Competition and coordination strategies of shared electric vehicles and public transportation considering customer travel utility

With the rise of the sharing economy and the concept of “green environmental protection and low-carbon travel,” the emerging project of shared electric vehicles is booming. However, the accompanying coordination problem between shared electric vehicles and public transportation system needs to be urgently solved. In reality, customers’ choice of travel mode is influenced by their own travel perceived utility. Thus, this paper will discuss the competition and coordination problem between shared electric vehicles and public transportation system from the perspective of customer travel utility. Considering the travel cost and comfort in the customer travel utility, the game models of shared electric vehicle and public transportation system in different scenarios are established by using competitive game and cooperative game. Then, the equilibrium solutions under different scenarios are obtained by solving the models. The analysis results show that shared electric vehicles would bring some beneficial improvements to the transportation system under certain circumstances. Furthermore, public transportation system should adopt a coordinative strategy with the shared electric vehicles to promote the total customer travel utility for the entire system. It is worth considering the improvement of the service quality of shared electric vehicle and public transportation, which would affect the rate of increasing in the total customer travel utility.


Introduction
During the past decades, buses, subways, and other public transportation (PT) as well as private transportation have played an important role in urban transportation. In recent years, with the rise of the sharing economy and the concept of "green environmental protection and low-carbon travel," the emerging project of shared electric vehicle (SEV) is developing rapidly, which has caused some changes in people's travel mode. SEV is a transportation sharing model that provides car rental services to customers in areas with dense human activities, such as commercial centers, residential communities, and public transportation stations, to meet customers' demand for short-time car use (Zhang et al. 2011;Li et al. 2016a, b;Kevin et al. 2020). Compared with PT, SEV provides greater comfort and flexibility to traveling regardless of route and timetable constraints (Liu et al. 2020). Different from private transport, SEV uses more environmentally friendly energy (Casals et al. 2016;Zhang et al. 2020) and reduces the waste of social resources (Liu et al. 2020) as well as people's travel costs such as vehicle purchase fee, gasoline, insurance, and other related costs (Xu et al. 2020). SEV uses electricity as the energy source to reduce greenhouse gas emissions, which pollutes less to the environment compared with the ordinary fuel vehicle (Casals et al. 2016;Zhang et al. 2020). Recently, an increasing number of people choose to travel by SEV. According to statistics from the "In-depth analysis report on China's shared car business model innovation and investment opportunities" released by the Foresight Industry Research Institute 2019, the scale of China's SEV market was 266 million dollars in 2017, and it was 561 million dollars in 2018, with a growth rate of 111%. In 2019, this market reached 1.021 billion dollars. It is predicted that the scale of China's SEV market will exceed 6.921 billion dollars by 2023 and the average compound growth rate in 2019-2023 is about 61.84% (In-depth analysis report on China's shared car business model innovation and investment opportunities 2019). With the rapid development of SEV, the travel modes of customers become increasingly diverse, and customers need to choose among these travel modes during their travel. In reality, SEV has been already starting to affect PT gradually. On the one hand, SEV attracts customers with its innovative travel methods, travel flexibility, and other factors, which is a noteworthy competitor with PT. On the other hand, SEV and PT coordinate to meet the diverse needs of customer travel. For example, customers can use PT to the adjacent SEV parking station and then use SEV to reach the destination. Thus, for the development of both sides, the coordination problem between SEV and PT needs to be solved urgently, which has gradually attracted the attention of academia and industry (Galatoulas et al. 2018;Jin et al. 2020).
With the gradual rise and development of SEV, there has been an upsurge of scholars' attention on the research of operational problems of SEV. The majority of the studies focus on the following three problems: rebalancing or scheduling problem (Bruglieri et al. 2014;Weikl and Bogenberger 2015;Li et al. 2016a, b;Correia and Antunes 2017), charging problem (Dong et al. 2014;Avci et al. 2014;Neubauer and Wood 2014;Wu 2019;Paula et al. 2020), and station selection and station planning (He et al. 2017;Kaspi et al. 2016;Mak and Shen 2014;Ang et al. 2012;Brandstätter et al. 2017). After years of research on the above-mentioned problems, it has been found that the impact of other transportation modes on SEV as well as the coordination problem between them should be considered. Therefore, it is increasingly concerned in recent studies with the coordination problem and the choice between different travel modes in the transportation system (Liu 2012;Hao and He 2013;Zhang and Chen 2018;Chen et al. 2016), which can be summarized as the choice and coordination under the competitive conditions (Guo et al. 2000). Research on the coordination problem between SEV and other modes of transportation is rare, but some researches on coordination between other modes of transportation have drawn more attention. In view of such problems, relevant researches are conducted from two aspects. The first aspect is the competitive relationship between private transportation and PT. Yong and Huang (2016) studied customers' choice between subway and driving under different charging strategies. Ci et al. (2017) used lotka-volterra model to describe the evolution of the balance between PT and private transportation. Liu (2012) put forward the concept of "ecological niche" and pointed out that the "ecological niche" of various travel modes has overlapping competitive relations. Some scholars pointed out that the research in this aspect is not universally applicable, for not all travelers have private transportation (Guo et al. 2000). The second aspect is the coordination problems among strong accessibility, which refers to buses, subways, taxis, shared bikes, etc. Zhang and Li (2014) set up the utility function to study the competitive relationship between the rail transit and buses in the case of collinearity, based on the travel time and cost as the principal variables. Ma et al. (2007) replaced the utility function with the generalized cost function and used the Logit model to explore the passenger flow sharing rate of the rail transit and buses. Ahmadreza et al. (2017) analyzed the competition of New York City's taxi and bike-sharing system (BSS) in terms of travel time through the data survey of New York City and believed that BBS system was more time saving in traffic congestion areas. Cantarella and De (2005) adopted the multi-layer feed forward network model to select and predict the mode of transportation. These have informed the research of coordination problems between SEV and other transportation.
