The effects and conflicts of co-opetition in a rail-water multimodal transport system

Abstract

This study investigates a rail-water multimodal transport system composed of a railway company, a liner company, and an emerging multi-modal operator. Based on the co-opetition game, we discuss decision strategy preference and conflict from a multi-stakeholder perspective to optimize individual profit and system efficiency. It is found that although the invasion of multi-modal operators into the market poses a threat to competition, their service effort directly affects the market demand and promotes the profits of each carrier. The free-riding and market expansion effects triggered by service effort interact with each other. However, multi-modal operators can cope with the negative impact of the free-riding effect through the service strategy and promote system efficiency optimization. Specifically, discussing each carrier's decision-making preferences for maximizing profits, we find that the multi-modal operators’ strategy can achieve the Pareto optimal of triple-win, and the system efficiency is also optimal simultaneously.

Introduction

The transportation industry consumes a lot of fossil energy. According to the report “Global Energy Overview: Carbon Dioxide Emissions in 2020”1 issued by the International Energy Agency, the transportation industry consumes 60% of global oil, and carbon emissions account for a large proportion. China has clarified that “emission peak will be achieved by 2030 and carbon neutrality will be achieved by 2060”, which puts forward new requirements and challenges for transportation companies. Some companies have focused on green investment, such as new energy substitution and technology upgrading. For example, the world’s largest dual-fuel propulsion container ship built by Jiangnan Shipyard has been launched, and China National Railway Group Co., Ltd. has promoted the green transportation mode by promoting e-tickets and enhancing electrification. The transportation field can not only implement low-carbon technology to cope with climate change in terms of hardware but also construct a low-carbon transportation system from the perspective of optimizing transportation structure and improving operational efficiency (Genc, 2021). Multimodal transportation integrates various modes of transportation to provide door-to-door transportation services and optimizes freight routes while reducing turnover operations. It is one of the low-carbon transportation organization forms that contribute to the realization of carbon emission reduction targets in the field of transportation (Kabadurmus & Erdogan, 2020).

Two types of companies mainly undertake rail-water multimodal transport business in the market.2 One part is emerging multi-modal operators that invade the market and adopt asset-light operational strategies by integrating upstream transport business. Such as Sino-Singapore Interconnection (Chongqing) Logistics Development Co., Ltd., which integrates railway and water transportation to provide multimodal transport business.3 Another part is traditional liner companies which develop new service businesses to fight fierce competition with multi-modal operators in the transportation market. Such as, Maersk cooperated with the Russian Railway Company to carry out transcontinental rail-water multimodal transport business. After entering the market, multi-modal operators have to face the competitive pressure of traditional liner companies and have to decide their service competition strategies. This paper considers two service strategies of multi-modal operators and provides the following practical cases for each strategy. Case1: Ex-ante service strategy. CIMC4 and Maersk5 explicitly put forward warehousing, “door-to-door” distribution, bulk cargo LCL, and other services about their multimodal services on their official websites. It indicates that some companies will publish information such as service types and degrees in advance. Case 2: Ex-post service strategy. SF Multimodal Transport Company,6 Sino-Singapore Logistics Development Co., Ltd., etc., did not provide service commitment to the shipper before undertaking transportation and provided customized services to the shipper according to its transportation needs.

The service investment of the multi-modal operators simplifies the turnaround of cargo transportation and the visa of transportation documents, improves the transportation experience of the shipper, and expands the basic transportation needs of the market. However, the cost of the service effort of the multi-modal operators is borne by themselves, so their service overflow provides an opportunity for liner companies to free-riding behavior (Wang et al., 2021; Oliveira et al., 2022). In this regard, the following research questions are raised: (a) Can service effort drive higher profits? How does the interaction between the free-riding effect and the market expansion effect, triggered by service effort, affect decision making? (b) What is the role of multi-modal operators in the market? Do they cause conflict, or do they promote the simultaneous development of the carriers in the rail-water multimodal transport system? (c) What is the optimal strategy in a multi-agent co-opetition game? Is there a strategy to achieve Pareto optimality with the dual objectives of maximum company profit and optimal system efficiency?

