Feasible approximation of matching equilibria for large-scale matching for teams problems

TL;DR

This work develops a scalable numerical framework to compute ε-approximate matching equilibria for the matching for teams problem, accommodating a large number of agent categories and non-discrete type measures. It introduces a parametric LSIP formulation with test-function sets, enabling computable primal/dual bounds and provable convergence to true equilibria as the sub-optimality bound vanishes. The authors establish complexity bounds and construct approximate equilibria through reassembly and binding, ensuring convergence under mild Lipschitz conditions. A Euclidean-space variant provides explicit CPWA test-function constructions with controllable errors. Numerical experiments on business-location distribution, 2-Wasserstein barycenters, and high-dimensional 1D type spaces demonstrate high-quality approximate equilibria and sub-optimality estimates far tighter than theoretical guarantees, with favorable scalability compared to MMOT-based methods.

Abstract

We propose a numerical algorithm for computing approximately optimal solutions of the matching for teams problem. Our algorithm is efficient for problems involving large number of agent categories and allows for non-discrete agent type measures. Specifically, we parametrize the so-called transfer functions and develop a parametric formulation, which we tackle to produce feasible and approximately optimal primal and dual solutions. These solutions yield upper and lower bounds for the optimal value, and the difference between these bounds provides a sub-optimality estimate of the computed solutions. Moreover, we are able to control the sub-optimality to be arbitrarily close to 0. We subsequently prove that the approximate primal and dual solutions converge when the sub-optimality goes to 0 and their limits constitute a true matching equilibrium. Thus, the outputs of our algorithm are regarded as an approximate matching equilibrium. We also analyze the computational complexity of our approach. In the numerical experiments, we study three matching for teams problems: a business location distribution problem, the Wasserstein barycenter problem, and a large-scale problem involving 100 agent categories. We showcase that the proposed algorithm can produce high-quality approximate matching equilibria, provide quantitative insights about the optimal city structure in the business location distribution problem, and that the sub-optimality estimates computed by our algorithm are much less conservative than theoretical estimates.

Figures (6)

• Figure 1: Experiment 1 -- The railway line, the locations of the train stations in the city, and the probability density functions of $\mu_1,\ldots,\mu_N$.
• Figure 2: Experiment 1 -- The values of the lower bound $\alpha_{\mathsf{MT}}^{\mathsf{LB}}$, the upper bounds $\hat{\alpha}_{\mathsf{MT}}^{\mathsf{UB}}$, $\tilde{\alpha}_{\mathsf{MT}}^{\mathsf{UB}}$, the sub-optimality estimates $\hat{\epsilon}_{\mathsf{sub}}$, $\tilde{\epsilon}_{\mathsf{sub}}$ computed by Algorithm \ref{['alg:mt-tf']}, and their a priori upper bound $\epsilon_{\mathsf{theo}}$.
• Figure 3: Experiment 1 -- The probability measures $\hat{\nu}$ and $\tilde{\nu}$ computed by Algorithm \ref{['alg:mt-tf']}.
• Figure 6: Experiment 2 -- The probability density functions of $\mu_1,\ldots,\mu_{20}$.
• Figure 7: Experiment 2 -- The values of the lower bound $\alpha_{\mathsf{MT}}^{\mathsf{LB}}$, the upper bounds $\hat{\alpha}_{\mathsf{MT}}^{\mathsf{UB}}$, $\tilde{\alpha}_{\mathsf{MT}}^{\mathsf{UB}}$, the sub-optimality estimates $\hat{\epsilon}_{\mathsf{sub}}$, $\tilde{\epsilon}_{\mathsf{sub}}$ computed by Algorithm \ref{['alg:mt-tf']}, and their a priori upper bound $\epsilon_{\mathsf{theo}}$.

Theorems & Definitions (32)

• Definition 1.1: Matching equilibrium
• Theorem 1.2: Characterization of matching equilibria
• Definition 2.1: Moment-based relation $\: \overset{ \text{$\mathcal{G}$}} { \underset{ \text{$\varsigma$}} { \text{$\sim$} } } \:$
• Theorem 2.2: Strong duality
• Definition 2.3: Global minimization oracle
• Theorem 2.4: Computational complexity
• Remark 2.5
• Definition 2.6: Reassembly
• Definition 2.7: Binding
• Theorem 2.9: Approximate matching equilibria