G Greenwald and Linda Hamilton Krieger. Implicit bias: Scientific foundations. California Law Review, 94(():945-90) 967, 2006.
[40] Ariane Hegewisch and Asha DuMonthier: The gender wage gap: 2015: Annual earnings differences by gender, race, and ethnicity. Report, Institute for Women's Policy Research, 2016. Available at Institute for Women's Policy Research website.
[41] Richard J. Hermstein and Charles Murray. The Bell Curve: Intelligence and Class Structure in American Life. Free Press, New York, 1994.
[42] Sylvia Ann Hewlett, Laura Marshall, and Lauren Sherbin. How diversity can drive innovation. Technical report, Center for Talent Innovation, New York, NY, 2013.
[43] Lily Hu, Nicole Immorlica, and Jennifer Wortman Vaughan. The disparate effects of strategic manipulation. FAT, pages 259-268. ACM, 2019.
[44] Jon Kleinberg, Himabindu Lakkaraju, Jure Leskovec, Jens Ludwig, and Sendhil Mullainathan. Human decisions and machine predictions. The quarterly journal of economics, 133(1):237-293, 2018.
[45] Jon M. Kleinberg and Manish Raghavan. Selection problems in the presence of implicit bias. In ITCS, volume 94 of L/Plcs, pages 33:1-33:17. Schloss Dagstuhl Leibniz-Zentrum für Informatik, 2018.
[46] Junpei Komiyama and Shunya Noda. On statistical discrimination as failure of social learning: multi-armes bandit approach. Management Science, 69())3333133350, 2023. Accessed: 2025-05-06.
[47] Ratul Lahkar and Saptarshi Mukherjee. Optimal large population Tullock contests. Oxford Open Economics, 2:odad003, 05 2023.
[48]
Khen Lampert. Meritocratic Education and Social Worthlessness. Palgrave Macmillan, 2012.
[49]
Tamar Lewin. A New SAT Aims to Realign With Schoolwork, March 2014, httpps:/wwww..... times wwt.comm////// 2ttp//2204//3///660 http://////////aaaaaaa/////aa/aa -nw.----o-o-ome-----a-aa--- -aaa--aoounndddb---ooloooo by-cll ge board. html
sat
[50] Bo Li, Zenan Wu, and Zeyu Xing. Optimally biased contests with draws. Economics Letters, 226: 11 1076, 2023.
[51] Richard J. Lipton, Evangelos Markakis, and Aranyak Mehta. Playing large games using simple strategies. In EC, pages 36-41. ACM, 2003.
[52] Xuyuan Liu and Jingfeng Lu. Optimal prize-rationing strategy in all-pay contests with incom- plete information. International Journal of Industrial Organization, 50:57-90, 2017.
[53] Karen S Lyness and Madeline E Heilman. When fit is fundamental: performance evaluations and promotions of leper-level female and male managers. Journal of Applied Psychology, 91(4):777,
[54] Daniel Markovits. The Meritocracy Trap. Penguin Press, New York, 2019.
[55] McKinsey Global Institute. Unlocking the full potential of women in the u.s. economy. Technical report, McKinsey & Company, New York, NY, 2011.
[56] Stephen McNamee and Robert K. Miller Jr. The Meritocracy Myth. Rowman & Littlefield, 2004.
[57] Smitha Milli, John Miller, Anea D. Dragan, and Moritz Hardt. The social cost of strategic classification. In FAT, pages 230-239. ACM, 2019.
13

---

[58] Corinne A. Moss-Racusioe John F. Dovidio, Victoria L. Brescoll, Mark J. Graham, and Jo Han- delsman. Science faculty's 's subtle gender biases favor male students, Proceedings of the National Academy of Sciences, 109(41): 1464))):64444679999 2012.
[59] Eunmi Mun and Naomi Kodama. Meritocracy at work?: Merit-based reward systems and gender wage inequality. Social Forces, 100(4):1561-1591 June 2022. Publisher Copyright: @ 2021 The (uthor(s) 2021. Published by Oxford University Press on behalf of the University of North Carolina at Chapel Hill. All rights reserved.
[60] Audrey Murrell. The privilege, bias, and diversity challenges in college admissions, 5 2019.
[61]
National Center for Education Statisties. The condition of education 2020. Technical report. U.S. Department of Education, 2020.
[62] Wojciech Olszewski and Ron Siegel. Large contests. Econometrica, 84(2):835 854, 2016.
[63] June O' O'Neill. The gender gap in wages, circa 2000. American Economic Review, 93(2):309-314, May 2003.
[64] Scott Page. The Diversity Bonus: How Great Teams Pay Off in the Knowledge Economy. Princeton University Press, Princeton, NJ, 2017.
[65] Pew Research Center. The enduring grip of the gender pay gap, 3 2023.
[66] Edmund S Phelps. The statistical theory of racism and sexism. The american economic review, 62(4):659-661, 1972.
[67] Robert S Pindyck and Daniel L Rubinfeld. Microeconomics, Prentice Hall, 1997.
[68] Press Information Bureau, Government of India. JEE (Advanced) 2024: wesd statistics of candidates registered and qualified. https://pib. http://pbb.g...... gov. www..i//reesseeeeesssee.eeeeeeee aspx? RID=2023653 2024. Accessed: 2025-05-07.
[69] Elaine D Pulakos. Selection assessment methods. Society for Human Resource Management (SHRM) Foundation, 2005. United States America.
[70] Manish Raghavan, Solon Barocas, Jon M. Kleinberg, and Karen Levy. Mitigating bias in algorithmic hiring: evaluating claims and practices. In Mireille Hildebrandt, Carlos Castillo, Elisa Celis, Salvatore Ruggieri, Linnet Taylor, and Gabriela Zanfir-Fortuna, editors, FAT* '20: Conference Fairness, Accountability, and Transparency, Barcelona, Spain, January 27-30, 2020, pages 469-481, Barcelona, Spain, 2020. ACM.
[71] Robert B. Reich. The Future of Success. Knopf, New York, 2000.
[72] Lauren A. Rivera. Pedigree: How Elite Students Get Elite Jobs. Princeton University Press, Princeton, NJ, 2015.
[73] Michael J. Sandel. The Tyranny of Merit: What's Become of the Common Good? Farrar, Straus and Giroux, 2020.
[74] Thomas C Schelling. The Strategy of Conflict. Harvard University Press, Cambridge, MA, 1981.
[75] Jessica Schieder and Elise Gould. Women's work and the gender pay gap: How discrimination. societal norms, and other forces affect women's 's occupational choices and their pay, 2016.
[76] Christian Seel and Cédric Wasser. On optimal head starts in all-pay auctions. Economics Letters, 124(2):211-114, 2014.
[77] The Journal of Blacks in Higher Education. Black students remain underrepresented at the nation's 's most selective colleges and universities, 2023.
[78] Robert D. Tollison. Rent-seeking: A survey. Kyklos, 35(4):575-602, 1982.
[79] Gordon Tullock. The welfare costs of tariff's, monopolies, and theft. Western Economic Journal, 5(3):224-233 1967.
14

---

[80] U.S. Bureau of Labor Statistics. Median earnings for women in 2021 were 83.1 percent of the median for men, 2022.
[81] Joseph Walker. Meet the New Boss: Big Data, September 2012. https://wws. wsj com/ ar ticles/SBi
[82] Cédric Wasser and Mengxi Zhang. Differential treatment and the winner's effort in contests with incomplete information. Games and Economic Behavior, 138:90-111, 2023.
[83] Lyss Welding. Women in higher education: 5 key facts & statistics, 2023.
[84] Christine Wennerås and Agnes Wold. Nepotism and Sexism in Peer-Reviews Nature, 387(6631):3411-44 May 1997.
[85] Rick Wicklin. Implementing the truncated normal distribution SAS. SAS Blogs, 2013. Available online: https://blogs. sas. wtt.comcc/c///eeee http:///tteenttmmmm////////////0 treuccccte normal-in- sas. sas.html
[86] Jazzlin Yee. Opinion: Females are underrepresented in computer science, 2022.
[87] Feng Zhu. On optimal favoritism in all-pay contests. Journal of Mathematical Economics, 95:102472, 2021.
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Contents
Introduction
Model and metries
Theoretical results: Nash equilibrium and metries for large n
Analysis: metric behavior and intervention design
5
Conclusion, limitations, and future work
10
Detailed related work
18
A.I Comparison of the two-group contest with relevant models
19
B Illustrative examples for the two-group case
20
Analysis of finite NE policies in the uniform distribution case
21
Empirical analysis
21
Theoretical analysis
22
Other bias models and analysis of metrics for their Nash equilibrium
23
D. Other bias models
23
D.2 Analysis of metrics for Nash equilibrium in the truncated normal distribution case
27
Proof of Theorem 3.1; two-group contest
28
Technical overview
28
E.1.1
NE policy for the undiffcrentiated contest for finite and obstacle in extend-
ing
29
E.1.2
A conjectured NE policy in two-group contests for large n
29
1.3
Proving convergence to the conjectured NE policy
31
E.2
A more comprehensive version of Theorem 3.1 convergence form
32
Proof of Theorem
E.3.1
Proof of Lemma
solution uniqueness
33
E.3.2
Bounding winning probability
................................
34
E.3.3
Proof that ((n) is approximate NE
36
................................
E.3.4
Completing the proof of Theorem
................................
36
Comparing with a distributional two-group contest
37
E.5
An alternative proof using infinite contest
38
Omitted details for uniform distribution analysis from Sections 3 and
40
Additional details to Section
44
G.1
Case study estimating perceived bias from JEE Advanced 2024
44
16

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G.2
Robustness analysis for findings in Section
45
G.3
An illustrative example: interpreting and applying the model
46
Details for alternative interventions!
46
17

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A Detailed related work
Meritocratic selection process, pay gap, and statistical discrimination, feedback loop. In the social sciences, there is large body of work that studies meritocratic selection processes and their lim-
itations; see [56] 54 73 and the references therein. [3] [59] discuss the pay gap in meritocratic systems, shedding light on how merit-based reward systems and gender wage gaps intersect. [38,, in an extensive line of work, discusses the gender pay gap and addresses the economic and social factors contributing to wage disparities between men and women. Another line of research focuses on studying statistical discrimination feedback loops, which model how firms update their beliefs about group quality over time, reinforcing disparities [3 66 21 [3 S 46). For instance, [3] emphasizes how the cost of individualized assessment incentivizes reliance on priors, which can become self-fun folfffuli and reinforce structural inequality. [ models a profit-maxmmizing employer who faces noisy signals of productivity and rationally uses group-level statistics, leading to persistent wage gaps even with equal underlying abilities. [21] show that pessimistic beliefs about group's s productivity can result in tougher standards, reduced investment incentives, and discriminatory equilibria.
[ extend this to two-sided settings where firms and workers both act on noisy beliefs, reinforcing low- inwiineetteet, low-opportunity equilibria. A key distinction, as we understand it, is that classical models of statistical discrimination typically generate disparities through imperfect and group-dependent beliefs about identical underlying abilities. In contrast, our framework allows perfect, unbiased information at the institutional level and identical selection criteria for all candidates. We focus instead valuation asymmetries that is, differences in the perceived benefit of success across groups grdups-and and show that these differences alone can lead to disparities in effort and representation, even under meritocratic selection.
All-pay auctions and Tullock contests. In game theory, there is a significant body of literature that study the setting in which every agent knows their private valuations and the distribution of other agents. Specifically, (76) study a "biased" 2-agent contest in which the designer is allowed to give a "headstart" to the effort of one agent. This headstart can be interpreted as differing merits of the agents, which corresponds to the initial abilities in our model. They characterize the optimal design for maximizing the expected highest effort or total effort of agents. In this case, the bias is introduced by the designer, rather than inherent in the system. [82] study the undiffcrentiated case for a single winner, while the contest designer is allowed to select the contest success function (CSF) based on agents' efforts. Their main focus is on studying the optimal design of the CSF that maximizes the total expected effort. The main difference from our model is that they consider bias in the efforts instead of in the valuations.
investigates all-pay auctions. For instance, [76 52 82]
The all-pay auction with complete information has also been well-studied. Unlike the setting this paper, these works assume that the valuations of all agents are known. 7 initiated the study of an n-agent k-winner all-pay auction and provided a complete characterization of the NE distribution. A line of research investigates the optimal design for maximizing the total expected effort/revenue, including imposing a multiplicative bias on the effort of agents [32 36 or introducing an additional
headstart 35 33
34
36
87].
model the probability of winning based on relative effort without direct costs for participation, whereas an all-pay auction requires all agents to pay their bid amounts regardless of winning, with only the highest bidder(s) securing the prize. [62] study the dynamics of large contests, where significant number of agents compete. Such contests pose unique analytical challenges and offer insights into the behavior of agents in mass competition scenarios.
Tullock contests
30
The works of [32 [ 34 also investigate how the design of contests can be optimized to maximize revenue, considering factors like bias in efforts, headstarts, and the structure of the CSF, Across these studies, common theme is the characterization of Nash equilibrium strategies within the context of different contest models, and identifying designs that encourage maximal effort or revenue.
Strategic classification and ranking. Another related direction is strategic learning, which mainly includes strategic classification [14 43 [ and strategic ranking (29]. In strategic classification, agents can exert effort to alter their features to achieve higher values according to the published classifier. The designer's aim is to select a classifier that is robust to the manipulation of inputs by strategic agents. However, in this setting, agents' efforts are influenced solely by the published classifier, with no competition among them. In strategic ranking problems, agents' payoffs depend on their post-ranking, which is determined by a combination of their prior rankings and efforts. While
18

---

there is competition in this problem, all agents have the same valuation, which is different from our model.
Models of bias in valuations. Several works have modeled group-level biases based on empirical
Additive and multiplicative skews in the valuations have also been modeled [45, 10.. 45 consider valuations v >> 0 of the advantaged group distributed according
observations 4 10
45
28
18
to the uniform or Pareto density and, for the disadvantaged group, they model the output as v/3 for some fixed 3 2 1. We consider a class of bias models inspired by this model, the in our case corresponds exactly to 1/3. The implicit variance model of (28] models differences in the amount of noise in the valuations for individuals in different groups. Here, the output estimate is drawn from Gaussian density whose mean is the valuation (which can take any real value) and whose variance depends on the group of the individual being evaluated: The variance is higher for individuals in the disadvantaged group compared to individuals in the advantaged group. [18] approach to model how group-wist valuation distributions can be
proposes an optimization-base obtained by tuning parameters such "information constraints" or "risk aversion".
A.1
Comparison of the two-group contest with relevant models
To the best of our knowledge, our model is novel and has not been studied in the literature. Below, we compare our model with the most relevant models. Firstly, [ examines specific case of our model with n = 2, k = 1, and a = 0.5, demonstrating the existence of a unique NE under certain conditions. While their analysis is limited to two-player scenario, our model generalizes this by considering any number of players and allowing for multiple winners. (29) propose another two-group contest model.
However, key distinction in model is the consideration of asymmetric valuation distributions across groups, whereas (29) introduces bias through the cost of effort, assuming symmetric valuations for all agents. This asymmetry in valuation distributions in our model adds complexity the analysis.
Another related work is (27], which explores an all-pay auction with two groups. In their model, agents' abilities are symmetrically distributed, and those in the advantaged group may receive additional rewards with equivalent bids. Despite the symmetric strategic environment in [27], our model features asymmetric valuation distributions between groups, resulting in asymmetric strategic environment. This asymmetry introduces further computational challenges for deriving the NE; details can be found below.
detailed comparison with 127. We provide a detailed comparison between our two-group model and that in (27). The primary distinction is that their model results in a symmetric strategic environment, while ours creates an asymmetric one. Below, we provide further details on this difference.
In the model of (27), each agent belongs to the target group with probability y or to the non-target group with probability 1 - independently of the other agents. The ability of an agent is then drawn i.i.d. from distribution F if they belong to the target group, and from G they belong to the non-target group. As a result, the ability of each agent is drawn identically and independently from the joint distribution ,FF + (1 G. Let H be the CDF of this joint distribution. The probability that a given ability v is among the top & abilitics is then given by Σi (" H())(1 H (()) (v))"=1-
which is the same for each agent, thereby resulting in symmetric strategic environment.
In our model, consider a simplified case where Pa a point mass at Then, F1 and F2 correspond to the cumulative distribution functions (CDFs) of p1 and p2, which represent the valuation densities of G and G2, respectively. The probability that, for an agent in G₁, a given valuation v is among the top l valuations is given by:
Σ Σ ("
( n
(" FROO
1000 Fa(v))an-5
In contrast, the probability for agent in G2 is given by
1 - J - -
Σ ΣΣ n1
-
an
These two expressions differ whenever P1 # P2, leading to asymmetry the strategic environment. This asymmetry significantly complicates the computation of the order statisties for the (k 1)-th
19