To solve the above problems, we construct a rail-water multimodal transport system based on the carriers’ co-opetition relationship, composed of an upstream railway company, a downstream liner company, and a terminal multi-modal operator. We consider the combined decision of multi-modal operator’s procurement mode and service strategy. The influence of the free-riding effect and market expansion effect caused by the service effort of multi-modal operator on carriers and system efficiency is deeply analyzed.

In analyzing the equilibrium results, some interesting conclusions are obtained, which are summarized as follows. First, we clarify the role of multi-modal operators and find that multi-modal operators invest service effort to develop a competitive advantage in terms of traffic volume. From the perspective of profit analysis, the market expansion effect brought by service effort drives the overall profit of the rail-water multimodal transport system. Secondly, we find that the negative impact of the free-riding effect can be countered through strategy formulation. Counterintuitively, liner companies’ free-riding behavior caused by service effort may damage their profits. Under certain circumstances, the profits of the multi-modal operators will rise with the appearance of free-riding behavior. Third, as the decision-making subject, the multi-modal operators can bring a triple-win situation by choosing ex-post service. The railway companies and liner companies can also maximize profits in multi-modal operators’ decisions. Finally, it is verified that the equilibrium strategy can promote the optimal efficiency of the rail-water multimodal transport system.

This study provides critical operational strategies and contributions to the literature. First, few studies have highlighted the conflicting relationships between decisions from multiple-stakeholder and system perspectives to the best of our knowledge. This paper is a theoretical pioneer in discussing how multi-modal operators can enhance the profits and efficiency of a rail-water multi-modal transport system based on a multi-stakeholder co-opetition relationship. Second, we have examined the optimal strategy for multi-modal operators based on profit and efficiency perspectives. The results suggest that the service efforts of multi-modal operators are conducive to efficiency gains but that over-investment may have a negative impact. Third, discussing the equilibrium results leads us to some novel ideas that provide theoretical guidance for business operations and policy formulation. These findings will contribute to the optimization and development of other rail-waterway intermodal systems facing similar dilemmas. Moreover, they have important management implications for the logistics industry to reduce emissions. In practice, several reasons have driven the research. Firstly, there is ample experience to show that rail-water intermodal transport has a huge role in global freight transport. Secondly, emerging intermodal operators are invading downstream markets and competing for services. Finally, the business strategy of intermodal operators buying capacity directly from rail and liner companies is widely adopted, such as the Hub Group.7

The organization of this paper is as follows. Section 2 reviews the literature about multimodal transport. Section 3 provides the model settings, assumptions, and notations. Section 4 discusses the service input value of multi-modal operators and the influence of related effects. Section 5 compares the equilibrium results under different operational strategies and determines the optimal strategy for multi-modal operators. Section 6 expands the basic model from demand uncertainty, cross-competition, and spillover-free perspectives. In Sect. 7, this study puts forward conclusions and future research. To explain more clearly, we put the certificate in the appendix.

Literature review

Our work mainly involves three types of literature: multimodal freight transportation, service effort strategies, and the marketing of transportation services. Next, we compare and contrast our work with these studies to highlight our contributions.

Multimodal transportation is to transport goods from the starting point to the destination through various modes of transportation. Since this process requires coordination among multiple carriers, there are many problems worthy of study that has attracted the attention of the theoretical circles. Xie et al. (2017) constructed a sea-rail multimodal transport model considering delivery strategy and designed a bilateral repurchase contract to achieve the goal of balanced freight transportation. They showed that Nash equilibrium exists in the game, and supply chain contract design can improve coordination. Sarhadi et al. (2017) constructed a three-layer mathematical model to design a framework from the perspective of “defense-attack-defense” in order to reduce the negative impact caused by service interruption. Then they discussed the best defense plan through the heuristic algorithm. With the continuous expansion of multimodal transport application scenarios, some scholars started to discuss based on the industry background. Algaba et al. (2019) assumed that multimodal transport participants launched a joint transportation card and introduced a transportation network colored graph combined with a game-theoretical approach to design two schemes of color averaging and cost proportional distribution to solve the problem of profit distribution among companies. Demir et al. (2016) considered the uncertainty of travel time in the green multimodal transport service network and designed the multimodal transportation route decision of multiple commodities through the sample average approximation method. They discussed the multimodal transportation plan under different objective constraints according to travel time and demand uncertainty. The existing research on multimodal transport focuses on solving the optimal solution problems such as transportation, loading, and unloading through operational planning optimization. Very little literature analyzes the competition between multi-modal operators and other carriers from the perspective of the multi-party game.

