# Game Analysis of Different Transportation Modes in a Corridor Considering Carbon Emission Costs
## 3 Game Model
### Dynamic Game Model
So,  
\[u_{k}(p_{k}^{i})=\alpha_{1}\times e^{\frac{-(p_{k}^{i})^{2}+\alpha_{k}\times p _{k}^{i}}{p_{k}}}+(\alpha_{3}\times d_{k}+\alpha_{2}\times s_{k}+\alpha_{5} \times c_{k})\times e_{k}. \tag{9}\]  
Transposing Equation (9) leads to Equation (10):  
\[u_{k}(p_{k}^{i})-\Delta_{k}^{hf}=\alpha_{1}\times e^{\frac{-(p_{k}^{i})^{2}+ \alpha_{k}\times p_{k}^{i}}{p_{k}}}. \tag{10}\]  
Supposing that when the consignor's utility to the mode of transport \(k\) reaches the maximum, the optimal solution of the consignor's utility function freight rate is \(p_{k}^{hs}\). From the concept of pure strategy Nash equilibrium [23], for any transportation mode \(\forall k\in N\), if \(p_{k}^{ys}\) exists, it is right that all other modes of transport have a pure strategy tariff combination \(p_{-k}^{*}=\left\{p_{j}^{*}\middle|j\in N,j\
eq k\right\}\), and the optimal response pure strategy satisfies the formula (11)  
\[F_{k}(p_{k}^{ys})\geq\ F_{k}(p_{k}^{i},p_{-k}^{*})\ i=1,2,\cdots,m_{k} \tag{11}\]  
\(\left\{p_{k}^{ys}\mid\ k\ \in\ N\ \right\}\) is called the pure strategy Nash equilibrium of the game between transportation modes. \(p_{-k}^{*}=\left\{p_{j}^{*}\middle|\ j\ \in N\,j\ \
eq\ k\ \right\}\) is the optimal response strategy for all transportation modes except transportation mode \(k\). Obviously, the game problem studied in this paper is a finite game, so according to the existence theorem of the Nash equilibrium, if there is no pure strategy Nash equilibrium, then there must be a mixed strategy Nash equilibrium. This is assuming that operators of various modes of transport have a probability distribution in their respective tariff strategies \(\omega_{k}=(\omega_{k}^{1},\omega_{k}^{2},\cdots\omega_{k}^{i}\cdots,\omega_{ k}^{m_{k}})\)[24]. \(\omega_{k}^{i}\) means for all \(i=1,2,\cdots,m_{k}\), the probability that the operator of transportation mode \(k\) chooses the freight rate \(p_{k}^{i}\). And for any probability distribution, \(\omega_{k}\) should satisfy the following constraint, \(\forall k\in N\):  
\[\omega_{k}=\left\{\omega_{k}\mid 0\ \leq\ \omega_{k}^{i}\leq\ 1\,\sum_{i=1}^{m_{k}} \omega_{k}^{i}=1\ \right\}i=1,2,\cdots,m_{k} \tag{12}\]  
Define the set of income \(F_{k}=(p_{k}^{1},p_{k}^{2},\cdots,p_{k}^{m_{k}})\) is in a mixed strategy combination \(\omega_{k}=(\omega_{k}^{1},\omega_{k}^{2},\cdots\omega_{k}^{i}\cdots,\omega_{ k}^{m_{k}})\forall k\in N\), and the expectation is  
\[E_{k}(\omega_{k})=E_{k}(\omega_{k}^{1},\omega_{k}^{2},\cdots\omega_{k}^{i} \cdots,\omega_{k}^{m_{k}})=\omega_{k}\bullet F_{k}=(p_{k}^{1},p_{k}^{2},\cdots,p_{k}^{m_{k}})^{T}=\sum_{i=1}^{m_{k}}\omega_{k}^{i}f_{k}(p_{k}^{i}) \tag{13}\]  
If there is a \(\omega_{k}^{*}\), its combination of mixed tariff strategy for all other modes of transport \(\omega_{-k}^{*}=\left\{\omega_{j}^{*}\mid j\ \in N\,j\ \
eq\ k\ \right\}\) is the optimal response mixed tariff strategy, and it satisfies the following formula  
\[E_{k}(\omega_{k}^{*})\geq\ E_{k}(\omega_{k},\omega_{-k}^{*})\forall\ \omega_{k}\in\ \omega_{k} \tag{14}\]  
So, we can say that \(\left\{\omega_{k}^{*}\mid\ k\ \in\ N\ \right\}\) is the mixed strategy Nash equilibrium of the game. In summary, the dynamic game model of the express freight corridor is established as the following:  
\[G\ =[N\,(P_{k}),(F_{k}),(U_{k})] \tag{15}\]First of all, various transportation modes conduct a complete information static game with the goal of maximizing their own generalized benefits:  
\[\max f_{k}(p_{k}^{y*},p_{-k}^{*})\] \[\left\{ \begin{array}{c}s.t.\ p_{k}^{i}\in P_{k}\\ 0<p_{k}^{\min}<\ p_{k}^{i}<\ p_{k}^{\max}\\ 0<\ q_{k}^{i}<\ Q\\ i=1,2,\cdots,m_{k},\forall k\in N\end{array} \right. \tag{16}\]  
Then, the consignor observes the strategies of various modes of transport with the goal of maximizing his own utility and participates in the game. \[\max u_{k}(p_{k}^{h*},p_{-k}^{*})\] \[\left\{ \begin{array}{c}s.t.\ p_{k}^{i}\in P_{k}\\ i=1,2,\cdots,m_{k},\forall k\in N\end{array} \right.