Adversarial Linear-Quadratic Mean-Field Games over Multigraphs

In this paper, we propose a game between an exogenous adversary and a network of agents connected via a multigraph. The multigraph is composed of (1) a global graph structure, capturing the virtual interactions among the agents, and (2) a local graph structure, capturing physical/local interactions among the agents. The aim of each agent is to achieve consensus with the other agents in a decentralized manner by minimizing a local cost associated with its local graph and a global cost associated with the global graph. The exogenous adversary, on the other hand, aims to maximize the average cost incurred by all agents in the multigraph. We derive Nash equilibrium policies for the agents and the adversary in the Mean-Field Game setting, when the agent population in the global graph is arbitrarily large and the "homogeneous mixing"hypothesis holds on local graphs. This equilibrium is shown to be unique and the equilibrium Markov policies for each agent depend on the local state of the agent, as well as the influences on the agent by the local and global mean fields.