The market competition has gradually transformed into service competition, so the academic circles have discussed the service effort strategy. Duan et al. (2021) introduced the fourth-party sales manager to analyze the impact of their sales effort on the supply chain. Chen et al. (2017) discussed retailers’ service effort effectiveness. Our study also relates to free-riding behavior caused by service effort in the market. Wu et al. (2004) discussed the impact of the free-riding effect on the company’s profits under the background of information service. Shin et al. (2007) showed that the free-riding effect had become a necessary mechanism in the market, which can cope with the aggressive reaction of competitive companies and reduce the price competition intensity. Kuksov and Liao (2018) analyzed the influence of free-riding behavior between manufacturers and retailers under endogenous contracts. When manufacturers endogenous contract types, retailers restrict this behavior, leading to a decline in profits. The research on service effort strategy focuses on effectiveness, but few consider service effort strategy and free-riding behavior simultaneously. We start from the dynamic perspective of the behavior simultaneously. We start from the dynamic perspective of the event sequence and discuss the interaction mechanism among the sequence under different service strategies, carriers, free-riding effect, and market expansion effect.

Marketing is the first and most crucial step in developing transportation services, especially multimodal services (Han et al., 2018). Many multi-modal operators expanded the potential shipper market through service innovation, such as quantity discount contracts and advertisements, or increased profits margins by reducing costs and increasing efficiency to gain competitive advantages. Wang et al. (2017) considered the employment decision of freight forwarders. Song et al. (2018) considered the competing game between liner companies and two heterogeneous ports. Koza (2019) combined service scheduling with freight rate allocation and constructed a branch pricing algorithm considering the constraints of container transit time and fuel consumption. Nowak et al. (2019) discussed the relationship between the carrier’s route and profits under the quantity discount contract. Wang and Meng (2019) studied the optimal pricing problem of terminal enterprises under the influence of congestion. Zheng and Luo (2021) constructed vertical and horizontal integration models of the shipping market, considering the impacts of transportation economies of scale and service substitution competition. Minimal studies consider the game relationship between different carriers and multimodal service providers. Here, we consider that each of the two modes of transportation undertakes the transportation in a specific area instead of different carriers in the same area. In addition, because the integration of rail-water multimodal transportation has the advantages of low carbon and convenience, it will attract more shippers to enter the market. We also incorporated this actual change into our model.

After clarification, it is found that there is little discussion on the service effort of professional carriers under the rail-water multimodal transport system, especially the interaction mechanism of the market expansion effect and the free-riding effect caused by the service effort. Therefore, in view of the competition in the terminal market between multi-modal operators and liner companies in the vertically integrated rail-water multimodal transport system, we innovatively consider the timing of service effort decision as a competitive means in the decision making of the rail-water multimodal transport system, investigate how multi-modal operators can improve their profits and system efficiency through the combination design of procurement mode and service strategy.

Model setting

The rail-water multimodal transport system studied in this paper is composed of a railway company, a liner company, and a multi-modal operator. Railway company (R) provides rail transportation service to the shipper, and the price for delivering the goods to the port is $w$ (Song et al., 2020; Jiang, 2021). Liner company (OS) can not only provide water transportation service but also extend its business upstream to integrate railway transportation. The price and volume of the rail-water multimodal transportation provided by OS are $p_{\mathit{os}}^{}$ and $q_{\mathit{os}}^{}$, respectively.

