## 4 Problem Decomposition and Solution
### Multiuser Computation Resource Allocation Game  
We introduce game theory and construct a centralized framework to obtain an appropriate solution. Accordingly, we formulate the optimization problem as a multi-TaV task offloading game \(\mathcal{G}=\{\mathcal{N},A_{n},U_{n}\}\), where \(\mathcal{N}\) is the TaVs' set, \(A_{n}=\alpha_{n}\otimes\beta_{n}\otimes\theta_{n}\otimes\rho_{n}\) represents TaVn's strategy space, and \(U_{n}\) is the utility function of TaVn. As in studies [17; 27], the impact of a certain decision-making action on the whole system is measured by the marginal utility theory, and the utility function of a game participant is denoted by  
\[U_{n}(a_{n},a_{-n})=\mathrm{T}_{n}(a_{n},a_{-n})+\sum_{i\
eq n}\left(\mathrm{T }_{i}(a_{i},a_{-i})-\mathrm{T}_{i}(a_{i},a_{-i\setminus n})\right), \tag{22}\]  
where \(a_{-n}\) is the TaVs' action profile, except TaVn, \(\mathrm{T}_{i}(a_{i},a_{-i})\) is the benefit of TaVi when TaVn takes action, and \(\mathrm{T}_{i}(a_{i},a_{-i\setminus n})\) is the benefit of TaVi when TaVn takes no action. Therefore, the first term and the second term of Equation (22) denote the benefit of TaVn and the influence on other TaVs brought by [PERSON]'s action, respectively. The formulated game \(\mathcal{G}\) reaches a NE while \(a^{*}=\left(a_{1}^{*},a_{2}^{*},\ldots,a_{N}^{*}\right)\) if and only if no Tav can increase its utility when changing its action unilaterally while other TaVs keep their decisions unchanged, namely,  
\[U_{n}(a_{n}^{*},a_{-n}^{*})\geq U_{n}(a_{n},a_{-n}^{*}),\forall n\in\mathcal{N}, \forall a_{n}\in A_{n},a_{n}\
eq a_{n}^{*}. \tag{23}\]  
In addition, if and only if there exists a potential function \(\Theta\) such that [28]  
\[U_{n}(\widetilde{a}_{n},a_{-n})-U_{n}(a_{n},a_{-n})=\Theta(\widetilde{a}_{n},a _{-n})-\Theta(a_{n},a_{-n}),\forall n\in\mathcal{N},\forall a_{n}\in A_{n}, \forall\widetilde{a}_{n}\in A_{n}, \tag{24}\]  
a game is an EPG, where \(a_{n}\) and \(\widetilde{a}_{n}\) are available actions from TaVn's strategy space. Equation (24) means that the fluctuation of the two functions is the same while any Tav changes its action. Theorem 4.2: _The proposed multi-TaV task offloading game is an EPG with at least one pure-strategy NE point, and the optimal combination of offloading strategy and computation resources allocation consists of a pure-strategy NE point of \(\mathcal{G}\)._  
Proof of Theorem 4.2.: Motivated by [17], to make the potential function have physical significance, the potential function is designed as  
\[\Theta(a_{n},a_{-n})=\sum\limits_{n\in\mathcal{N}}\mathrm{T}_{n}(a_{n},a_{-n}), \tag{25}\]  
which is equivalent to the aggregate values of all TaVs. When TaVn changes its decision from \(a_{n}\) to \(\widetilde{a}_{n}\), the change in individual utility function can be denoted by  
\[U_{n}(\widetilde{a}_{n},a_{-n})-U_{n}(a_{n},a_{-n})\] \[=\mathrm{T}_{n}(\widetilde{a}_{n},a_{-n})+\sum\limits_{i\
eq n} \left(\mathrm{T}_{i}(a_{i},\widetilde{a}_{-i})-\mathrm{T}_{i}(a_{i}, \widetilde{a}_{-i\backslash n})\right)-\left(\mathrm{T}_{n}(a_{n},a_{-n})+ \sum\limits_{i\
eq n}\left(\mathrm{T}_{i}(a_{i},a_{-i})-\mathrm{T}_{i}(a_{i}, a_{-i\backslash n})\right)\right). \tag{26}\] \[=\mathrm{T}_{n}(\widetilde{a}_{n},a_{-n})-\mathrm{T}_{n}(a_{n},a_{ -n})+\sum\limits_{i\
eq n}\left(\mathrm{T}_{i}(a_{i},\widetilde{a}_{-i})- \mathrm{T}_{i}(a_{i},a_{-i})\right)+\sum\limits_{i\
eq n}\left(\mathrm{T}_{i }(a_{i},a_{-i\backslash n})-\mathrm{T}_{i}(a_{i},\widetilde{a}_{-i\backslash n })\right)\]  
Whether or not the action of TaVn changes, the influence on other TaVs maintaining their decisions is the same regardless of TaVn's decision, i.e., \(\mathrm{T}_{i}(a_{i},a_{-i\backslash n})=\mathrm{T}_{i}(a_{i},\widetilde{a}_ {-i\backslash n})\).