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effort compared to the symmetric ones. E.g., for strategies and $2, let Fs, (v) denote the CDF of efforts sy(e) when e ~ Py- The cumulative distribution of the (k - 1) effort from an agent in G1 is then given by:
Pr/c*= v|
(
Σ
Σ
n-1 n-1-1
Σ
Fsx (o) (v) Ps2 (v))an-) (o)(1)(1 (
n2 1
i
(=(1-c)n-
In contrast, the cumulative distribution of the (k - 1)-th effort ex from an agent in G2 is given by:
Pr|e u
Σ n2 - 1 n 1 { Fei (1)
Σ
- Fer (1) Fsz (v))an
i
==
In the symmetric ones (81 = = s), the computation simplifies to:
( n- ("T ) -
Prfe" S =
(=11c)n
Thus, the calculus and approximations for the two-group contest is significantly more difficult, making it harder to arrive the equilibrium policies than in the contest with symmetric strategic environment.
B Illustrative examples for the two-group case
In this section, we present a two-agent example with a biased valuation distribution illustrate both the difficulty of computing the Nash equilibrium (NE) policy and the sigmificant impact of valuation bias on the contest outcome.
e = 0.5. Let the density P1 of agent be the uniform distribution on Ω [0, 1] and of agent 2 be the p-biased version of supported on ft [0. pl- Let the density Pa be a point mass at Let Ag S2g + R2 be the NE policy that maps valuation by effort A₂ (vg)). We assume A is monotonically increasing on the domain Qt.
In this example, if agent puts in effort wins if the effort of agent is smaller than e. Thus, its Similarly, if agent (e)e 2 puts in effort e, its winning probability P2 = (e) and payoff is π2 e; A1) = AT "e)( e.
winning probability P: A5 (e)4 and its payoff is (1 e; A2) (c)
P
Then, by the stability condition I.. we have (m (=) 0, implying that
de
A2(A5
and
=
Solving this gives the following explicit forms:
P tt1 and Vo E 92, A2(u) = p- 1+1/p Ve E Ωt Ai(v) = 1+1
(7)
Specifically, when (the unbiased case), we have A = A = A, which simplifies the stability y²
condition to A'(v) v, yielding the NE policy A(v) 2.
Also note that for E U2
A1 1111 < sp 1/10 + 1
Az(v)
which implies that (v) S A2(v). Thus, agent 2 is more inclined put in greater effort than agent 1 for identical valuations.
Here, assume A5 p Az(p) <c.
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Imagine an institute that is unaware of the bias in valuations across two agents and thus applies the unbiased NE policy A(u) v² to predict the contest outcome. For instance, would predict the average revenue
RV(A, m)
/ Pr Pe
u 2
22 > dt 0.25.
where the merit function is m(t) = However, under p-biased valuation distribution, the true average revenue is RVV Tīp-īj. which decreases monotonically with p. This implies that the institute could overestimate its expected benefit RV by fraction of
0.25 RV, 1-P
RV.
2p
which amounts to approximately 13% when p 0.8. This example underscores the importance of studying asymmetric valuations and highlights the relevance of our proposed metrics for analyzing their impacts.
We also observe that even for this simple two-agent example, the stability condition is considerably more complex than in the undifferentiated case. In more general settings-such as those involving multiple spots, non-uniform valuation densities, or non-trivial ability densities-the explicit forms of NE policies for two-group contest become even more complicated, making direct computation and explicit analysis impractical.
C Analysis of finite NE policies in the uniform distribution case
In this section, introduce a dynamic algorithm (Algorithm[[) to approximate the finite NE policies, and perform a statistical comparison between the finite and infinite cases. Additionally, provide a theoretical analysis of the closeness between the NE policies and associated metrics in the finite and infinite cases.
the uniform distribution example from Section 3 running example,
C.1 Empirical analysis
Dynamics for computing finite NE policies. Recall that we consider P1 = p Unif(O, 1]. P2 P2 Unif(0, pi for p [0, 1]. and Pa 0. Algorithm presents dynamic procedure to approximately compute the finite-population NE policies.
We initialize each group's policy 8t with smoothness variant of the infinite NE policy (Lines 3-4), then iteratively update these policies over N steps and return the final output approximation of the finite NE (Lines 5-38). At each iteration t:
1. We first update the effort set E, based on the policies ( ((-1) from the previous iteration (Line 6). Since the action space is continuous, we restrict agents choose efforts only from this finite set E,,
2. Next, we update the policy for group Gj using the policy 82 (t-1) from the previous iteration (Lines 7-21). The computation is performed over finite set ((1) of discrete valuation levels (Line 1). For each valuation v, we determine the best-response effort that maximizes the agent's 's expected payoff by computing winning probabilities through a convolution of binomials (Lines 9-19). Specifically, we set P1 = v in Lines 9 and 12, consistent with the monotonicity constraint enforced in Line 11. Finally, Line 21 updates the policy using carefully chosen step size a to ensure convergence.
3.
We then update the policy for group G2 based on the policy af (t-1) from the previous iteration (Lines 7-21). This process mirrors that of G,, with the main difference lying in the computation of winning probabilities for each effort in E, due to the asymmetric valuation distributions.
The resulting policies 8, o are defined on discrete valuation grids. To obtain continuous policies, we interpolate them by connecting adjacent valuation points with straight lines, resulting in lineaar approximations.
21

---

Choice of hy hyperparameters. In our simulations, we set the valuation resolution m, = 1011, effort
resolution me = 101, total number of iterations T 500, and step sizes a 022 (() always set n = 122 2. which means = 0.5.
107 We
Metrics.
For each iteration I, we compute the following metric to evaluate the updated policies "t"
A (1,,)_ Eveveo | (1) (v) - (t- ((-11( (v) Eeevoo (v) (2- ( [v)
m,
m
which quantifies the average policy update for group Ge at iteration t. Intuitively, a decreasing A (f,r) indicates convergence of the policy sequence Se However, since we work with discretized valuation and effort sets, we do not expect A (f.t) vanish entirely.
Results. Figures 5 6 and present the evolution of equilibrium policies for population sizes
with fixed parameters p 0.8 and 0.2. Although all runs begin with smoothed version of the infnite--opullatoon NE, the dynamics vary significantly with population size. For small (e.g., nt = 20), we observe noticeable fluctuations in early iterations, particularly in group G2, whose valuation distribution is more concentrated. By / = 500, both policies stabilize, though they retain visible irregularities due stochasticity in rank-based feedback. Even though all runs begin with a smooth initialization based on the anfnite-popplattion NE, the dynamics unfold differently depending small n (e.g., n 20). we observe noticeable fluctuations in the early iterations, especially in group G2, whose valuation distribution is more concentrated. At t = 500, the policies stabilize but retain some irregularity, reflecting noise the agent-level ranking and feedback structure.
n E 200, 600, T200) across four time snapshots E {50, 150,
population size. For
As n increases, both groups' policies become smoother and stabilize more quickly. By n 600, the effort policies align closely with the infinite NE, and further updates beyond 300 are negligible. These trends are confirmed by the convergence plots in Figure which show sharp reduction in the fn-norm policy update (f.t) with increasing n. Group G1 consistently converges faster than G2, a pattern attributable to its broader valuation support and greater flexibility in effort choice. Overall, the results illustrate that the infinite- pooptt--ooolltioon equilibrium is a good predictor even for moderately sized finite systems, while also quantifying the transient effects and instability that emerge in low-n regimes.
Interestingly, also observe from these plots that when is small (n 20, 200), s2(v) s (1),, while for larger values of n (n = 600, 1200), 82 < 81 (v). In the subsequent subsection, we will provide theoretical analysis to explain the underlying reasons for this behavior.
C.2 Theoretical analysis
We begin by presenting theoretical evidence for the alignment between finite and infinite NEs, relationship that is observed empirically. In the proof of Theorem 3.1 (see Section Ep, we show that for any finite n, En NE policy s( stated in Theorem 3.1 "O( /lo n/n)-coo to the policy for infinite n and is "O(v ma/))-lloss to an NE policy. Specifically, when P1,P2 are uniform distributions and Pa is a point mass at 0, for any constant a, the closeness between s(a) and can be directly translated into the policy form: g [n) = 0 for v < t - O1 /logn/n) and ((()) = t for OO ((Voog n/n). For general P2, Pa. we note that the O( / /log m/nn)closeness depends on the concept of densities, which is more complex. Corollary 3.2 from (0 implies that the NE policy must be symmetric within each group. Let 81 represent the policy for G1 and 82 for G2. The above analysis indicates that the closeness between 81, 82, and s (from Equation (3) is expected to be bounded by O( (Tog n/n)).
In the following, we analyze the empirical observations regarding the scaling of $1 and s2. Intuitively, si (v) increases from approximately 0 to approximately t as v increases from Vlog n/n / to t t+ / /log n/n. If an agent in G2 exerts an effort of t(1 /logn/n) the agent's winning probability could exceed 95%. This observation motivates the choice of setting sz(u) = t(1 Vlog n/n). Vlog h//nnn/95// As result, if t(1
rather than 0, to generate positive profits when t(1
22

---

(logg/n)/990 /95% S t, i... log 0.05, then s2(t) 2 t(1 Vlogn/n 2 s1(v). Therefore, log 0.05, it holds that s2(x) (v) for E Q2. Conversely, when is
when n is small and
large, Jogn becomes small, and 81 (v) s2(v), as agents in G2 consistently receive lower payoffs n than those in G1 when 51 = 52. This behavior explains the observed scaling between 81 and 82, as discussed in Section C.I
Finally, we discuss the impact on metries when the number of agents is finite. Recall that in the finite case, we can assume NE policies and for group G1 and G2, respectively. As discussed above, when n is not too small, we have 1) > 82 and 2) 81, 82, and are O( log n/n)- close. Since 81 > 82 and they converge to the same policy as n grows, the representation ratio E
R1 (A)
[ decreases with while R2(A) = E [G1l
[SnGa]] [G2
increases with n, where S is
R2(A) increases as the (random) winning set. Consequently, the representation ratio rR RA(
grows. Since the gap between and 82 is bounded by O O(ylogn//m this results in an increase of O(y n/n) in (A) and a similar decrease of O OoVlog n//n) in R, (A) compared the infinite case. Thus, the increase (R( (A) should be bounded by O( log n n/n). similar quantitative analysis applies to the social welfare ratio r.s (A) and average revenue (A,
Policies at Iteration 50
Policies at iteeration 150
Effort e
Effort
0.5
D4
5-6
0.8
0.0
0.2
0.6
1.0
Valuation
Valuation
Herrtion
Iteratioo
6.4
S
0.8
51
$2
S2
Effort e
Effort
0.1
ca
Valuation
Valuation
Figure 5: Evolution of group effort policies over time for = 20, 0.8, and c 0.2.
D
Other bias models and analysis of metrics for their Nash equilibrium
D.1 Other bias models
natural extension of Pı Unif 1] in Section 2 is when P1 is the density of the uniform < oo) and pa the density of the uniform distribution on a2 [pa, pb]. Then p2(v) plo-aj and again, En-pp [v] - p 2 p En--p [v]. More generally, one might consider a density P1 that is supported on a domain Q = (a, oo]. along with a p-biased density defined as (v) PP(("" for p (0, 1] and [pa, oo].
distribution on an interval (a, b] (0
24d
Besides the uniform distribution case, we consider valuations coming from truncated normal distribution supported on [0. 1]. Formally, let p1 be the density truncated normal distribution N N(, o²) on the interval Ω [0, 11, where y lies within (0, 1) and > O. Let p2 be the density of
23

---

Algorithm 1 Dynamics for Computing Finite NE Policies for the Uniform Distribution Case
Input: Group sizes 14, 12 2 1, selection rate E (0, 1), bias E [o, 11, valuation steps m. 2 1, interger T 2 1, and two sequences of step sizes {a? (E((T),E((12) Output: Policies 81, 82 for G1 and 22, respectively
effort steps m.
1: Initialize grids V(2) + linspace(O, 1, my), ((2) + linspace(O, p, my), and
linspacc(O,, me). and I: f- [c(n1 + n2)
2: Compute a + tit
n2
1-
3: Compute threshold
0 (1 c)
otherwise,
pax +
4: Initialize $1
1
82 (v)
(0)
Tta-Buc-oT
Smoothness of the infinite NE in Proposition 4.1
5: for to 7 do
6: Eg EU 8t
(v) DE v(0, (E
Updating effort set
7:
last. +- 0
8:
for = 1 to my do
0 Ut v((1, best, +- lastes f- 1 - V and p2 +- max max weveense! (or)) D-ve
9:
(beste) f- Σx=1 Ei= (15) (ppp(pp(((11)))))))))) - -p:) ( - p1 p1)m2--at) for all E E, with last, do P1 t u and P2 max MEVIW):!! 11 (v)) p-v
10:
(e) - E Eb=o
-1 (m ((m3)))2 p1 21)
14:
pay t (e)
15:
if pay > (best)) v - best, then
beste te end if end for
17:
18:
19:
I (v) + beste, last. + beste
end for
$1 + s/t (t-1) + a ( -s11-11 11-
21:
Update policy for G1 at iteration
22:
last t- 0 for i to m. do
v - v,(,) beste +- laste, P1 f- max /" - V and P2 x- p-v
24:
(beste) + Ii-I Ei_o ((%) pf m-)) (m=-)) 0 01 - p1 - laa
25:
for all Es with last. do
P1 t max (" 1 and P2 & p-v'
(e) Ed- Eiso (L p1) m)) b 1 P1 - -
28:
29:
pay +- (e) -
30:
pay > ((2) v beste then beste
31:
32:
end if end for
33:
34:
T22 (v) +- beste, last, + besta
35:
end for
(t-1) + d2 (T2 82 (1-1)]
36:
82
Update policy for G2 at iteration
37:
end for
(T)
(T)
return Policies 81 < 81
82 + 82
24

---

Tolicies at heration
Policies at iteratiin 150
06
$1
S1
Effort
Effort
0.4
51-
6.-
00-
64 Valuation
Valuation
Iteratin
118
S1
20-
51
65
$2
52
Effort
C4
Effort
0.4
03-
08
Valuation
Valuation
Figure Evolution of group effort policies over
200, p = 0.8, c 0.2.
Policies at Iteration
Policies at Iteraaiin 150
00
S
51
08
S2
S2
Effort
Effort
C0
00-
54
Valuation
Valuation
86
51
51
65
S2
012
S2
Effort
Effort
0.1
21-
C8
Valuation V
Valuation
Figure 7: Evolution of group effort policies over
600. p = 0.8,
25

---

Petrries at neration
Poliries iteration 150
C5-
$2
0.5
52
Effort
Effort
212
0.0
02
64
0-4
0.0
0.0
0.
06
0.
1.
Valuation
Valuation
Policies Ieeratio 300
Policies at Itteaatio 500
67
0.-
S1
S
S2
S2
Effort
64
Effort
0.5
60
00
Valuation
Valuation
Figure 8: Evolution of group effort policies over time for n = 1200, p == 0.8, and = 0.2.
Policy convergence when
20
Policy convergence when 200
S
51
18-7
S2
S
10
20-2
400
Iteration
Iteration
Policy convergence when n 600
Policy convergence when n 1200
51
S2
10
20-4
51
52
120
200
420
200
320
300
Iteration
Iteration
Figure 9: Convergence of group-wise policy updates A (f.t) for different population sizes with fixed parameters p 0.8 and 0.2.
26

