Minimizing Instability in Strategy-Proof Matching Mechanism Using A Linear Programming Approach

TL;DR

This paper introduces a linear programming framework to design strategy-proof matching mechanisms that minimize instability, achieving lower violations than existing methods in small markets and providing insights for larger markets.

Contribution

It formulates the mechanism design as a linear program to minimize instability under strategy-proofness, offering new optimal and approximate solutions for various market sizes.

Findings

Reduced average instability to one third of RSD in 3x3 markets

Any two-sided strategy-proof mechanism has at least two blocking pairs in small markets

Proposed extension decreases blocking pairs by about 0.25 on average in larger markets

Abstract

We study the design of one-to-one matching mechanisms that are strategy-proof for both sides and as stable as possible. Motivated by the impossibility result of Roth (1982), we formulate the mechanism design problem as a linear program that minimizes stability violations subject to exact strategy-proofness constraints. We consider both an average-case objective (summing violations over all preference profiles) and a worst-case objective (minimizing the maximum violation across profiles), and we show that imposing anonymity and symmetry when the number of agents in both sides are the same can be done without loss of optimality. Computationally, for small markets our approach yields randomized mechanisms with substantially lower stability violations than randomized sequential dictatorship (RSD); in the $3\times 3$ case the optimum reduces average instability to roughly one third of RSD. For deterministic mechanisms with three students and three schools, we find that any two-sided strategy-proof mechanism has at least two blocking pairs in the worst case and we provide a simple algorithm that attains this bound. Finally, we propose an extension to larger markets and present simulation evidence that, relative to sequential dictatorship (SD), it reduces the number of blocking pairs by about $0.25$ on average.