Weighted Age of Information-Based Scheduling for Large Population Games on Networks

In this paper, we study a multi-agent game between $N$ agents, which solve a consensus problem, and receive state information through a wireless network, that is controlled by a Base station (BS). Due to a hard-bandwidth constraint, the BS can concurrently connect at most $R_{d} < N$ agents over the network. This causes an intermittency in the agents' state information, necessitating state estimation based on each agent's information history. Under standard assumptions on the information structure, we separate each agent's estimation and control problems. The BS aims to find the optimum scheduling policy that minimizes a weighted age of information based performance metric, subject to the hard-bandwidth constraint. We first relax the hard constraint to a soft update-rate constraint and compute an optimal policy for the relaxed problem by reformulating it into an MDP. This then inspires a sub-optimal policy for the bandwidth constrained problem, which is shown to approach the optimal policy as $N \rightarrow \infty $. Next, we solve the consensus problem using the mean-field game framework. By explicitly constructing the mean-field system, we prove the existence of a unique mean-field equilibrium. Consequently, we show that the equilibrium policies obtained constitute an $\epsilon $ -Nash equilibrium for the finite-agent system.