# Game Analysis of Different Transportation Modes in a Corridor Considering Carbon Emission Costs
## 3 Game Model
### Payoff of Four Modes of Transportation  
As participants in the game, the four modes of transportation will always pursue the maximization of their respective benefits. Let the set of transportation modes be \(N=(1,2,3,4)\); \(k=1,2,3,4\) correspond to general-speed railway, HSR, air and highway, respectively. The set of freight rates given by the operator of the transportation mode is \(P_{k}=\left\{p_{k}^{1},p_{k}^{2},\cdots,p_{k}^{m_{k}}\right\}\). The initial freight rate set of the game can be given according to the actual freight rate in the current market and its reasonable floating range. \(p_{k}^{i}\) is a specific selected freight rate in set \(P_{k}\). Through the foregoing summary and analysis, it can be concluded that, in a broad sense, the revenue of transportation modes is equal to the revenue minus the cost, in which the cost includes transportation cost and risk cost.  
At the same time, based on the strategic goal of green sustainable development, from a government perspective, in order to achieve carbon peaking and carbon neutrality goals, a carbon tax policy can be implemented to regulate the market. As far as transportation is concerned, the carbon emissions of different transportation modes are different, so the emission tax should be added to the cost of the transportation modes. Under the government's macro control, the market should be the main body, and consignors should choose the most favorable mode of transportation for themselves. The revenue function of express freight mode \(k\) is established as follows, \(\forall k\in N\):  
\[f_{k}(p_{k}^{i})=p_{k}^{i}\times q_{k}^{i}-b_{k}\times q_{k}^{i}\times L_{k}-(g _{k}+w_{k}\times(t_{k}^{sh}-t_{k}^{x})\times\lambda_{OD}+Tax_{k})\;i=1,2,\cdots,m_{k} \tag{5}\]  
\[0\;<p_{k}^{\min}<\;p_{k}^{i}<\;p_{k}^{\max} \tag{6}\]  
where \(q_{k}^{i}\) represents the freight volume shared when the operator of the transportation mode \(k\) chooses a pure strategy rate \(p_{k}^{i}\). \(L_{k}\) represents the distance of transportation mode \(k\). \(\rm g_{k}\) represents the constant cost of the transportation mode \(k\). \(\rm b_{k}\) represents the unit variable cost of the transportation mode \(k\). \(w_{k}\) represents the delay rate of transportation mode \(k\). \(t_{k}^{sh}\) represents the actual transport time of the transportation mode \(k\). \(t_{k}^{x}\) represents the stipulated transport time. \(Tax_{k}\) represents emission tax. \(p_{k}^{\min}\) represents the upper limit  
Figure 2: Relationship between freight rate and utility.  
of freight rates and \(p_{k}^{\max}\) represents the lower limit of freight rates. The corresponding equation is \(f_{k}(p_{k}^{i})\in\ F_{k}=(p_{k}^{i}|k\in N,i=1,2,\cdots,m_{k})\).  
In this paper, the Logit model is used to solve the freight volume \(q_{k}^{i}\) of transportation mode \(k\). The form of the Logit model when combined with the problem in this paper is as follows, \(\forall k\in N\):  
\[q_{k}^{i}=Q\ \times\frac{\exp(u_{k}(p_{k}^{i}))}{\sum\limits_{k}\exp(u_{k}(p_{ k}^{i}))}\ i=1,2,\cdots,m_{k} \tag{7}\]  
where \(Q\) is the total amount of goods in all modes in the express freight corridor. \(u_{k}(p_{k}^{i})\) is the utility of the consignor on mode \(k\) when the price of mode \(k\) is \(p_{k}^{i}\).