Game theory and the frequency selective interference channel - A tutorial

The success of unlicensed broadband communication has led to very rapid deployment of communication networks that work independently of each other using a relatively narrow spectrum. For example the 802.11g standard is using the ISM band which has a total bandwidth of 60 MHz. This band is divided into 12 partially overlapping bands of 20 MHz. The success of these technologies might become their own limiting factor. The relatively small number of channels and the massive use of the technology in densely populated urban metropolitan areas will cause significant mutual interference. This is especially important for high quality real time distribution of multi-media content that is intolerant to errors as well as latency. Existing 802.11 (WiFi) networks have very limited means to coordinate spectrum with other interfering systems. It would be highly desirable to improve the interference environment by distributed spectral coordination between the different access points. Another scenario is that of centralized access points such as 802. 16 (WiMax) where the resources are allocated centrally by a single base station.

Similar situation is facing the advanced DSL systems such as ADSL2+ and VDSL. These systems are currently limited by crosstalk between the lines. As such the DSL environment is another example of highly frequency selective interference channel. While the need to operate over interference limited frequency selective channels is clear in many of the current and future communication technologies, the theoretical situation is much less satisfying. The capacity region of the interference channel is still open (see [1] for short overview) even for the fixed channel two user case. In recent years great advances in understanding the situation for flat channels under weak interference have been achieved. It can be 1 School of Engineering, Bar-Ilan University, Ramat-Gan, 52900, Israel. 2 Faculty of EEMCS, Delft University of Technology. This work was supported by Intel Corporation and by the Netherlands Foundation of Science and Technology. e-mail: leshema@eng.biu.ac.il .

shown that in this case treating the interference as noise is almost optimal. On the other hand for the medium strength interference as is typical in the wireless environment, the simplest strategy is by using orthogonal signaling, e.g. TDMA/FDMA, for high spectral efficiency networks, or CDMA for very strong interference with low spectral efficiency per user. Moreover, sequential cancelation techniques that are required for the best known capacity region in the medium interference case [2] are only practical for small number of interferers. The interference channel is a conflict situation, and not every achievable rate pair (from informational point of view) is actually reasonable operating point for the users. This conflict is the reason to analyze the interference channel using game theoretical tools. Much work has been done on competitive game theory applied to frequency selective interference channel, with the early works of Yu et al. [3] and subsequent works of Scutari et al. (see [4] and the references therein). A particularly interesting topic is the use of generalized Nash games to the weak interference channel [5], and the algorithm in [6] which extends the FM-IWF to iterative pricing under fixed rate constraint.

The fact that competitive strategies can result in significant degradation due to the prisoner’s dilemma has been called the price of anarchy [7]. In the interference channel case, a simple case has been analyzed by Laufer and Leshem [8], who characterized the cases of prisoner’s dilemma in interference limited channels. To overcome the sub-optimality of the competitive approach we have two alternatives: Using repeated games or using cooperative game theory. Since most works on repeated games concentrated on flat fading channels, we will mainly concentrate on cooperative game theoretic approaches. One of the earliest solutions for cooperative games is the Nash bargaining solution [9]. Many papers in recent years were devoted to analyzing the Nash bargaining solution for the frequency flat interference channel in the SISO [10], [11], MISO [12], [13] and MIMO cases [14]. Interesting extensions for log-convex utility functions appeared in [15]. Another interesting application of the bargaining techniques discussed here is for multimedia distribution networks. Park and Van der Schaar [16] used both Nash bargaining and the generalized Kalai-Smorodinski solutions [17] for multimedia resource management. Another alternative cooperative model was explored in [18] where, the cooperation between rational wireless players was studied using coalitional game theory by allowing the receivers to cooperate using joint decoding.

In the context of frequency selective interference channels much less research has been done. Han et al. [19] in a pioneering work, studied the Nash Bargaining under FDM/TDM strategies and total power constrain

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