## 3 Analysis of Port Competition Mode and Equilibrium Result
The objective functions for port A and port B are as follows:  
\[Max\ \pi_{A}=(p_{A}-C_{A}^{port})\frac{P_{B}-P_{A}+C_{B}^{sea}-C_{A}^{sea}+C_{bB}^ {land}+b\sum\limits_{m=1}^{M}w_{m}\Big{(}\delta^{m}+\varepsilon\frac{1}{ \overline{V}^{m}}\Big{)}+H_{B}}{2\sum\limits_{m=1}^{M}w_{m}\Big{(}\delta^{m}+ \varepsilon\frac{1}{\overline{V}^{m}}\Big{)}+H_{A}+H_{B}} \tag{13}\]  
\[Max\ \pi_{B}=(p_{B}-C_{B}^{port})\frac{P_{A}-P_{B}+C_{A}^{sea}-C_{B}^{sea}-C_{bB} ^{land}+(2-b)\sum\limits_{m=1}^{M}w_{m}\Big{(}\delta^{m}+\varepsilon\frac{1}{ \overline{V}^{m}}\Big{)}+H_{A}}{2\sum\limits_{m=1}^{M}w_{m}\Big{(}\delta^{m}+ \varepsilon\frac{1}{\overline{V}^{m}}\Big{)}+H_{A}+H_{B}} \tag{14}\]  
By calculating the first-order partial derivatives in Equations (11) and (12) and further simplifying the derivation, the optimal pricing of ports at their respective profit maximization can be obtained as follows (* represents the optimal value in the equilibrium):  
\[P_{A}^{*}=\frac{C_{B}^{sea}-C_{A}^{sea}+C_{bB}^{land}+(2+b)\sum\limits_{m=1}^ {M}w_{m}\Big{(}\delta^{m}+\varepsilon\frac{1}{\overline{V}^{m}}\Big{)}+2H_{B} +H_{A}+2C_{A}^{port}+C_{B}^{port}}{3} \tag{15}\]  
\[P_{B}^{*}=\frac{C_{A}^{sea}-C_{B}^{sea}-C_{bB}^{land}+(4-b)\sum\limits_{m=1}^ {M}w_{m}\Big{(}\delta^{m}+\varepsilon\frac{1}{\overline{V}^{m}}\Big{)}+2H_{A} +H_{B}+2C_{B}^{port}+C_{A}^{port}}{3} \tag{16}\]  
Substituting \(P_{A}^{*}\) and \(P_{B}^{*}\) into the port demand function, the equilibrium cargo demand for port A and port B can be calculated as follows:  
\[Q_{A}^{*}=\frac{C_{B}^{sea}-C_{A}^{sea}+C_{bB}^{land}+(2+b)\sum\limits_{m=1}^ {M}w_{m}\Big{(}\delta^{m}+\varepsilon\frac{1}{\overline{V}^{m}}\Big{)}+H_{A} +2H_{B}+C_{B}^{port}-C_{A}^{port}}{6\sum\limits_{m=1}^{M}w_{m}\Big{(}\delta^{m}+ \varepsilon\frac{1}{\overline{V}^{m}}\Big{)}+3H_{A}+3H_{B}} \tag{17}\]\[\pi_{A}=(p_{A}-C_{A}^{port})\cdot Q_{A} \tag{9}\]0  
By substituting the equilibrium cargo demand for port A (Equation (17)) and the optimal pricing of port A (Equation (15)) into Equation (9), the maximum profit function of port A can be obtained as follows:  
\[\pi_{A}=(p_{A}-C_{A}^{port})\cdot Q_{A} \tag{9}\]1  
Likewise, by substituting the equilibrium cargo demand for port A (Equation (18)) and the optimal pricing of port A (Equation (16)) into Equation (10), the maximum profit function of port B can be obtained as follows:  
\[\pi_{A}=(p_{A}-C_{A}^{port})\cdot Q_{A} \tag{9}\]2  
**Proposition 1**. : _Compared with port A, the higher the maritime transportation cost of port B and the cross-border land transportation cost of goods transported to port B, the higher the cargo transportation demand and the equilibrium price of port A, and the lower the cargo transportation demand and the equilibrium price of port B, resulting in a decrease in the competitiveness of port B._  
Proof. : By calculating the first-order partial derivative of the optimal pricing and equilibrium cargo demand of port A and port B, that is, \(\frac{\partial P_{A}^{*}}{\partial C_{B}^{sea}-C_{A}^{sea}}=0\), \(\frac{\partial Q_{B}^{*}}{\partial C_{B}^{sea}-C_{A}^{sea}}=0\), \(\frac{\partial P_{B}^{*}}{\partial C_{B}^{land}}=0\), \(\frac{\partial Q_{B}^{*}}{\partial C_{b}^{land}}=0\), \(\frac{\partial Q_{B}^{*}}{\partial C_{b}^{land}}=0\) (\(i=A,B\)), the following can be obtained:  
\[\pi_{A}=(p_{A}-C_{A}^{port})\cdot Q_{A} \tag{9}\]3  
\[\pi_{A}=(p_{A}-C_{A}^{port})\cdot Q_{A} \tag{9}\]4  
This indicates that the equilibrium price and equilibrium demand of port A increase with the increase in \(C_{B}^{sea}-C_{A}^{sea}\), while the equilibrium price and equilibrium demand of port B decrease with the increase in \(C_{B}^{sea}-C_{A}^{sea}\). The equilibrium prices and equilibrium demand of port A increase with the increase in \(C_{bB}^{land}\), while the equilibrium price and equilibrium demand of port B decrease with the increase in \(C_{bB}^{land}\).