---

Density Pp
Plot of Ey-p, for 0.5, o=
8
P-0
0.5
00011
0.4
6
P-033
Expected value
5
0.3
p=05
4
0.2
p-07
2
0.1
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
Blas p
(a) Density for various
(b) Expected value of
Figure 10: Statistics for truncated normal distribution with 0.5, 0.1.
Plot of to for various with 0.5
0.60
0.55
Threshold
0.50
0.45
0=1
0.40
0.35
0,3
0.0.4
0.30
0.5.
0.25
0.0
0.2
0.4
0.6 0.8
1.0
p
Figure 11: Plots of versus o for various with 0.1 for the truncated normal distribution. The dotted line 0.9 corresponds the undifferentiated contest with density p pt.
truncated normal distribution (p o²) on the interval Q2 [0, 1]. Since the bias multiplicative, the domain of PI, 82 = 1], does not influence the assessment of the contest's 's results. Note that o( PM )-002
Ep-p2[v] PJ
where o(z) is the probability density function of the standard
- normal distribution N(O, 1) and ((x) is its cumulative distribution function. The expectation of P2 does not decrease linearly with p as in the uniform case, but closely approximates a linear function and monotonically decreases with This is motivated by real-world settings where the valuations (such as pay or SAT scores) exhibit a truncated normal distribution [85). Other variants of distributions include piecewise-lince polynomial (such Pareto), and log-normal distributions, along with their biased versions.
We implicitly assume that the bias parameter is fixed and identical for all agents in G2 above. However, p could be noisy and non-identical to agents. For instance, let Pp density supported on [0, 1]. We assume each agent G2 has an individual bias p; i.i.d. drawn from Pp. and its valuation drawn from the pi-biased density of pt. Then pa is supported on f2 U4 and satisfies that for any Ωn
P2(U) = 1 Lo
D.2 Analysis of metrics for Nash equilibrium in the truncated normal distribution case
In this section, we do similar analysis as in Section 4 for the case that P1 is truncated normal distribution Nuu, o²) supported on (0, 1), P2 is a p-biased truncated normal distribution N (pu, o²) 0.5 and ensures that the density function is narrowly focused around the mean and the expected value of pa
supported on (0, 1], and pa is point mass at O. We choose
0.1. This selection
is approximately pn; see Figure 10 for illustration. Note that analogues to Proposition 4.1 is the
27

---

F-00
3et ot
wfeeeee C-0A 2-15
0000
11006.10
210
350
1.4
302
8.11
66
12
00
05
12
(a) PR (4) V.S.
(b) Px (R(4) (4) v.S.
(c) (A) v.s.
(d) 15 (A) v.s.
Figure 12: Plots illustrating the group-wise social welfare (A)) and the social welfare ratio (s
as functions of the parameters c, and a for the truncated normal distribution. By default, we set (p, C, C,
(0.8,0.1, 0.5). A dotted line within these plots indicates the threshold at which (s) = 0.8.
solution of the following equation:
o0 d("-²) 2l (- + a
)(" D B(L-0H)
Φ 1-
(1 (1-a) a+)
=1-c.
(8)
+ (1-=)) (
PM
Unlike the uniform distribution case, hard to derive closed-form expressions for metrics on the outcomes of the contest. However, can do numerical computations and plot solution t, representation ratio (A) and group-wise social welfare Sz(A) together with social welfare ratio rs rs(A) (A) in Figures [1 and 12 respectively. All plots exhibit monotonic behavior similar to that observed with the uniform distribution. Next, we highlight some distinctions with the uniform distribution.
No inflection point. A notable feature of the truncated normal distribution is its lack of an inflection point. This trait is observed not only for but also in the behaviors of rx (A) and rs(A). This difference arises because the domain f2 = [0, 1] remains consistent across all values of p.
Representation ratio. Figure 12 shows that to achieve representation ratio TR (A) 0.8, it is necessary to adjust to a minimum 0.979 or increase to at least 0.862. The need to elevate c is more pronounced than the required 0.5 observed with the uniform distribution in Figure ( This difference arises because the truncated normal distribution tends to be more focused around its mean, leading higher number of agents in G1 possessing valuations greater than the expected value R: 0.4 of p2.
E Proof of Theorem 3.1
two-group contest
In this section, begin by providing more detailed technical overview of the proof of Theorem 3.1 (Section [..tionnE...... Next, we present a more comprehensive version of 73. including the explicit
form of (m) (Theorem E.2 in Section E.2). Finally, we provide the proof of Theorem E.2 (Section
E.3)
E.1
Technical overview
We present an overview of the proof of Theorem 3.1
which characterizes an NE policy for the two-group contest. Recall that, there are n agents belonging to one of the two disjoint groups G1, G2 with | (1 a)n and |G an. The valuations of agents in G1 come from the density P1 supported on 22 and those of G2 come from the density y2 supported on Ωg Each agent has initial ability drawn from the density Pa- The selectivity the contest a constant < e < Theorem + co, there is an NE policy for the agents which is determined by threshold t E (2) U.S (22) + P. when the (Ω₂ U Uf2) + Q2 a connected subset of R>o-
3.1 asserts that,
For ease of analysis, we first consider the simple case where Pa is a point mass at 0, so that agents' policies only depend on their valuations. We then show that the extension straightforward. We start by first quickly showing how to compute the NE policy in the special case when P1 P2 for finite n and why this approach does not extend the two-group setting of interest E.I.I. In Section 1.2 we show that even though there are major challenges in extending the one-group case, leads us the right form of the NE policy for the two-group case as n + oo: single threshold function that defines the strategies of agents in both G₁ and G2. This also explains how we arrive Equation (2) that characterizes the threshold Finally, in Section F...3] we present the approach to formally argue about and prove the convergence of the finite n two-group contest to
general Pa is
1S..ttio/E...I)..
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this pair of NE policies. This analysis also reveals why the conjectured policy is a Nash equilibrium. With all the background, we conclude Theorem 3.1
E.1.1
NE policy for the undifferentiated contest for finite n and obstacle in extending it
Here, we consider the special case when P1 P2 p is the density of the uniform distribution on [0, 1]. and Pa point mass at O. The argument for other densities similar. First note that by symmetry among agents, it is reasonable to assume that, at equilibrium, each agent will follow the same policy s. Moreover, since the domain of p1 is nonnegative reals, it is reasonable to assume that the NE policy is monotone in an agent's 's valuation. The following calculations show that both symmetry and monotonicity hold.
Recall that an agent i with valuation e is selected if the effort s(v) is among the top e fraction of efforts. Thus, assuming that all agents follow s, agent i is selected if and only if there are at least (1 - c)n distinct agents j with s(z,) < s(v). (We ignore the issue that there may be ties for this discussion.) Since is a monotone function, s(n)) < s(v) holds and only if vy < u. Thus, the probability of selection of this agent is
n ("T1) ed -))----
P((s(o):A.i)= =
Σ j=(1-c)n
Hence, its expected payoff is (e, P(( s(v). (See Section 2 for notation). key observation is that the calculation of (s()):: A_) only depends on density Pi and v, and is independent of the choice of 8, under the monotonicity and symmetry assumptions on If is to be an NE, then it must satisfy that for any other effort e, (u( A-) 2 m,(,, e; A-.). This follows from the condition that the derivative with respect to v, T((U,, (v);A_ O. As noted above, (() does not depend on s, so we get a simple differential equation
(s(v)); A_))
involving the derivative of s' s() = (E (1-
Thus.
s(v) = (1 c)
Σ
()) 1)
(9)
(=(1-)n+1
the unique NE for the undifferentiated contest and it that this is monotone. For general density p, apply similar analysis to obtain that
s(D) = Qp(u) -- fr Op(x) dr,
(10)
2
where Qv(v)= >E ("T)) - (1
F,(o))=-1- (1- for any E Ω. We also note that the computation of s can become significantly more complicated for a general density Pa. as it requires considering the stability condition for two partial derivatives:
da(v.a) de and @s(v.a)
da
We now attempt to extend the analysis above to the two-group contest. Since each agent in each group uses the same valuation density, we can still hope for a symmetric and monotone NE policy for each group; say 81 for G1 and 82 for G2. However,
longer assume that 81 82. There are simple examples (see Section B) for which it can be shown that 81 * 82- This considerably complicates the calculation of probability P. for the ith agent getting selected since the order of agents' valuations may differ from that of agents' efforts. For instance, it is now possible that for agent i Gi and agent j G2, Vi < Uj but 81 (v:) 82 (vz). Thus, P. must depend on functions si and 82, instead of only depending on density p = and v as in the undifferentiated case. Thus, it is no longer possible to write simple differential equation as in the undifferentiated case.
E.1.2 A conjectured NE policy in two-group contests for large n2
Since it seems intractable to find an NE policy for the two-group case, we study whether the situation becomes casier when n is large. This hope rooted in the observation that for the undifferentiated contest when n is large, the NE policy s(v) defined in Equation (9) converges to threshold functiion.
To see this, recall that for the uniform distribution, s(v) (1 - c) ンー (1-())) () 70 - v)"-j,
Since s(v) is the probability associated with sum of i.i.d. random variables, it follows from the
29

