# Game Analysis of Different Transportation Modes in a Corridor Considering Carbon Emission Costs
## 3 Game Model
### Dynamic Game Model
\tag{17}\]  
In Formula (17),  
\[f_{k}(p_{k}^{i})\geq p_{k}^{i}\times q_{k}^{i}-b_{k}\times q_{k}^{i}\times L_{ k}-(g_{k}+w_{k}\times(t_{k}^{sh}-t_{k}^{x})\times\lambda_{OD}+Tax_{k}) \tag{18}\]  
\[f_{k}(p_{k}^{i})\leq p_{k}^{i}\times q_{k}^{i}-b_{k}\times q_{k}^{i}\times L_{ k}-(g_{k}+w_{k}\times(t_{k}^{sh}-t_{k}^{x})\times\lambda_{OD}+Tax_{k})\]  
\[u_{k}({p_{k}}^{i})\geq\alpha_{1}\times e^{\frac{-(p_{k})^{2}+\alpha_{k}\times p _{k}^{i}}{\rho_{k}}}+\Delta_{k}^{hf} \tag{19}\]  
\[u_{k}({p_{k}}^{i})\leq\alpha_{1}\times e^{\frac{-(p_{k}^{i})^{2}+\alpha_{k} \times p_{k}^{i}}{\rho_{k}}}+\Delta_{k}^{hf}\]  
\[i=1,2,\cdots,m_{k},\ \forall k\in N \tag{20}\]  
In theory, when \(p_{k}^{h*}\) and \(p_{k}^{y*}\) are equal, it is considered that the game reaches equilibrium and the equilibrium solution is \(p_{k}^{*}=p_{k}^{h*}=p_{k}^{y*}\). In that case, we suppose that the optimal sharing rate of transportation mode \(k\) when the game reaches equilibrium is \(r_{k}^{*}\), and the transportation volume shared by transportation mode \(k\) is \(q_{k}^{*}\), when the game is in equilibrium:  
\[r_{k}^{*}=q_{k}^{*}/\ Q\ \times 100\% \tag{21}\]  
The solution to the game is the combination of the optimal strategies of all participants, that is, the state when \(p_{k}^{y*}\), the optimal freight rate of the various modes of transportation games is theoretically equal to \(p_{k}^{h*}\), the freight rate with the maximum utility of the owner. Nevertheless, equality is difficult to realize. Hence, the difference between \(p_{k}^{y*}\) and \(p_{k}^{h*}\) is close to the infinitesimal value instead of equal to zero. In the equilibrium state of the game, no player has the motivation to change his strategy. According to the previous analysis and the actual situation, when consignors choose the mode of transportation for the purpose of maximizing their own utility, because they want to obtain high-quality service, the utility will increase with the increase in freight charges. But there must be a cap \(p_{k}^{h*}\) for the freight charges. Beyond this cap, the satisfaction of the consignor will decrease with the increase in freight charges. Therefore, various modes will not make the freight rate rise indefinitely, and they will not reduce the freight rate indefinitely because of the cost. In the game, the owner must choose one of the modes of transportation to transport the goods. Therefore, there is an optimal solution to the game problem in this paper, and the game can reach equilibrium. In this model, the constraint is non-convex, so it is difficult to solve. According to the characteristics of the model, this paper takes the proportion of freight rate in the utility of the consignor as weight. According to the characteristics of the problem studied by the model, the advantage is that the appropriate iteration step can be determined. The process of solution is the process of constantly adjusting the freight rate strategy of the transportation mode to gradually approach the game equilibrium, which is also the process of the dynamic game. The solving steps are as follows:  
Step 1: Set the iteration number \(x=0\), give the initial tariff policy set \(P_{k}^{0}\) of transportation mode \(k\), and assign a quite small positive value to the elements in the vector \(\varepsilon\ =\{\varepsilon_{k}|k\ \in N\,\}\). Step 2: For a given set of tariff strategies \(P_{k}^{x}\), use Equation (9) to obtain the utility value \(u_{k}^{x}\) and the optimal solution \(p_{k}^{lx}\) of the freight owner to the mode \(k\), then calculate the freight volume \(q_{k}^{x,i}\) of the mode \(k\) according to the logit model and finally, use Equation (5) to obtain the generalized revenue value \(f_{k}^{x}\) and the optimal solution \(p_{k}^{ys}\). Step 3: Judge whether the convergence condition \(\left|p_{k}^{hs}-\ p_{k}^{ys}\right|\leq\varepsilon_{k}\) is satisfied or not. If it is satisfied, the optimal solution \(p_{k}^{ys}\) or \(p_{k}^{hs}\), the freight volume \(q_{k}^{x}\) and freight volume sharing rate \(r_{k}^{x}\) are all output and algorithm stops.