# Characteristic Analysis and Decision Model of Lane-Changing Game for Intelligent Connected Vehicles
## 4 Game Model
### Game Equilibrium Analysis  
The evolutionary game theory originated from biological evolution, which can dynamically analyze the evolutionary process of a group and clarify the evolutionary mechanism of which strategy to choose. With the cross development of game theory and the transportation field, many researchers apply game theory to study a vehicle interaction model [27; 28; 29]; compared with other game theories, the evolutionary game takes into account the characteristics of imperfect rationality of game subjects and reflects the evolutionary process of decision making more objectively. The evolutionary game reflects the evolutionary trends of different strategies through dynamic evolution equations, and the expected payoffs of the LV choosing to change lanes and choosing not to change lanes are:  
\[E_{lc}^{LV}=(2y-1)\left(\alpha P_{eff}^{LV}+(1-\alpha)P_{safe}^{LV}\right) \tag{10}\]  
\[E_{nlc}^{LV}=-\alpha P_{eff}^{LV}+(1-\alpha)P_{safe}^{LV} \tag{11}\]  
The overall expected payoff of the LV based on the payoff matrix is:  
\[E_{total}^{LV}=xE_{lc}^{LV}+(1-x)E_{nlc}^{LV}=2\left(\alpha P_{eff}^{LV}+(1- \alpha)P_{safe}^{LV}\right)xy-2x(1-\alpha)P_{safe}^{LV}-\alpha P_{eff}^{LV}+(1 -\alpha)P_{safe}^{LV} \tag{12}\]  
where \(x\) is the probability that the LV chooses to change lanes and \(1-x\) is the probability that the LV chooses not to change lanes. The evolutionary dynamic equation for the LV selection lane-change decision is:  
\[F_{LV}=\frac{\partial x}{\partial t}=x\left(E_{lc}^{LV}-E_{total}^{LV}\right) =2x(1-x)\left[\alpha yP_{eff}^{LV}+(1-\alpha)(y-1)P_{safe}^{LV}\right] \tag{13}\]  
Similarly, the expected payoffs of the RV's choice of giving way and not giving way are:  
\[E_{gw}^{RV}=-\beta P_{eff}^{RV}+(1-\beta)P_{safe}^{RV} \tag{14}\]  
\[E_{ngw}^{RV}=(1-2x)\left(\beta P_{eff}^{RV}+(1-\beta)P_{safe}^{RV}\right) \tag{15}\]  
The expected payoff of the RV with a probability of choosing to give way and with a probability of choosing not to give way is:  
\[E_{total}^{RV}=yE_{gw}^{RV}+(1-y)E_{ngw}^{RV}=-2\beta P_{eff}^{RV}y+(2 xy-2x+1) \left(\beta P_{eff}^{RV}+(1-\beta)P_{safe}^{RV}\right) \tag{16}\]  
where \(y\) is the probability that the RV chooses to give way and \(1-y\) is the probability that the RV chooses not to give way. The evolutionary dynamic equation for the RV selection yield decision is:  
\[F_{RV}=\frac{\partial y}{\partial t}=y\left(E_{gw}^{RV}-E_{total}^{RV}\right) =2y(1-y)\left[(x-1)\beta P_{eff}^{RV}+(1-\beta)xP_{safe}^{RV}\right] \tag{17}\]  
The strategy combinations of the LV and RV are (0, 0), (0, 1), (1, 0), and (1, 1), which are obtained by Equations (13) and (17); the corresponding strategy combinations are (do not change lane, do not give way), (do not change lane, give way), (change lane, do not give way), \begin{table}
\begin{tabular}{l c c} \hline \hline  & **RV** \\ \cline{2-3}
**LV** & **Give Way** & **Do Not Give Way** \\ \hline Change lanes & \(\alpha P_{eff}^{LV}+(1-\alpha)P_{safe}^{LV}\) & \(-\alpha P_{eff}^{LV}-(1-\alpha)P_{safe}^{LV}\) \\ Do not change lanes & \(-\alpha P_{eff}^{LV}+(1-\alpha)P_{safe}^{LV}\) & \(-\alpha P_{eff}^{LV}+(1-\alpha)P_{safe}^{LV}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Payoffs of LV in different strategy profiles. and (change lane, give way). In addition to the above four strategy combinations, there is another set of solutions for \(\left(P_{eff}^{RV}/\left(P_{eff}^{RV}+(1/\beta-1)P_{safe}^{RV}\right)\), \(P_{safe}^{LV}/\left((\alpha/1-\alpha)P_{eff}^{LV}+P_{safe}^{LV}\right)\right)\). The Jacobi matrix is constructed for the dynamic game system and expressed as:  
\[\text{Jacobi}= \begin{pmatrix}\frac{\partial F_{LV}}{\partial x}&\frac{ \partial F_{LV}}{\partial y}\\ \frac{\partial F_{RV}}{\partial x}&\frac{\partial F_{LV}}{\partial x}\end{pmatrix}  =