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 a large number of agent categories and allows for the measures describing the agent types to be non-discrete. Specifically, we parametrize the so-called transfer functions and develop a parametric version of the dual formulation. Our algorithm tackles this parametric formulation and produces feasible and approximately optimal solutions for the primal and dual formulations of the matching for teams problem. These solutions also yield upper and lower bounds for the optimal value, and the difference between the upper and lower bounds provides a direct sub-optimality estimate of the computed solutions. Moreover, we are able to control a theoretical upper bound on the sub-optimality to be arbitrarily close to 0 under mild conditions. 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. Through numerical experiments, we showcase that the proposed algorithm can produce high-quality approximate matching equilibria and is applicable to versatile settings, including a high-dimensional setting involving 100 agent categories.