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

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 version of the dual formulation, which we tackle to produce feasible and approximately optimal solutions for the primal and dual formulations. These solutions yield upper and lower bounds for the optimal value, and the difference between these bounds provides a direct 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 theoretical computational complexity of our parametric formulation as well as the sparsity of the resulting approximate matching equilibrium. In the numerical experiments, we study three matching for teams problems: a problem of business location distribution, the well-known 2-Wasserstein barycenter problem, and a high-dimensional problem involving 100 agent categories. Through the numerical results, we showcase that the proposed algorithm can produce high-quality approximate matching equilibria in these settings, 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.