Travel utility and travel price are often considered by most scholars when examining the above-mentioned problems. Travel utility serves as the customer satisfaction with the travel modes, which could directly affect customers' travel choice (Whalen et al. 2013;Sprumont et al. 2017). The interest in considering and studying the customer travel utility and preference in the travel choice problem has grown rapidly over the past two decades. Mokhtarian and Salomon (2001) and Redmond and Mokhtarian (2001) were the first to describe the travel liking, a new concept in transportation.
Currently, many scholars have demonstrated that individuals might be satisfied in performing the travel and the degree of customer satisfaction with the travel modes would directly affect their choice (Banister 2008;Chorus 2012;Ettema et al. 2013;Russell and Mokhtarian 2014). Even if relatively new, the travel satisfaction concept shares some common traits with the well-known concept of utility (Sprumont et al. 2017), which is considered as a measure of preference among different alternatives belonging to the same choice set for a decision maker (Ben-Akiva and Lerman 1985;McFadden 1980). Meanwhile, Whalen et al. (2013) also observed satisfaction feelings during travel. Travel perceived utility is considered as a preference measure that the decision maker chooses between different travel modes. Therefore, a growing number of scholars begin to study travel utility. The results of a travel survey implemented by the University of Luxembourg in 2012 show that the utility function fitted by this survey is positively correlated with the stated commuting satisfaction. The number of publications that provide empirical research on determinants of travel perceived utility is growing rapidly. St-Louis et al. (2014) made an interesting classification among the travel perceived utility determinants, where the internal factors and external factors are opposite. Internal or nonmode-specific factors are related to individual characteristics, such as an individual's lifestyle, preference, and attitude. External factors or mode-specific attributes are related to travel time, cost, comfort, availability etc. At the same time, the selected travel mode has an important effect on the travel utility (Sprumont et al. 2014). Páez and Whalen (2010) found that the mode of travel with the highest utility for travelers is realized by bicycle or walking, while the mode with medium utility is realized by driving or taking taxis, and the mode of lowest utility is achieved by bus or subway. Some scholars also mentioned that the low utility reported by PT users may be related to the frustration of being crowded and uncomfortable (St-Louis et al. 2014;Páez and Whalen 2010).
Previous studies have significantly advanced the research on operational problems of SEV from different perspectives and provided references for this paper. However, there are still some gaps in the existing research as follows.
(1) Most of the existing studies only consider the competitive relationship between different transportation modes, whereas few of them consider the coordinative relationship between them. It is not sound because there is not only competition, but also coordination between different transportation modes in reality. For example, a customer could choose from the following three methods to travel: the first is to take the bus; the second is to drive a SEV; the third is to take the bus to an adjacent SEV station and then transfer a SEV to reach the destination. Thus, on such a journey, the relationship between the bus and SEV is both competitive and coordinative. This paper will consider the transfer behavior of customers to research the coordination problem between PT and SEV. That is more in line with the reality. (2) Most of the existing research considers the price equilibrium or time equilibrium resolved by converting travel time into travel costs, while the customer's choice is also affected by other factors such as the space comfort besides the price or time cost. In real life, customers often show a dislike to the travel mode that is crowded. Therefore, this paper will introduce the customer travel comfort by calculating the degree of space congestion of travel mode. For example, as the number of customers in the bus increases, the perceived comfort of customer decreases, which could further affect the choice of travel mode. Based on the abovementioned analysis, it is essential to study the coordination and competition problem between SEV and PT from the perspective of customer travel utility, composed of travel comfort and price. That is the motivation of our study.
This paper aims to solve the competition and coordination problem between SEV and PT. The transportation system is considered as a closed system and the number of vehicles and customers is constant for the convenience of analysis and calculation. First, the competition and coordination problem between SEV and PT is analyzed by considering the transfer behavior of customers between different travel modes. The equilibrium state and interactive relationship between SEV and PT could be obtained by comparing and analyzing the following three scenarios: (a) only PT in transport system, (b) competition between SEV and PT, and (c) competition and coordination between SEV and PT. Then, this paper considers the customer travel utility from the perspective of the space comfort. In addition, there is a correlation between the space comfort and travel price. Specifically, the travel price affects the number of customers, which affects the space comfort in turn. Therefore, the impact of travel price and customer quantities on the customer travel utility is considered in this paper. Furthermore, digital simulation experiments are conducted to present the validity of the conclusions obtained from three scenarios.
The rest of this paper is organized as follows. "Methodology" not only gives the basic assumptions of this problem and establishes the customer travel utility function. More importantly, it proposes three different scenarios of this problem, which refer to PT, competition between SEV and PT, and competition and coordination between SEV and PT in the transportation system. The optimal equilibrium results for the three scenarios are solved as well as the related discussions and propositions are given in "Results and discussions." In "Simulation experiment," digital simulation experiments are conducted. The paper concludes with a summary and an outlook in "Conclusion." The proofs of all the propositions are given in the Appendix.

