# Dynamic Investigations of Shared Bicycle Operators' Competition Based on Profit Maximization
## 3 Model Description  
In this section, we concentrate on an economic model defined by two shared bicycle operators, \(T_{1}\) and \(T_{2}\). The competition is predominantly observed in the last mile of commuter journeys, exemplified by the rivalry between Hellobike and Meituan Bike near subway stations, bus stops, and residential complexes. Such competition is characterized as a duopoly market game. The key distinction between these two shared bicycles is the disparity in comfort. Additionally, to attract more users, the shared bicycle operator \(T_{2}\) utilizes discount strategies. In addition, we assume that the shared bicycle operators involved in the game can be considered as noncooperative games with complete information. The APPs provided to users are public, and both sides can easily obtain information about each other's deposits and fees. It assumed that there is no collusion between the two parties, and their competition is noncooperative. The services provided by both parties are highly substitutable. Both provide users with dockless shared bicycles, allowing users to unlock and use the bikes through a mobile APP, picking up and dropping off at any time and place, to meet their own short-distance transportation needs. Assume that the generalized travel cost of passengers choosing shared bicycle operator \(T_{i}(i=1,2)\), denoted as \(G_{i}\), consists of travel time cost \(k\), comfort loss cost \(\beta_{i}(\beta_{1}<\beta_{2})\), and fare \(p_{i}\). Specifically, the generalized travel cost for passengers choosing \(T_{2}\) will remove the discounts \(S\) provided by operator 2 to attract more passengers. Then, \(G_{i}\) can be expressed as  
\[G_{1}=p_{1}+k+\beta_{1} \tag{1}\]  
\[G_{2}=p_{2}+k+\beta_{2}-S \tag{2}\]  
In line with the literature [30], we adopt the utility function as shown in the Equation (3):  
\[U(x_{1},x_{2})=x_{1}+x_{2}-0.5\Big{(}x_{1}^{2}+2 gx_{1}x_{2}+x_{2}^{2}\Big{)} \tag{3}\]  
where \(x_{i}\) is the number of passengers who chose shared bicycle operator \(T_{i}\). \(g\in[0,1)\) is used to measure the degree of substitutability between services provided by two shared bicycle operators. If \(g=1\), then the services are perfect substitutes and \(g=0\) is the case of independent markets. Hence, parameter \(g\) can be considered as indicating the degree of substitutability between operators in a particular market sector. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Citation** & **Year** & **Model** & **Innovation Point** & **Limitation** \\ \hline [27] & 2018 & Bertrand model & Government regulation & No simulation \\ \hline [13] & 2019 & Cournot oligopoly competition model & Case study & Static model \\ \hline [22] & 2020 & Stackelberg/Cournot model & Models vary by stage of competition & Price is consistent \\ \hline [28] & 2021 & Expand the Bertrand model & Bertrand trioligarch model & Neglecting users \\ \hline [24] & 2022 & \begin{tabular}{c} Dynamic differential game \\ model/Hotelling game model \\ \end{tabular} & Advertising/loyalty reward & Disregarded service quality \\ \hline [23] & 2022 & \
\begin{tabular}{c} Equilibrium model of bike-sharing \\ service networks \\ \end{tabular} &
\begin{tabular}{c} Government \\ incentives/loyalty reward \\ \end{tabular} & Static model \\ \hline [12] & 2022 & – & Smart contract & Operator homogeneity \\ \hline [3] & 2023 & Hotelling game model & Government regulation & Static results \\ \hline [29] & 2023 & spontaneous evolutionary game model & Government regulation & Neglecting users \\ \hline This paper & – & nonlinear duopoly game model & Discounts/comfort & No deposit/advertising revenue \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Comparison of duopoly game theory of shared bicycles.