Fixed Points and Stochastic Meritocracies: A Long-Term Perspective

TL;DR

The paper analyzes long-run fairness in meritocratic, scarcity-driven admissions using a stylized inter-generational two-group, two-type model with capacity $C=(2N)\alpha$. It develops a deterministic transition operator $\mathcal{T}$ and studies two regimes: Equal Advantage (EA), where no group-specific post-admission boosts exist, and Affinity Advantage (AA), where the leading group gains an affinity boost $\epsilon$ for non-attendees. Under EA, there is a unique fixed point $x_A=x_B=\alpha p$ in the under-subscribed regime, and parity emerges in the long run though stochastic fluctuations can cause temporary separations, especially for small $N$. Under AA, a small $\epsilon$ can produce enduring disparities, with a threshold $\tilde{\epsilon}=\tfrac{2\alpha(1-p)}{(1-2\alpha)}$ determining whether the fixed point yields sustained separation (over-subscribed) or under-subscribed persistence. The authors corroborate these findings with a richer simulation model using continuous abilities and Gamma-distributed gains, demonstrating robustness and highlighting policy implications for dynamic fairness interventions in resource-scarce, merit-based systems.

Abstract

We study group fairness in the context of feedback loops induced by meritocratic selection into programs that themselves confer additional advantage, like college admissions. We introduce a novel stylized inter-generational model for the setting and analyze it in situations where there are no underlying differences between two populations. We show that, when the benefit of the program (or the harm of not getting into it) is completely symmetric, disparities between the two populations will eventually dissipate. However, the time an accumulated advantage takes to dissipate could be significant, and increases substantially as a function of the relative importance of the program in conveying benefits. We also find that significant disparities can arise due to chance even from completely symmetric initial conditions, especially when populations are small. The introduction of even a slight asymmetry, where the group that has accumulated an advantage becomes slightly preferred, leads to a completely different outcome. In these instances, starting from completely symmetric initial conditions, disparities between groups arise stochastically and then persist over time, yielding a permanent advantage for one group. Our analysis precisely characterizes conditions under which disparities persist or diminish, with a particular focus on the role of the scarcity of available spots in the program and its effectiveness. We also present extensive simulations in a richer model that further support our theoretical results in the simpler, stylized model. Our findings are relevant for the design and implementation of algorithmic fairness interventions in similar selection processes.

Figures (12)

• Figure 1: Sample trajectories of the fraction of high-types in each group (top) and the fraction of students belonging to each group out of all college admissions (bottom) over time. Left (over-subscribed, $(X_A(0),X_B(0))=(0.1,0.7)$) and right (under-subscribed, $(X_A(0),X_B(0))=(0.1,0.4)$) panels illustrate different starting points for the system. Parameter combination for runs: $p=0.90$, $q=0.40$, $\alpha=0.30$. In both cases, the system converges to $x_A=x_B=\alpha p=0.27$ (top), with each group occupying half the college seats in the long run (bottom).
• Figure 2: Average maximum separation $\mathbb{E}[\max_{t\le T}|\Delta(t)|]$ achieved versus population size $N$ for different capacity parameters $\alpha$ ($T=100, p=0.90, q=0.40$). The left panel shows the trend for smaller $N$ ($10 \le N \le 100$). The right panel highlights that the trend continues for large $N$ --- the expected maximum level of separation achieved over $T$ steps continues to shrink and approaches zero. At low capacities, the extent of separation is significant even for moderate $N$.
• Figure 3: Sample trajectories showing the impact of adding a small affinity advantage ($\epsilon=0.03$): in the EA model (left), parity is restored, while in the AA model (right), the same advantage produces persistent separation under identical starting conditions.
• Figure 4: Equilibrium separation ($\Delta$, solid, blue, with markers) and the lower bound on separation (dashed, orange) from Theorem \ref{['thm:affinity']} as a function of the affinity advantage $\epsilon$. The vertical line marks the threshold value $\tilde{\epsilon} = \frac{2\alpha(1-p)}{(1-2\alpha)}$. Parameter combinations: $T=500, \alpha=0.30, p=0.90, q=0.0$. For very small $\epsilon$, the lower bound predicted by the theorem is vacuous, but it approximates the true equilibrium separation better for larger $\epsilon$ while matching it exactly for $\epsilon \geq \tilde{\epsilon}$. Note that separation is meaningful even for small $\epsilon$.
• Figure 5: Depiction of how affinity advantage ($\mu_\epsilon$) affects ability distributions and admission outcomes over time. We compare two identical initial groups (group A and group B) under no advantage (left, $\mu_\epsilon = \sigma_\epsilon= 0$) and with advantage (right, $\mu_\epsilon = 0.20, \sigma_\epsilon = 0.10$) at time steps $t={1, 10, 20, 99}$. The vertical red line indicates the admission threshold. On the right, the ability distribution for one group progressively stochastically dominates the other.

Theorems & Definitions (24)

• Definition 1
• Theorem 1
• proof
• Definition 2: Fixed Point
• Theorem 2
• proof
• Remark 1: Applications to real-world scenarios
• Definition 3: $\eta$-parity
• Theorem 3
• proof