Methodology
This section is divided into two parts. The first part covers the basic assumptions and the customer travel utility function. Based on the first part, different competition and coordination scenarios between PT and SEV are described in the second part.

Basic assumptions and the travel utility function
Based on the reality and previous studies (Chen et al. 2016;Brandstätter et al. 2017;He et al. 2017;Jin et al. 2020), the basic assumptions and the customer travel utility function are given in this part.
Basic assumptions Assumption 1. The system is a closed system. It means that the number of PT, SEV, and customers in the system are constant and they do not increase or decrease as the system flows. Meanwhile, the travel chain is simplified into three parts, namely the departure station, the transit station, and the terminal station.
Assumption 2. The transportation system has two subsystems: PT and SEV, among which only buses and subways are considered in PT since both are the two most common ways in the urban transportation.
Assumption 3. The transportation resources in the system are abundant. To be specific, road resources are sufficient, and customers will not encounter traffic congestion during the travel. Meanwhile, the parking space at the destination is adequate, which ensures that SEV entering the destination can park normally.
Assumption 4. In the SEV subsystem, the power and the number of SEV are sufficient. In other words, customers who use the SEV from any stations can successfully reach their destination and each customer demand would not be missed.
Assumption 5. Customers are homogenous. Each customer has the same utility function for the same travel mode.
Assumption 6. When multiple customers use the same SEV at the same time, they are treated as one. For example, if a couple share the same SEV, they will be regarded as one customer.

