[Paper Review] Approximately Stable Matchings with Budget Constraints

Yasushi Kawase, Atsushi Iwasaki|arXiv (Cornell University)|Nov 17, 2017

Game Theory and Voting Systems6 citations

TL;DR

This paper introduces a novel framework for two-sided matching with budget constraints, where hospitals face fixed wage budgets rather than quota limits. It proposes two mechanisms—one sacrificing strategy-proofness to achieve the best possible approximation bound of $\frac{1}{1 - \overline{s}}$, and another strategy-proof for doctors in proportional utility markets with a bounded approximation ratio—both efficiently computing approximately stable matchings via a sequential choice function based on contract size ordering.

ABSTRACT

This paper considers two-sided matching with budget constraints where one side (firm or hospital) can make monetary transfers (offer wages) to the other (worker or doctor). In a standard model, while multiple doctors can be matched to a single hospital, a hospital has a maximum quota: the number of doctors assigned to a hospital cannot exceed a certain limit. In our model, a hospital instead has a fixed budget: the total amount of wages allocated by each hospital to doctors is constrained. With budget constraints, stable matchings may fail to exist and checking for the existence is hard. To deal with the nonexistence of stable matchings, we extend the "matching with contracts" model of Hatfield and Milgrom, so that it handles approximately stable matchings where each of the hospitals' utilities after deviation can increase by factor up to a certain amount. We then propose two novel mechanisms that efficiently return such a stable matching that exactly satisfies the budget constraints. In particular, by sacrificing strategy-proofness, our first mechanism achieves the best possible bound. Furthermore, we find a special case such that a simple mechanism is strategy-proof for doctors, keeping the best possible bound of the general case.

Motivation & Objective

• To address the nonexistence of stable matchings under budget constraints in two-sided matching markets.
• To extend the matching with contracts framework to handle approximately stable matchings under multiplicative utility improvements.
• To design efficient mechanisms that satisfy budget constraints exactly while minimizing the maximum utility gain from blocking coalitions.
• To explore trade-offs between strategy-proofness and approximation quality in budget-constrained matching.
• To establish tight bounds on the approximation ratio for both general and proportional utility settings.

Proposed method

• Introduces $\alpha$-approximation as a relaxation of stability, allowing hospitals to deviate only if utility improves by a factor $\alpha$.
• Proposes Algorithm 2 and Algorithm 4 for non-strategy-proof mechanisms using a size-ordered choice function that greedily selects contracts by size under budget constraints.
• Designs Algorithm 5 as a sequential, order-invariant, size-monotone choice function that selects contracts in increasing order of size to maximize budget utilization.
• Applies the generalized Deferred Acceptance (DA) mechanism with these choice functions to compute approximately stable matchings in polynomial time.
• Proves that the choice function in Algorithm 5 is $\frac{1}{1 - \overline{s}}$-approximate, where $\overline{s}$ is the maximum fraction of a contract’s wage relative to the hospital’s budget.
• Establishes strategy-proofness for doctors in the proportional utility case using Algorithm 5, while maintaining a bounded approximation ratio.

Experimental results

Research questions

• RQ1 Can we design a mechanism that computes an approximately stable matching under budget constraints while satisfying the budget exactly?
• RQ2 What is the best possible approximation ratio achievable when sacrificing strategy-proofness in additive utility settings?
• RQ3 Is there a strategy-proof mechanism for doctors that still achieves a bounded approximation ratio in proportional utility markets?
• RQ4 How does the choice function’s structure (e.g., size-monotonicity, sequentiality) affect the approximation guarantee?
• RQ5 Can we derive tight upper and lower bounds on the approximation ratio for both general and proportional utility models?

Key findings

• The non-strategy-proof mechanism achieves an approximation ratio of $\frac{1}{1 - \overline{s}}$, which is optimal and matches the lower bound.
• For proportional utility markets, the strategy-proof mechanism achieves an approximation ratio of $\frac{1}{1 - \overline{s}}$, matching the best-known upper bound.
• In the proportional case, a separate mechanism achieves a better approximation ratio of 1.62, which is tight as both upper and lower bounds are 1.62.
• The choice function defined in Algorithm 5 is proven to be sequential, order-invariant, size-monotone, and $\frac{1}{1 - \overline{s}}$-approximate.
• The generalized DA mechanism with Algorithm 5 is strategy-proof for doctors and runs in $O(|X|\log|X|)$ time.
• The paper establishes that the bound $\frac{1}{1 - \overline{s}}$ is tight for the general additive case, and no better bound is possible under the same conditions.