Multi-modal operator (MT) provides two integration modes: centralized procurement mode (Mode C) and decentralized procurement mode (Mode D). Under mode C, the price for MT to purchase the rail-water multimodal transportation service integrated by OS is $t_{}^C$, and the price at which the shipper purchases the service through MT is $p_M^C$, and the volume is $q_M^C$. Under mode D, MT orders rail transportation services from R and water transportation services from OS at $w_M^D$ and $t_{}^D$, respectively. The price for the shipper to purchase the rail-water multimodal transportation service integrated by MT is $p_M^D$, and the volume is $q_M^D$. The model framework can be described as in Fig. 1.

Fig. 1.

Structure of the rail-water multimodal transport system

Due to the limitation of traffic volume in the multimodal transport industry, and transportation companies focus more on traffic volume and profits, the Cournot model is adopted in this study to describe the competitive relationship between OS and MT. The inverse demand functions of MT and OS are:

$$p_{\mathit{os}}^{}=a+\lambda v-q_{\mathit{os}}^{}-q_M^{}$$ 1

$$p_M^{}=a\left(1+\theta \right)+v-q_{\mathit{os}}^{}-q_M^{}$$ 2

Among them, $a$ is the potential market traffic volume of OS. As commonly done in the literature (Li et al., 2019a, 2019b; Yao & Liu, 2005), the potential market traffic volume of OS is normalized to 1 to simplify the calculation, $a=1$. This assumption is relaxed during the discussion of the extended model, and it is discussed according to the market potential transportation demand equals $a$. We find that the conclusion of the extended model is basically consistent with the main model. The results show that standardizing requirements to 1 in this paper has no effect on subsequent analysis. The degree of service effort of the MT is $v$. As a professional service company, MT is committed to promoting multimodal transport, which is a convenient and efficient “door-to-door” transportation mode. They can expand the potential market demand by establishing portal websites through putting in multimodal transport advertisements and building comprehensive freight rate systems, so the potential market traffic volume of MT is $a\left(1+\theta \right)$. The positive spillover degree of service effort to OS is $\lambda \left(0<\lambda <1\right)$, which is because MT’s service effort make more shippers know about rail-water multimodal transportation. However, due to the service effort, the free-riding behavior of OS is generated. Therefore, $\lambda$ can also indicate the free-riding effect brought by the service effort of MT.

To mitigate the dual competitive pressure of OS’s business competition and free-riding behavior, MT, as the primary marketing input, can further strengthen its competitive advantage by changing its service strategy. One of the academic views on this matter is that multimodal transport companies should adopt an ex-ante service strategy and promise the service effort with the help of the first-mover advantage. On the one hand, it affects the pricing of railway and liner companies, and on the other hand, it influences the shipper’s choice of purchase by anchoring effect. Another point of view is that the multimodal transport companies should fully grasp the critical information of the upstream and downstream carriers’ volume and pricing before deciding the degree of service effort. That is, the ex-post service strategy is more conducive to MT’s participation in the competition. There are four operational strategies for MT to choose from:

Strategy Cl

MT chooses the centralized procurement mode and purchases rail-water multimodal transportation services from OS. At the same time, after entering the market, they first announce the degree of service effort.

Strategy Dl

MT chooses the decentralized procurement mode and orders transport services from R and OS, respectively. At the same time, MT promises the degree of service effort to the shipper first.

Strategy Cr

MT announces the degree of service effort after the rate of railway transportation and water transportation are determined. It chooses the centralized procurement mode and purchases integrated transportation services from OS.

Strategy Dr

MT chooses the decentralized procurement mode and orders transport services from R and OS, respectively. In the final stage, the degree of service effort is decided by MT.