Travel utility function
Based on the utility theory, the customer perception utility functions of PT and SEV travel modes are established. The utility function is expressed by the customer's perception of the space comfort, which is affected by the number of customers. In addition, the number of customers is affected by the travel price. The relationship between the customer travel utility of taking SEV or PT and the number of customers is shown in Fig. 1, and the customer travel utility functions of SEV and PT are presented as follows, where k, a, and b refer to the perceived coefficients of travel comfort representing the customer's sensitivity to travel comfort, and these coefficients are constants (Cantarella and De 2005;Páez and Whalen 2010), k > 0, a > 0, and b > 0. The values of these coefficients need to be determined by fitting the relevant data of the customer's travel utility survey, including offline survey results or online polling and commenting. Q S and Q P are the number of customers who travel by SEV and PT in the function expression, respectively. Q ′ refers to the maximum number of SEV.
As shown in Fig. 1, the utility functions of the two subsystems are different. As the number of customer increases, the total travel utility of taking SEV first increases linearly and then remains constant. When customers use SEV, everyone is independent and does not affect each other. According to Assumption 5, the total travel utility of using SEV is a linear function, which is positive and can be superimposed. Therefore, the first half of total travel utility function of taking SEV increases linearly. Since the number of SEV is finite and appropriate, some customers could not take SEV and the total customer travel utility would not change when the number of customers exceeds the maximum number of SEV. Therefore, the latter half of the total travel utility function is constant. The total travel utility function of taking PT is a quadratic function with an opening downward. When customers use PT for travel, each person is not independent, which indicates that customers will gradually feel crowded as the number of customers in PT vehicles increases slowly. Therefore, the total travel utility of taking PT gradually increases as the number of customer increases. That is in accordance with the law of diminishing marginal utility. When the number of customers inside a PT vehicle reaches a certain limit, Q ″ , customers feel crowded, the travel utility of each customer would become negative, and the total travel utility would begin to decline. Even as the number of customers reaches a higher limit, the customers become intolerable and the total travel utility becomes negative.
Customer quantity and the price are not independent of each other. Customer quantity changes in the opposite direction with the travel price. That means when the price rises, the number of customers decreases (Mokhtarian and Salomon 2001). Therefore, the relationships between travel price and the number of customers conform to the Demand Theory  Fig. 2. In addition, the travel demand functions of SEV and PT are as follows.
where Q all is the total number of customers, which is constant according to Assumption 1. p ′ and p S are the price of PT and SEV in the whole journey, respectively. Based on Assumption 1, it can be assumed that the travel chain is settled and the travel price of using PT, p P = p ′ , is a determined value (Chen et al. 2016;Ci et al. 2017). In addition, the travel price of taking SEV, p S , is a variable. The price of taking PT is the lowest bound on that of SEV since the vicious competition with PT is not considered. Meanwhile, when p S increases, the number of customers, Q S , would decrease, so it is a decreasing function (Mokhtarian and Salomon 2001). Besides, β and c are the coefficients of price elasticity of demand and these coefficients are constants (Mokhtarian and Salomon 2001;Zhang and Chen 2018), β > 0 and c > 0. The value of coefficients needs to be obtained by fitting the relevant data of the customer's travel records.

Competition and coordination between SEV and PT
In this section, customers in the system could take SEV or PT to travel. To better analyze the impact of SEV on PT under different conditions, three different scenarios are considered. The first scenario is only PT in transportation system. The second scenario is only competition between SEV and PT. The third scenario contains both competition and coordination between SEV and PT. Meanwhile, this section constructs the total customer travel utility in the system under different scenarios based on the different competitive and cooperative relationship between SEV and PT. Subsequently, the different models for three scenarios are built up. These models would be solved in the next section and the differences of different scenarios' equilibrium results are discussed. In addition, the flowchart of this section is shown in Fig. 3.

Scenario 1: no SEV
In this scenario, customers can only use PT to travel. The travel process of customers is shown in Fig. 4, where station A is the departure station and station C is the terminal station. It should be pointed out that customers may ride bicycles or walk to neighboring stations to take PT due to exercise or avoiding overcrowding in reality. Since it is a short distance, the customer travel utility and demand functions do not change significantly, and this case is treated as the same with that only taking PT.
The total number of customers, Q all , is a determined value in the transportation system. Owing to only PT in the transportation system, customers have no choice but to use PT from station A to station C. In this scenario, Q P = Q all . Therefore, the total customer travel utility, U(Q all ), is shown as follows.
U(Q all ) is the total customer travel utility in Scenario 1. This paper considers maximizing the total customer travel utility, i.e., max U(Q all ), to derive the optimal equilibrium of the transportation system.

Scenario 2: competition between PT and SEV
The travel process of customers in this scenario is shown in Fig. 5. Since there are PT and SEV in the transportation system, customers can choose either PT or SEV from station A to station C. The number of customers who travel by PT and SEV are Q P and Q S = Q all − Q P , respectively. Meanwhile, the price of taking SEV is p = p S . In addition, the customer travel utility and demand function of PT or SEV are Eqs. (1)-(4). Based on the above-mentioned analysis, the total customer travel utility in scenario 2, U(Q P , Q s ), is shown as follows.
where Q P = βp ′ − cp and Q S = Q all − Q P = Q all − βp ′ + cp. U(Q P , Q s ) is the total customer travel utility in Scenario 2. When both PT and SEV reach equilibrium in the transportation system, the total customer travel utility can reach its maximum and could not change (Galatoulas et al. 2018). Thus, this paper considers maximizing the total customer travel utility, i.e., max U(Q P , Q s ), to derive the optimal equilibrium in this scenario.

