Analysis and Design of Cognitive Radio Networks Using Game Theory
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Modeling and
Problem Formalization

Analysis Objectives

definition :

a path may terminate in a stable point, but under different conditions a path may enter an infinite loop. There may also be points in the action space that are fixed
points of the decision update rule but are unstable so that any small perturbation in initial conditions drives the network away from the point. Each of these concepts is illustrated in the example interaction diagram shown in Figure 2.2 where paths are shown by the arrows and fixed points are labeled as “NE”.

Figure 2.1: A three radio interaction diagram with three steady tates (NE1, NE2, and NE3) and adaptation paths.

This conceptual interaction diagram illustrates the four different analysis questions we identified in Chapter 1 that we would like to answer when analyzing the interactions of a network of cognitive radios.
1. What is the expected behavior of the network?
2. Does this behavior yield desirable performance?
3. What conditions must be satisfied to ensure that adaptations converge to this
behavior?
4. Is the network stable?

Establishing Expected Behavior

assumes that expected behavior of a cognitive radio network is equivalent to its steady-state behavior. Accordingly, establishing expected behavior is concerned with addressing the following issues:

• Existence – Does the system have a steady state?
• Identification – What are the specific steady states for the system?

definition :

Establishing Expected Behavior

there are two specific issues that we would like to address:
• Desirability – How “good” are the steady states of the algorithm?
• Optimality – Does an optimal action vector exist and how close do the steady states come to achieving optimal performance?
There are many different ways of identifying whether or not an action vector is a “good” steady state, but we will make the assumption that the network designer has some objective function, J : A-->R
that he/she wishes to maximize or minimize (perhaps total system goodput or spectrum utilization).

Convergence Conditions

Even if we demonstrate that a cognitive radio network has desirable steady states, it is important to identify the conditions (decision rules, passive operating environments, initial conditions) under which paths converge.

definition :

Network Stability

Wireless networks are stochastic, not deterministic. Accordingly, the cognitive radios’ observations will not be the deterministic functions and instead will be estimates of their operating environment. Because these are only estimates, the radios will frequently make adaptations that appear to be mistakes to the analyst. While this research assumes the radios’ estimates and errors are unbiased, there is the concern of stability as small perturbations could potentially lead to undesirable behavior. Because of this concern, this research addresses the following analytical issues with respect to a network decision rule’s steady state(s):
• Lyapunov stability – After a small perturbation, will stay the system within a bounded region about the steady state?
• Attractivity – After a small perturbation, will the network converge back to the steady state?