  Federated learning (FL) has been proposed as a popular learning framework to
protect the users' data privacy but it has difficulties in motivating the users
to participate in task training. This paper proposes a Bertrand-game-based
framework for FL in wireless networks, where the model server as a resource
buyer can issue an FL task, whereas the employed user equipment (UEs) as the
resource sellers can help train the model by using their local data. Specially,
the influence of time-varying \textit{task load} and \textit{channel quality}
on UE's motivation to participate in FL is considered. Firstly, we adopt the
finite-state discrete-time Markov chain (FSDT-MC) method to predict the
\textit{existing task load} and \textit{channel gain} of a UE during a FL task.
Depending on the performance metrics set by the model server and the estimated
overall energy cost for engaging in the FL task, each UE seeks the best price
to maximize its own profit in the game. To this end, the Nash equilibrium (NE)
of the game is obtained in closed form, and a distributed iterative algorithm
is also developed to find the NE. Simulation result verifies the effectiveness
of the proposed approach.