Scenario 3: competition and coordination between PT and SEV
In Scenario 3, PT and SEV have both competitive and coordinative relationship, which means customers in the transportation system can use both PT and SEV to travel. Transfer behavior of customers is considered in this scenario by selecting a station, station B, as the transfer station. There are three travel routes for customers, which are shown in Fig. 6. Customers could take PT or SEV from station A to station C directly. Moreover, customers could take PT to station B and then transfer to take SEV to station C. In reality, most customers tend not to change their travel modes after driving or taking a car like SEV or taxi (St-Louis et al. 2014;Páez and Whalen 2010), which is not considered in this paper.
Since there is a certain distance between station A and station B, the customer travel utility from station A or station B is significantly different, which cannot be ignored at this point. The demand function and customer travel utility of taking SEV to station C directly are expressed as follows.
where p A is the price of taking SEV from station A, Q S is the number of customer who takes SEV from station A, and η is the proportionality coefficient influenced by the distance between station A and station B and the service quality of SEV (Zhang and Li 2014;Ma et al. 2007;Páez and Whalen 2010), 0 < η ≤ 1. The customer travel utility of taking SEV from station B to C is η times as much as that from station A to B; thus, the customer travel utility of taking SEV to station C directly can be regarded as the sum of the two parts.
For customers taking PT at station A then transferring SEV at station B, their travel utility is the sum of the utility of taking PT from station A to B and that of taking SEV from station B to C. Therefore, the demand function and customer travel   utility of taking PT first and then transferring SEV to station C are expressed as follows.
where Q P and Q 0 S are the number of customers who take PT from station A and take SEV from station B, respectively. p B is the price of taking SEV from station B. U(PtS) is the customer travel utility of this travel route.
Since the transfer behavior of customers is considered, some customers taking PT from station A could transfer SEV at station B and others continue to take PT to station C. Therefore, the number of customers who still take PT to station C and the customer travel utility of taking PT to station C directly are presented as follows.
where the proportionality coefficient, μ, is influenced by the distance between station A and station B, and the service quality of PT, 0 < μ ≤ 1 (Zhang and Li 2014;Ma et al. 2007;Páez and Whalen 2010). The customer travel utility of taking PT from station B to C is μ times as much as that from station A to B; thus, the customer travel utility of taking PT to station C directly can be regarded as the sum of these two parts.
In summary, customers can choose one of the three routes to travel, so the total customer travel utility in Scenario 3, U all Q P ; Q S ; Q 0 P ; Q 0 S À Á , is the sum of the three routes travel utility.
U all Q P ; Q S ; Q 0 P ; Q 0 S À Á is the total customer travel utility in Scenario 3. Similar to Scenario 2, when both PT and SEV reach equilibrium in the transportation system, the total customer travel utility reaches its maximum and could not change (Galatoulas et al. 2018). Different from Scenario 2, customers in Scenario 3 have transfer behavior, which does not affect the fact that the total customer travel utility maximizes in the equilibrium. Thus, this paper considers maximizing the total customer travel utility, i.e., max U all Q P ; Q S ; Q 0 P ; Q 0 S À Á , to derive the optimal equilibrium in this scenario.

Results and discussions
Two parts are contained in this section. One part is the results of different scenarios between SEV and PT, and the other part discusses the variations of the results in different competition and coordination between SEV and PT.

Results
In the previous section, three different scenarios of competition and coordination between SEV and PT were mentioned and modeled, which is resolved to obtain the optimal equilibrium state of the transportation system.

Scenario 1: no SEV
Taking the derivative of Eq. (5) and making its derivative function equal to 0, it could be obtained that the optimal number of customers taking PT is Q all * ¼ a 2b and the maximum total customer travel utility of the transportation system is U Q all * À Á ¼ a 2 4b in Scenario 1.

Scenario 2: competition between PT and SEV
It could be obtained the optimal state of transportation system for Scenario 2 by taking the derivative of Eq. (14) and making its derivative function equal to 0. The optimal numbers of customers using PT and SEV are Q * P ¼ a−k 2b and Q * S ¼ Q all − a−k 2b , respectively. In addition, the optimal travel price of taking SEV is p * ¼ a−k−2bQ P þ2bβp 0 2bc . In this scenario, the maximum total customer travel utility of the transportation

