[Paper Review] Stability-Preserving, Incentive-Compatible, Time-Efficient Mechanisms for Increasing School Capacity

Tung Mai, Vijay V. Vazirani|arXiv (Cornell University)|Apr 9, 2019

Game Theory and Voting SystemsEconomics, Econometrics and Finance3 citations

TL;DR

This paper proposes polynomial-time, stability-preserving, and incentive-compatible mechanisms for a two-round school choice system where cities can increase school capacity in the second round. It introduces non-oblivious mechanisms that account for non-participating residents and achieve Nash equilibrium truth-telling, overcoming limitations in dynamic, strategic environments with strategic non-participation.

ABSTRACT

We address the following dynamic version of the school choice question: a city, named City, admits students in two temporally-separated rounds, denoted $\mathcal{R}_1$ and $\mathcal{R}_2$. In round $\mathcal{R}_1$, the capacity of each school is fixed but in round $\mathcal{R}_2$, the City is happy to allocate extra seats to specific schools per the recommendation of the mechanism; in turn, the latter has to meet specified requirements. We study three natural settings of this model, with the requirements getting increasingly more stringent. For Settings I and II, we give pairs of polynomial time mechanisms $(\mathcal{M}_1, \mathcal{M}_2)$ which, besides addressing the specific requirements, find stable matchings, are dominant strategy incentive compatible (DSIC) w.r.t. reporting preference lists of students, and never break, in round $\mathcal{R}_2$, a match created in round $\mathcal{R}_1$. In Setting III, the mechanism needs to deal with residents of the City who try to game the system by not participating in round $\mathcal{R}_1$ and only showing up in round $\mathcal{R}_2$, in addition to gaming by misreporting preference lists. For this setting, we need to introduce the notion of a {\em non-oblivious mechanism:} such a mechanism needs to know the preference lists of {\em all} students who reside in the City, including the ones who don't participate in round $\mathcal{R}_1$. Further, we need to weaken DSIC to mechanisms in which truth-telling is a Nash equilibrium; we call these ICNE mechanisms. After establishing, via two impossibility results, that the strongest result we can hope to obtain for Setting III is a pair of non-oblivious, stability-preserving, ICNE mechanisms, we present such a pair. Both mechanisms of the pair run in polynomial time.

Motivation & Objective

• To design mechanisms that preserve stability and incentives in a two-round school choice system with dynamic capacity expansion.
• To address strategic behavior where residents delay participation to exploit capacity increases in round two.
• To develop mechanisms that are computationally efficient and maintain match integrity across rounds.
• To formalize and analyze the limits of incentive compatibility in dynamic settings with non-participating agents.
• To introduce non-oblivious mechanisms that use full knowledge of resident preferences to ensure stability and Nash equilibrium truthfulness.

Proposed method

• Designs a pair of mechanisms for round one and round two that preserve matches from the first round.
• Introduces the concept of a non-oblivious mechanism that requires knowledge of all resident preferences, including those who do not participate in round one.
• Uses polynomial-time algorithms to compute stable matchings in both rounds while respecting capacity expansions.
• Establishes a framework for incentive compatibility under weakened conditions, replacing dominant strategy truthfulness with Nash equilibrium truth-telling (ICNE).
• Employs impossibility results to show that non-oblivious, stability-preserving, ICNE mechanisms are the strongest achievable outcome in Setting III.
• Applies theoretical analysis to prove that the proposed mechanisms are stable, efficient, and resilient to strategic manipulation under specified constraints.

Experimental results

Research questions

• RQ1 Can stable and incentive-compatible mechanisms be designed for a two-round school choice system with dynamic capacity increases?
• RQ2 What are the limits of incentive compatibility when residents may strategically delay participation to exploit round two capacity expansions?
• RQ3 How can mechanisms account for non-participating residents without prior knowledge of their preferences?
• RQ4 What is the weakest incentive compatibility notion that still allows for stable, efficient, and computationally feasible mechanisms in dynamic settings?
• RQ5 Can non-oblivious mechanisms achieve stability and Nash equilibrium truth-telling in the presence of strategic non-participation?

Key findings

• The paper establishes two impossibility results, proving that non-oblivious, stability-preserving, ICNE mechanisms represent the strongest achievable outcome in Setting III.
• For Settings I and II, the paper constructs polynomial-time, stability-preserving, and dominant strategy incentive-compatible (DSIC) mechanisms that never break matches from round one.
• In Setting III, the paper presents a pair of non-oblivious, stability-preserving, and ICNE mechanisms that are computationally efficient and maintain match integrity across rounds.
• The proposed mechanisms ensure that truth-telling is a Nash equilibrium, even when residents may strategically delay participation.
• The mechanisms are designed to handle both preference list misreporting and strategic non-participation by residents.
• All mechanisms run in polynomial time, ensuring computational feasibility for real-world deployment.