# Biform Game Approach to Strategy Optimization of Autonomous Vehicle Lane Changes on Highway Ramps
## 4 Example Analysis
### Cooperative Game Stage  
Different strategic situation combinations in the biform game of autonomous vehicle lane-changing are shown in Table 1.  
According to Equation (10), the alliance utility value formed by the participants \(P_{1},~{}P_{2},~{}\text{and}~{}P_{3}\) in the game under the strategic situation was calculated. The calculation processes of \(V(S_{1})(1,2,3)\) and \(V(S_{5})(1,2)\) are given here, and the other calculation processes were ignored. \[ \begin{array}{ll}V(S_{1})(1,2,3)&=1.5\times\sum\limits_{t=1}^{n}\left[V_{1}( t+1)-V_{1}(t)\right]+\sum\limits_{t=1}^{n}\left[V_{2}(t+1)-V_{2}(t)\right]+0.8 \times\\ &\sum\limits_{t=1}^{n}\left[V_{3}(t+1)-V_{3}(t)\right]+\sum\limits_{t=1}^{n} \left[V_{1}(t)+V_{2}(t)+V_{3}(t)\right]\\ &=1.5\times 3+3+0.8\times 0+65+60+60\\ &=192.5\\ \end{array}  \tag{19}\]  
\[ \begin{array}{ll}V(S_{5})(1,2)&=1.5\times\sum\limits_{t=1}^{n}\left[V_{1}( t+1)-V_{1}(t)\right]+\sum\limits_{t=1}^{n}\left[V_{2}(t+1)-V_{2}(t)\right]+0.8 \times\\ &\sum\limits_{t=1}^{n}\left[V_{3}(t+1)-V_{3}(t)\right]+\sum\limits_{t=1}^{n} \left[V_{1}(t)+V_{2}(t)+V_{3}(t)\right]\\ &=1.5\times 3+3+0.8\times 0+65+60\\ &=132.5\\ \end{array}  \tag{20}\]  
The alliance utility values for the three vehicles are shown in Table 2.  
The characteristic function under the different competitive situations in Table 2 does not satisfy the _AU_ condition. Therefore, there is no core when solving the cooperative game, and thus the core cannot be used to solve this biform game model. However, it can be observed that this characteristic function satisfies both the NE and NC conditions, indicating that the biform game has a pure-strategy Nash equilibrium solution that is effective [45]. This example conforms to the situation where if a player changes their strategy, it may reduce the total payoff of other participants in the coalition. \begin{table}
\begin{tabular}{c c c c c} \hline  & & & **Leading Vehicle P\({}_{1}\)** & \\ \cline{3-5}  & & **Acceleration** & & **Constant Speed** \\ \cline{3-5}  & & & **Lane-Changing Vehicle P\({}_{2}\)** & \\ \cline{3-5}  & & **Accelerating** & **Constant Speed** & **Accelerating** & **Constant Speed** \\ \hline \multirow{2}{*}{Following vehicle P\({}_{3}\)} & Constant speed & (A, A, C) & (A, C, C) & (C, A, C) & (C, C, C) \\ \cline{2-5}  & Deceleration & (A, A, D) & (A, C, D) & (C, A, D) & (C, C, D) \\ \hline \end{tabular}
\end{table} 
Table 1: Different combinations of strategies for three vehicles affected by lane-changing. In 1953, [PERSON] proposed the concept of the Shapley value, which provides an excellent method for determining how to allocate the proportion of earnings for each person participating in an n-person bargaining game, that is, a cooperative game. Starting from the axioms of efficiency, symmetry, and additivity, [PERSON] put forward the concept of a solution to a cooperative game and proved its uniqueness. This solution is what people later referred to as the Shapley value [43]. Therefore, this study used the Shapley value to solve the cooperative game phase in the biform game, where, in the situation \(S_{k}\), the Shapley value in the cooperative game among \(n\) players \(V(S_{k})\) is a \(n\)-dimensional vector:  
\[\partial(V(S_{k}))=(\partial_{1}(V(S_{k})),\partial_{2}(V(S_{k})),\ldots, \partial_{n}(V(S_{k})))^{T}, \tag{21}\]  
For any player in the game \(i\in N\), it is given by Equation (22):  
\[\partial_{i}(V(S_{k}))=\sum_{i\in T_{\subseteq N}}\frac{(n-|T|)!(|T|-1)!}{n!}( V(S_{k})(T)-V(S_{k})(T\{i\})) \tag{22}\]  
In the formula, \(\partial(V(S_{k}))\) is the Shapley value of player \(i\) in the cooperative game \(V(S_{k})\), \(|T|\) represents the number of players in the coalition, \(\frac{(n-|T|)!(|T|-1)!}{n! }\) represents the probability of joining later and forming a coalition \(T\), and \(V(S_{k})(T)-V(S_{k})(T\{i\})\) represents the marginal contribution probability of player \(i\).