There are two kinds of costs in rail-water multimodal transportation. One is the procurement cost, such as the cost of rail space ordered by OS, the cost of integrated services purchased by MT. The second category is transportation cost. As the primary carrier, R or OS will pay for fuel and labor when they deliver goods to their destinations. The railway company is located upstream of the rail-water multimodal transport system, so there is no procurement cost. The train runs at a fixed frequency, and the transportation costs such as personnel and fuel are similar to fixed inputs. Moreover, this paper focuses on the game competition between liner company and multi-modal operator, and the main body of the discussion is not the railway company. Therefore, in the basic model analysis, it is assumed that the railway transportation cost is zero. The cost expenditure of OS includes procurement and transportation. The purchasing cost of OS is reflected in the profit function through the paid railway transportation rate $w$, while its transportation cost in the basic model is nominalized to zero. Furthermore, for the robustness of the model, the transportation costs of OS are considered in the extended model. By comparison, it is found that the conclusions of the extended part are basically consistent with the basic model. Next, we discuss the profit function and decision sequence of the event in the four operational strategies.

In strategy Cr, as the main body of transportation upstream, the railway company first decides its own transportation rate $w_{\mathit{os}}^{\mathit{Cr}}$. Secondly, OS integrates railway transportation to provide integrated service to MT at the rate of $t_{}^{\mathit{Cr}}$. At last, the decision-making service level of MT is $v$. According to Li (2019) and other studies, it will be assumed that the service cost of MT is the quadratic function of the degree of service effort as $\frac{v^2}{2}$. At the same time, MT and OS compete in the market to decide their own traffic volume $q_M^{\mathit{Cr}}$ and $q_{\mathit{os}}^{\mathit{Cr}}$, respectively. Decision models are as follows:

$$\underset{w_{\mathit{os}}^C\ge 0}{max}{\pi }_R^{\mathit{Cr}}\left(w_{\mathit{os}}^{\mathit{Cr}}\right)=w_{\mathit{os}}^{\mathit{Cr}}\left(q_{\mathit{os}}^{\mathit{Cr}}+q_M^{\mathit{Cr}}\right)$$ 3

$$\underset{t_{}^C\ge 0}{max}{\pi }_{\mathit{os}}^{\mathit{Cr}}\left(\left(t_{}^{\mathit{Cr}}\right|q_{\mathit{os}}^{Cr\ast }\right)=\left(p_{\mathit{os}}^{\mathit{Cr}}-w_{\mathit{os}}^{\mathit{Cr}}\right)q_{\mathit{os}}^{\mathit{Cr}}+\left(t_{}^{\mathit{Cr}}-w_{\mathit{os}}^{\mathit{Cr}}\right)q_M^{\mathit{Cr}}$$ 4

$$\left\{\begin{array}{c}\underset{q_M^C\ge 0,v\ge 0}{max}{\pi }_M^{\mathit{Cr}}\left(q_M^{\mathit{Cr}},v\right)=\left(p_M^{\mathit{Cr}}-t_{}^{\mathit{Cr}}\right)q_M^{\mathit{Cr}}-\frac{v^2}{2}\\ \underset{q_{\mathit{os}}^C\ge 0}{max}{\pi }_{\mathit{os}}^{\mathit{Cr}}\left(q_{\mathit{os}}^{\mathit{Cr}}\right)=\left(1+\lambda v-q_{\mathit{os}}^{\mathit{Cr}}-q_M^{\mathit{Cr}}-w_{\mathit{os}}^{\mathit{Cr}}\right)q_{\mathit{os}}^{\mathit{Cr}}+\left(t_{}^{\mathit{Cr}}-w_{\mathit{os}}^{\mathit{Cr}}\right)q_M^{\mathit{Cr}}\\ \end{array}\right)$$ 5