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Chernoff bound that as n + oc, s(t) + 1 for any v > 1 - and s(n) + 0 for any V < 1 This argument is not specific to the uniform distribution and extends to any density p1 with NE policy defined in Equation (1O). In particular, if Fi denotes the CDF of pl, the limiting NE is given by s(v) = 'T (1 (1 v 2 Fr (1 -c) and s(t) = 0 otherwise. The threshold F (1-c) guarantees that the expected fraction of agents that put in a nonzero effort is 1 - (1 - c)) = c. The rationale for s(v) = Fi- (1-o c) when is above the threshold is twofold: 1) agents would not exert effort beyond their valuation, ensuring s( (1 - c)) - FT - c), and 2) agents with valuations below Fi (1 - c) are disincentivized from participating. leading to s(F) (1 (1- - c)) 2 FT (1 c).
Thus, one may hope that in two-group contest, to two threshold functions 81 and 82, each with a corresponding threshold 4.. While this assumption allows us give an explicit form for the probability P. of agent getting selected, this expression is quite complicated and, importantly, depends on & and f2.. Thus, we are unable to obtain conditions that determine i and i2 from the NE condition.
oc, the NE policy might similarly converge
Going back to the setting when pi is the density of the uniform distribution over [0. 1] and p2 = P2 is the density of the uniform distribution over [0, pl. first observe that as p - 0, l2 + 0. Thus, one would expect & to be more than tg. We now argue that counterintuitive to the above observation, > t2 cannot lead to an NE. To see this, first observe that when t > t2. if agent puts in effort t1, then will get selected. Thus, the probability of an agent in G₃ getting selected is 1 - Fi (f1). Hence, - fewer than en agents in G1 get selected. Thus, agents in G1 getring selected will find that putting in effort slightly larger than l2 instead of & suffices to ensure their effort is larger than all agents G2, and consequently, they will still be selected. Through this reduction in effort, they can gain an additional payoff of l 122 which violates the stability condition. A similar argument holds for G2 when 1 - (tz) > --a Thus, t > t2 leads to instability. Similarly, we can argue that & < 1/ also leads to instability. Thus, in case the NE policies for G1 and G2 are thresholds, it must be the case that f iz when is large. However, it is not clear how this can hold given that the domains of P1 and P2 are different.
if - Fi (11)
To explore this, we consider a scenario when a = 0.5, P1 is the density of the uniform distribution over [0, 1] and p2 is the density of the uniform distribution over [0, 0.5] (p- = 0.5). With high probability, there would be more than 0.1n agents from G1 whose valuation is larger than 0.5. Hence, if we set = 0.1, no agent from G2 will have any incentive to put in an effort while for agents in G a threshold of 4 0.8 suffices. The key observation is that even though the two policies are different, the policy of G2 is just O. This suggests that both policies can be seen as restrictions of the same threshold function to their respective domains.
Now we show how to compute the threshold The idea is reduce the two-group contest to an undifferentiated one whose density = (1 a)pi + ap2, supported on the domain Ωn U S2. In this undifferentiated contest, as n + co, it likely that (1 a -f-ftaction of agents with valuation come from p1 and a a--fracionn of agents come from p2- This suggests that these two contests are increasingly
indistinguishable as n grows, leading to the same limiting threshold t = Fp (1 c). Thus, threshold t the solution of the equation (1 a) Fi (v) + (F2(v) = 1 c the one denoted in Equation (2) This argument can be extended to general Pa. resulting in the following lemma.
Lemma E.1 (Unique solution). Let a. E 1), Pi be a density supported on the domain 22 S R20 p2 be a density supported on the domain O2 C R20, and Pa is a density supported domain Pip C Rgz. If (S4 U 122) + Pt is connected and each density pt, P2.Po Po is positive at any point of its domain, then there exists a unique solution 1 E Ωi U S2 for the following equation: (1-a)Fi(s) + aF2(C) = 1 - c, where for any Ç E Rzo, Fc(S) = Prespe ly + + c).
the
The assumption is naturally met in cases such as the uniform distribution and the truncated normal
distribution discussed in Section 2 Since Fę is a CDF, must be strictly monotonic across its domain
Ω2 + L2. For any Ç. E (9, U82,) + St with < < Ç', if (1 F) (c) = (1 - a) Fi(') FF2(S"), we must have both Fe(()= = (C") holds. Then (4.C') n ( UfA)) + 2) = 6. which contradicts the connected domain assumption. Hence, (1) + a aF(())
is strictly monotonic across the domain (Ω2 U U12)) + S2 which ensures the uniqueness of solution f. The proof can be found in Section E.3.I
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E.1.3 Proving convergence to the conjectured NE policy
Now we outline how to prove that, as n - oo, the NE policy for the two-group contest converges to the threshold policy corresponding to as guaranteed by Lemma [E.]] To do so, first, we have to make it precise what convergence means. Towards this, we revisit the undifferentiated contest. While we argued that in this case, as n + oc, the NE policy tends to threshold function, recall that we used the explicit form of the NE policy for finite n. Unfortunately, since we do not have an explicit form for the two-group contest (for finite n), we need a strategy for the undifferentiated case that works without the knowledge of the explicit NE for finite
Let A be the NE policy of the undifferentiateed contest as n + oc. characterized by function s which is a threshold with parameter f. It suffices to prove that for policy A and every > 0 there is an ne such that for 2 Tg. A is an E-NE. However, we find that there exists an > such that for any n 2 1, A is not an e-NE policy. We revisit the simple example of uniform distribution discussed above, which e = 0.5, pi the density of the uniform distribution on [0, 1]. and Pa is point mass 0.5. Then, in expectation, 0.5n agents have valuations at least 0.5 and put in effort 0.5. As n + oo, it follows from symmetry that the probability that fewer than 0.5m agents with valuation 2 approaches approximately 0.5. Thus, an agent i with a valuation v = 0.2 and putting in an effort e = 0.01 would have about a 0.5 probability of being selected, i.c.,
at O. Recall that the threshold / 1
(: A)+ & 0.5. Since s(v) 0, the probability (e): A = O. Thus, we have
(u, e; A_:) - (v, (v): A) = Pe(e; A.) v - 0 Rd 0.2 * 0.5 0.01 > 0. This inequality implies that when E 0.08, for any 2 1, A is not e-NE policy.
To bypass this, we consider a sequence of proxies for A, denoted by A and characterized by threshold functions g(m), These proxies aim to ensure that the winning probability (e A_:) z 0 and hence, serve as an approximate NE policy (see Definition 3.1.. Therefore, we need A (x0 to satisfy two conditions:
1. (m) converges to A as n approaches infinity, i.e., lit (n)) = 8, and
2. The winning probability
under
A(n) approaches zero in the limit, i.e.,
lim Po(e; A An)) 0.
Ensuring Pi(e; A (n) ~ 0 essentially involves guaranteeing that the probability of having fewer
than en agents with valuation 2 is negligible. Specifically, for the uniform case, by adjusting the threshold by Vlog n/n = o11, we define the policy (1)) as follows: s(*)
if v > t - v /logn/n and s(n) (u) = O otherwise. By concentration, this s(x)) ensures that the probability A A(")) is bounded above by / 1/n (Lemma E.4). This bounded probability ensures A()) be an /I/n-NE policy (Lemma E.6 This concludes the proof that the policy A is an NE in the large limit for the uniform distribution case for one group.
Finally, we adapt this new proof technique of constructing (1)) to the two-group case with general densitics pl and p2. Mirroring the strategy employed in the uniform distribution case, we would like to shift the threshold of ((1) to ensure that, in expectation, a (c+ + /log //lggn/n)-fracee -frcciinn of agents put
and E.2 for the in effort t. To satisfy this, we define the following threshold An (Definitions E. policy ((n)
A == min (FT - - (F2 (F2(1)-11(1))))
1/lo logg//))
Consequently, we define the policy s(1) as follows: s(m)(1) = if v 2 -- An and 5lm) (v) 0 otherwise (see Theorem E.2) We find that lim An = t, indicating that s(a) converges to s. Crucially, such a A,, ensures that (1 a) (A.,) + aF, < 1 - (c + Vlog n/n), thereby of agents, in expectation, putting in effort Using this property, we establish a bound for the winning probability P. analogous to the undifferrentatedd contest: Pi(e; (m)) < -(-o)) - if < t (see Lemma E.4 The factors n -1-a) and n derive from concentration bounds applicable to group G1 and G2, respectively. Furthermore. this minimal winning probability guarantees that (1)) is an e,E with lim en = 0 (Lemma R.6). The extension to the general Pa is straightforward. The main difference is that the initial ability
maintaining at least (c+ Vlog na/nnn)/aacctiii
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may already surpass the threshold t, in which case the agent does not need to exert any effort to be selected. This is characterized by the amount of effort s( (v, a) max {t - a. 0 if u 2 t- A..
To summarize, we first arrived at policy A that is characterized by function which is parameter- ized by threshold defined by Lemma E.1 and then we constructed sequence of "proxies" A(") that converge to as n + op. Moreover, we construct a sequence El,...,ãa En,... with limit 0 such that A(m) is an E -N policy for every n. This implies that A is NE policy as n - oc. Thus, the overview above allows us to prove Theorem 3.1
E.2 A more comprehensive version of Theorem 3.1 convergence form
Now show how to construct series of policies {A(*), In that approach A, the NE policy from Equation in Theorem 3.1
- The most technical part will be to prove that ((m) acts as an -NE policy, where en - 0 with increasing n.
Suppose (22 U Q2) + S2 is connected and let t E (Ω4 U + S2 be unique solution of the equation (1 - a)) Pi aF2(C = 1 (the uniqueness of is ensured by Lemma E. Since
E (0, 1), we have that either (t) 0 or (2(t) 0. Accordingly, we define a threshold ne as follows.
Definition E.1 (Threshold n²). Given a value & Ω U U22 we define a threshold ne as follows:
both Fi(1) > 0 and F2(t) > let ny be the smallest integer such that Fe(t) for = 1,2.
V
log n Rt
Fit) > 0 and (t) 0, let be the smallest integer such that Fi(t)
log the
The > 0.
If (it) = 0 and F2(t) > 0, let ny be the smallest integer such that F2(t)
log ne
> 0.
0 - cc. Also note that is monotonically decreasing to across the domain Q2 U 02. This value of n, useful for defining An. which is essential for the construction of policy A
Note that such n, is finite and always exists since
log
Definition F.2 (Threshold A,). Let 2 be integer. We define threshold An, follows:
both (t(t)= and (t) > let
PT (F)(( V log n ), F2 (F2(1)
log
A, min
n
Ff > 0 and F2(1) 0, let A, = (F)(( log
Otherwise if (t = 0 and (t) 0, let An = Fr (F2(()
log
The requirement that n n4 ensures the proper definition of A.. As the number of agents n grows indefinitely, the term logn approaches leading A, to converge towards The threshold function ((n) defined as Equation [T], designed as a threshold function, incorporates A,, as its threshold. The convergence of A, to as + is crucial for ensuring that Ann) gradually aligns with the policy A over large populations.
We ready to provide the formal statement of Theorem 3.1
Theorem E.2 (Two-group contest: Large limit). Let E (0,1). For 1,2, let pe be density supported on a domain Q4 S R-2- Let m R20 TR be merit function that strictly increasing. Suppose Ω2 U Or is connected and each density Pa is positive at any point of domain
'If the inverse function F
logn yields multiple values, it is defined to be the maximum of
these values.
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Qr Lett E 92 U U12 be unique solution of the equation (1 ()) + (F)(( = 1 - (the uniqueness of is ensured by LemmalE.I]. Let T4 be defined as in Definition E. Let n 2 n4 be an integer and let An be defined as in Definition E.2 Define
(1,)) max a, 0,
ifu An if 2 A.
(11)
gives rise to a policy for the two-group contest: Under this policy, agent i E G1 uses the restric-
tion A!!)) = s(n) while each agent j E G2 uses the restriction A!T)) = ll We have lim g(n) 8, where the threshold function defined as in Equation Moreover, the sequence of policies A ) Alo+t)), .... satisfies the following property:
VA > 0, 3gg 2 ne, s.t. Vn 2 ne: A is an c-NE policy.
(12)
This theorem establishes how an NE policy for the two-group contest approaches limit as the number of agents, n, grows indefinitely. reveals that the sequence of policies {A((),, Jn, not only converges to A but also aligns with an NE policy for the two-group contest. Thus, it validates the assertion made in Theorem 3.1 that A serves as an NE policy for the two-group contest the limit as + 00.
E.3 Proof of Theorem 122
We provide overview of the proof, summarized as follows.
1. In Section E.3.1
we prove Lemma E.1 for the uniqueness of solution t that decides the
threshold function
2. In Section E.3.2 we bound the winning probabilities Pe(e; A!)) under policy A sum- marized by Lemma E.4 Its proof relies on the winning probability for the undifferentiated contest (Lemma whose computation is via auxiliary function defined in Definition
3. In Section E.3.3 we apply Lemma E.4 to prove that ((n) is approximate NE (Lemma E.6
4. Finally in Section E.3.4 we show that Theorem E.2 corollary of Lemma E.6
For simplicity, we first assume that Pa is point mass O, such that policies depend solely on valuations. In this case, s(v, a), Po(e; a, A_;., < a, e; A-;) are simplified to s(u), A_,), x(u, e; A.;) respectively. At the end, we will show how to extend this general Pa-
E.3.1 Proof of Lemma
E.1 solution uniqueness
Instead of proving Lemma E.1 we directly prove for the general multi-group case. Let GI,..., Gm
be m 2 2 groups where each G2 has size n, = Ogn and valuation distribution pe on the domain Ωt R2o- We have ae E (0, for every E E [m] and Eie(m) 0g = 1. We have the following lemma that generalizes Lemma B.I
Lemma E.3 (Unique solution for multiple groups). Suppose (e) Qa is connected and each density Pe and Pa is positive at any point of its domain. There exists a unique solution t Uegimg for the equation Eie(m] Fe({))
c. where for any Ç E Rzo. Fe(()
Prospetacpa [U + < C].
Proof: Fix é E [m]. Recall that we expand the domain of every CDF Fe to Rzo. We have the following properties for Fe:
1. Fr(-) is non-decreasing across the domain Rzo, i.e., for any S.C' E R2o with ( < S' Fe(C) < Fe(C") holds.
2. (-(-) strictly monotonous across the domain Qe + Ωzg i.e., for any S, E Qg + Q2 with
Ç < 5' and (5.5)) n (At + Qa)) * 0, we have Fe(v) < Fe(1').
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Define a function 9 R20 - R20 such that for any Ç E Rzo. 9(S) = - [semm] aeFe(s). Since g(-) is convex combination of Fe(.)'s, we know that g(-) is also non-decreasing across the domain Rz0- Moreover, since (Ure(m) Ω2t) + Ptg is connected, for any 5,5' E (Ueeim) ste) + Ω2 with < < ç. there must exist at least one & E [m/] such that (c( < (C( and ( + Q2). This implies that g()) is strictly monotonous across the domain (Ueg(m) (te) + Saa
Now let L and U denote the infimum and the supremum of domain (Urg(m) Qt) + Qt respectively. We have = gL) < 1 - < g(U) =1. 1. Thus, there must exist a unique point t E (Uke(m) Qt) + Ste such that (() = 1 - c. This completes the proof.
E.3.2 Bounding winning probability
We first have the following lemma that bounds the winning probability under policy A Lemma E.4 (Bounding winning probability). For every integer n 2 ntz we have
Vi E [n]] Pi(e; A 1 ife 2 t; and P:(e: S n= (1-aa ife
For preparation, define the following function that is useful for computing the winning probability (e A-.) for the undifferentiated contest.
Definition E.3 (Function for computing winning probability). Given integers n, 2 1 and density pr supported on
Ω _ R20 we denote a function ((a.k) :Ω + Rz0 to be for any v E Q
i=- i ! Ri( (1- (i-)
a-1
(
Σ
Σ
B(n - 1,,, (13)
where B(n, k, x) (2) (011=)=--t w) k is the Bernstein polynomial.
independent and identically distributed (i.id.) values u1,,... -n- from distribution pl, the value ranks among the top k values in the set {.i,.... 1n-1. e. Given its algebraic significance, the function o (a..k) (.) is monotonically increasing to across the domain Q. This means that as v increases, the probability of v being in the top l also increases. Also note that for any integers n, n' 2 1 with n < n',
By definition, 2f (2,)) (u) represents the probability that, when sampling
Q(n.k(()) (v) 2 Q(n'
(14)
This means that as the number of agents increases, is less likely to be in the top A. This function can be used compute (e; for the undifferentiated contest in the following sense.
Lemma E.5 (Computation of winning probability for the undifferentiated contest). Let n, k 2 1 be integers and pı be a density supported on Q S Rzo- Let A = (Ai...., An) be a symmetric policy for the undifferentiated contest satisfying that every A₃ is strictly monotonically increasing to across the domain Q. Then for every E [m) and Q2 we have P.(A.(u); A k)kkk) (v).
Proof: By symmetrie, we only need to prove the lemma for i = n, i.e. proving ()mx)(()) Let U1...., Un-1 be i.i.d. samples from p1- Since A; is strictly monotonically increasing to v across the domain Q2 we note that the sequence U, .i...., should have the same order as the sequence A, (v),, A (vi)....., "...,,An-(
Pa (A),
Hence, (v) is among the top l of [AI(vI)..... An. (1nn 1), (v)} if and only if v is among the top of {UI,.... Un- v). By the definition of winning probabilities and Definition B.3 this implies that
(v) A..) = o (n.k) (v). This completes the proof of Lemma E.5
Now we are ready to prove Lemma
Proof[: [: of Lemma E.4 It suffices to prove for the case that both (t(t) > 0 and Fz(t) > 0. Proof for the other two cases identical. By Definition E.2 we have
log n
log n
A = min F (Fi(t)
Fz (F2(t)=
-
n
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Let event E E((.c) be that there are at least (1 (1( a.n agents in G1 \ { that put in effort larger than e; and let EX that there are at least (1 11-PPp F2(()) an agents in G2 {i} that put in effort larger than e. Note that
(1 (i(t)) - a)n + (1 F2(t)) on = cn.
When 2 1, we have
R(e;A (m)) 2 Pr
El
E
2 Pr
E E(n.e)
Pr
Eg ((..))
Then to prove Pi(e; A = 1. it suffices to show that Pr
D(ni))
= Pr
Eg (n.))
0. Note that
by policy ((), the maximum effort put in by agent is < Hence, no agent can put in effort
larger than e, which implies that Pr E(me)) = Pr E2 (m.e) = 0. This completes the proof of (e(e; (
when 2
When < t, note that if both E( and E2 E/ happen, there are at least & agents that put in effort f. Since events E(m,e) and E are independent, we have
Pi(e; A (1 S Pr p(ne)) F((.ee) = Pr
p Pr [n(ne))
Then to prove Pi(E; (
S n a + n (1-a), suffices to show that Pr [((i)e) 2 1 (1--a (2-a)
and Pr
E [f
2 1 - n a
We first bound E(nee)
Note that there are least (1 - at) agents in G1 U [i). Also, note that an agent j E G1 {i) puts in effort (y, if and only if their valuation vj 2 A, holds. Now consider the undifferentiated contest among G1 U {i} with k1 = (1 - (1) a)n and density
P1. By Lemma E.5 have
Pr
D (m,e)
1 2 (((GU))))
(Lemma E.5
((1-a))..k)) (An)) P1
(Ineq. (4
log n
2 1 (())a))n,A)( (FT
(Defn. of A,, (15)
(1-a)n-1
B((1 3((1-a)n -Ij. log
Σ
(Eq.
n
j=(1 - a)n 10m)n-1--0 1-
Let X1, X, be (1 - a) m i.i.d. random variables where each X; = 0 with probability (1)
log and otherwise X; 1. We note that (1-o)=11
-j=(1-a)m--kk B((1 a)n 1, Fi(()
log
is equivalent to the probability that Eie(n X; S hi 1. Also note that
E
Σ
X; (11 a)n 1) (1- F1(1) +
log 72
(16)
iE((--)a-11
Then by the Chernoff bound, we have
(1 Q-ajn-1
log n
Σ
B((1 a)n - F.)
j= (1---ann---
Pr
Σ
X
[E((1-a)n-1]
log
< Pr
Σ
X
Σ
X, (1 a)n V
(Eq. (6] and Defn. of k1)
n
iEl(1-a)n-
(e(1-a)n-1]
201 un In
<
(1-0)=-I
<n (1-aa.
(Chernoff bound)
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Combining with Inequality 13 we can also prove Pr
we prove that Pr E 2 1 11-n-(1-a) By similar argument,
n_a Overall, we prove that Pi(e;A s n a + 1
when < t. This completes the proof of Lemma E.4
E.3.3 Proof that A is approximate NE
Based on Lemma E.4 we are now ready to prove the approximate degree of (()) to be an NE policy.
Lemma E.6 (A()) is approximate NE). For any n ne. A is an NnNNE policy, where en (n-2 + n (1(1)))) + +1
An.
Proof: Fix 8 1,2, i E Gg, and u,e E Ze. We discuss the value Te(U, e; A!
T1 A (0) (v); A ). By Lemma E.4 we know that
(u A
eife 2 t; and eu S (n - +n ---00=0) e i < t.
Then v A.s we have (u; A (v); (3) = (,, ( 0. which implies that
Ti e; 1 - A A (m) (v):
(1m) V (n
(1--0 tn e (n +n (1 if f
0-e50
if 2 t
(m) (v); A = ( ( Otherwise if v 2 A.. we have (u, A
which implies that
Te(u, v; ( - (u, A (Tth
(n + (1 e if < t if
t t-
Note that when v ž An and to
(n~ +n (1 al) ) + l. u (n _ + n (1--))) +1 An.
Overall, we conclude that the following inequality always holds:
(T e; A (a)) - ,. A (v); ( < (n~ + a + t - -n = En.
This verifies that (1) is an NNE policy for the two- two-group contest.
E.3.4 Completing the proof of Theorem
E.2 Assume Pa is point mass at O. We first prove that lim s(t)) = s. This is a direct corollary of the fact that lima-xo An t. Consequently, for any E 2, there exists n. such that for any integer 2 nee g(0) (v) s(v) holds.
Proof of Theorem
A(n) is a -NE policy, where limp-400 L, we have
By Lemma E.6
(n (n~a + n (1-a))) + - An. Since
lim En +00
lim (n a +n (1-a)) ) An, +t A,, =0,
n-
This completes the proof of Equation 12
Uniqueness of A. To prove that A is the unique NE, we first recall Corollary 3.2 of (20) that says that a subset of symmetric agents should have the same policy in an NE. Thus, assuming A' is an NE policy for the two-group contest as n + R2o. all agents i E G1 use a common threshold policy 81, and those in G2 use 82- Suppose the threshold for 8y is 4.- We next prove that 4 = f22 which implies that 51 82. When t > 12, if agent puts in effort ł1, then it will get selected. Thus, the probability of an agent in G1 getting selected is (t,). Hence, 1 - (t1) < --m fewer than en agents in G, get selected. Thus, agents in G, getting selected will find that putting in effort slightly larger than le instead of 4 suffices to ensure their effort is larger than all agents in G2, and consequently, they will still be selected. Through this reduction effort, they can gain an additional payoff of t tg. which violates the stability condition. A similar argument holds for G2 when
36

