# A Power Control Algorithm Based on Chicken Game Theory in Multi-Hop Networks
## 2 System Model
\begin{array}{ll}P_{r}(i)&=\sum\limits_{j=0}^{i-1}P_{t}(j)d_{j,i}^{-\alpha}1 0^{-\frac{\tau_{j,j}}{10}}\sum\limits_{l=0}^{L-1}\left|\xi_{j,i}(l)\right|^{2 }\\ &=\sum\limits_{j=0}^{i-2}P_{t}(j)d_{j,i}^{-\alpha}10^{-\frac{\tau_{0,1}}{10}} \sum\limits_{l=0}^{L-1}\left|\xi_{j,i}(l)\right|^{2}+P_{t}(i-1)d_{i-1,j}^{- \alpha}10^{-\frac{\tau_{0,-1}}{10}}\sum\limits_{l=0}^{L-1}\left|\xi_{i-1,j}(l) \right|^{2}\end{array}  \tag{4}\]  
The summed transmit power \(P_{total}\) is calculated by totaling the transmit powers as follows:  
\[P_{total}=\sum\limits_{l=0}^{n-1}P_{t}(i) \tag{5}\]  
Because \(P_{r}(i)=P_{req}\), by using TPC, the transmit power \(P_{t}(i-1)\) of port \(\#i=1\) is derived via  
\[P_{t}(i-1)=\frac{P_{req}-\sum\limits_{j^{\prime}=0}^{i-2}P_{t}(j^{\prime})d_{ j^{\prime},j}^{-\alpha}10^{-\frac{\tau_{j^{\prime},j}}{10}}\sum\limits_{l=0}^{L-1} \left|\xi_{j^{\prime},i}(l)\right|^{2}}{d_{i-1,j}^{-\alpha}10^{-\frac{\tau_{i,- 1}}{10}}\sum\limits_{l=0}^{L-1}\left|\xi_{i-1,i}(l)\right|^{2}} \tag{6}\]  
Once the power \(P_{r}(i)\) received from previous ports \(\#0\)-\(\#(i-1)\) at port \(\#i\) is larger than the required received power \(P_{req}\), i.e., \(P_{r}(i)\geq P_{req}\), port \(\#(i-1)\) can be fired from the constructed route, i.e., \(P_{t}(i-1)\) = 0. The transmit power \(P_{t}(i-2)\) of port \(\#(i-2)\) is given as:  
\[P_{t}(i-2)=\frac{P_{req}-\sum\limits_{j^{\prime}=0}^{i-3}P_{t}(j^{\prime})d_{ j^{\prime},j}^{-\alpha}10^{-\frac{\tau_{j^{\prime},j}}{10}}\sum\limits_{l=0}^{L-1} \left|\xi_{j^{\prime},i}(l)\right|^{2}}{d_{i-2,j}^{-\alpha}10^{-\frac{\tau_{i,- 2}}{10}}\sum\limits_{l=0}^{L-1}\left|\xi_{i-2,i}(l)\right|^{2}} \tag{7}\]  
Applying this route modification method, the number of these hops decreases, and consequently, the time of the delay also reduces. The average normalized power \(P_{norm}\) using MHMRC diversity [9] is described as the average total transmit power following the route normalized by that of a single-hop case, i.e.,\(E[P_{total}]/E[P_{single-hop}]\), where \(P_{total}\) is calculated by Equation (5) and \(P_{single-hop}\) is calculated by Equation (1) with \(i=0\) (mobile terminal) and \(j=n\) (central port). Therefore, \(P_{norm}\) is derived as follows:  
\[P_{norm}=\frac{E[P_{total}]}{E[P_{single-hop}]}=\frac{E[\sum\limits_{i=0}^{n-1 }\left(\frac{1-\sum\limits_{j=0}^{i-1}\frac{P_{t}(j)}{r_{t0}}\bar{t}_{j+1}^{- \alpha}10^{-\frac{\eta_{j}+1}{10}}\sum\limits_{l=0}^{l-1}\left|\xi_{i,j+1}(l) \right|^{2}}{\bar{t}_{j+1}^{-\alpha}10^{-\frac{\eta_{j}+1}{10}}\sum\limits_{l=0 }^{l-1}\left|\xi_{i,j+1}(l)\right|^{2}}\right)]}{E[\frac{1}{d_{0,n}^{-\alpha}10 ^{-\frac{\eta_{0}+1}{10}}\sum\limits_{l=0}^{l-1}\left|\xi_{0,n}(l)\right|^{2} }]} \tag{8}\]  
The sending power \(P_{t}(j)\) is derived from Equation (3) for port #\(j=0\) and from Equation (6) for ports #\(j=1\)-\(n-1\), displacing \(i-1\) by \(j\). \(P_{t}(j)\) is derived recursively from Equation (6). Due to \(P_{t}(0)\propto P_{req}\), it can be easily accepted that \(P_{t}(j)/P_{req}\) is not a function of \(P_{req}\). Consequently, \(P_{norm}\) is an independent variable relative to \(P_{req}\). In Equation (8), considering the fact that two operations of expectation and addition have been done, the algorithm complexity of \(P_{norm}\) is O(n\({}^{3}\)). Figure 2 draws the normalized average power as a function of \(N\) with \(\varrho\) as a factor for \(\alpha=3.5\), \(\sigma=7\) dB, \(L=2\), and \(K=50\). It can be seen that the MHMRC (Multi-Hop Maximum Ratio Combining) [9] reduces the sum transmit power for all \(\varrho\) values. The power decrease by MHMRC is bigger when \(\varrho\) reduces. As for \(\varrho\geq 0.5\), the MHMRC diversity gain reduces and becomes very small once \(\varrho=1\). While \(\varrho=1\), both the routing construction channel and the data's communication channel experience the same fading. The data's communication routing is equal to the minimum transmit power routing.