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

Tohya Sugano

TL;DR

The paper tackles minimizing instability in two-sided strategy-proof one-to-one matching by formulating a linear program that enforces exact strategy-proofness while minimizing stability violations, under either an average-case or worst-case objective. It introduces symmetry and anonymity reductions to make the problem tractable and provides both deterministic and randomized mechanism results for small markets, including a 3×3 case where two blocking pairs are unavoidable with deterministic SP. For randomized mechanisms in 3×3, the optimized designs substantially reduce average instability relative to randomized sequential dictatorship, and the authors propose scalable generalizations to larger markets with simulations showing improvements over sequential dictatorship. The framework offers a computational path to best-possible mechanisms under incentive constraints and can extend to broader matching settings beyond the studied cases.

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.

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

TL;DR

The paper tackles minimizing instability in two-sided strategy-proof one-to-one matching by formulating a linear program that enforces exact strategy-proofness while minimizing stability violations, under either an average-case or worst-case objective. It introduces symmetry and anonymity reductions to make the problem tractable and provides both deterministic and randomized mechanism results for small markets, including a 3×3 case where two blocking pairs are unavoidable with deterministic SP. For randomized mechanisms in 3×3, the optimized designs substantially reduce average instability relative to randomized sequential dictatorship, and the authors propose scalable generalizations to larger markets with simulations showing improvements over sequential dictatorship. The framework offers a computational path to best-possible mechanisms under incentive constraints and can extend to broader matching settings beyond the studied cases.

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 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 on average.

Paper Structure

This paper contains 21 sections, 4 theorems, 35 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Related Work
  3. Preliminaries
  4. Stability
  5. Strategy-Proofness
  6. Anonymity
  7. Symmetry
  8. Non-wastefulness
  9. Individual Rationality
  10. The Optimization Problem for Matching Problems
  11. Results
  12. The Case $n=m=2$
  13. The Case $n=m=3$
  14. Deterministic Matching Mechanisms
  15. Randomized Matching Mechanisms
  16. ...and 6 more sections

Key Result

Theorem 1

For any two-sided strategy-proof randomized matching mechanism $f$, there exists a two-sided strategy-proof and anonymous randomized matching mechanism $g$ such that, for both objective functions $A$ and $B$, the objective value of $g$ is no larger than that of $f$.

Figures (2)

  • Figure 1: Comparison of Algorithm \ref{['alg3']} and SD
  • Figure 2: Comparison of anonymized and symmetric mechanisms

Theorems & Definitions (8)

  • Theorem 1
  • Theorem 2
  • Lemma 1
  • proof
  • Lemma 2
  • proof
  • proof
  • proof