Beyond Stability: Improved Efficiency Guarantees for $\alpha$-Stable Matchings

Stable matching mechanisms are fundamental to market design but face an inherent tension between stability and social welfare optimality. We study a natural relaxation of stability, termed $\alpha$-stability, which models agents as willing to deviate only when the potential improvement is sufficiently large. Under $\alpha$-stability, no pair of agents can deviate and improve their valuations by more than a factor of $1/\alpha$, with $\alpha \in (0,1]$. We provide a complete characterization of the stability-efficiency tradeoff under asymmetric valuations. This tradeoff depends on the degree of asymmetry $\mu \in (0,1]$, which bounds the ratio between agents' valuations for any pair. Our results show that relaxing stability can substantially improve achievable efficiency guarantees. We further present a polynomial-time algorithm that computes an $\alpha$-stable matching attaining the best possible efficiency guarantee. For $\alpha \le \mu/(\mu+1)$, our algorithm achieves 1-efficiency; for larger $\alpha$, it computes an $\alpha$-stable matching achieving at least $(1/\alpha)\cdot \mu/(\mu+1)$ of the optimal social welfare. Remarkably, our algorithm inflates the values of an optimal matching and then applies the Gale-Shapley algorithm to the modified instance. Finally, we show that computing an optimal $\alpha$-stable matching is NP-hard, even under slight relaxations of stability, i.e., for $\alpha$ close to 1.