A stackelberg game for pricing uplink power in wide-band cognitive radio networks

We study the problem of pricing uplink power in wide-band cognitive radio networks under the objective of revenue maximization for the service provider and while ensuring incentive compatibility for the users. User utility is modeled as a concave function of the signal-to-noise ratio (SNR) at the base station, and the problem is formulated as a Stackelberg game. Namely, the service provider imposes differentiated prices per unit of transmitting power and the users consequently update their power levels to maximize their net utilities. We devise a pricing policy and give conditions for its optimality when all the users are to be accommodated in the network. We show that there exist infinitely many Nash equilibrium points that reward the service provider with the same revenue. The pricing policy charges more from users that have better channel conditions and more willingness to pay for the provided service. We then study properties of the optimal revenue with respect to different parameters in the network. We show that for regimes with symmetric users who share the same level of willingness to pay, the optimal revenue is concave and increasing in the number of users in the network. We analytically obtain achievable SNRs for this special case, and finally present a numerical study in support of our results.