Section: New Results

Application of Game Theory and Distributed Optimization to Wireless Networks

In wireless networks, channel conditions and user quality of service (QoS) requirements vary, often quite arbitrarily, with time (e.g. due to user mobility, fading, etc.) and users only have a very limited information about their environment. In such context optimizing transmission while taking power consumption into account is extremely challenging. We apply game theory technique to MIMO wireless network using OFDM or OFDMA where multi-path channels can be handled efficiently

In [25] , [9] , we show that distributed power allocation in heterogeneous OFDMA cognitive radio networks can be modeled as a game where each user equipment in the network engages in a non-cooperative game and allocates its available transmit power over subcarriers to maximize its individual utility. The corresponding equilibrium (Debreu, an extension of Nash Equilibrium) can be characterized with fractional programming and we provide sufficient conditions for computing such equilibria as fixed points of a water-filling best response operator. Using such approach can however be quite slow and is very sensitive to delay and information uncertainty (it may not converge). Therefore, we explain in [17] how signal covariance matrices in Gaussian MIMO multiple access channel can be learnt in presence of imperfect (and possibly delayed) feedback. The algorithm we propose is based on on the method of matrix exponential learning (MXL) and it has the same information and computation requirements as distributed water-filling. However our algorithm converge much faster even for large numbers of users and/or antennas per user and in the presence of user update asynchrony, random delays and/or ergodically changing channel conditions. Yet, since the system may evolve over time in an unpredictable fashion (e.g. due to changes in the wireless medium or the users' QoS requirements), static solution concepts (such as Nash equilibrium) may be no longer relevant and users must adapt to changes in the environment “on the fly”, without being able to predict the system's evolution ahead of time. Hence, we focus on the concept of no-regret : policies that perform at least as well as the best fixed transmit profile in hindsight. In  [31] and  [41] , we provide a formulation of power control as an online optimization problem and we show that the FM dynamics lead to no regret in this dynamic context. In [40] we apply this approach energy efficient transmission in MIMO-OFDM systems and we show through numerical simulations that, in realistic network environments even under rapidly changing channel conditions, users can track their individually optimum transmit profile, achieving gains of up to 600in energy efficiency over uniform power allocation policies.

We also apply this technique to multi-carrier cognitive radio systems. Such systems allow opportunistic secondary users (SUs) to access portions of the spectrum that are unused by the network's licensed primary users (PUs), provided that the induced interference does not compromise the PUs' performance guarantees. In [14] , we introduce a flexible spectrum access pricing schemes such that the corresponding Nash equilibrium is unique under very mild assumptions and satisfies the performance constraints. In addition, we derive a dynamic power allocation policy that converges to equilibrium within a few iterations (even for large numbers of users) and that relies only on local—and possibly imperfect—signal-to-interference-and-noise ratio measurements. In [24] , we draw on exponential learning techniques to design an algorithm that is able to adapt to system changes “on the fly”, i.e. such that the proposed transmit policy leads to no regret even under rapidly changing network conditions.