In strategy Cl, MT first decides the degree of service effort $v$. Then, R and OS set their respective rates as $w_{\mathit{os}}^{\mathit{Cl}}$ and $t_{}^{\mathit{Cl}}$. Finally, MT and OS respectively provide rail-water multimodal transportation services to the market and decide their traffic volumes $q_M^{\mathit{Cl}}$ and $q_{\mathit{os}}^{\mathit{Cl}}$, respectively. Decision models are as follows:

$$\underset{v\ge 0}{max}{\pi }_M^{\mathit{Cl}}\left(\left(v\right|q_M^{Cl\ast }\right)=\left(1+\theta +v-q_{\mathit{os}}^{\mathit{Cl}}-q_M^{\mathit{Cl}}-t_{}^{\mathit{Cl}}\right)q_M^{\mathit{Cl}}-\frac{v^2}{2}$$ 6

$$\underset{w_{\mathit{os}}^C\ge 0}{max}{\pi }_R^{\mathit{Cl}}\left(w_{\mathit{os}}^{\mathit{Cl}}\right)=w_{\mathit{os}}^{\mathit{Cl}}\left(q_{\mathit{os}}^{\mathit{Cl}}+q_M^{\mathit{Cl}}\right)$$ 7

$$\underset{t_{}^C\ge 0}{max}{\pi }_{\mathit{os}}^{\mathit{Cl}}\left(\left(t_{}^{\mathit{Cl}}\right|q_{\mathit{os}}^{Cl\ast }\right)=\left(1+\lambda v-q_{\mathit{os}}^{\mathit{Cl}}-q_M^{\mathit{Cl}}-w_{\mathit{os}}^{\mathit{Cl}}\right)q_{\mathit{os}}^{\mathit{Cl}}+\left(t_{}^{\mathit{Cl}}-w_{\mathit{os}}^{\mathit{Cl}}\right)q_M^{\mathit{Cl}}$$ 8

$$\left\{\begin{array}{c}\underset{q_M^C\ge 0}{max}{\pi }_M^{\mathit{Cl}}\left(q_M^{\mathit{Cl}}\right)=\left(1+\theta +v-q_{\mathit{os}}^{\mathit{Cl}}-q_M^{\mathit{Cl}}-t_{}^{\mathit{Cl}}\right)q_M^{\mathit{Cl}}-\frac{v^2}{2}\\ \underset{q_{\mathit{os}}^C\ge 0}{max}{\pi }_{\mathit{os}}^{\mathit{Cl}}\left(q_{\mathit{os}}^{\mathit{Cl}}\right)=\left(1+\lambda v-q_{\mathit{os}}^{\mathit{Cl}}-q_M^{\mathit{Cl}}-w_{\mathit{os}}^{\mathit{Cl}}\right)q_{\mathit{os}}^{\mathit{Cl}}+\left(t_{}^{\mathit{Cl}}-w_{\mathit{os}}^{\mathit{Cl}}\right)q_M^{\mathit{Cl}}\\ \end{array}\right)$$ 9

In strategy Dr, the rates of ordering railway transportation service by OS and MT are $w_{\mathit{os}}^{\mathit{Dr}}$ and $w_M^{\mathit{Dr}}$, respectively. Then the rate of water transportation service provided by OS to MT is $t_{}^{\mathit{Dr}}$. At last, MT provides rail-water multimodal transportation services to compete with OS and decides their traffic volume to be $q_{\mathit{os}}^{\mathit{Dr}}$ and $q_M^{\mathit{Dr}}$ respectively. Decision models are as follows:

$$\underset{w_{\mathit{os}}^{\mathit{Dr}}\ge 0,w_M^{\mathit{Dr}}\ge 0}{max}{\pi }_R^{\mathit{Dr}}\left(\left(w_{\mathit{os}}^{\mathit{Dr}},w_M^{\mathit{Dr}}\right|q_{\mathit{os}}^{Dr\ast },q_M^{Dr\ast }\right)=w_{\mathit{os}}^{\mathit{Dr}}{q}_{\mathit{os}}^{\mathit{Dr}}+w_M^{\mathit{Dr}}{q}_M^{\mathit{Dr}}$$ 10

