# Spacing Allocation Method for Vehicular Platoon: A Cooperative Game Theory Approach
## 4 Methodology Based on Cooperative Game Model for Platoon  
The cooperative game theory focusses on the method of allocating benefits to participants according to their marginal contribution. When giving the headways of vehicles in platoon, it could be considering as a spacing allocation which make cooperative game model sometimes preferable. Figure 3: Center gathering principle. Figure 2: Velocity match principle. In this section, a cooperative game-based platoon spacing policy is proposed. The cooperative game model is directly relative to the information follow topology of the platoon, which is shown in Figure 4. The information of the leading vehicle (vehicle 0) is available for all following vehicles (vehicle 1-n). Additionally, the following vehicles can obtain the position, velocity and acceleration of the nearest vehicles. The vehicles will determine their desired spacing with its previous vehicle according to the information. To simplify the problem, the following assumptions are made:  
1. CAVs in platoon are assumed to be able to achieve autonomous acceleration and deceleration;
2. CAVs in platoon are assumed to have wireless communication capabilities. 3. For the controller design, it was assumed that there is no time delay in the process of the communication system. 4. The cooperative game of the platoon is a balanced cooperative game, which mean that the core of the problem is non-empty. 5. The vehicles in platoon are arranged and cannot skip a vehicle to form a separate coalition, which means that the available coalition only should be {0}, {0,1},..., {0, 1,..., \(i,\)..., \(n-1\), \(n\)}. The profit of cooperative game here is the desired spacing which should be distributed to the participants. The allocation of spacing is considered as a transferable utility (\(TU\)) game, and the definition is shown as followed. Cooperative game of spacing allocation for platoon can be described as a pair (\(N\), \(v\)) consisting of a non-empty and finite set of vehicles \(N\) and coalition function \(v\): \(2^{N}{\rightarrow}R\), \(v(\emptyset)=0\). Subsets of \(N\) are called coalitions, and \(v(K)\) is called the worth of coalition \(K\). The spacing profit of a coalition could be obtained from proposed spacing policy based on three principles. Assuming that the key indicators \(\omega_{v}\) and \(\omega_{c}\) are the same to simplify the model. While the vehicles are non-cooperative, they won't care velocity and center gathering. Therefore, if a coalition only has one participant, what they care is avoiding collision. Besides, the last following vehicle in platoon cannot follow the centering position of two vehicles. Thus, the characteristic function of the game model is given in Equation (10). The profits of all other coalitions are zeros. \[\left\{ \begin{array}{c}v[\varnothing]=0\\ v[\{1\}]=J_{a1}+J_{v1}\\ v[\{1,2\}]=J_{a1}+J_{v1}+J_{c1}+J_{a2}+J_{v2}\\ v[\{1,2,3\}]=J_{a1}+J_{v1}+J_{c1}+J_{a2}+J_{v2}+J_{c2}+J_{a3}+J_{v3}\\ \ldots\\ v[N]=\sum\limits_{i=1}^{n}J_{ai}+\sum\limits_{i=1}^{n}J_{vi}+\sum\limits_{i=1 }^{n-1}J_{ci}\end{array} \right. \tag{10}\]  
where \(N\) denotes the grand coalition composed of \(n\) vehicles. **Definition 1**. : _The game \(v\in G^{N}\) is monotonous, if all coalition \(S\) and \(T\) satisfy \(S\subset T\), and there is \(v(S)\leq v(T)\)  
Figure 4: Information flow topology of the platoon. **Definition 2**. : _The game \(v\in G^{N}\) is super-additivity, if all coalition \(S\) and \(T\) satisfy \(S\cap\ T=\varnothing\), and \(v(S\cup\ T)\geq v(S)+v(T)\)._  
**Definition 3**. : _The game \(v\in G^{N}\) is convex, if all coalition \(S\) and \(T\), and \(v(S)+v(T)\leq v(S\cup\ T)+v(S\cap\ T)\)._  
The proposed cooperative game of platoon is convex. The proof is shown in Equation (11). \[ \begin{array}{c}v(N)-[v(N/n)+v(n)]\\ =\sum\limits_{i=1}^{n}J_{ai}+\sum\limits_{i=1}^{n}J_{vi}+\sum\limits_{i=1}^{n- 1}J_{ci}-\begin{bmatrix}\sum\limits_{i=1}^{n-1}J_{ai}+\sum\limits_{i=1}^{n-1}J_ {vi}+\sum\limits_{i=1}^{n-2}J_{ci}+J_{an}+J_{vn}\\ =J_{cn-1}\geq 0\end{bmatrix}\end{array}  \tag{11}\]