Scenario 3: competition and coordination between PT and SEV
Taking the derivative of Eq. (14) and making its derivative function equal to 0, it could be obtained the optimal state of transportation system for Scenario 3. In Scenario 3, the optimal number of customers taking SEV to station C directly is B, the optimal number of customers taking PT is Q * Þ and the optimal number of customers transferring SEV in station B is Q The optimal number of customers taking PT to station C directly is Þ . In addition, the optimal travel price of taking SEV from station A and station B are , respectively. In this scenario, the maximum total customer travel utility of t h e t r a n s p o r t a t i o n s y s t e m i s U all Q P ; Q S ; Q

Discussions of equilibrium results for different scenarios
In order to compare the equilibrium results in different scenarios more clearly and visually, the equilibrium results are summarized in Table 1. It can be visualized from Table 1 that equilibrium results are significantly different in three scenarios. And it would bring certain beneficial improvements to the transportation system after SEV enters under certain circumstances. In addition, the maximum total customer travel utility can be seen in Scenario 3. Therefore, after SEV enters the transport system, PT should adopt a coordinative and competitive strategy to ensure the maximization of total customer travel utility for the entire system.
Comparing Scenario 1 and Scenario 2, we can summarize the following propositions and all of the proofs are given in the Appendix.
Proposition 1. When SEV enters the transportation system and does compete with PT, the optimal number of customers using PT is reduced.
Proposition 2. When the total customer quantity overs a certain level, Q all > a 2b − k 4b , the total customer travel utility in the only competitive scenario is higher than that when only PT is present in the system, and vice versa. From Proposition 1 and Proposition 2, it is obvious that when a new transportation mode, SEV, is added to the original transportation system, the equilibrium state of the transportation system will be changed. Meanwhile, SEV attracts some customers in the transportation system, making the number of customers who use PT to travel decrease. However, only when the total number of customers in the transportation system reaches a certain value, the total customer travel utility of the whole transportation system increases as SEV enters the transportation system.
Proposition 3. As the number of customers using SEV does not reach a certain level, SEV should be subsidized to promote the total customer travel utility of the whole system. And on the contrary, the result is reversed.
Comparing the equilibrium results of Scenario 2 and Scenario 3 and analyzing the difference between the two of them, the propositions are summarized as follows and all of the proofs are given in the Appendix.
Proposition 4. When μ < η, the optimal number of customers taking PT in the only competition scenario is higher than that in the competitive and coordinative scenario, Q * P > Q 0 * P , and vice versa.
Further conclusions can be drawn from Proposition 4. The values of μ and η are influenced by the distance from station A to station B and the service quality of the travel modes (Zhang and Li 2014;Ma et al. 2007;Páez and Whalen 2010). Therefore, when station B in the transport system is settled, the values of μ and η are influenced only by the service quality. As μ < η, PT should promptly improve its service quality to attract more customers. In addition, the price of taking SEV from station A in Scenario 3 is lower than that in Scenario 2, p A < p S , and the reduction is 1þ2μ . Based on the demand function of SEV, when the price of taking SEV decreases, it could result in an increase in the number of customer taking SEV and a decrease in the number of customer The optimal number of customers (PT) The optimal number of customers (SEV) The number of customers (Transfer) The total customer travel utility The optimal price (SEV) taking PT at station A. It explains why the optimal number of customers traveling by PT decreases in Scenario 3.
Proposition 5. The total customer travel utility in the coordinative and competitive scenario is higher than that in the only c o m p e t i t i v e s c e n a r i o , i . e . , U all Q P ; Q S ; Q 0 P ; Q 0 S À Á > U all Q P ; Q S ð Þ. With Proposition 5, it is clear that the total customer travel utility in the coordinative and competitive scenario is higher than that in the only competitive scenario. Therefore, SEV and PT should adopt a strategy of both competitive and coordinative with each other to maximize the total customer travel utility of the whole system. Proposition 6. The variety of total customer travel utility, is strictly increased as the parameter η increases, while its increase or decrease depends on the different value of the parameter μ.
Further conclusions can be drawn from Proposition 6. The values of μ and η are influenced by the service quality of the travel modes as stations in the transportation system are settled (Zhang and Li 2014;Ma et al. 2007;Páez and Whalen 2010). It is wise to improve the service quality of SEV to further increase the total customer travel utility throughout the transportation system. However, it is important to notice that improving the service quality of PT might result in a decrease in the total customer travel utility of the entire transportation system.

