[Paper Review] Perturbation Robust Stable Matching

Royi Jacobovic|arXiv (Cornell University)|Dec 24, 2016

Game Theory and Voting Systems14 references3 citations

TL;DR

This paper introduces a novel notion of social optimality in stable matching called $ν$-perturbation robust stable matching, which balances social cost and resilience to agent departures. It proposes an $ϵ(n^4)$-time algorithm to compute this matching under stability constraints, using a probabilistic model of agent exit and a trade-off between expected cost and regret from re-matching.

ABSTRACT

A well known result states that stability criterion for matchings in two-sided markets doesn't ensure uniqueness. This opens the door for a moral question with regard to the optimal stable matching from a social point of view. Here, a new notion of social optimality is proposed. Its novelty is its ability to take into account the possibility of agents to leave the matching after it has already been established. To formalize this real-life scenario, this work includes a well-defined probability model and a social cost function that maintain the general guidelines of leaving-agents situations. Finally, efficient algorithms to optimize this function are developed either under stability constraint or without it.

Motivation & Objective

• To address the lack of uniqueness in stable matchings by introducing a new social optimality criterion that accounts for agent departures.
• To model real-world scenarios where agents may leave after matching, causing costly re-matching.
• To develop an efficient algorithm that optimizes a trade-off between social cost and expected regret from re-matching after departures.
• To extend existing stable matching frameworks by incorporating perturbation robustness under probabilistic agent exit.
• To provide a computationally feasible solution for market designers facing uncertainty about agent retention.

Proposed method

• Proposes a probabilistic model where each agent $ω_0$ has a known exit probability $p_{ω_0}$, and $φ$ denotes no departure.
• Defines a cost function $\psi(\cdot;\nu)$ that combines expected squared social cost and expected squared regret from re-matching.
• Uses Feder's algorithm to compute the optimal matching $\mu_{\omega_0}$ for each potential departure $\omega_0$.
• Reduces the perturbation-robust matching problem to an assignment problem via a cost function $f(\omega_1, \omega_2)$.
• Applies Edmonds-Karp-Hopcroft algorithm to solve the resulting assignment problem in $\mathcal{O}(n^3)$ time.
• Establishes that the full algorithm runs in $\mathcal{O}(n^4)$ time by precomputing relevant matchings and evaluating $f(\cdot)$ efficiently.

Experimental results

Research questions

• RQ1 How can a stable matching be optimized to minimize both social cost and expected regret from agent departures?
• RQ2 What is the computational complexity of finding a matching that balances stability and robustness to perturbations from agent exits?
• RQ3 Can a perturbation-robust stable matching be computed efficiently under probabilistic exit models?
• RQ4 How does the stability constraint affect the social cost in the presence of potential agent departures?
• RQ5 What is the impact of using different norms (e.g., $L_1$, $L_\infty$) on the robustness and efficiency of the matching?

Key findings

• The paper presents a new social cost function that balances expected social cost and expected regret from re-matching after agent departures.
• The proposed $\nu$-perturbation robust stable matching is defined as the solution to a convex optimization problem over stable matchings.
• An $\mathcal{O}(n^4)$-time algorithm is developed to compute the perturbation-robust stable matching under stability constraints.
• The relaxed version of the problem—without stability—can also be solved in $\mathcal{O}(n^4)$ time via an assignment problem formulation.
• The method reduces the robust matching problem to an assignment problem using a carefully defined cost function $f(\omega_1, \omega_2)$.
• The algorithm is efficient and scalable under the assumption that at most one agent may leave, with running time dependent on the number of possible exit subsets.