Analysis and Design of Cognitive Radio Networks Using Game Theory
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Analysis and Design of Cognitive Radio Networks Using Game Theory |
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Basic Game-Theoretic Links
Example(1) : The Cognitive Radios’ Dilemma
Suppose two cognitive radios are operating in the same environment and are attempting to maximize their throughput. Each radio can implement two different waveforms – one a low-power narrowband waveform, the other a higher power wideband waveform. If both radios choose to implement their narrowband waveforms – action vector (n,N) – the signals will be separated in frequency and each radio will achieve a throughput of 9.6 kbps. If one of the radios implements its wideband waveform while the other implements its narrowband waveform – action vectors (n,W) or (w,N) – then interference occurs with the narrow band signal achieving a throughput of 3.2 kbps and the wideband signal a throughput of 21 kbps. If both radios implement wideband waveforms, then each radio experiences a throughput of 7 kbps. These waveforms can be visualized in the frequency domain as shown in Figure 4.1 and represented in matrix form as shown in Figure 4.2. Without going into the analysis of this game, the insightful reader may already anticipate that this algorithm tends to lead to less than optimal performance.
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Figure 4.1: Frequency domain
representation of waveforms in the Cognitive Radios’ Dilemma |
Figure 4.2: The Cognitive Radios’ Dilemma in Matrix |
Example(2) : The Cognitive Radios’ Dilemma
Two cognitive radios are operating in the same environment and are attempting to independently achieve the highest possible voice quality. Each radio can implement three different waveforms - an FM, an AM, and a spread spectrum waveform. The general dynamics of this game are illustrated in Figure 4.3 where one randomly selected radio adapts its waveform in each cycle with the her radio’s waveform held constant for the stage (cycle). The combination of the adaptation by one radio and the continued waveform by the other radio specify an action vector. Via the outcome function, this action vector determines an outcome. In this case, the radios observe their SINR which based on the radios’ application-determined orientation and goal (voice quality) specifies a utility for that SINR. Based on their observations and inferences about the future, the radios’ cognition cycle would then determine their next action in the repeated game.
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figure 4.3 : A Repeated Two Player Cognitive Radio Game |
definition : Nash Equiliberium(NE)
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Example(3) : Identifying the NE of Cognitive Radios’ Dilemma
For action vector (n,N), either radio can improve its performance by choosing a wideband waveform (21>9.6). For action vector (n,W) or action vector (w,N), the radio with the narrowband waveform can improve its performance by changing to a wideband waveform (7 > 3.2). For action vector (w,W), neither radio can improve its performance by switching to a narrowband waveform (3.2 < 7), thus (w, W) is identified as an NE for the game.
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figure 4.4 :The Cognitive Radios’ Dilemma. This game has a unique NE at (w, W). |
NE identification for a game is an NP-complete problem [Neel_04a]. It is possible to reduce this search process for some special games, but not all games satisfy the special properties of the Potential Games . Alternately, some analysts are forced to turn to simulations – the very step we’re intent on minimizing. For example, [Ginde_03] used an exhaustive simulation that ran for days to show that a GPRS network employing joint rate-power adaptations had four NEs, even though the modeled system included only 7 players.
Example(4) : Existence of a NE in a Power Control Game
In a power control game, radios adjust their power levels in an attempt to maximize some utility function, typically some function that balances SINR or throughput against power consumption or battery life. For this example, consider the power control algorithm presented in [Goodman_00] which considers a single cell where the mobiles are adapting their transmit power levels in an attempt to maximize the utility function given in:
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which is an expression of throughput for a FSK waveform divided by transmit power pi. In this expression, throughput is a function of the data rate, R, the packet length, L, and the received SINR of player i’s signal, where is calculated as:
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where W is the bandwidth of the transmitted signal, gk is the gain of the kth mobile to the base station, pk is the transmit power of mobile k and s is the noise power at the base station. This can then be modeled as a normal form game by the tuple <N, P, {ui}>, where P is the power (action) space formed by the Cartesian product of the sets of power levels available to each player. This game has at least one pure strategy NE.
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