Simulation experiment
Numerical simulation experiments are conducted for Scenario 3 to further analyze the impact of μ and η on the total customer travel utility in this section. When station B in the transportation system is settled, the values of μ and η are influenced only by the service quality of PT and SEV (Zhang and Li 2014;Ma et al. 2007;Páez and Whalen 2010). Two cases are considered based on Proposition 6. In case 1, set a = 10, b = 0.05, k = 1, and Q all = 200 and it can be obtained that μ < 1 2b þ k 1þη ð Þ a −1. Based on Eq. (16), line charts of sensitivity analysis under different values of μ and η are shown in Fig. 7.
According to Fig. 7, it could be clearly seen that the total customer travel utility increases at a slower rate as η increases when the value of μ is fixed. Conversely, the total customer travel utility increases at a faster rate with increasing of μ when the value of η is fixed. Meanwhile, it can be intuitively reflected from Fig. 7 that different values of η have less impact on the change of total customers travel utility than μ. It is consistent with Proposition 6.
Then, a contrast experiment is set up to research the impact of μ and η on customer travel utility better. In case 2, set a = 10, b = 0.6, k = 1, and Q all = 200 in this controlled experiment and this case only changes the value of b so that the value of The difference between Fig. 7 and Fig. 8 can be clearly seen. The conclusion drawn from Fig. 8 is opposite with that of Fig. 7. It is consistent with Proposition 6. In other words, these numerical experiments demonstrate the validity of Proposition 6.
Finally, the three-dimensional map is drawn to better display the impact of μ and η on customer travel utility. Based on Eq. (14), sensitivity analysis under different values of μ and η are shown in Fig. 9.
It can be intuitively reflected from Fig. 9 that when μ < 1 2b þ k 1þη ð Þ a −1, the total customer travel utility increases with increasing of μ at a rapid rate, while the increase of η has less effect on the increase in customer travel utility. On the contrary, when μ > 1 2b þ k 1þη ð Þ a −1, these effects are opposite.
Since the values of μ and η are influenced only by the service quality of PT and SEV, when μ < 1 2b þ k 1þη ð Þ a −1, PT should promptly and emphatically improve its service quality and SEV should maintain or slowly improve its service quality, and vice versa.

Conclusion
The competition and coordination problem between SEV and PT is solved in this paper by considering the customer travel utility. Considering the travel price and comfort in the customer travel utility, the game models of SEV and PT in three different scenarios are set up and the optimal equilibrium results are obtained. Some conclusions are drawn by comparing and analyzing these optimal equilibrium results and the numerical simulation experiments are conducted to prove the validity of them. This paper researches competition and coordination problem between SEV and PT by considering the transfer behavior of customers. Based on the reality, the diversity of customer travel makes transportation modes no longer a mere competition, which has rarely been considered in the previous research. In addition, this paper considers the customer travel utility from the perspective of the space comfort, which is realistic and enriches the study of the travel utility factors. Previous research has rarely considered the impact of space comfort on the customer's travel choice, while the customers would resent the crowded travel in reality and change their travel choice as a result. The evolution of the transportation system is investigated in this paper from the initial existence of only PT to the existence of the competition between SEV and PT, and then to the existence of the competition and coordination between SEV and PT. The following conclusions are drawn by comparing and analyzing the equilibrium of different scenarios. First, after SEV enters the transportation system, PT should adopt the strategy of both competition and coordination to maximize the total customer travel utility of the whole system. Meanwhile, after the position of the transit station in the travel chain is determined, it is worth considering the improvement of the service quality of SEV and PT. Second, when the number of customers taking SEV does not reach the optimal equilibrium result, it is necessary to subsidize SEV at this point to attract more customers and thus ultimately promoting the total customer travel utility. Finally, as SEV enters the transportation system, the optimal  Fig. 9 The impact of μ and η on customer travel utility number of customers taking PT in competitive and coordinative scenario is greater than that of in competitive scenario when the proportionality coefficient of taking PT is higher than that of taking SEV. Therefore, it is wise for SEV to promote its service quality to attract more customers. In the same vein, it is wise for PT to promote its service quality to attract more customers.
In future studies, we will consider the coordination problem between SEV and PT under the open travel chain, where customers and SEV are allowed to enter or leave the travel chain. It is worthwhile to consider the three-way coordination problem of SEV, PT, and taxi in the future research. Taxi is one of the most common ways of public transportation used by customers, while there are a few differences between Taxi and other public transportation like bus and subway. In addition, the coordination problem between SEV and PT with road congestion will be considered.