---

1 1- (1(t1) > I-a Thus, A' is not an NE when t > t2. Similarly, we can prove that A' is not an NE when 1 < t2. Thus, we must have 1 = 12 = and 81 $2 S.
(f (t) + af2(()) > 1 1- then fewer than en agents put in a non-zero effort and get selected. Thus, an agent with valuation t' < t, has a willingness to put in an effort E slightly larger than instead of 0. Through this increase in effort, it can gain additional payoff of t' - Thus, A' is not an NE. Similarly, we can prove that A' is not an NE if (1 - a Fi + F= < c. Thus, for an NE policy, must be the solution of (1 a Fi + 2F2(1) =
By Lemma E.1 the equation
(1 ai + a (2)) = 1 c has a unique solution. Thus, A' = A, which proves the uniqueness.
Extension to general Pa- For general Pa. the solution t of Equation (3) represents a score that c-fraction of agents can achieve without making their expected payoff negative (u + a 2 t). The proof is almost identical to that when Pa is a point mass at 0, except that the effort an agent with v + a 2 is willing to put in should be s(v, a) = max(t - a, O} instead of t. This is because t represents the score that the agent aims to reach, rather than the effort itself.
Overall, we complete the proof of Theorem E.2
Remark E.7 (Extension of Theorem 3.1.. Using the same proof technique, Theorem 3.7 can be extended to handle multiple groups and non non-identical effort costs. Let G Gm represent m 2 2 groups, where [Gel = Ogei and the valuation density for each group is pt over the domain S2 S R2o- Each og E (0, 1) satisfies the condition Lee(m) [tg = Recall that Pa represents the initial ability density over the domain The C <Rzo, and we introduce an additional effort cost density Pa over the domain Sh C R2 Each agent i E [n] knows its type 0; = (Vi, Us, Ks) and selects an effort level e, O. The agent's score is given by m(e, + at), and their expected payoff is P.u.- RRgg, It is important to note that agents' costs of effort Re may vary and affect only their expected payoff, not their score.
In this extended multi-group contest, for E E [m], we extend CDF Fe to be Fe(() = PIu--paarpa.ne [ + a S 4] Now suppose domains Ure(m Ste Da. On are connected and den- sities pe, Pa.Pr. are positive at any point of their own domains. Let be the unique solution to the equation L(e(m) Fe) = 1 - c. The infinite NE policy (3) extends to:
s(v. a) == 0 if V +a < t and s(v) = max (t 0} if ta2t.
E.4 Comparing with a distributional two-group contest
Recall that the technical challenges for Theorem 3.1 are mainly caused by the asymmetric strategic environment across groups. To avoid asymmetry, one may consider the following variant of the two-group contest. Note that for simplicity, we also assume that Pa is a point mass at O.
Definition E.4 (Distributional two-group contest). Let n 2 k > 1 be integers, E (0, 1), p E (0, 1), and pt be a density supported on domain S2g S for E= 1, Let each agent [n] belong to G1 with probability 1 - a and belong to G2 with probability a independently. Let the valuation of each agent in G₁ be drawn i..dd from Pr, and the valuation of each agent in G2 be drawn ii.d. from pz. Assume that each agent i E Ge (E = 1,2) knows nı. n22, k, P1. P2, the group it belongs to and its valuation, and has to choose a policy A; : 9t + R20 to maximize its expected payoff. The goal of the distributional two- wo-group contest is to compute the NE policy satisfying Equation (.)
The main difference from the two-group contest is that this distributional variant's group identity is random and the valuation density of each agent is identical, say = -1 a)p1 + ap22 Thus. using a similar argument as an undifferentiated contest, it is easy to verify that the NE policy of the distributional two-group contest is identical to that of an undifferentiated contest with density pr. Consequently, in the infinite n case, the NE policy of the distributional two-group contest is identical to that of the two-gon wwo-group contest, say A in Theorem 3.1 Then one may wonder whether this distributional two-group contest can also be used to simplify the proof of Thhorem 3.1 as the infinite contest does. Below, we show that this is not the case and discuss the essential differences between the two models. For simplicity, we call the two-group contest Model I and call the distributional two-group contest Model II.
Model distinction. Firstly, Model itself is of relevant interest as it captures real- examples in which group sizes are well understood, while in Model II the group sizes are random variables.
37

---

Convergence distinction. Though Model I and Model II share the same NE policy A in the infinite case, we would like clarify that our main convergence result (Theorem E.2) cannot be inferred simply from knowing that the limit of the NEs of the two models is the same. To put it in simplest terms, consider two sequences ai....; an The proof of convergence for the first sequence does not necessarily provide any information about the convergence of the second sequence. Therefore, the convergence result for our model cannot be simply inferred from prior work.
... and bi...., bn... ... that converge to the same limit
point z.
Moreover, the analysis of Model relies heavily on symmetric policies for all agents (enabled precisely by the fact that group sizes are random), allowing the use of order statistics of Pt. In contrast, in Model I, we expect asymmetric policies across groups. For example, consider two-agent case with A 1: Agent I's valuation follows a uniform distribution on [0, 0.5), while Agent 2's valuation follows a uniform distribution on [0.5, 1]. In Section B we show that the NE policy for this example must be asymmetric. Any symmetric policy A would result in a near-zero winning probability for Agent I, leading to a negative expected payoff and implying that is not an NE. This negative payoff arises from the asymmetric strategic environment faced by Agents and 2, where the density of the highest valuation among other agents differs for each agent. Consequently, the order of winning probabilities of agents (P,) can differ from the order of valuations (2t). posing significant mathematical challenge for determining the NE. E.g., for strategies 81 and 82, let Fs, (v) denote the cumulative distribution of efforts sz(v) when pc- The cumulative distribution of the (k- - t) effort e from an agent in G1 is then given by:
Analysis distinction.
Prfe" < vl
n-1 1-1
Σ
Σ
n1 (
n - 1-
Far Fe -Fa (v)(11 Fss (b))amn
i
(1(-))nn-
Compare this to the expression for the symmetric case
n-1 Σ ("T (=1-e)n
Pr(e S
Thus, the calculus and approximations for the expression for the two-group contest are significantly more difficult, making it much harder to arrive at the equilibrium policies than for Model II.
E.5 An alternative proof using an infinite contest
co for the two-group contest. To increase the understanding of the infinite case, we propose the following infinite version of the two-group contest, where every real number in the interval (0, ai corresponds to agent in G1 and in the interval (1 (1-ax ar 1] corresponds to an agent in G2. For simplicity, we still assume that Pa is a point mass at O. Formally, provide the following definition.
Recall that Theorem 3.1 studies the case of
Definition E.5 (Infinite contest). Let k 2 1 be integers, E (0, 1), E (0,1), and pt be density supported on a domain Q2 C. R2 for L. = 1,2. Let every real number in the interval [0, - al correspond to an agent in group G1, and in the interval (1 a., correspond to an agent in group G2- For & E (1, 2}, let the valuation of every agent in G3 draw ii.d from Pe. Assume that each agent E Ge (f = 1, 2) knows k, Pl. P2, the group it belongs to and its valuation, and has to choose
policy A; 2g + Rz0 to maximize its expected payoff.
There are countless agents in this infinite contest. Also, note that G₁ contains (1 a -frrcctionn of agents while G2 contains the remaining a-fraction. Below, we show how to use this infinite contest to hypothesize the NE policy A for the two-group contest defined in Theorem 3.1 It mainly consists of two steps: Showing that two-group contests converge to the infinite contest as n + oo and computing NE for the infinite contest.
Showing two-group contests converge to the infinite contest as n + oo. We first show that the infinite contest is the limit of two-group contests. Let 9n denote two-group contest as defined in
38

---

6. the two-group contest with an NE policy Ag. Let g denote the infinite game as defined in Problem E.5 We view 9n as a collection of density functions of agents, with the i-th agent represented by the real number t-1 I-I- Agent belongs to group 11 if 1 S i S (1 - a and to group G2 otherwise. From this viewpoint, we propose the following theorem.
Theorem E.8 (Two-group contests converge to the infinite contest). 9n converges to g in the following sense: For any & >> 0, there exists sufficiently large no such that for all n. 2 no-
For any t [0,1-al] al. 1S5 dx b
S i.e., the difference in the fraction of agents in G1 associated with real number at most between g and gn. is at most E.
For any t (1 a. 1], Is dr |ieGo:
i.e., the difference in the fraction of agents in G2 associated with real number at least between g and go. is at most e.
n
Note that the agents in In be captured by uniform distribution Pa real numbers (i [n]). The convergence conditions in the theorem state that the limit of pan is the uniform density
over [0, 1}, where represents the density of agents in g.
Proof:[of Theorem
E.8
Let noo = 11 Then we have _ For any t E [0, 1 - a]. we have
i G1 s
S/
(n 1)t
dr
and
(n- 1)t + 1
St+ 1-t
n
n
n
We conclude that
L dx KiEGi #
s1)1
( t 1 1
1E0.1-a]
1
n2c
< <max
[n'
n
n
(1EG2:
Similarly, for any t E (1 a, 1], can prove that Je dr the proof of Theorem E.8
< E. This completes
n
Computing NE for the infinite contest. follows from Theorem E.8 that the limit of the two wooggropp contests In is the infinite contest g. Then, assuming the NE policy gn. is A we can infer that the limit of (m)) is the NE policy of the infinite contest. Thus, to hypothesize the NE policy for gn as + oo, it suffices to compute the NE policy for the infinite contest.
We first observe that the strategic environment for all agents in the infinite contest is the same, i..., the probability that a given valuation e is among the top o-fraction is the same for all agents. This undifferentiated contest (except for the different domains of valuation densities), leading to a symmetric NE policy. Formally, we provide the following lemma that shows that A is exactly the unique NE policy for the infinite contest.
property reduces the infinite contest
Lemma E.9 (The infinite contest). Suppose Ω U Zg is connected and each density pt is positive at any point of domain Qz. Then policy defined in Equation 33) is the unique NE policy for the infinite contest.
Proof: We first note that for any agent (whether in G1 or G2), the probability that a given valuation is among the top c-fraction is given by:
(v()) = lim Σ E
((0))(1-(0)))*
yo
i=(1-o)n
where F is the CDF of the joint density (1 - a)p + ap2. Recall that is the unique solution to the equation
F(c) = (1- (1)a)Fiivv + aF2(e)
1
39

---

i.e., t (1 c). Then through straightforward calculation, it follows that (1(v) for v > t and pı (v) for v < t. Since the winning probability function pi is identical for all agents, the strategic environment for all agents in the infinite contest is the same. Recall that by Corollary 3.2 of 20), symmetric agents will use symmetric policy in an NE. Thus, we can assume an increasing symmetric policy $2 U + R20 for all agents.
By the equilibrium condition, for any valuation v and effort E,
(i(U)0 s(v) 2 (e))
By similar argument as for Equation IT] (undifferentiated contest), we can compute that s(v) for v > and s(v) = 0 for v < t. This turns out to be the threshold function defined in Equation (3)
Thus, the policy A, where each agent restricts to its valuation the domain, is indeed the unique NE
for the infinite contest. This completes the proof of Lemma E.9
Using the infinite game to provide an alternative proof of Theorem E.2 As shown in Lemma
E.9 instead of relying on observations from the finite case as in Section E.1.2 we can use this infinite contest to hypothesize the NE policy A for the two-group contest in the infinite n case.
Once have solid guess for the NE policy A using the infinite contest, we need show that remains an NE as n - c0. While this approach simplifies the initial hypothesis, the challenge remains in proving that A is indeed NE. Similar our current proof of Theorem 3.1 must construct a series of proxies that converge to A and increasingly approximate an NE policy, As
detailed in Section E.2
this step remains technically challenging.
Overall, using the infinite contest could provide an alternative proof of Theorem E.2 Moreover, the symmetric strategic environment of this infinite contest can provide deeper insights into why the NE policy remains symmetric, even when valuations are asymmetric across groups.
F Omitted details for uniform distribution analysis from Sections 3 and 4
Theorem F.1 (Restatement of Theorem 3.2). Assume p2(v) 1/Pi (5) for some p E (0, 1] and Pa
is mass point at O. Let policy A be defined in Theorem 3.1 characterized by being the unique solution of Equation 4 Then for any density P1,
(t/p))))) (R(A)= = (-) Ts(A) = -." (x-t)xx(o)d
S (v)pp))ate))
and RV(A, m)) = mm().
Moreover (A) is monotonically increasing w.r.t. p. c, and a, while RV(A, m) is monotonically increasing w.r.t. and monotonically decreasing w.r.t. c and a. for any merit function m.
Proof: We discuss three metrics separately.
Metric RV(Am)) m). Recall that A is a threshold function characterized by t. Thus, agents with the sum of valuation and initial ability v >> t get spots. Since p. point mass at 0, we know that the score of each selected agent is exactly t. Thus, the average revenue RV(A, m) m(t).
Next, prove the monotonicity of RV(A, m). Since m is monotonically increasing, we only need to prove the monotonicity of with respect to p, c, a. Recall that t the solution of Equation 4 which can be rewritten
(1-a) + an. (
Let ((S) (1 (1) + (F)
Then & is the solution of f(() 1.
Since Fi(S) is monotonically decreasing function of p. we know that f(C) is also monotonically decreasing function of p. Thus, the solution t increases with p. Since Fa(S - pu) Fa(ç v), f(C) an increasing function of a. Also note that f()) is an increasing function of c. Thus, as c or a increase, solution decreases.
Overall, we prove that RV(A, () monotonically increasing w.r.t. p and monotonically decreasing w.r.t. and a, for any merit function m.
40

---

Metric PR (A). Recall that Ft is cumulative density function (CDF) of the sum of valuation and initial ability such that for any Ç E Rzo. Fi(c) = Pr [U 1 a S {). Thus, F3 is the CDF of Ps when Pa is point mass at 0. Then the linearity of expectation yields:
E, [|S n G1ll = (1 a)n(1 Fi(t)) and e,, Il G2ll = an(1 F2(t)).
This translates
[[R] au n
(t) and ([2) Bu, || F2(().
E
a)
an
Since p2(U) = pr 2 (A)) S (A)).
we know that F2(t) = Fi(t/p) and Fi(t) & F2(t). Thus, we have
As n + oo, suffices to prove that
E [
E (A)]
1 F2(1) 1 1- F((UP)
(A()) + min
[A) E (A)]
1 1- Fi()) 1 1- (i()
We first note that |S Gel is highly concentrated at au [[S [[SIISnnG. Gill since all agents G2 are i.i.d. Concretely, the following inequality holds for any & > 0 by the Chernoff bound:
Pr [ - a. [[S Gell 2 t. E,,,a, [|S Gelll - 2e
Hence, for t ol /t), we have Pr [||S Gel E n 2 t Ee a. |S Gell + 0. This
implies that as n > DC,
(2(AA) R1(4)
}
[A)] (AR) [A)] E f
(R( = Ev,
min
- min
R2 (A) R2(A)
which completes the proof of the formula of r' (A).
Next, we prove the monotonicity of YR (A) with respect to Recall that is monotonically increasing
with p. We know that 1 (4) is monotonically decreasing function of p. Since [2) (1-))))(1-F)( we know that 1 - (t()) is monotonically increasing with p. Thus, rx (A) =
1-Fi(t/) is an increasing function of p. 1-F((1)
Metric (str) (A). By a similar argument as for FR (R(,,,, (A), we first have that as n - oo,
E (A)] B[S)(A))
rs(A) - min E (4) ([S]
Also note that
1
(Il selected) w - e:)] -w-)))]= 1.00 (u-D)e(v)de
[[S) Gel iEG,
Then we have
[A)) = 1 / T0 (u t) ()pp(")de
1 (pu (pp ([)]
000
1.t (1-(t))p2())d Since p2(v) = 1:22(5) we have
Combining the above all, we obtain that (s( =
1: (o-t))p((r)d
(A( = fe (v 100-1)p220 (v))
P p (u (
J (v t)pı(e)dd
Je (u (u)dd
Next, we analyze the monotonicity of rs( (A) with respect to p. Let g = fe (v-x)pp (x) which is monotonically decreasing of x. We have rs(A) p-g(t/p) Since / is monotonically increasing with p. (t) is also monotonically increasing with Thus, to prove that rs(A) monotonically increasing with p. it suffices to prove that t/p is monotonically decreasing with p. Recall that have shown that F, (t/p) = (t()) is monotonically decreasing with p. This implies that t/p is indeed monotonically decreasing with p. completing the proof.
s(t)
Overall, we have completed the proof of the theorem.
41