$$\underset{t_{}^{\mathit{Dr}}\ge 0}{max}{\pi }_{\mathit{os}}^{\mathit{Dr}}\left(\left(t_{}^{\mathit{Dr}}\right|q_{\mathit{os}}^{Dr\ast }\right)=\left(p_{\mathit{os}}^{\mathit{Dr}}-w_{\mathit{os}}^{\mathit{Dr}}\right)q_{\mathit{os}}^{\mathit{Dr}}+t_{}^{\mathit{Dr}}{q}_M^{\mathit{Dr}}$$ 11

$$\left\{\begin{array}{c}\underset{q_{\mathit{os}}^{\mathit{Dr}}\ge 0}{max}{\pi }_{\mathit{os}}^{\mathit{Dr}}\left(q_{\mathit{os}}^{\mathit{Dr}}\right)=\left(1+\lambda v-q_{\mathit{os}}^{\mathit{Dr}}-q_M^{\mathit{Dr}}-w_{\mathit{os}}^{\mathit{Dr}}\right)q_{\mathit{os}}^{\mathit{Dr}}+t_{}^{\mathit{Dr}}{q}_M^{\mathit{Dr}}\\ \underset{q_M^{\mathit{Dr}}\ge 0,v\ge 0}{max}{\pi }_M^{\mathit{Dr}}\left(q_M^{\mathit{Dr}},v\right)=\left(p_M^{\mathit{Dr}}-t_{}^{\mathit{Dr}}-w_M^{\mathit{Dr}}\right)q_M^{\mathit{Dr}}-\frac{v^2}{2}\\ \end{array}\right)$$ 12

In strategy Dl, MT decides the degree of service effort, then the rates of rail transportation service decided by the railway company for MT and OS are $w_M^{\mathit{Dl}}$ and $w_{\mathit{os}}^{\mathit{Dl}}$ respectively. Next, OS decides that the water transportation rates for MT are $t_{}^{\mathit{Dl}}$. At last, MT and OS compete in the market to decide their own traffic volume $q_M^{\mathit{Dl}}$ and $q_{\mathit{os}}^{\mathit{Dl}}$, respectively. Decision models are as follows:

$$\underset{v\ge 0}{max}{\pi }_M^{\mathit{Dl}}\left(\left(v\right|q_M^{Dl\ast }\right)=\left(1+\theta +v-q_{\mathit{os}}^{\mathit{Dl}}-q_M^{\mathit{Dl}}-t_{}^{\mathit{Dl}}-w_M^{\mathit{Dl}}\right)q_M^{\mathit{Dl}}-\frac{v^2}{2}$$ 13

$$\underset{w_{\mathit{os}}^{\mathit{Dl}}\ge 0,w_M^{\mathit{Dl}}\ge 0}{max}{\pi }_R^{\mathit{Dl}}\left(\left(w_{\mathit{os}}^{\mathit{Dl}},w_M^{\mathit{Dl}}\right|q_{\mathit{os}}^{Dl\ast },q_M^{Dl\ast }\right)=w_{\mathit{os}}^{\mathit{Dl}}{q}_{\mathit{os}}^{\mathit{Dl}}+w_M^{\mathit{Dl}}{q}_M^{\mathit{Dl}}$$ 14

$$\underset{t_{}^{\mathit{Dl}}\ge 0}{max}{\pi }_{\mathit{os}}^{\mathit{Dl}}\left(\left(t_{}^{\mathit{Dl}}\right|q_{\mathit{os}}^{Dl\ast }\right)=\left(1+\lambda v-q_{\mathit{os}}^{\mathit{Dl}}-q_M^{\mathit{Dl}}-w_{\mathit{os}}^{\mathit{Dl}}\right)q_{\mathit{os}}^{\mathit{Dl}}+t_{}^{\mathit{Dl}}{q}_M^{\mathit{Dl}}$$ 15

