TABLE OF CONTENTS The techniques presented in Section 15.3 provide us with tools that allow us to answer many of our analysis questions, but there are still some noticeable limitations. Establishing the existence and uniqueness of a fixed point for an evolution function says little about convergence or stability. Finding an appropriate Lyapunov function can provide valuable convergence and stability information, but finding that Lyapunov function is largely a hit-or-miss affair. Contraction mappings provide many of the results that we desire, but are encountered infrequently. Although Markov models can do an excellent job of handling the nondeterministic nature of one of the most promising cognitive radio adaptation algorithms genetic algorithms solving for a network's transition matrix can be a daunting task. Finally, for all of these approaches, analyzing one decision rule says little about the performance of related decision rules. It would be nice if, given the application-specific goal of a cognitive radio, we were able to immediately predict what decision processes would be required to achieve the desired level of network performance.
Frequently, progress in analysis occurs by introducing additional information. In this case, introducing cognitive radios' goals allows us to apply techniques from game theory to gain additional insights. As we will see in the remainder of this chapter, game theory has a number of advantages over more traditional techniques. A number of readily identifiable game models allow us to simultaneously identify the existence of steady states, convergence criteria, and stability of cognitive radio...
