## 5 Game-Based Energy-Efficient Underwater Communication Mechanism and NIP Problem
### The Optimal Mechanism and Problem Formulation  
Based on economic game theory, we propose a payment mechanism to incorporate overhead. Therefore, the game-based, energy-efficient underwater communication mechanism can be expressed as minimizing the total overhead of the optimization layer SNs. We use \(a_{i}\in\{0,1\}\) to denote whether SN i chooses mode \(a\), i.e., transmits data directly to the BS, where \(a_{i}=1\) means that mode \(a\) is selected, otherwise it is not selected, and the corresponding strategy can be represented by \(\mathbf{a}=\left\{a_{i}\right\}_{i\in\mathcal{I}}\). We use \(b_{i,j}\in\{0,1\}\) to denote whether SN i chooses mode \(b\) and transmits data to SN j for its assistance, where \(b_{i,j}=1\) means that SN i chooses mode \(b\), otherwise it does not choose. The corresponding strategy can be represented by \(\mathbf{b}=\left\{b_{i,j}\right\}_{i\in\mathcal{I},j\in\mathcal{J}}\). Similarly, we use \(d_{i,j}\in\{0,1\}\) to denote whether SN i chooses mode \(d\) to process the data task, i.e., assisted by SN j after calculating. As above, \(d_{i,j}=1\) means selection, otherwise it means no selection, and the corresponding strategy can be represented by \(\mathbf{d}=\left\{d_{i,j}\right\}_{i\in\mathcal{I},j\in\mathcal{J}}\). Since the optimization layer has only one datum per SN and cannot be split, we have the constraint as  
\[\sum_{j=1}^{J}\left(b_{i,j}+d_{i,j}\right)+a_{i}\leq 1,i\in\{1,2,\ldots 1\}. \tag{23}\]  
Therefore, the total overhead of all SNs in the optimization layer can be denoted as  
\[\sum_{i=1}^{I}\left\{a_{i}C_{i}^{a}+\sum_{j=1}^{J}\left(b_{i,j}C_{i,j}^{b}+d_ {i,j}C_{i,j}^{d}\right)+[1-a_{i}-\sum_{j=1}^{J}\left(b_{i,j}+d_{i,j}\right)]C_ {i}^{c}\right\} \tag{24}\]  
It can be simplified as  
\[\sum_{i=1}^{I}\{\sum_{j=1}^{J}[b_{i,j}\left(C_{i,j}^{b}-C_{i}^{c}\right)+d_{ i,j}\left(C_{i,j}^{d}-C_{i}^{c}\right)+a_{i}(C_{i}^{a}-C_{i}^{c})+C_{i}^{c}]\} \tag{25}\]  
At the same time, since the data size accepted by each SN in the assisted layer is limited in each round of task processing, we have the constraints as  
\[\sum_{i=1}^{J}\left(b_{i,j}D_{i}+d_{i,j}D_{i}^{c}\right)\leq Z_{j},j\in\{1,2, \ldots,J\}, \tag{26}\]where \(Z_{j}\) is the maximum data size that can be accepted by the assisted layer SN j during a single round of transmission. We set the following delay constraints to ensure its efficiency in terms of transmission time. \[\sum_{j=1}^{J}\left[b_{ij}\left(T_{i,j}^{b}-T_{i}^{c}\right)+d_{i,j}\left(T_{i,j }^{d}-T_{i}^{c}\right)\right]+a_{i}(T_{i}^{a}-T_{i}^{c})+T_{i}^{c}\leq T_{i}, \forall i, \tag{27}\]  
where \(T_{i}\) is the maximum delay acceptable for transmitting SN i data to BS. Apparently, with satisfying the limit of the number of accepted tasks per round for each SN in the upper layer and the delay constraint of SN i, the optimal mechanism can be written as  
\[\underset{\mathbf{a},\mathbf{b},\mathbf{d}}{min}\sum_{i=1}^{J}\left\{\sum_{j=1}^{J}\left[b _{i,j}\left(C_{i,j}^{b}-C_{i}^{c}\right)+d_{i,j}\left(C_{i,j}^{d}-C_{i}^{c} \right)\right]+a_{i}(C_{i}^{a}-C_{i}^{c})+C_{i}^{c}\right\}, \tag{28a}\] \[s.t.\sum_{i=1}^{I}\left(b_{i,j}D_{i}+d_{i,j}D_{i}^{c}\right)\leq Z_{j},j\in \{1,2,\ldots,J\},\] (28b) \[\sum_{j=1}^{J}\left(b_{i,j}+d_{i,j}\right)+a_{i}\leq 1,i\in\{1,2, \ldots I\},\] (28c) \[\sum_{j=1}^{J}\left[b_{i,j}\left(T_{i,j}^{b}-T_{i}^{c}\right)+d_{i,j}\left(T_{ i,j}^{d}-T_{i}^{c}\right)\right]+a_{i}(T_{i}^{a}-T_{i}^{c})+T_{i}^{c}\leq T_{ i},\forall i,\] (28d) \[a_{i},b_{i,j},d_{i,j}\in\{0,1\},\forall i,j. \tag{28e}\]  
The objective function (28a) aims to minimize the total overhead of SN in the optimization layer. The constraint (28b) ensures that the total data size transmitted by all SNs in the optimization layer to SN j in each round of transmission does not exceed its maximum acceptable data size. The constraint condition (28c) indicates that the optimization layer adopts only one way per SN to transmit data. In addition, the constraint condition (28d) represents the delay constraint.