---

Pott of verious 0.5
for
0.8
Tioeshold
C+04 C+05
Timestofd
0.6
1.2
cmgs
0.4
0.2
0.0
0.0
0.2
0.4
0.6
0.8
1.0
0.0
0.2
0.4
0.6
0.8
1.0
Blas
Blas p
(a) Pa is uniform on [0, 1]
(b) Pa is a point mass at O
Figure 13: Plots of I versus p for various with a = 0.5 for the uniform distribution.
note that when fixing p. both (R(A) and (A( are functions of Since is monotonically decreasing with respect to c and o, we only need investigate the monotonicity of P (A) and rA()) with respect t. Proposition 4.7 demonnsttsss that (A(() and (A) are monotonically decreasing with I, and hence, monotonically increasing with and a. Below, we provide an example with P1 show that this monotonicity does not always hold.
Remark F.2 (Monotonicity of (A) and (A(A) w.r.t. C, a. By Theorem 3.2
Let > 0 be sufficiently small number and p 0.5. Let p1 be supported on 92 |0, 2] such that Pr (u) = 1 - for u E 0.5] U [1.5,2] and (v = E for E (0.5, 1.5). Then (1 = (1 E)U
for 0.5). 0.5 - - Eo for E (0.5,1.5) and (1 E)V 2e 1. By Theorem 3.2
have FR () 1- (t(pp) Then in this case, (A( = I-(1-e)74
1-(1-c)/2 2 when I. 0.25; while
I- (1)
FR (A) = (-(1-))22-000 computation can be done for (A).
1-(1-e)73 se 1. Thus, (A( is not monotonically decreasing with t. A similar
By a similar argument as for Theorem 3.2 we provide the following formulas of metrics for general densities. The computation is straightforward and we omit here.
Theorem F.3 (Metries in general). Let policy A be defined as in Theorem 3.1 characterized by t being the unique solution of Equation (2) Then for any densities pt, P2, and Pa.
TR(A) = min
Pa(O) 1-)((() FL(F) [-F2(t)
Sa,
v)ddud f.a Sn,
v))ddd vidud J. Jo, (o v)dook
(A( = min
J. Ja,
RV(A) 1-2 led So, u (e)) v)dudy
+ a S. Sng m(max(t - + v. (v))) - v)dodC.
In the following, we complete the analysis from Sections and 4 for the case where p1 and pa are uniform densities. The visualization of for them can be found in Figure [3 We first give the proof of Equation (5) for the case that Pa is uniform.
Proposition F.4 (Complete version of Equation (5). Let a, (0,1) and p (0,1). Let pr be
uniform on [o, 1), p2 be uniform on [O, p), and Pa be uniform on 1]. Let t E [o, 2 be the solution
to the equation fo (1 a) min(1, (S +)+ + S p min(1, (5 +d
Then if
0 <<c c
1-a
2
2c
2
2e
a
I-a
1=
a/p
- a a(1 a) p)-
a + a/p
1 1- 1-a+a/p
P2
V
I-a
if 1-aa < <cs 1
2
2'
2 - 0 + a/p
22c - a a a11 a) - -P/TpP
2e-lta
a-a+al
1-a a
2 + (1 ap 2c)
p 2c 2c-1ta
1-a
1-a
42

---

2 <0sL
V V + (1- (2 + up 2c)
pc at V
+
a
1-a
2(1-a
2(1 -c)
a228(1-a)
02
a/p"
2(1-a)
Proof: Let f(() = 16 min (5 2)
1 minnn (S }d We first note
that f(C) is monotone increasing function with ʃ(0) = 0 and ((2) = 1. By analyzing the value of
(i }) we also have the following equation:
0,
0 0. v,
min
v,
" ' S a 1,
1,
v+15652
Accordingly, we know that
[1 - a)({ - v) 10 + (s - l5
a
055SP. P.
&
(1 - a)(S- v) 1 +- -- -v ld.
P5<SL,
f(()
p
(1 a) ( (s-1+ (S ((-v) (6-1) + a
( - (5 (6
15(51+A
(1= (5-1+ (6-1) ta,
1+pS6s2.
Thus, f is a seeceewise-polynoone function of Ç. Solving f(C) = 1 - c results in Corollary
Proposition F5 (Restatement of Proposition 4.1). Let pr be uniform on [0, 1}. P2 be uniform on
[0, pl. and Pa be a point mass Let A be the NE policy for the two-group contest as n + 00. Then
a and = p(1- (11-)) c) I-a
1= 1 i <
if 1
op +
(A( = 0 if < 1 I-a and rre (A) p apap a a
ifp 1 I-a-
a app
ap a 1)2
rs(A) = 01p
--a and rs(()= (a- cp)
is 2 -
Moreover, RV(A, = m(t) for any merit function m(.); (A( and rs(A) are monotonically increasing functions of parameters p, p,c,a.
Proof: Note that (V.A,m) mff) directly from Theorem 3.2
Computation of t. Note that (e) e and (v) min 1, p.. Let g(u) (1- -a)
min {1. We note that g()) is a piece-wise linear function of with inflection point p. Plugging into the equation, we obtain that =1- - T-o which is an inflection point of t. Then if solution > p. we have that is the solution of the equation (1 - aiv + = 1 implying that I-o The condition for this case is p < I-o
e Otherwise if solution S p. we p(1-c) The
have that is the solution of the equation (1 a)e + a condition for this case is 2 1- - This completes the proof for t.
c, implying that t p-apta
Analysis for (A). By Theorem 3.2 we have TTR (A) 1-min verify the explicit form (A).
By the form of t, we can
1-t
Note that when 2 I we have
(A(A) p apap+aa
(1- (1-c)(1 - p) ap + cp
ap + cp
43

---

Let
f(p,c,a)=1 =1 (1 - () -p
a--ap- cp
Define the auxiliary functions:
(p, G, a) (1-c)(1- (1- Y(Occa)) aa op +p
Then when p 2 I-a we have
X
f( =1 y and ((..c))
The partial derivatives w.r.t. P, C, a are:
0f
(1 - CJ (c-
c(1 - c)
2 0,
0p
Y
Y2
0ʃ
(1 - (1-p)Y + PX
20,
dc
y2
05 1 -
0.
da Y2
Thus, TR (A) is monotonically increasing with p. p,C, when p 2 1 - r-a Moreover, the threshold
1 e is monotonically decreasing with a. Thus, we conclude that (A(A) is a monotonically increasing function of parameters Or.
Analysis for rA(A). By similar argument for rR(A), we can obtain the formulas of rs A) () (4)2. Thus, rs(A) is monotonically using Theorem 3.2 and the form of t. Note that (A) = prR increasing with p, a.
Overall, we have completed the proof of the proposition.
G Additional details to Section 4
In this section, we first illustrate details for how to estimate perceived bias from JEE Advanced 2024 (Section G.1 Then we provide robustness analysis for key findings in Section by varying and e (Section G.2 Next, we give an illustrative example for the practical use of our model, including how to make interpretable predictions and policy interventions (Section G.3) Finally, we provide omitted details for alternative interventions in Section 4
G.1 Case study - estimating perceived bias from JEE Advanced 2024
To illustrate our framework in a high-stakes meritocratic setting, we calibrate the model using data from JEE Advanced 2024, the entrance examination for admission the Indian Institutes of Technology (ITss). The
statistics were published by the Government of India's
Press Information Bureau 68):
Group
Candidates Appeared
Qualified
Male
139,180
40,284
Female Total
41,020
7,964
180,200
48,248
Model calibration. We define the disadvantaged group as female candidates and the advantaged group as male candidates. From the data:
Proportion of disadvantaged applicants:
41,020
2J 0.228
180,200
44

---

Selection rate for the entire applicant pool:
48,248
~ 0.268
180,200
Admit rate for each group:
Female admit rate 7,964
40,284
~ 0.194,
Male admit rate
0.289
41,020
139,180
Observed representation ratio:
0.194
Tabs
2 0.671
0.289
Solving for the bias parameter p. Using the closed-form expression for the representation ratio in the uniform-valuations model:
(1 (1- c) p) CR(P,C,a)= = =1-
a ap - cp
we plug FR = 0.671, c = 0.268, and a = 0.228 to solve for p:
(1 - 0.268)
0.671 1
0.228 0.228p 0.268p
Simplifying both sides:
0.329 = 0.732(1 p)
0.228 + 0.04p"
0.329 + 0.04p) = 0.732 - p).
Compute both sides:
0.0749 + 0.01316p = 0.732 - 0.732p.
Bring all terms to side:
0.6571
0.74516p 0.6571
p 0.74516
0.882.
The inferred bias parameter is:
p Rd 0.882,
which reflects perceived disadvantage for female candidates: they value qualification outcomes at roughly 88.2% of their male counterparts' valuation, consistent with the observed gender gap in qualification rates. This example demonstrates how our model can be applied to quantify bias in selection systems using real-world statistics.
G.2 Robustness analysis for findings in Section 4
In this section, we assess whether our core conclusions in Section 4 depend on the specific parameter settings. To verify robustness, we conducted additional simulations varying a and e beyond the default values. Below we summarize our findings:
Metric robustness across group sizes. We varied a from 0.5 to 0.3 (to represent smaller disadvan- taged groups) and recalculated the representation ratio Fx. rR(AA) and welfare ratio (A) across grid of disparity levels (p) and selection rates (c): see Figure [4 The overall trends remain consistent: for example, rx (A) < 0.2 still holds for = 0.1 and 0.85, confirming that strategic behavior amplifies underrepresentation highly selective settings.
Robustness of intervention takeaways. We varied e from 0.228 (derived from the JEE Advanced data) to 0.1 and a from 0.268 to 0.5 and re-evaluated intervention strategies. Figure T5 optimal interventions for various threshold т. The overall trends remain consistent. For instance, in Figure [5(a) when S 0.87, increasing access (raising c) remains more cost-effective. In contrast, when 0.87, improving group valuation (increasing /) becomes more impactful. This confirms that the recommendation to prioritize access vs. valuation interventions depending on the disparity level remains valid across reasonable choices of a and
45

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82
(a) PR A) v.S.
(b) PR (A) v.S.
(c) r v.5. P
(d) Ts rs( v.s.
Figure 14: Plots of the representation ratio ræ (A) and the social welfare ratio rs (A) as parameters p and c
vary for Proposition[4.] 4.1 with default settings of (p, c, a. = (0.8, 0.1, A dotted line in these plots indicates the threshold at which Fr (A) = 0.8 or r (AA) = 0.8.
AL
(a) When c 0.1
(b) When a 0.5
Figure 15: Plot of optimal interventions (Ap, A.) for various т
G.3 An illustrative example: interpreting and applying the model
We provide a concrete example to illustrate how our theoretical framework can be used to diagnose and compare policy interventions.
Interpretable diagnostics. Suppose a policymaker observes persistent underrepresenttatin of disadvantaged group (e.g., female students) in selective admissions process. Given data on the overall selection rate c, group size a, and the observed representation ratio n (A), the policymaker can use our model to infer the implied valuation gap parameter p (as shown in Section G.1 p. This parameter summarizes how much lower the disadvantaged group perceíves the value of success relative to the advantaged group.
Although p is not directly observable, its interpretation transparent: attributes behavioral dispar- ities (such as lower effort investment) to structural differences in perceived incentives rather than innate ability. In this way, the model provides a normative reading of observed disparities as equilibrium responses to valuation asymmetries.
Policy interventions. Once the implied parameters are estimated, the policymaker can consider two classes of interventions:
improving the perceived value of success (e.g., through mentorship programs or financial aid), which effectively increases ni
Valuation-based interventions:
Access-based interventions: expanding the number of available slots, thereby increasing
Our framework allows simulation of the effects of each intervention on representation, welfare, and efficiency, enabling counterfactual comparisons under a fixed behavioral model. For example, when p is low, expanding access may yield larger gains representation, while when p already high, improving valuation can be more cost-effective.
Implementation challenges. Estimating parameters such p empirically is nontrivial and remains an open direction. would require auxiliary data sources (e.g., surveys, longitudinal effort-outcome data) or structural assumptions about the effort cost function. Nonetheless, once such estimates are available, our framework provides transparent scaffold for interpreting behavioral disparities and evaluating the relative effectiveness of competing policy interventions.
G.4 Details for alternative interventions
Below we provide more detailed theoretical analysis for alternative potential interventions discussed in Section[4
46

