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Analysis and Design of Cognitive Radio Networks Using Game Theory | ||
Steady States of Potential Games & Optimality
Main Theorem: Nash equilibrium existence and finite potential games All finite potential games have at least one NE
Theorem(5-4-1):
The global maximizers of the potential function, V, may merely be a subset of the all NE in a game.
Optimality
In general, little can be said about the optimality or desirability of the steady states of a
cognitive potential game. They need not be Pareto efficient, and they are not generally
maximizers of a design objective function. For example if we are trying to maximize the
sum utilities of the radios, the steady-state of the cognitive radios’ dilemma is
undesirable.
However, when the potential func tion is also the network objective function, i.e., V J = ,
then if V admits a global maximum, then there exists a NE that is optimal. Further, as
deterministic unilateral play increases the value of V with each iteration, it is safe to say
that the stable steady-state of the network will give better performance than the initial
state of the network. While game necessarily has numerous steady states, if the network’s
designer is attempting to minimize total network interference, then the steady-states will
generally be desirable and performance will improve with each adaptation.
Alternately, it is a relatively straight forward task for a cognitive radio designer to move
the steady-state of an exact potential game, a* to some other desired valid state, a* by
exploiting the linear space properties of exact potential games. The procedure for doing
so is as follows.
Introduce an identical interest network cost function, NC(a) which is found by solving follow equaion for NC(a):
where V(a) is the original game’s potential function. In other words, at the desired action tuple, the network cost function should have the negative slope of the potential function’s slope. After solving for NC(a), “charge” the radios this cost function. This can either be done by directly incorporating NC into each radio’s reasoning process or as a cost imposed by the network on the radios. Thus the modified utility function for each radio takes the form shown in:
Since the original game was an exact potential game and NC(a) is an identical interest (coordination) game, the modified game is also an exact potential game with an exact potential given by
To minimize the creation of new NE, it is suggested that the function be of low order or a piecewise function. Ideally, V should be a concave function and NC a linear function as this combination preserves both the concavity and the uniqueness of the potential maximizing NE.
example: effect of linear function in simple figure:
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