## 4 Distributed Real-Time Computation Offloading Algorithm Based on MDPSO
### Modeling the RCO Problem Based on Multi-Player Static Game
Therefore, we will analyze the existence of Nash equilibrium solutions for the given \(N\)-player static game problem \(\mathcal{G}\). Firstly, the number of players participating in this finite-strategy static game \(\mathcal{G}\) is denoted as \(N\). Let the \(n\)-th player's pure channel strategy available at time \(t\) be represented as  
\[\text{P}_{n,t}=\{A_{1}^{n},A_{2}^{n},...,A_{F}^{n}\},n\in\text{N}_{d}. \tag{26}\]  
The strategy space composed of pure strategies for \(N_{d}\) players is  
\[\text{P}_{t}=\text{P}_{1,t}\times\text{P}_{2,t}\times...\times\text{P}_{N,t}. \tag{27}\]  
Any strategy combination in strategy space A is \[p_{t}=\left(\text{A}^{1}_{j_{1},t},\text{A}^{2}_{j_{2},t},...,\text{A}^{N}_{j_{ j_{k},t}}\right),\text{A}^{n}_{j_{n},t}\in\text{P}_{n,t},n=1,2,...,N. \tag{28}\]  
Denote  
\[\bar{\text{P}}^{n-1}_{t}=\text{P}_{1,t},\text{P}_{2,t},...,\text{P}_{n-1,t}(1<n \leq N),\bar{\text{P}}^{n-1}_{t}=\Phi(n=1), \tag{29}\]  
\[\bar{\text{P}}^{n+1}_{t}=\text{P}_{n+1,t},\text{P}_{n+2,t},...,\text{P}_{N,t}( 1\leq n<N),\bar{\text{P}}^{n+1}_{t}=\Phi(n=N). \tag{30}\]  
Apparently  
\[\text{P}_{t}=\left\{ \begin{array}{ll}\text{P}_{1,t}\times\bar{\text{P}}^{n+ 1}_{t},&n=1\\ \bar{\text{P}}^{n-1}_{t}\times\text{P}_{n,t},&n=N\\ \bar{\text{P}}^{n-1}_{t}\times\text{P}_{n,t}\times\bar{\text{P}}^{n+1}_{t},&1< n<N.\end{array} \right. \tag{31}\]  
Without loss of generality, when \(1<n<N\), the following exists:  
\[ \begin{split}&\forall p^{n-1}_{t}\in\bar{\text{P}}^{n-1}_{t},p^ {n-1}_{t}=\left(\text{A}^{1}_{j_{1},t},\text{A}^{2}_{j_{2},t},...,\text{A}^{n- 1}_{j_{n-1},t}\right),\text{A}^{k}_{j_{k},t}\in\text{P}_{k,t},k=1,2,...,n-1,\\ &\forall p^{n-1}_{t}\in\bar{\text{P}}^{n+1}_{t},p^{n+1}_{t}=\left(\text{A }^{1}_{j_{n+1},t},\text{A}^{2}_{j_{n+2},t},...,\text{A}^{N}_{j_{N},t}\right),\text{A}^{k}_{j_{k},t}\in\text{P}_{k,t},k=n+1,n+2,...,N.\end{split}  \tag{32}\]  
According to the utility function \(\Gamma_{n}(v^{d,t}_{n})\) of the \(n\)-th player, if there exists the following:  
\[p^{*}_{t}=\left(p^{n-1*}_{t},\text{A}^{n*}_{j_{n},t},p^{n+1*}_{t}\right), \tag{33}\]  
let  
\[\Gamma_{n}(p^{*}_{t})=\Gamma_{n}\Big{(}p^{n-1*}_{t},\text{A}^{n*}_{j_{n},t},p^ {n+1*}_{t}\Big{)}\geqslant\Gamma_{n}\Big{(}p^{n-1*}_{t},\text{A}^{n}_{k,t},p^ {n+1*}_{t}\Big{)},\forall\text{A}^{n}_{k,t}\in\text{P}_{n,t}. \tag{34}\]  
Then, \(p^{*}_{t}\) is referred to as a pure-strategy Nash equilibrium of the finite-strategy static game \(\mathcal{G}\). Building upon the above, we can continue the derivation; let  
\[ \begin{split}& T_{i,t}(p^{n-1}_{t},p^{n+1}_{t})=\Bigg{\{}(p^{n-1} _{t},A^{n}_{j_{k},t},p^{n+1}_{t})|\max_{A^{n}_{k}\in\text{P}_{n,t}}U_{i}(p^{n-1 }_{t},A^{n}_{k,t},p^{n+1}_{t})\Bigg{\}},i=1,2,\cdots,N\\ & T_{i,t}=\cup\limits_{(p^{n-1}_{t},p^{n+1}_{t})\in\bar{\text{P}} ^{n-1}_{t}\times\bar{\text{P}}^{n+1}_{t}}T_{i}(p^{n-1}_{t},p^{n+1}_{t})=\\ &\{(p^{n-1}_{t},A^{n}_{j_{k},t},p^{n+1}_{t})|\max_{A^{n}_{k}\in \text{P}_{n,t}}U_{i}(p^{n-1}_{t},A^{n}_{k},p^{n+1}_{t})\},\\ &\forall(p^{n-1}_{t},p^{n+1}_{t})\in\bar{\text{P}}^{n-1}_{t}\times \bar{\text{P}}^{n+1}_{t}|\;i=1,2,\cdots,N.\end{split}