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Introducing preference heterogeneity. Recall that the institution applies group-specite merit mappings of the form: for group G2 (L = 1, and for score s, me(s) = - ye for group-specific parameters Ze, Ve 2 O. We have the following generalized theorem of Theorem 3.1
Theorem G.1 (Generalization of Theorem 3.1; Introducing preference heterogeneity). Let o,c E (0,1). For f 1,2, let Pt be density supported a domain Ω 20 Let Pe be a density supported on a domain Sh S Ry0. For i = 1,2, let my be a merit function defined above. For L = 1, 2. let Fe be cumulative density function (CDF) of the sum of valuation and initial ability such that for any Ç E Rzo, Fe(C) Premppeaape [xeU + ye + S C). Suppose (ats2 + y1) U (2-5222 32)) + Stg connected and densities pr, P2s Po are positive at any poins of their own domains. Lett t be the unique solution to the equation
- a + (F2(5) =1 -1-c
Then the threshold functtoon of the NE policy defined Eq.
(3) extends to be: for 1,2
se(v,a) = =
ye
and se(v,a) -maxx {
( (- ye
a, 0 if 1 y1
It
zt
It
Moreover, the threshold in for G2 is monotonically decreasing with 22 3/2-
The proof for is almost identical to that of Theorem 3.1 We only need to note that for any agent i E G2 (e- = 1, 2) with valuation-ability pair (vi, a:) E Pt x Saz if its valuation ii 2 f-we di. then its merit must be
Proof:
me(se a;)) = xe- (a: max (1- az of) + ve 1e22
which is within the top c-fraction and makes the agent get selected.
Regarding the monotonicity of t-pa note that as 32 or 1/2 increases, (e(() decreases. Then the .²2
solution i must increase, resulting in higher threshold I-l1 for group G1. This reduces the fraction 21 of agents in G1 to get selected, and consequently, increases the fraction of agents in G2 to get selected. Then the threshold f-W2 must decrease, which completes the proof.
Note that when #1 = x2 = 1 and y1 = y/2 = O, this theorem is exactly Theorem 3.1 and hence, is a generalization. This theorem implies that by increasing 22, 1/, more agents in G2 are willing to put in efforts due to lower valuation threshold 8-v2, This supports the properties discussed in Section 4
22
Setting group-specific selection rates. Assume that the institution selects c-fraction of agents from G₁ and G2 independently. The model decomposes into two independent within group contests, each with its own Nash equilibrium.
For the disadvantaged group G2, let F22 denote the CDF of its combined signal + a. The equilibrium threshold l2 under group-specific capacity is the unique solution to:
(2(12) =
In contrast, under uniform selection rate appllied to the full population (original two-group contest), the common threshold solves:
(1 (-a))((++ -1
Since G2 has lower valuations by assumption, we typically have F2(C) 2 Fi({) for all ć, which implies la < l. That is, the disadvantaged group faces a lower selection bar under group-specific quotas.
As result, agents in G2 exert more effort on average under per-group capacities compared the uniform-c case. This is because effort is increasing in the probability of selection, which improves when the threshold is lowered.
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Reviews:
[{"review_id": "Ba8lC0PHy8", "reviewer": "Reviewer_H86j", "summary": "To examine how selection rates can diverge across groups even under group-blind selection criteria, the authors model a contest setting in which agents rationally choose effort based on their expected benefit from selection. In this setup, selection depends on a combination of ability and effort, but only effort is chosen strategically. When the expected benefit of selection is lower for one group—modeled as a compressed distribution of valuation (captured by the parameter rho)—members of that group exert less effort on average, resulting in lower selection rates. The authors characterize equilibrium behavior in this setting and analyze how group disparities in selection, welfare, and average “quality” of selected candidates depend on key parameters: the degree of valuation compression (rho), the selectivity of the institution (cc), and the share of disadvantaged candidates in the population (alpha).", "questions": "1.\tClarify the meaning of the feedback loop- the abstract and introduction refer to a behavioral “feedback loop” as a core motivation, but the model appears static: group-based disparities arise from fixed, exogenous differences in valuations. Could you clarify whether the feedback loop is meant as part of the formal model or just the broader motivation? If the latter, it would help to revise the language to avoid suggesting a dynamic or endogenous mechanism. extensions (e.g., beliefs updating over time or feedback via algorithmic guidance tools). \n2.\tposition the paper relative to the statistical discrimination literature- the model closely resembles ideas from the economics literature on statistical discrimination. These works emphasize feedback between group behavior and institutional inference. Your model differs in that the institution is passive and group-blind, but that distinction is not clearly drawn. I recommend engaging with this literature more explicitly to clarify what is novel about your setup and what simplifying assumptions are being made. \n3.\tBetter motivate the setup and assumptions- for example, the assumption that one group derives less value from selection (modeled via a compressed valuation distribution) is plausible in some domains, but less so in others. I would encourage you to (a) clarify when this assumption holds, (b) discuss when it might be reversed, and (c) consider how the model might be adapted to allow for heterogeneity in valuations or outside options.\n4.\tIdeally, you would be able to say something about how the analysis would change if you had a strategic firm, even if you do not formally solve the model.", "limitations": "yes", "rating": 5, "confidence": 4, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "Strengths-\n\n-\tThe paper introduces an elegant and intuitive framework to capture how group-based differences in the perceived value of success (landing a job, receiving a loan, etc.) can affect effort and, in turn, selection outcomes, even when the selection rule is group-blind. While this mechanism will feel familiar to economists as a classic asymmetric-contest setup, it offers a valuable perspective for a CS/ML audience, where fairness analyses often focus narrowly on constraints and selection rules. The model highlights that disparities can arise endogenously when valuations differ across groups.\n-\tThe Nash equilibrium solution, where effort is exerted only by individuals whose valuation plus ability exceed a common threshold, is analytically clean and well suited to quantifying disparities. The resulting inequalities in selection rates and payoffs are both interpretable and empirically relevant.\n-\tThe authors clearly explain the role of key parameters in the model (such as valuation bias rho, selection rate c, and group proportion alpha) and how changes in these parameters affect the fairness metrics.\n-\tThe paper is well-motivated by real-world phenomena, such as wage gaps and unequal returns to opportunity, that suggest group-based differences in the perceived value of “winning” are empirically plausible and normatively important.\n-\tOverall, the paper brings a less-studied dimension of fairness into focus: how group disparities in incentives or perceived benefit can drive unequal outcomes even in the absence of explicit discrimination or biased selection.\n\nWeaknesses \n\n-\tTo some extent, the paper’s strength is also its weakness: while the analysis is elegant and clean, it may be too stylized to capture important real-world dynamics. In particular, it abstracts away from richer feedback mechanisms and strategic institutional behavior that are well-studied in the economics literature on statistical discrimination.\n-\tFor example, the paper repeatedly invokes the idea of a “feedback loop,” but it is unclear what that loop entails. The model is static, and group disparities arise entirely from exogenous differences in valuations. That is, differences in selection rates do not feed back into future valuations, institutional behavior, or perceptions, making the use of “feedback” more rhetorical than actually part of the model.\n-\tSurprisingly, the paper does not engage with the economics literature on statistical discrimination, despite addressing similar questions. While the authors may not come from an economics background, their model aligns more naturally with this tradition than with many of the social science works cited. Foundational work such as Phelps’s “The Statistical Theory of Racism and Sexism” (1972), Arrow’s “The Theory of Discrimination” (1973), Coate and Loury’s “Will Affirmative-Action Policies Eliminate Negative Stereotypes?” (1993), and Craig and Fryer’s “Complementary Bias: A Model of Two-Sided Statistical Discrimination” (2013) examine settings where disparities emerge endogenously from a feedback loop between agent behavior and institutional inference. These models typically feature rational expectations equilibria in which institutions interpret noisy signals differently across groups, reinforcing group-level behavioral patterns.\n-\tMore broadly, engaging with this literature would not only help position the contribution more clearly but also point to natural extensions. For instance, the current model assumes that institutional payoff is a direct function of observable scores. However, in many settings, especially in hiring or lending, payoff depends on unobserved or latent characteristics like ability (or creditworthiness), and effort merely increases the signal. Modeling the institution as one that tries to infer ability from scores (and does so differently for each group) would allow the paper to explore how strategic inference might perpetuate or mitigate disparities. It would also bring the institutional actor more clearly into focus, rather than treating it as a passive selector.\n-\tAnother potential limitation is the assumption that the disadvantaged group has lower valuations. While this is plausible in some setting, for example, where minorities being treated poorly at work (created a lower value to the job, assuming job climate is part of the evaluation), there are others (like lending) where the opposite may be true: marginalized groups might derive higher value from access to opportunity (loans or jobs), due to lack of outside options or greater marginal returns. It would be helpful for the authors to clarify when this compressed valuation assumption is realistic and when it might not be.\n-\tRelatedly, the model assumes that differences in valuation are problematic, but it is worth considering when they may reflect legitimate preference heterogeneity. For example, if one group places more value on work-life balance and therefore exerts less effort for high-paying jobs, should we interpret the resulting selection rate disparities as unfair? The paper might benefit from more discussion of when group differences in effort due to valuation gaps are normatively troubling.\n-\tThe assumption of a threshold strategy—where agents either exert just enough effort to be selected or none at all—also limits applicability. In many real-world contests (like job applications, school admissions), individuals may over-prepare or exert continuous effort for marginal improvements, especially when downstream rewards like salary or financial aid are continuous rather than binary. This opens the door to modeling richer selection environments, such as multi-stage contests or settings with heterogeneous reward structures.\n-\tOverall, while these issues do not undermine the core contribution, the paper would benefit from a more nuanced discussion of its simplifying assumptions, clearer positioning relative to existing models of discrimination and effort, and acknowledgment of avenues for future work.", "was_revised": true, "final_justification": "I appreciate the author comments and continue to believe this is a strong paper that should be accepted for publication.", "rebuttal": "Thank you for your thoughtful and constructive feedback. We are grateful for your recognition of the paper’s modeling contributions, and we appreciate your detailed suggestions for further improvement, particularly regarding assumptions and connections to prior work. Below we address your questions:\n\n> 1. Clarify the meaning of the feedback loop? ... \n\nThank you for raising this point. To clarify: the “feedback loop” is not part of the formal model, but rather serves as a motivating narrative for why disparities in perceived value can matter. The model itself is static—group-based disparities arise from fixed, exogenous differences in perceived post-selection value, with no endogenous belief updating or institutional dynamics.\n\nWhat we intended to highlight was the following phenomenon: when one group expects lower returns (e.g., due to underrepresentation or systemic inequities), its members rationally exert less effort, which then leads to continued underrepresentation—even under group-blind selection.\n\nWe agree that the phrase “feedback loop” may suggest dynamic or time-evolving mechanisms that our model does not capture; we will remove it from the abstract and introduction and explicitly state that the model is static.\n\n>2. position the paper relative to the statistical discrimination literature ...  \n\nThank you for the references on statistical discrimination. Below we give a brief comparison. We will add a dedicated subsection in the final version to clarify connections and distinctions.\n\n- [Phelps 1972] models a profit-maximizing employer who faces noisy signals of productivity and rationally uses group-level statistics, leading to persistent wage gaps even with equal underlying abilities.\n- [Arrow 1973] emphasizes how the cost of individualized assessment incentivizes reliance on priors, which can become self-fulfilling and reinforce structural inequality.\n- [Coate and Loury 1993] show that pessimistic beliefs about a group’s productivity can result in tougher standards, reduced investment incentives, and discriminatory equilibria.\n- [Craig and Fryer 2013] extend this to two-sided settings where firms and workers both act on noisy beliefs, reinforcing low-investment, low-opportunity equilibria.\n\nA key distinction, as we understand it, is that classical models of statistical discrimination typically generate disparities through **imperfect** and **group-dependent** beliefs about identical underlying abilities. In contrast, our framework allows **perfect, unbiased** information at the institutional level and identical selection criteria for all candidates. We focus instead on **valuation asymmetries**—that is, differences in the *perceived benefit* of success across groups—and show that these differences alone can lead to disparities in effort and representation, even under meritocratic selection.\n\nTo our knowledge, this offers a complementary behavioral mechanism to those studied in the statistical discrimination literature—one that centers the agent's internal valuation rather than institutional inference. \n\n> 3. Better motivate the setup and assumptions ... encourage you to a) clarify when this assumption holds, b) discuss when it might be reversed, and c) consider how the model might be adapted to allow for heterogeneity in valuations or outside options. \n\nThank you for the suggestions. To clarify our setting: the core model assumes no outside option, group-blind selection based on a common score, and a binary reward for selection. While the model can be generalized to include outside options or heterogeneous selection (see more details below), even in this core form, valuation asymmetry can arise; not from individual preferences, but from group-dependent beliefs about the value derived post-selection. These beliefs may be shaped by structural inequality, lived experience, or algorithmic signals.\n\nWe now address each part of your comment:\n\n**a) When lower valuations are plausible:** Our model applies to settings in which disadvantaged groups perceive lower returns to selection due to structural or informational asymmetries. As described in the introduction (lines 39–54), wage gaps persist across gender and race even after controlling for qualifications, and algorithmic systems often return lower-valued recommendations for marginalized groups. Such disparities shape expectations about the benefits of success and can depress effort before selection. These are the environments our model aims to capture.\n\n**b) When the asymmetry might reverse:** Indeed, in some settings (particularly in access-to-opportunity domains such as credit, housing, or education) disadvantaged groups may face higher marginal benefits from selection due to fewer outside options or steeper returns. Our model accommodates such cases by reversing which group has the compressed valuation distribution. However, our motivating focus and analysis remains on the scenario of lower perceived returns among disadvantaged groups.\n\n**c) Preference heterogeneity and outside options:** While our core model assumes homogeneous evaluations and no outside option, the framework can be extended to derive insights about this setting that institutions can strategically reduce the disparities caused by heterogeneity in valuations. For example, consider the setting where institutions may apply group-specific merit mappings of the form: for group $G_\\ell$ ($\\ell = 1,2$) and for score $s$, \n$$m_\\ell(s) = x_\\ell \\cdot s + y_\\ell$$  \nfor group-specific parameters $x_\\ell, y_\\ell \\geq 0$. One can interpret $x_\\ell$ as a *scaling factor* or \"handicap\", and $y_\\ell$ as an *offset* or \"headstart\" (see Appendix A and [75]).\n\n- **Differences to homogeneous case:** For the homogeneous case, we show that 1) both representation ratio $r_{\\mathrm{R}}(A)$ ($A$ is the NE policy defined in Theorem 3.1) and the social welfare ratio $r_{\\mathcal{S}}(A)$ are monotonically increasing functions of bias parameter $\\rho$, selectivity $c$ and disadvantaged fraction $\\alpha$; 2) in highly selective environments (e.g., $c=0.1$), a slight bias ($\\rho\\leq 0.85$) leads to high disparities ($r_{\\mathrm{R}}(A)\\leq 0.2$ and $r_{\\mathrm{S}}(A)\\leq 0.2$); 3) Increasing $\\rho$ and $c$ are two intervention ways to reduce disparities. \nBy the above extension, the findings for the homogeneous case changes to be: 1) $r_{\\mathrm{R}}(A)$ and $r_{\\mathcal{S}}(A)$ are not only increasing functions of $\\rho,c,\\alpha$, but also increasing functions of $x_2,y_2$ and decreasing functions of $x_1,y_1$; 2) By selecting merit parameters with $x_2 > x_1$ and $y_2 > y_1$, disparities can remain low (e.g., $r_{\\mathrm{R}}(A)\\geq 0.8$ and $r_{\\mathrm{S}}(A)\\geq 0.8$) even in highly selective environments; 3) Besides increasing $\\rho$ and $c$, increasing the value of $x_2,y_2$ is the third intervention way to reduce disparities. \n\n- The intuition behind these changes is that agents from $G_2$ are easier to select with the same score $s$ compared to the homogeneous case due to boosted merits, increasing their investments and resulting in higher selection from $G_2$. This intuition can also be verified mathematically since our techniques extend naturally to this setting and yield a NE characterization similar to Theorem 3.1:\n\n**Theorem.**  Let $\\alpha$ be the fraction of $G_2$ in the population and $c$ is the selectivity. For $\\ell = 1,2$, let $p_\\ell$ be a valuation density of $G_\\ell$ and $m_\\ell$ be a merit function defined above. Then there exists a unique $t$ such that the group-specific effort policies are as follows: for an agent from group $G_\\ell$ and any valuation $v$, \nif $v < \\frac{t - y_\\ell}{x_\\ell}$, then $s_\\ell(v) = 0$; otherwise, $s_\\ell(v) = \\max \\left( \\frac{t - y_\\ell}{x_\\ell}, 0\\right)$.\n\nMoreover, the threshold $\\frac{t - y_2}{x_2}$ for valuation $v$ is monotonically decreasing with $x_2,y_2$.\nThis theorem implies that by increasing $x_2,y_2$, more agents in $G_2$ are willing to put in efforts due to a lower valuation threshold $\\frac{t - y_2}{x_2}$, consistent with the above intuition.  \n\nAdditionally, our framework can incorporate **outside options**.  Let each agent in group $G_\\ell$ receive a reservation payoff $\\lambda_\\ell \\ge 0$ if she is *not* selected.  Because this payoff is earned only when losing, a higher $\\lambda_\\ell$ lowers the marginal benefit of exerting effort, working in the *opposite* direction of the merit parameters $x_\\ell$ and $y_\\ell$. Hence, increasing $\\lambda_1$ (the outside option for the advantaged group) depresses their effort incentives and can be used as an additional intervention for narrowing representation and welfare gaps.\n\nWe will clarify points (a) and (b) in the main body and add the insights along with accompanying results mentioned for \\(c) in the appendix.\n\n> 4. ... strategic firm, even if you do not formally solve the model.\n\nThank you for the suggestion. While our core model assumes a passive institution applying a fixed, group-blind selection rule, the extended framework mentioned above (in Question 3c) naturally accommodates *strategic* settings—where the institution selects group-dependent mappings of the form $m_\\ell(s) = x_\\ell \\cdot s + y_\\ell$ to influence contest outcomes.\n\nIn this view,  $x_\\ell$ and $y_\\ell$ serve as *strategic choices* that determine how agent scores are mapped to merits across groups. Given our closed-form characterization of the NE in this setting, one can formulate optimization problems, e.g., \n\n$\\max_{x_1, x_2, y_1, y_2} \\mathcal{S}(A(x_1, x_2, y_1, y_2))$ subject to $r_{\\mathcal{R}}(A(x_1, x_2, y_1, y_2)) \\geq \\tau$,\n\nwhere $\\mathcal{S}(A)$ denotes the social welfare of the selected set under NE policy $A$, $r_{\\mathcal{R}}(A)$ is the representation ratio, and $\\tau \\in [0,1]$ is a fairness threshold.\n\nWhile this lies beyond the scope of our current analysis, our results make such design problems tractable, and we will note this as an avenue for future work.\n\nThanks again for your detailed feedback."}, {"review_id": "5kyrpqOWW1", "reviewer": "Reviewer_uD1e", "summary": "This paper studies a contest with two demographic groups. The groups differ only in their perception of how valuable winning the contest is. This is interpreted as the disadvantaged group having a lower valuation of the win, and how that impacts different performance criteria, including the representation ratio, the welfare ratio, and the institution’s revenue.", "questions": "1. How is the model or its results reflecting on the underlying “structural” and societal issues?