## 4 Static Game Analysis
### Game Profit Calculation  
Players \(F_{1}\) and \(F_{2}\) finally select or give up taking bus \(k_{1}\) under the influence of the bus information, which generates four strategy combinations, as shown in Table 1.  
Under different strategy combinations, players \(F_{1}\) and \(F_{2}\) have different profits. The profit values of \(F_{1}\) and \(F_{2}\) under each strategy combination is calculated according to Formula (4). (1) P\({}_{1}\)(S,S), that is, both \(F_{1}\) and \(F_{2}\) choose bus \(k_{1}\). The expressions of their profits, \(U_{1}\) and \(U_{2}\), are shown in Equation (5). \[U_{1}=U_{2}=\frac{E_{k_{1}}}{q_{k_{1},i-1}^{\text{on}}+q_{i}^{\text{wait}}}, \tag{5}\]  
where \(q_{i}^{\text{wait}}\) denotes the passengers waiting for the bus at platform \(i\). The corresponding profit combination of P\({}_{1}\)(S,S) is: (\(\dfrac{E_{k_{1}}}{q_{k_{1},i-1}^{\text{on}}+q_{i}^{\text{wait}}}\),\(\dfrac{E_{k_{1}}}{q_{k_{1},i-1}^{\text{on}}+q_{i}^{\text{wait}}}\)). (2) P\({}_{2}\)(G,G), that is, both \(F_{1}\) and \(F_{2}\) choose to travel on bus \(k_{2}\). The expressions of their profits, \(U_{1}\) and \(U_{2}\) are shown in Equation (6). \[U_{1}=U_{2}=\frac{E_{k_{2}}}{q_{k_{2},i-1}^{\text{on}}+q_{i}^{\text{wait}}} \tag{6}\]  
The corresponding profit combination of P\({}_{2}\)(G,G) is (\(\dfrac{E_{k_{2}}}{q_{k_{2},i-1}^{\text{on}}+q_{i}^{\text{wait}}}\),\(\dfrac{E_{k_{2}}}{q_{k_{2},i-1}^{\text{on}}+q_{i}^{\text{wait}}}\)). (3) P\({}_{3}\)(S,G), that is, \(F_{1}\) chooses bus \(k_{1}\) and \(F_{2}\) chooses bus \(k_{2}\). Because the congestion perception of an individual on a bus is equal to the crowd perception of the group on the bus, the congestion perception of the players on the bus can be obtained by calculating the congestion degree on the bus. Assume that the proportion of passengers who finally choose bus \(k_{1}\) among the waiting passengers on platform \(i\) is \(\alpha\). Therefore, the number of people who finally take bus \(k_{1}\) is \(\alpha\cdot q_{i}^{\text{wait}}\), and \(F_{1}\) belongs to this group. The number of passengers who give up choosing bus \(k_{1}\) is \((1-\alpha)\cdot q_{i}^{\text{wait}}\), and \(F_{2}\) belongs to this group. The expressions of their profits, \(U_{1}\) and \(U_{2}\), are shown in Equations (7) and (8). \[U_{1}=\frac{E_{k_{1}}}{q_{k_{1},i-1}^{\text{on}}+\alpha\cdot q_{i}^{\text{ wait}}} \tag{7}\]  
\[U_{2}=\frac{E_{k_{2}}}{q_{k_{2},i-1}^{\text{on}}+(1-\alpha)\cdot q_{i}^{\text{ wait}}} \tag{8}\]  
The corresponding profit combination of P\({}_{3}\)(S,G) is (\(\dfrac{E_{k_{1}}}{q_{k_{1},i-1}^{\text{on}}+\alpha\cdot q_{i}^{\text{wait}}}\), \(\dfrac{E_{k_{2}}}{q_{k_{2},i-1}^{\text{on}}+(1-\alpha)\cdot q_{i}^{\text{wait}}}\)). (4) P\({}_{4}\)(G,S), that is, \(F_{1}\) chooses to travel on bus \(k_{2}\) whereas \(F_{2}\) chooses bus \(k_{1}\). Similar to the profit calculation method for P\({}_{3}\)(S,G), it is assumed that the proportion of passengers who finally give up bus \(k_{1}\) is \(\beta\). Finally, the expressions of profits \(U_{1}\) and \(U_{2}\) for players \(F_{1}\) and \(F_{2}\) are obtained by Equations (9) and (10). \[U_{1}=\frac{E_{k_{2}}}{q_{k_{2},i-1}^{\text{on}}+\beta\cdot q_{i}^{\text{ wait}}} \tag{9}\] \begin{table}
\begin{tabular}{c c c c} \hline \hline  & & \multicolumn{2}{c}{\(F_{1}\)} \\ \cline{3-4}  & & S & G \\ \hline \multirow{2}{*}{\(F_{2}\)} & S & P\({}_{1}\)(S,S) & P\({}_{3}\)(S,G) \\ \cline{2-4}  & G & P\({}_{4}\)(G,S) & P\({}_{2}\)(G,G) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Game strategy player combinations. \[U_{2}=\frac{E_{k_{1}}}{q_{k_{1},i-1}^{\text{on}}+(1-\beta)\cdot q_{i}^{\text{wait}}} \tag{10}\]  
The corresponding profit combination of P4(G,S) is \((\frac{E_{k_{2}}}{q_{k_{2},i-1}^{\text{on}}+\beta\cdot q_{i}^{\text{wait}}},\)\(\frac{E_{k_{1}}}{q_{k_{1},i-1}^{\text{on}}+(1-\beta)\cdot q_{i}^{\text{wait}}})\). The profit combinations of P3(A,R) and P4(R,A) vary with \(\alpha\) or \(\beta\).