Appendix
The proof of Proposition 1.
Comparing the optimal number of customers using PT in Scenario 2 with Scenario 1, it can be found that a−k 2b < a 2b owing to a > 0, b > 0, and k > 0. And the number of customers reduced is ΔQ P ¼ k 2b . The proof of Proposition 2. The change in the total utility is . W h e n k + 4bQ all − 2a > 0, it can be obtained ΔU > 0. Therefore, when Q all > a 2b − k 4b , U Q * P ; Q * S À Á > U Q P * À Á . In other words, after SEV enters the system and only competes with PT and Q all > a 2b − k 4b , the total customer travel utility increases. Similarly, when Q all ¼ a 2b − k 4b , U Q * P ; Q * S À Á ¼ U all Q * P À Á . When Q all < a 2b − k 4b , U Q * P ; Q * S À Á < U all Q * P À Á . In summary, the magnitude of the total utility of Scenario 1 and Scenario 2 depends on the relationship between Q all and a 2b − k 4b in the transport system.
The proof of Proposition 3. In Scenario 2, it can be known that the optimal numbers of customers using PT and SEV are Q * P ¼ a−k 2b and Q * S ¼ Q all − a−k 2b , respectively. In addition, the maximum total utility of the system is U Q * P ; Q * Since Q all is a determined value in the transport system, when the number of customers taking SEV does not reach the equilibrium result, i.e., Q S < Q all − a−k 2b , it could be obtained that Q P > a−k 2b . In this case, the total utility is less than that in t h e e q u i l i b r i u m , U Q P ; Q S ð Þ< U Q * P ; Q * . Thus, SEV needs to be subsidized to lower the price of taking SEV to attract more customers and ultimately promote the total customer travel utility of the whole system. Similarly, when Q S ¼ Q all − a−k 2b , it could be obtained Q P ¼ a−k 2b and U Q P ; Q S ð Þ¼U Q * P ; Q * S À Á . SEV does not need to be subsidized to upset the optimal equilibrium of the whole system. When Q S > Q all − a−k 2b , it can haveQ P < a−k 2b and U Q P ; Q S ð Þ< U Q * P ; Q * S À Á . Therefore, SEV does not need to be subsidized, but PT needs to be subsidized to improve the service quality to attract more customers and ultimately promote the total customer travel utility of the whole system.
In summary, the total customer travel utility of the system varies with the relationship between Q S and Q all − a−k 2b . When Q S < Q all − a−k 2b , the total customer travel utility of the whole system is decreasing. Therefore, it is recommended to subsidize SEV to promote the total customer travel utility of the whole system in this case. On the contrary, the result is reversed.
The proof of Proposition 4. As the travel utility function coefficients are determined in Scenario 3, it can be easily found that the size relationship is affected by u and η. For example, when μ < η, Q * P −Q > 0. Therefore, it could be obtained that the optimal number of customers served by PT is reduced and the optimal price of using SEV at station A is reduced compared to that in Scenario 2. Meanwhile, the amount of change in the optimal number of customers taking PT is k η−μ ð Þ 2b 1þμ ð Þ . Similarly, when μ = η, Q * P ¼ Q 0 * P , and p S = p A . When μ > η, Q * P < Q 0 * P , and p S < p A . Further conclusions can be drawn from the above. The values of μ and η are influenced by the distance from station A to station B and the service quality of the travel modes (Zhang and Li 2014;Ma et al. 2007;Páez and Whalen 2010). Therefore, when station B in the transport system is settled, the values of μ and η are influenced only by the service quality. As μ < η, PT should promptly improve its service quality to get more customers. In addition, the price of taking SEV from station A in Scenario 3 is lower than that in Scenario 2, p A < p S , and the reduction is 1þ2μ Based on the demand function of SEV, when the price of taking SEV decreases, it could result in the increasing number of customer taking SEV and the decreasing number of customer taking PT at station A. It explains why the optimal number of customers traveling by PT decreases in Scenario 3.
The proof of Proposition 5.
Since μ ¼ k 1þη ð Þ a −1, it can be obtained that ∂ΔU ∂μ ≡0. Therefore, ΔU is not changed as the value of μ changes. At this time, ΔU is a constant value, and this constant value is ΔU = ηkQ.
(2) When μ ¼ 1 2b þ k 1þη ð Þ a −1 > 0, ΔU decreases as the value of μ increases. It means that μ ¼ 1 2b þ k 1þη ð Þ a −1 is the only maximum point of ΔU, and ΔU max ¼ ηkQ þ Author contribution Zhiyong Zhang analyzed the research problem, wrote the original draft based on the formal analysis, and finished the revision. He was also responsible for the drawing part of the paper and for some calculation with software. Xiao Zhang had provided the methodology support and reviewed and edited the writing, and also funded this manuscript. All authors read and approved the final manuscript.