\n\n2. The paper considers the representation ratio to be the min of the representation ratios of the groups. Isn’t it always the case that the disadvantaged group ends up being underrepresented in the outcome? And if not, why not?\n\n3. It seems that the model can be used to answer an additional question: if the institution sets per-group capacities (i.e., a group specific selection rate c), then how would the effort the disadvantaged group exert compare to the one when they are under a uniform c rate? (even keeping the valuations asymmetric). Is this correct? It would be interesting to know more about this aspect as well. \n\n4. Section 8, line 299, mentions some default values when some parameters are not being varied. It is assumed then that the corresponding results sensitive to these choices? For instance, is the takeaway at the end of the section (“This suggests that expanding access is more impactful under high disparity, while improving group valuation is better when gaps are narrower.” ) independent of the \\alpha and c selected based on the JEE Advanced data?\n\n5. Typos: line 127, a missing comma in the definition of the representation ratio. Line 247: “is to that” --> “is that”", "limitations": "Yes", "rating": 5, "confidence": 2, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 4, "significance": 3, "originality": 3, "strengths_and_weaknesses": "**Strengths**:\n- The paper provides a new perspective on the issue of disparities in competitive selection processes (like school admissions), by considering asymmetric valuations. It is also different from the literature on strategic classification because of competition between agents, and therefore complements that line of work well, too. \n- The paper is quite well-written. The intuitive walkthrough of the main theorem is clear and informative. The sensitivity analysis corollaries and experiments are well-thought. \n- Several of the provided insights from the theoretical analysis have the potential to be informative for questions of contest/institution design and interventions to encourage underrepresented minorities to consider exerting effort for competitive opportunities. \n\n**Weaknesses**:\n- Some of the statements in the introduction and their alignment with the results are unclear. Specifically, the paper does open by noting that structural barriers are a key driver of disparity in representation rates during competitive selection processes, including “unequal access to resources that enhance merit”. The closest element of the model to these, to me, is p_a, the distribution of initial ability. However, this is taken to be group-independent throughout the paper. Beyond this, it is not clear if there are elements in the model that are capturing the “structural” rather than “behavioral” aspects. If no such \"structural\" elements exist in the model, then that makes it unclear why “This paper reconciles structural and behavioral explanations”, as stated on top of page 2. \n\n- Some additional requests for clarification and questions are added below.", "was_revised": true, "final_justification": "The rebuttal has been useful in providing clarifications to my questions, especially the additional discussions and results provided around group-specific selection rates and sensitivity analysis for the experiments. I believe this is a technically solid paper with interesting insights, and accordingly maintain my score at Accept.", "rebuttal": "We thank you for your detailed and encouraging review. We are especially grateful for your recognition of our framework’s real-world applicability. Below we address your specific questions.\n\n>How is the model or its results reflecting on the underlying “structural” and societal issues?\n\nThank you for this thoughtful question. We appreciate the opportunity to clarify how our model captures the distinction between structural and behavioral explanations.\n\nIn our framework, structural disparities are represented by group-dependent valuation distributions—that is, differences between $p_1(v)$ and $p_2(v)$ that reflect external and systemic inequalities. These may stem from unequal access to opportunity, differences in marginal returns, labor market discrimination, or broader societal narratives about value.\n\nBy contrast, behavioral disparities emerge endogenously in our model: given these structural valuation gaps, agents rationally adjust their effort levels. Even when ability distributions $p_a$ are identical across groups, the perceived return to effort is lower for disadvantaged groups, leading to systematically lower effort—and thus to disparities in selection rates.\n\nIn this way, our model illustrates a concrete feedback from structural conditions to behavioral outcomes. This is what we mean by reconciling the two perspectives: our analysis shows how externally imposed valuation asymmetries can, even under unbiased and group-blind selection, result in behavioral underrepresentation.\n\nWe will clarify this interpretation in Section 1 of the final version.\n\n\n> The paper considers the representation ratio to be the min of the representation ratios of the groups. Isn’t it always the case that the disadvantaged group ends up being underrepresented in the outcome? And if not, why not?\n\n\nThank you for the question. You are correct that in the infinite population limit—our main analytical regime—the disadvantaged group is always underrepresented due to lower valuations, and the representation ratio reflects this.\n\nIn the finite-$n$ case, however, randomness can occasionally lead to equal or even higher representation for the disadvantaged group, though this becomes vanishingly rare as $n \\to \\infty$.\n\nWhile the qualitative direction of underrepresentation may not be surprising, our contribution lies in quantifying how strategic behavior amplifies these disparities under varying selection pressures and valuation gaps. This allows us to make precise, interpretable predictions and compare interventions within a unified framework.\n\nWe will clarify this in the final version.\n\n\n> It seems that the model can be used to answer an additional question: if the institution sets per-group capacities (i.e., a group-specific selection rate c), then how would the effort the disadvantaged group exert compare to the one when they are under a uniform c rate? (even keeping the valuations asymmetric). Is this correct? It would be interesting to know more about this aspect as well.\n\nThank you for the insightful question. Yes, your interpretation is correct: if the institution sets **group-specific selection rates**—i.e., separate capacity constraints for each group—then the model decomposes into two **independent within-group contests**, each with its own Nash equilibrium.\n\nFor the disadvantaged group $G_2$, let $F_2$ denote the CDF of its combined signal $v+a$. The equilibrium threshold $t_2$ under group-specific capacity $c$ is the unique solution to:\n$$\nF_2(t_2) = 1 - c.\n$$\n\nIn contrast, under a **uniform** selection rate $c$ applied to the full population, the common threshold $t$ solves:\n$$\n(1-\\alpha) F_1(t) + \\alpha F_2(t) = 1 - c.\n$$\n\nSince $G_2$ has lower valuations by assumption, we typically have $F_2(\\zeta) \\geq F_1(\\zeta)$ for all $\\zeta$, which implies $t_2 < t$. That is, the disadvantaged group faces a **lower selection bar** under group-specific quotas.\n\nAs a result, agents in $G_2$ exert **more effort** on average under per-group capacities compared to the uniform-$c$ case. This is because effort is increasing in the probability of selection, which improves when the threshold is lowered.\n\nWe agree that this highlights an interesting direction for intervention, and we will include a remark about this in the final version.\n\n\n> Section 8, line 299, mentions some default values when some parameters are not being varied. It is assumed then that the corresponding results sensitive to these choices? For instance, is the takeaway at the end of the section (“This suggests that expanding access is more impactful under high disparity, while improving group valuation is better when gaps are narrower.” ) independent of the \\alpha and c selected based on the JEE Advanced data?\n\nThank you for pointing this out. We agree that it is important to assess whether our core conclusions depend on the specific parameter settings (e.g., $\\alpha$, $c$) derived from the JEE Advanced data.\n\nTo verify robustness, we conducted additional simulations varying $\\alpha$ and $c$ beyond the default values. Below we summarize our findings:\n\n- **Metric robustness across group sizes:** We varied $\\alpha$ from 0.5 to 0.3 (to represent smaller disadvantaged groups) and recalculated the representation ratio $r_{\\mathcal{R}}(A)$ and welfare ratio $r_{\\mathcal{S}}(A)$ across a grid of disparity levels ($\\rho$) and selection rates ($c$). The overall trends remain consistent: for example, $r_{\\mathcal{R}}(A)\\leq 0.2$ still holds for $c=0.1$ and $\\rho \\leq 0.85$, confirming that strategic behavior amplifies underrepresentation in highly selective settings (cf. Lines 314–317).\n- **Robustness of intervention takeaways:** We varied $c$ from 0.228 to 0.1 and $\\alpha$ from 0.268 to 0.5 and re-evaluated intervention strategies. The takeaway remains robust:  \n  - When $\\tau \\leq 0.87$, increasing access (raising $c$) remains more cost-effective.  \n  - When $\\tau > 0.87$, improving group valuation (increasing $\\rho$) becomes more impactful.  \n  This confirms that the recommendation to prioritize access vs. valuation interventions depending on the disparity level remains valid across reasonable choices of $\\alpha$ and $c$.\n\nWe will include these robustness checks and associated plots in Appendix G of the final version.\n\nWe thank you again for your thoughtful engagement, which has helped us clarify and strengthen the contributions of our work."}, {"review_id": "ke18zidSCb", "reviewer": "Reviewer_rBtT", "summary": "This paper investigates strategic asymmetries arising from valuation differences in settings where group sizes are known and fixed, such as admissions and hiring. The authors propose a formal framework to model these strategic interactions, aiming to provide interpretable predictions and support data-grounded policy interventions. The paper presents a theoretical analysis, including definitions, theorems, and discussions of metrics.", "questions": "1. Could you elaborate on the specific aspects of your framework that enable \"interpretable predictions\" and \"data-grounded policy interventions\"? Perhaps provide a brief, concrete example of how a policy maker could use your framework's outputs.\n2. Could you briefly summarise in the main text (e.g., in the introduction or related work section) the precise novel contribution of your work especially in contrast to the most closely related prior research?", "limitations": "Yes", "rating": 5, "confidence": 3, "soundness": null, "presentation": null, "contribution": null, "strengths": "", "weaknesses": "", "quality": 3, "clarity": 3, "significance": 3, "originality": 3, "strengths_and_weaknesses": "### Strengths\n* S1 – The paper addresses a highly relevant and important problem concerning strategic behavior in critical application areas like admissions and hiring. The proposition that valuation differences lead to strategic asymmetries in fixed-group settings is compelling and aligns with real-world observations. I believe this is a very important and underresearched area of algorithmic fairness literature, focus on formal neutrality and static metrics leave much underexplored in fair selection. By focusing on valuation gaps, the work complements the dominant literature on classifier bias and post-hoc fairness metrics. \n\n* S2 – Clear Presentation of Theoretical Framework: While the mathematical depth is significant, the paper is generally well-written, well-cited, and the notation is clear. The definitions and theorems are set out lucidly, making the overall structure and conceptual flow of the theoretical framework understandable, even without deep expertise in game theory. The metrics introduced in Section 4 are clearly defined and appear appropriate for the problem context, drawing on established concepts in the relevant literature.\n\n### Weaknesses\n* W1 – While the authors claim their work is the first to study strategic asymmetries arising from valuation differences in settings with known and fixed group sizes (a compelling claim if true), the main body of the paper does not sufficiently articulate how its specific contributions differentiate it from existing literature. Although Appendix A provides detailed related work, a concise explanation of the paper's novelty should be integrated into the main text to immediately clarify its unique contribution for the reader. Despite reviewing supplementary material it still isn't perfectly clear to me what exactly is unique about your method. \n\n* W2 – The paper states that its \"framework enables interpretable predictions and supports data-grounded policy interventions.\" However, it does not explicitly elaborate on how this framework achieves these goals. While concrete guidance on parameter estimation (e.g. recovering ρ from survey or wage data) is deferred to future work, more detailed discussion or concrete examples illustrating the path from theoretical insights to practical policy implications would strengthen this claim.\n\n* W3 - Real-world fit is uncertain. The theory assumes very large applicant pools and that every person always chooses the effort level that maximises their payoff. In practice, many selection rounds involve only a few dozen or a few hundred candidates, and people often make decisions based on limited information, habits, or simple rules of thumb. Because of this, the neat patterns the model predicts might look quite different when applied to smaller, messier real-world settings.", "was_revised": true, "final_justification": "I am confident with my positive score and appreciate the detailed rebuttal from the reviewers. I am confident that with the additional details from the rebuttal, the paper will be improved and should make a valuable contribution.", "rebuttal": "We thank you for your thoughtful, detailed, and insightful feedback. In response, we added an illustrative example to elaborate on the applicability of our framework and addressed your specific questions and a comment (W3).\n\n> Could you elaborate on the specific aspects of your framework that enable \"interpretable predictions\" and \"data-grounded policy interventions\"? Perhaps provide a brief, concrete example of how a policy maker could use your framework's outputs.\n\nThank you for this insightful question. We agree that the connection between our theoretical framework and its practical applications warrants clearer articulation. Below, we offer a concrete example to illustrate how our model supports interpretable predictions and can inform data-grounded interventions—while also noting what is required to operationalize it in practice.\n\n**Interpretable predictions:** Suppose a policymaker observes persistent underrepresentation of a disadvantaged group (e.g., female students) in a selective admissions process. Given data on the overall selection rate ($c$), group size ($\\alpha$), and the observed representation ratio $r_{\\mathcal{R}}(A)$, the policymaker can use our model (e.g., Lines 365–372) to infer an implied valuation gap parameter ($\\rho$). This parameter summarizes how much lower the disadvantaged group perceives the value of success relative to the advantaged group.\n\nAlthough $\\rho$ is not directly observable, its interpretation is transparent: it explains behavioral disparities (e.g., lower effort investment) as arising from structural differences in perceived incentives. This enables a normative interpretation of observed disparities—not simply as gaps in ability, but as strategic responses to valuation asymmetries.\n\n**Policy interventions:** With this diagnosis, the policymaker can consider and compare two classes of interventions:\n\n- **Valuation-based interventions**: improving perceived value of success (e.g., through financial aid, mentorship), effectively increasing $\\rho$.\n- **Access-based interventions**: expanding the number of available slots, thereby increasing $c$.\n\nOur framework (cf. Figure 3 and Section 8) allows one to simulate the effects of each intervention on representation and welfare, and to compare their relative cost-effectiveness. For example, when $\\rho$ is low, expanding access may be more impactful; when $\\rho$ is already high, improving valuation may yield better gains. This enables counterfactual comparisons under a fixed behavioral model.\n\nWe acknowledge that empirically estimating parameters—especially $\\rho$—is challenging and remains an open direction. It would require auxiliary data sources (e.g., surveys, longitudinal effort-outcome observations) or structural assumptions on effort cost. Nonetheless, our framework offers a transparent scaffold: once estimates are in place, it enables interpretable diagnostics and structured policy evaluation.\n\nWe will include this example and discussion in Appendix G of the final version.\n\n> Could you briefly summarise in the main text (e.g., in the introduction or related work section) the precise novel contribution of your work, especially in contrast to the most closely related prior research?\n\nThank you for the suggestion. We agree that a concise summary of our novel contributions relative to prior work should appear in the main text and will incorporate this in Section 1.\n\n**Our key contribution** is to show how group-level disparities can arise endogenously from **valuation asymmetries**, even when ability is identical, information is unbiased, and selection criteria are uniform. This provides a new structural explanation for representation gaps in competitive settings with fixed group sizes.\n\n**Compared to prior two-group contest models**:  \n- [1] studies a two-player contest with heterogeneous valuations; we generalize this to many agents and multiple selection spots.  \n- [24,27] analyze two-group contests but assume symmetric valuation distributions; we are the first to study equilibrium behavior under **asymmetric valuation distributions** across groups in such settings.\n\n**Compared to prior work on statistical discrimination**:  \n- Classical models [4,46] generate disparities via **biased beliefs** or **informational asymmetries** from the institution side, even when true abilities are equal.  \n- In contrast, our model assumes **perfect information** and **equal evaluation** for the institution, but introduces valuation asymmetries—different returns to the same success—that alone generate group disparities in effort and selection.\n\nWe will add this comparison to the introduction to clarify the novelty and position our work relative to the most closely related literature.\n\n\n> W3 - Real-world fit is uncertain. The theory assumes very large applicant pools and that every person always chooses the effort level that maximises their payoff. In practice, many selection rounds involve only a few dozen or a few hundred candidates, and people often make decisions based on limited information, habits, or simple rules of thumb. Because of this, the neat patterns the model predicts might look quite different when applied to smaller, messier real-world settings.\n\n\nThank you for raising this concern about the realism of the infinite-agent assumption. These assumptions are common in theoretical work for tractability, but as you rightly point out, they may not fully capture the messiness of real-world decision-making. However, in addition to using the large-$n$ limit to characterize Nash equilibrium policies, our results show that the infinite-population approximation is **quantitatively accurate** even at moderate population sizes.\n\nAs shown in Section 3 (Lines 229–236) and Appendix C.2, the value of the finite-$n$ Nash equilibrium policy is provably $O(\\sqrt{\\log n / n})$-close to that of the infinite-population policy. Furthermore, Figure 1 demonstrates that for population sizes as small as $n = 600$, the observed equilibrium policies closely track the infinite limit across all selection rates $c$. This supports the use of the infinite-$n$ model as a practical and robust approximation for many real-world settings involving hundreds of candidates.\n\nWhile real-world decision-making can of course involve noise, habits, or bounded rationality, our model aims to capture the **incentive-aligned baseline** under strategic behavior. We believe this remains a highly informative reference point, even when actual behavior deviates from full rationality.\n\nWe will clarify this justification in the final version.\n\nOnce again, we thank you for your detailed feedback and for helping us improve the quality and impact of our work."}]

Author rebuttal:

Meta-review: The paper studies game theoretic dynamics in contests where agents have valuation asymmetries (even when they have identical abilities). They show how group-level disparities arise from just this factor, even with unbiased information and uniform selection criteria over all groups, where they consider Nash equilibria as their solution concept. They make the key assumption that group sizes are fixed and known, which is the case in some selection contests in the real world (e.g., admissions and hiring). 

Strengths:
- The paper studies a well-motivated and interesting topic. Its model is clean and succinct, and the Nash equilibrium solution is analytically clean and fits the line of inquiry.
- The authors present their work well and provide high-level intuition of key results.

Weaknesses:
- Some of the reviewers brought up good points about insufficiently deep literature review (rBtT), ambiguous wording of claims (H86j, uD1e), and confusing terminology (all); the authors promised to address these in the camera-ready version.
- H86j ad rBtT aptly note that the stylized model is also probably not sufficient to capture real-world dynamics in the setting they consider. The authors rebut this by pointing out that they ran simulations to buttress their theoretical results.
Decision: Accept (spotlight)