Matching Market Mechanics: Supply & Demand Side Logic

matching_market_mechanics.md

Matching Market Mechanics: Supply & Demand Side Logic

This document details the agent-based modeling approach used to simulate the decentralized matching market of elite US college admissions.

1. The Demand Side (Student Agents)

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.

A. Stochastic Generation & Archetypes

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.

B. The Utility Function (List Building)

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:

$$U_{i,j} = \text{Prestige}i + \text{Fit}{i,j} + \text{Legacy}{i,j} + 5 \times \ln(P(\text{admit}){i,j})$$

  • Prestige: A tiered base score (Tier 1 = ~40 pts, Tier 5 = ~2 pts).
  • Fit: A matrix-based bonus (1–5 scale) matching archetypes to institutional "vibes."
  • Rational Expectation: Students estimate their own $P(\text{admit})$ using a simplified academic-only model. This prevents "average_strong" students from wasting all applications on HYPSM.

C. Strategic Round Selection

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.

2. The Supply Side (University Agents)

The supply side represents 30 institutional agents optimizing for "Class Composition," "Academic Prestige," and "Yield Stability."

B. Yield Protection (The "Tufts Syndrome")

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.

3. The Market Clearing (The Cascade)

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.

  1. Early Rounds: ~30% of the market clears.
  2. Regular Decision: The main "poaching" event occurs.
  3. The Waitlist Cascade:
    • Students who hold multiple acceptances choose their highest-utility option on May 1st.
    • This creates "vacuums" in the enrollment of lower-prestige schools.
    • Schools activate their Waitlist to pull students from "lower" tiers.
    • The simulation runs up to 5 cascade iterations until the market reaches a semi-stable equilibrium.

Reference: Calibrated from Common Data Sets (2024-2025), Chetty et al. (2023), and SFFA v. Harvard Trial Testimony (Arcidiacono).

Some sections containing simulation-specific implementation details have been omitted from this public version. The research data and analysis above is based on publicly available sources.