## 4 Game Model
### Forwarding Probability  
Forwarding candidate nodes would take the following three factors into consideration to form their forwarding probabilities:  
1. The ratio composed by the out-degree of their own and the average out-degree for all nodes. We denote it as \(O_{i}\)/\(O_{a}\). Here, \(O_{i}\) means the out-degree of node \(i\) and \(O_{a}\) means the average out-degree for all nodes in the sensor field. 2. The ratio composed by current energy and initial energy (\(E_{\textit{{\{current\}}}}\)/\(E_{\textit{{\{initial\}}}}\)) of nodes. We denote it as \(E_{\textit{{\textit{i(c)}}}}\). 3. The cosine value of the angle composed by the line linked candidate node \(i\) and source \(S\) and the line constituted by source node \(S\) and destination \(D\). It is shown in Figure 3 and we denote it as \(\cos A\). Figure 2: Construction of a forbidden region for a concave region. For any candidate node \(i\), it it easy to obtain \(E_{i(c)i}\). From Figure 3, we can deduce the cosine value of the angle between line \(iS\) and \(SD\) by Equation (1). Since we assume that the nodes are discrete and uniformly distributed, it is easy to obtain \(O_{i}/O_{a}\) for any node:  
\[cos\ A=(i\overline{iS}^{2}+i\overline{iD}^{2}-\overline{SD}^{2})\ /\ 2\ast i \overline{S}\ast i\overline{D} \tag{1}\]  
_Lemma 1_: The angle composed by the line constituted by candidate node \(i\) and source \(S\) and the line constituted by source node \(S\) and destination \(D\) is less than 90 degrees. _Proof_: In Figure 3, we can consider the node S as the source and construct a line from \(S\) to \(D\). We assume that there is a candidate node of \(S\) and we denote it as \(i\). In case the angle composed by the line constituted by candidate node _(i)_ and source _(S)_ and the line constituted by source node \(S\) and destination \(D\) is more than 90 degrees, the distance between \(i\) and \(D\) (\(iD\)) must be more than the distance between \(S\) and \(D\) (\(SD\)). In case \(iD>SD\) and here, \(iD\) is the distance from node \(i\) to destination \(D\), \(SD\) is the distance from source node \(S\) to destination \(D\), \(i\) must be the in-degree node of \(S\) and has no qualification to be the candidate node to participate in forwarding. This contradicts the supposition, and the Lemma is proven. \(\mathcal{LA}\) is the angle between the candidate node and the line \(SD\). We have concluded that the angle between the candidate node and the line \(SD\) is at the degree interval [0, 90] from Lemma 1. For the cosine value of \(\mathcal{LA}\), the cosine value is a decreasing function in the interval [0, 90]. In other words, \(\mathcal{LA}=0^{\circ}\), \(\cos\ A=1\); \(\mathcal{LA}\to 90^{\circ}\), \(\cos\ A\to 0\). _Lemma 2_: In terms of load balance and energy consumption, minimum angle routing has better performance than shortest path routing. _Proof_: In Figure 4, \(S\) is the source and \(D\) is the destination node. Line \(SD\) is the linear distance between \(S\) and \(D\). \(A_{SD}\) is the forwarding path selected by the minimum angle routing and \(G_{SD}\) is the forwarding path selected by the shortest path routing. \(A_{SD}=1\to 2\to 3\to 4\to 5\to 6\); \(G_{SD}=7\to 8\to 9\to 10\). According to the radio model [25], we can calculate the energy consumption of link \(A_{SD}\) and \(G_{SD}\). It is shown by Equations (2) and (3):  
Figure 3: Illustration for forwarding angle with source, destination and forwarding candidate node. node \(i\) possesses the mixed strategy based on pure strategy space \(S_{i}\). It means that node \(i\) would forward packets according to the probability _P(i)_. During the game progress, different strategies lead to different benefits for each participant. The benefit to participants is described by the payoff function. For any participant, the benefit has some relation with strategy.