# Dynamic Investigations of Shared Bicycle Operators' Competition Based on Profit Maximization
## 3 Model Description
### Cournot Model with Profit Maximization  
This section examines the demand competitive market among bike-sharing operators. This scenario involves both participants fully comprehending the oligopoly market, meaning the two bike-sharing operators have rational expectations, but this is highly unlikely [33]. In reality, the true nature of the duopoly's monopolistic evolution in the bike-sharing market can only be exhibited in a dynamic setting. Therefore, this article posits that operators are endeavoring to employ a more practical approach, that is, bounded rationality. This paper will analyze the marginal profits \(\partial\pi_{i}/\ \partial x_{i}\) of bike-sharing operators \(i\) under the framework of bounded rational expectations. The demand for \(T_{i}\) in \(t+1\) period is dictated by the marginal profit achieved in the \(t\) period. If the shared bicycle operator \(T_{i}\) achieves a marginal profit of \(\partial\pi_{i}/\ \partial x_{i}>0\) (\(<\)0) during period \(t\), then operator \(T_{i}\) will either increase (decrease) deployed in the subsequent period \(t+1\). If operator \(T_{i}\) achieves a marginal profit where \(\partial\pi_{i}/\partial x_{i}=0\) during period \(t\), then its deployed will remain unchanged in the following period \(t+1\). If the operator implements a bounded rationality adjustment mechanism, the dynamic equation for this game can be represented as (11)  
\[x_{i}^{C}(t+1)=x_{i}^{C}(t)+v_{i}x_{i}^{C}(t)\left(\frac{\partial\pi_{i}}{ \partial x_{i}^{C}}\right) \tag{11}\]  
where \(v_{i}>0\) represents the adjustment speed of shared bike operator \(i\), reflecting their responsiveness to the shared bicycle market. In this study, it is hypothesized that during each \(t\) period, bike-sharing operator \(T_{i}\) will anticipate the number of passengers that its competitors might serve in the subsequent \(t+1\) period, thereby determining its own potential maximum passenger count for the next \(t+1\) time period \(x_{i}^{C}(t+1)\).  
When two shared bicycle operators maximize their profits through quantity competition, and thus their objective functions can be modeled as  
\[\begin{cases}\pi_{1}^{C}(x_{1},x_{2})=\big{(}(1-k-\beta_{1})-x_{1}^{C}-gx_{2}^{C}- \mu_{1}-\mu_{2}D\big{)}x_{1}^{C}-f\\ \pi_{2}^{C}(x_{1},x_{2})=\big{(}(1-k-\beta_{2}+S)-x_{2}^{C}-gx_{1}^{C}-\mu_{1} -\mu_{2}D\big{)}x_{2}^{C}-f\end{cases} \tag{12}\]  
The conditions required to achieve optimal demand by maximizing the profit of operators are: \(\partial\pi_{1}^{C}/\partial x_{1}^{C}=0\) and \(\partial\pi_{2}^{C}/\partial x_{2}^{C}=0\). Therefore, the conditions for profit maximization can be obtained:  
\[\begin{cases}1-k-\beta_{1}-2x_{1}^{C}-gx_{2}^{C}-\mu_{1}-\mu_{2}D=0\\ 1-k-\beta_{2}+S-2x_{2}^{C}-gx_{1}^{C}-\mu_{1}-\mu_{2}D=0\end{cases} \tag{13}\]  
Then, we can construct and study the dynamic characteristics of the Cournot game as  
\[\begin{cases}x_{1}^{C}(t+1)=x_{1}^{C}(t)+v_{1}[(1-k-\beta_{1})-2x_{1}^{C}-gx_{2 }^{C}-\mu_{1}-\mu_{2}D]\\ x_{2}^{C}(t+1)=x_{2}^{C}(t)+v_{2}[(1-k-\beta_{2}+S)-2x_{2}^{C}-gx_{1}^{C}-\mu_{ 1}-\mu_{2}D]\end{cases} \tag{14}\]