# An Energy-Efficient Game-Theory-Based Spectrum Decision Scheme for Cognitive Radio Sensor Networks
## 3 Energy-Efficient Game-Theory-Based Spectrum Decision
### Coordination Mode (C Mode)
#### 3.2.3 Spectrum Decision Module of C Mode
For example, \(S_{i}\) with \(type(CH)_{i}\) = 1 selects its operating channel and backup channel randomly from the set of clean unlicensed channels, but it must not be the same as the operating channel. Notice that, although the channels are selected randomly, they are selected inside the set of idle licensed channels or clean unlicensed channels. EGSD-R refines Equations (11), (13) and (15) into:  
\[[(C_{i})_{f^{\prime}},(C_{i}^{\prime})_{f}]= \begin{cases}[B_{yu1},B_{yu2}]& \text{if }type(CH)_{i}=1\\ [A_{x1},B_{yu1}]&\text{if }type(CH)_{i}=2\\ [A_{xv1},A_{xv2}]&\text{if }type(CH)_{i}=3\\ [A_{xw1},A_{xw2}]&\text{if }type(CH)_{i}=4\end{cases}  \tag{17}\]  
In Equation (33), the following conditions should be satisfied:  
\[(B_{yu1},B_{yu2})\in(B_{clean})_{i} \tag{18}\]  
\[(A_{x1},A_{xv1},A_{xv2})\in(A_{idle})_{i} \tag{19}\]  
and:  
\[(A_{xw1},A_{xw2})\in\{(A_{idle})_{i}-(A_{excl})_{i}\} \tag{20}\]  
where the values of \((u1,u2,v1,v2,w1,w2)\) are random values with boundaries of \(1\leqslant u1\), \(u2\leqslant yk\), \(1\leqslant v1\), \(v2\leqslant xl\), and \(1\leqslant w1\), \(w2\leqslant h(R-1)\). For values \(yk\), \(xl\), and \(h(R-1)\), see Equations (21), (23) and (27). In EGSD-G, we proposed a game theory solution for the spectrum selection problem, which we called mixed strategy with lowest payoff elimination (LPE). The payoff is the channel holding time obtained from the Markov chain in the spectrum characterization stage. First, the payoff is sorted in a descending manner, with the top as the highest payoff:  
\[ \begin{array}{ll}payoff_{i}&=\left\{payoff_{x^{\prime}},payoff_{y}^{\prime} \right\}_{i}\\ &=\left\{sort((CHT_{x})_{f}),sort((CHT_{y}^{\prime})_{f})\Big{|}(CHT_{x1} \geqslant CHT_{x2}\geqslant...\geqslant CHT_{xl}),\right.\\ &\left.(CHT_{y1}^{\prime}\geqslant CHT_{y2}^{\prime}\geqslant...\geqslant CHT_ {yk}^{\prime})\right\}_{i}\end{array}  \tag{21}\]  
where \(payoff_{i}\) is the payoff of \(S_{i}\), \(payoff_{x}\) and \(payoff_{y}\) are the payoff of licensed channels and unlicensed channels, respectively, observed by \(S_{i}\). (The variable CHT written without frame information is CHTat the current frame, i.e., \(CHT_{x1}=(CHT_{x1})\)/, \(CHT_{x1}\
eq(CHT_{x1})_{\textit{firve}}\).) Then, we perform lowest-payoff elimination by deleting the channel with payoff lower than half of the maximum payoff:  
\[ \begin{split} cutpayoff_{i}&=\left\{cutpayoff_{x},cutpayoff_{y}^{\prime}\right\}_{i}\\ &=\left\{(CHT_{x1},CHT_{x2},...,CHT_{xlp}[CHT_{xlp}\geqslant \frac{1}{2}CHT_{x1},xlp\varepsilon\leq xl),\right.\\ &\quad\left.(CHT_{y1}^{\prime},CHT_{y2}^{\prime},...,CHT_{ylp}^{ \prime}\right|CHT_{ylp\varepsilon}^{\prime}\geqslant\frac{1}{2}CHT_{y1}^{ \prime},ylp\varepsilon\leqslant yk)\right\}_{i}\end{split}  \tag{22}\]  
where \(cutpayoff_{i}\) is the payoff without the eliminated payoffs, and the payoffs for licensed channels and unlicensed channels are stored in \(cutpayoff_{x}\) and \(cutpayoff_{y}\), respectively, observed by \(S_{i}\). The remaining channels on \(cutpayoff_{i}\) are called the candidate channels. Between these candidate channels, an operating channel and a backup channel are selected. As the payoff of a channel is higher, the probability of it getting selected as an operating/backup channel is higher. Channel selection depends on a probability distribution called selection game probability (SGP). The SGP of \(S_{i}\) is \((SGP)_{i}=\{(SGP)_{x},(SGP)_{y}\}_{i}\), where \((SGP)_{x}\) and \((SGP^{\prime})_{y}\) are the SGPs for licensed channels and unlicensed channels, respectively. Thus:  
\[ \begin{split}(SGP)_{x}=\left\{SGP(A_{x1}),SGP(A_{x2}),...,SGP(A_{ xlp})\middle|SGP(A_{x1})\geqslant SGP(A_{x2})\geqslant...\geqslant SGP(A_{xlp})\right\} \end{split}  \tag{23}\]  
\[ \begin{split}(SGP^{\prime})_{y}=\left\{SGP^{\prime}(B_{y1}),..., SGP^{\prime}(B_{ylpe})\middle|SGP^{\prime}(B_{y1})\geqslant SGP(B_{y2}) \geqslant...\geqslant SGP^{\prime}(B_{ylpe})\right\}\end{split}