[Paper Review] Exploring Strategy-Proofness, Uniqueness, and Pareto Optimality for the Stable Matching Problem with Couples

Andrew Perrault, Joanna Drummond+1 more|arXiv (Cornell University)|May 13, 2015

Game Theory and Voting Systems11 references3 citations

TL;DR

This paper uses a SAT encoding to exactly enumerate all stable matchings in the Stable Matching Problem with Couples (SMP-C), enabling empirical analysis of strategy-proofness, uniqueness, and Pareto optimality. It finds that while unique stable matchings do not become more common with larger markets, resident-optimal stable matchings do approach probability one as market size grows, and that residents can still manipulate even when a unique stable matching exists.

ABSTRACT

The Stable Matching Problem with Couples (SMP-C) is a ubiquitous real-world extension of the stable matching problem (SMP) involving complementarities. Although SMP can be solved in polynomial time, SMP-C is NP-Complete. Hence, it is not clear which, if any, of the theoretical results surrounding the canonical SMP problem apply in this setting. In this paper, we use a recently-developed SAT encoding to solve SMP-C exactly. This allows us to enumerate all stable matchings for any given instance of SMP-C. With this tool, we empirically evaluate some of the properties that have been hypothesized to hold for SMP-C. We take particular interest in investigating if, as the size of the market grows, the percentage of instances with unique stable matchings also grows. While we did not find this trend among the random problem instances we sampled, we did find that the percentage of instances with an resident optimal matching seems to more closely follow the trends predicted by previous conjectures. We also define and investigate resident Pareto optimal stable matchings, finding that, even though this is important desideratum for the deferred acceptance style algorithms previously designed to solve SMP-C, they do not always find one. We also investigate strategy-proofness for SMP-C, showing that even if only one stable matching exists, residents still have incentive to misreport their preferences. However, if a problem has a resident optimal stable matching, we show that residents cannot manipulate via truncation.

Motivation & Objective

• To investigate whether theoretical properties of the classical stable matching problem, such as uniqueness and resident-optimality, extend to the NP-complete Stable Matching Problem with Couples (SMP-C).
• To evaluate the empirical frequency of unique stable matchings and resident-optimal stable matchings as market size increases in SMP-C instances.
• To assess the performance and limitations of existing deferred acceptance-style algorithms (e.g., RP99, KPR) in finding stable and resident-optimal matchings.
• To explore whether strategy-proofness holds in SMP-C, particularly under conditions of unique or resident-optimal stable matchings.
• To develop and apply an enhanced SAT encoding that allows full enumeration of stable matchings and guarantees identification of resident Pareto optimal solutions.

Proposed method

• A recently developed SAT encoding (SAT-E) for SMP-C is extended with additional constraints to enumerate all stable matchings for any given instance.
• The extended SAT encoding includes a constraint to verify whether a given stable matching is resident Pareto optimal, and if not, to find a better one.
• The method enables exact computation of all stable matchings and resident Pareto optimal matchings, overcoming limitations of incomplete DA-style algorithms.
• Random problem instances of varying market sizes and couple proportions are generated to simulate real-world conditions and test theoretical conjectures.
• The approach allows for the systematic evaluation of strategy-proofness by checking whether residents can benefit from truncating preference lists.
• Performance of DA-style algorithms (RP99 and KPR) is benchmarked against the SAT-based enumeration to assess their completeness and optimality.

Experimental results

Research questions

• RQ1 Does the probability of a unique stable matching in SMP-C approach one as market size increases, as observed in the classical SMP?
• RQ2 Does the probability of a resident-optimal stable matching in SMP-C approach one as market size increases, despite the problem's NP-completeness?
• RQ3 Can residents manipulate the matching outcome via preference list truncation even when a unique stable matching exists?
• RQ4 Do existing deferred acceptance-style algorithms for SMP-C (e.g., RP99, KPR) reliably find stable matchings or resident-optimal matchings in practice?
• RQ5 Can the SAT-based enumeration method identify all stable matchings and resident Pareto optimal matchings where DA-style algorithms fail?

Key findings

• The percentage of SMP-C instances with a unique stable matching does not increase with market size, contradicting a conjecture based on classical SMP results.
• The percentage of instances with a resident-optimal stable matching increases with market size, supporting a new conjecture that this probability approaches one as market size grows.
• Even when a unique stable matching exists, residents still have incentive to misreport preferences, demonstrating that SMP-C is not strategy-proof.
• The RP99 algorithm has a higher success rate in finding resident-optimal matchings than KPR, but its overall failure rate (36.25% at 20% couples) is significantly higher than KPR’s (5.75% at 20% couples).
• KPR outperforms RP99 in solving the majority of problem instances, with a failure rate below 0.25% for markets with 10% or fewer couples.
• The SAT-based method successfully enumerated all stable matchings and identified all resident Pareto optimal matchings, even in cases where DA-style algorithms failed or missed optimal solutions.