Constant Regret Primal-Dual Policy for Multi-Way Dynamic Matching

We study a discrete-time dynamic multi-way matching model. There are finitely many agent types that arrive stochastically and wait to be matched. State-of-the-art dynamic matching policies in the literature require the knowledge of all system parameters to determine an optimal basis of the fluid relaxation, and focus on controlling the number of waiting agents using only matches in the optimal basis (Kerimov et al., 2021a,b; Gupta, 2021). In this paper, we propose a primal-dual policy that schedule matches for future arrivals based on an estimator for the dual solution. Our policy does not require the knowledge of optimal bases, and is the first to achieve constant regret at all times under unknown arrival rates. In addition, we show that when the arrival rates are known, the primal-dual policy achieves the optimal scaling as the lower-bound described in Kerimov et al. (2021a,b). Furthermore, we find that when the arrival rates are known, the primal-dual policy can significantly outperform alternative dynamic matching policies in numerical simulations