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In this paper, we consider a discrete-time multi-agent system involving $N$cost-coupled networked rational agents solving a consensus problem and acentral Base Station (BS), scheduling agent communications over a network. Dueto a hard bandwidth constraint on the number of transmissions through thenetwork, at most $R_d < N$ agents can concurrently access their stateinformation through the network. Under standard assumptions on the informationstructure of the agents and the BS, we first show that the control actions ofthe agents are free of any dual effect, allowing for separation betweenestimation and control problems at each agent. Next, we propose a weighted ageof information (WAoI) metric for the scheduling problem of the BS, where theweights depend on the estimation error of the agents. The BS aims to find theoptimum scheduling policy that minimizes the WAoI, subject to the hardbandwidth constraint. Since this problem is NP hard, we first relax the hardconstraint to a soft update rate constraint, and then compute an optimal policyfor the relaxed problem by reformulating it into a Markov Decision Process(MDP). This then inspires a sub-optimal policy for the bandwidth constrainedproblem, which is shown to approach the optimal policy as $N \rightarrow\infty$. Next, we solve the consensus problem using the mean-field gameframework wherein we first design decentralized control policies for a limitingcase of the $N$-agent system (as $N \rightarrow \infty$). By explicitlyconstructing the mean-field system, we prove the existence and uniqueness ofthe mean-field equilibrium. Consequently, we show that the obtained equilibriumpolicies constitute an $\epsilon$-Nash equilibrium for the finite agent system.Finally, we validate the performance of both the scheduling and the controlpolicies through numerical simulations.

Weighted Age of Information based Scheduling for Large Population Games on Networks