Source: matching_market_mechanics.md
This document details the agent-based modeling approach used to simulate the decentralized matching market of elite US college admissions.
The demand side represents ~4,000+ high-achieving student agents competing for a scarce resource (seats at 30 elite universities). Their behavior is governed by three primary sub-systems.
Students are generated using a Cholesky-decomposed bivariate normal distribution to model the correlation between GPA and SAT scores ($\rho = 0.65$). This ensures that while stats are correlated, "stat-mismatch" profiles (high GPA/low SAT) exist realistically.
Every student is assigned one of eight Archetypes, which act as behavioral profiles: - STEM/Humanities Spikes: High application counts (12–16), aggressive "reach" strategies, and specific fit-bonuses for technical or liberal arts colleges. - Recruited Athletes: Low application counts (4–8), massive "hook" multipliers, and lower academic thresholds. - Legacy/Development: High probability of "Donor" status (35%) and a guaranteed inclusion of their legacy school on their list. - First-Gen/URM: Modeled with specific essay quality boosts and income-stratified SAT offsets.
Students do not apply randomly. They use a Utility Model to select a target list of $K$ schools (where $K$ is lognormally distributed by archetype). The utility of a college $i$ for student $j$ is defined as:
$$U_{i,j} = \text{Prestige}i + \text{Fit}{i,j} + \text{Legacy}{i,j} + 5 \times \ln(P(\text{admit}){i,j})$$
Students optimize for the Early Decision (ED) Boost. The simulation assumes students are "Early-Aware": - They identify their #1 utility choice and apply ED (binding) or REA (restrictive) if the school offers it. - This creates a "Commitment Trap" that colleges exploit to manage their supply-side risk.
The supply side represents 30 institutional agents optimizing for "Class Composition," "Academic Prestige," and "Yield Stability."
Colleges evaluate applications in Logit Space (log-odds). This allows for additive "tips" without the exponential blow-up of raw percentages.
$$\text{Logit}(\text{admit}) = \frac{\text{AI Component} - 46}{20} + \text{Feeder Bonus} + \text{Hook Tips} - \text{Threshold}_i + \text{Noise}$$
Supply-side agents are defensive. To protect their "Yield" (the % of admitted students who enroll), certain schools apply a penalty to overqualified applicants who lack a significant hook:
$$\text{If } \Delta\text{AI} > 25 \text{ and No Hook} \implies \text{Score} \times (1 - \text{Strength})$$
This models the institutional fear that a "perfect" student is merely using the school as a safety and will likely poach a seat from a student more likely to attend.
Colleges manage the "Meltdown Risk" (under-enrollment) through ED Fill Rates: - High-prestige/High-yield schools (Harvard/Stanford) fill only ~20–30% early. - "Vulnerable" elite schools (Middlebury/Duke/Northwestern) fill 45–60% of their class early to lock in half their class before the Regular Decision "poaching" begins.
Because the US system is decentralized (unlike the UK's UCAS or a centralized Gale-Shapley match), the market "clears" through a series of waves.
Reference: Calibrated from Common Data Sets (2024-2025), Chetty et al. (2023), and SFFA v. Harvard Trial Testimony (Arcidiacono).