	Analysis and Design of Cognitive Radio Networks Using Game Theory
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Game theory

Convergence of NE

Accordingly, convergence is more frequently discussed in the context of myopic repeated games and the decision rules that guide the radios’ reactions to their observations of the network state.

Decision Rule Classes

1-Best Response Dynamic

2- Better Response Dynamic

3- Random Better Response Dynamic(a)

4- Friedman’s Random Better Response(b)

Stage Game Properties

Improvement Path Terminology

1- Path

2- Improvement Path

3- Finite Improvement Property

Theorem : Improvement Cycles and FIP

Example :Improvement Paths in the Prisoners’ Dilemma

Consider the 2x2 game shown in matrix representation in Figure 4.5. A complete listing of the improvement paths for this game is given in Table 4.1.

figure 4.5 : Prisoners’ Dilemma Game Matrix for Improvement Path Analysis

Table 4.1 : Improvement aths for Game Presented in Figure 4.1

4- Weak Finite Improvement Property (weak FIP)

Theorem : FIP and NE existence

All games with FIP have at least one Nash equilibrium.

Convergence Properties

The criteria for assured convergence for the classes of games, decision rules, and decision timings are summarized in Table 4.2. Note that the broadest convergence conditions hold under random and asynchronous timings and for the random better response rules. This implies two important results for myopic cognitive radio network design. First, cognitive radio networks should in general be designed so decision timings are randomized instead of synchronized – a good thing as synchronized algorithms generally do not scale as well as randomized algorithms due to the increased overhead inherent in the synchronization process. In general, the decision engines in cognitive radios should support decision rules that satisfy the class of decision rules of Definition 4.12 as that rule
supports the broadest range of convergence conditions. However, it should be noted that when any of the other classes of decision rules converge, it should be possible to design algorithms that converge faster than the class of decision rules of Definition 4.12. This design suggestion appears to be reasonable as Virginia Tech has built a cognitive radio whose decision engine implements a decision process in this class of algorithms [Rondeau_04].

Table 4.2 : Convergence Criteria

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