Two-Stage Stochastic Capacity Expansion in Stable Matching under Truthful or Strategic Preference Uncertainty

A team of researchers from INESC TEC and Polytechnique Montréal has published a significant paper tackling a core problem in algorithmic matching markets like school choice and medical residency placement. The issue is that in these "many-to-one" stable matching systems, capacity decisions (how many seats a school has) are made long before students submit their preferences. Traditional models assume students report their true preferences honestly, but in reality, applicants often game the system. They strategically misreport their top choices based on beliefs about their admission chances, creating what the researchers term "endogenous uncertainty."

The new research introduces a two-stage stochastic optimization framework. In the first stage, a central clearinghouse must decide on expanding school capacities, facing uncertainty about what preferences students will ultimately report. The second stage then runs the matching algorithm (like the student-optimal stable mechanism). To solve this complex problem, the team developed behavior-specific mathematical models and heuristic algorithms based on Lagrangian relaxation and local search. Their key innovation is using Sample Average Approximation (SAA) to handle the uncertainty, which significantly outperforms simpler methods like optimizing for the "average" scenario. Their simulations show this approach leads to better overall matching quality and more robust capacity plans that anticipate real-world strategic behavior.