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

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.