# Biform Game Approach to Strategy Optimization of Autonomous Vehicle Lane Changes on Highway Ramps
## 3 Establishment of Lane-Changing Model Based on Biform Games
### Biform Game Judgment Condition  
For the characteristic function of the biform game model, [PERSON] and [PERSON] proposed three conditions that need to be satisfied: the Adding-Up condition, the No Externalities condition, and the No Coordination condition.  
The following definition is given: \(N\backslash\{i\}\) is the set of all players \(\{1,\ldots,i-1,i+1,\ldots,n\}\) except player \(i\).  
**Definition 2**.: _Given a biform game \(\{S^{1},S^{2},\ldots,S^{n};V;a^{1},a^{2}\ldots a^{n}\}\), for any strategy profile \(s\in S\), the expression is_  
\[V(s)(N)-V(s)(N\backslash\{i\})\]  
_In the cooperative game \(V(s)\), this is called the additional value of player. It can be interpreted as follows: in a given cooperative game, the additional value of player is the difference between the overall value created by all players and the overall value created by the all the players other than player \(i\). That is, the allocation value of each player in the coalition is its marginal contribution value to the grand coalition. This value is also the value that player \(i\) adds to the entire game, which is the concept of the standard edge-set contribution cooperative game._  
**Property 1**.: _Adding-Up condition_  
_For any strategy profile \(s\in S\), assume that the core of \(V(s)\) is a non-empty subset. If_  
\[\sum_{i=1}^{n}\left[V(s)(N)-V(s)(N\{i\})\right]=V(s)(N) \tag{8}\]  
_then the biform game \(\{S^{1},S^{2},\ldots,S^{n};V;a^{1},a^{2}\ldots a^{n}\}\) is said to satisfy the Adding-Up property (Adding Up), simply referred to as the AU condition. This condition ensures that the core of the cooperative game exists and is unique. In the second-stage cooperative game, the sum of the additional values of the players is equal to the overall value created by the game._  
**Property 2**.: _No Externalities condition_  
_For any player \(i=1,\ldots,n,r^{i},s^{i}\in S^{i},\)\(s^{-i}\in S^{-i}\), assume that the core of \(V(s)\) is a non-empty subset. If_  
\[V\Big{(}r^{i},s^{-i}\Big{)}(N\backslash\{i\})=V\Big{(}s^{i},s^{-i}\Big{)}(N \backslash\{i\}) \tag{9}\]  
_then the biform game \(\{S^{1},S^{2},\ldots,S^{n};V;a^{1},a^{2}\ldots a^{n}\}\) is said to satisfy the No Externalities condition, simply referred to as the NE condition. It indicates that the strategy choice of each player \(i\in N\) does not affect the payoff of the remaining players in the coalition \(\{N\backslash\{i\}\}\)._  
**Property 3**.: _No Coordination condition_  
_For any player \(i=1,\ldots,n,r^{i},s^{i}\in S^{i},\)\(s^{-i}\in S^{-i}\), if and only if_  
\[V\Big{(}s^{i},S^{-i}\Big{)}(N)>V\Big{(}r^{i},s^{-i}\Big{)}(N) \tag{10}\]  
_then the biform game \(\{S^{1},S^{2},\ldots,S^{n};V;a^{1},a^{2}\ldots a^{n}\}\) is said to satisfy the No Coordination condition, simply referred to as the NC condition. It shows that when one player changes their strategy, the degree of its impact on the overall value created is independent of the strategy choices of other players, which ensures the existence of the solution of the non-cooperative game Nash equilibrium._