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Fueled by the rapid development of communication networks and sensors in portable devices, today many mobile users are invited by content providers to sense and send back real-time useful information (e.g., traffic observations and sensor data) to keep the freshness of the providers' content updates. However, due to the sampling cost in sensing and transmission, an individual may not have the incentive to contribute the realtime information to help a content provider reduce the age of information (AoI). Accordingly, we propose dynamic pricing for the provider to offer age-dependent monetary returns and encourage users to sample information at different rates over time. This dynamic pricing design problem needs to balance the monetary payments to users and the AoI evolution over time, and is challenging to solve especially under the incomplete information about users' arrivals and their private sampling costs. For analysis tractability, we linearize the nonlinear AoI evolution in the constrained dynamic programming problem, by approximating the dynamic AoI reduction as a time-average term and solving the approximate dynamic pricing in closed-form. Then, we estimate this approximate term based on Brouwer's fixed-point theorem. Finally, we provide the steady-state analysis of the optimized approximate dynamic pricing scheme for an infinite time horizon, and show that the pricing scheme can be further simplified to an ε-optimal version without recursive computing over time.

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In this study, the authors consider the large population dynamic games where each agent evolves according to a dynamic equation containing the input average of all agents. The long time average (LTA) cost that each agent aims to minimise is coupled with other agents' states via a population state average (PSA), which is also known as the mean field term. In order to design decentralised controls, the Nash certainty equivalence is introduced. It is shown that the resulting decentralised mean field control laws lead the system to mean-consensus asymptotically as time goes to infinity. The stability property of the mass behaviour and the almost sure asymptotic Nash equilibrium property of the optimal controls are also guaranteed, and the case with non-linear system dynamics is also discussed. In addition, the influence of inaccurate mean field information on individual agent is analysed. Finally, they investigate the socially cooperative formulation where the objective is to minimise the social cost as the sum of all individual LTA costs containing the PSA. In this case, they show that the decentralised mean field social controls are the same as the mean field Nash controls for infinite population systems.