Mean-Field Systems with Heterogeneous Subteams: Optimality of Cluster-Symmetric Independent Policies and Equivalence with Decentralized McKean-Vlasov Control of Cluster-Representative Agents

Abstract

Across science and engineering, mean-field methods have been a powerful and versatile approach for the analysis of systems of many interacting elements. However, common arguments used to characterize an infinite population limit can be quite restrictive from a modeling perspective by requiring that all agents be identical (i.e. symmetric, or homogeneous). In this paper, we consider large interactive particle systems under agent heterogeneity for a class of discrete time teams composed of finitely many species of agents, grouped into symmetric subteams, called clusters. In particular, for the class of discounted, partially exchangeable cost criteria considered, we establish the optimality of centralized joint policies which are exchangeable within each cluster and depend on the agent ensemble only up to the state empirical distribution over each cluster. Following this, a generalization of De Finetti's theorem is used to demonstrate the subsequential convergence of these optimal policies to one which is decentralized (depending on only the local state and distribution over each cluster) and symmetric within each subteam as the population size approaches infinity. This solution is shown to induce a sequence of asymptotically optimal policies for the finite population problems which retain their structure and decentralization. Furthermore, our analysis justifies the optimality of a decentralized McKean-Vlasov team representation involving coupled representative agents for each of the clusters, and establishes a verification theorem/value iterations for the mean-field limit. In this way, we provide an avenue for analyzing complex, cooperative systems with finite heterogeneity and set the stage for further research on learning algorithms.