  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 real-time 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 e-optimal version
without recursive computing over time.