$$\left\{\begin{array}{c}\underset{q_M^{\mathit{Dl}}\ge 0}{max}{\pi }_M^{\mathit{Dl}}\left(q_M^{\mathit{Dl}}\right)=\left(1+\theta +v-q_{\mathit{os}}^{\mathit{Dl}}-q_M^{\mathit{Dl}}-t_{}^{\mathit{Dl}}-w_M^{\mathit{Dl}}\right)q_M^{\mathit{Dl}}-\frac{v^2}{2}\\ \underset{q_{\mathit{os}}^{\mathit{Dl}}\ge 0}{max}{\pi }_{\mathit{os}}^{\mathit{Dl}}\left(q_{\mathit{os}}^{\mathit{Dl}}\right)=\left(1+\lambda v-q_{\mathit{os}}^{\mathit{Dl}}-q_M^{\mathit{Dl}}-w_{\mathit{os}}^{\mathit{Dl}}\right)q_{\mathit{os}}^{\mathit{Dl}}+t_{}^{\mathit{Dl}}{q}_M^{\mathit{Dl}}\\ \end{array}\right)$$ 16

Effect analysis

This section discusses the direct effects and the indirect effects caused by the service effort of the multi-modal operator. The logic of the analysis is as follows. From the perspective of direct effects, multi-modal operator invests in service effort and bear the costs. The choice of service strategy is discussed under the constraint of profit maximization. The service effort of the multi-modal operator induces two types of indirect effects. One is the market expansion effect, manifested as an increase in potential market demand of multi-modal operator. The other is the free-riding effect, expressed as a spillover of the multi-modal operator’s service effort, which enables the liner company to obtain a certain percentage of the increase in demand.

Since the model is a multi-stage game, the reverse induction method is used to discuss the equilibrium solution. The equilibrium solutions in different operational strategies are proved in the appendix. To ensure that the equilibrium solution can be obtained under each decision model, we set conditions on parameters such as market expansion effect, free-riding effect, etc.

Direct effect analysis

The multi-modal operator can make independent decisions and make decisions about their own service effort with the goal of maximizing revenue, which directly affects their own profits. This section discusses direct effects from the perspectives of profit and efficiency and compares the conflicts of strategies between the railway company and liner company. By comparison, it is found that multi-modal operator always chooses ex-post service, which has a positive effect on itself and is conducive to improving input–output efficiency. The railway company should track and align with the multi-modal operator’s strategy. There are similar cases in practice where MT and R collaborated. The Hub Group has a diverse range of partnerships with all Class-I railroads in North America.

Corollary 1

MT chooses the ex-post service strategy to maximize its profits. R’s preference of service strategy is consistent with that of MT.

A comparison of Fig. 2 shows that there is a high probability of a triple-win situation for the carrier MT choosing the ex-post service, whether under Mode C or Mode D. MT can maximize profits when it chooses the ex-post service strategy, while R’s preference of service strategy is consistent with MT. While OS also prefers to choose the ex-post service strategy in region triple-win. This is because, on the one hand, the ex-post service strategy can reduce the free-riding effect's negative impact; on the other hand, it can help MT and OS obtain large-scale traffic. Conversely, when the market expansion effect is small (as in OS-Cl or OS-Dl in Fig. 2), MT squeezes OS's market share through an ex-post service strategy, resulting in a decline in OS's profits and a failure to achieve a win–win-win situation.

Fig. 2.

Preference distribution of service strategy of each carrier

MT bears the input cost of transportation services. Although the profits can be maximized through the ex-post service strategy, the goal of the rail-water multimodal transport system is to optimize the operation efficiency, so what is the impact of service input on the output efficiency? Similar to studies such as Thompson et al. (1997) and Niu et al. (2019), which discuss the efficiency of MT service inputs. Firstly, from the perspective of MT, we analyze the influence of service input on its own efficiency, which is expressed as $E_M^{}=\frac{2{\pi }_M^{}}{v^2}$. $E_M$ represents the service input efficiency of MT. Secondly, the efficiency of service input is analyzed from the perspective of the rail-water multimodal transport system, that is, the influence of service input of MT on the system, which is defined as $E_{}^{}=\frac{2{\pi }_{\prod }^{}}{v^2}$ Through calculation, we find that the distribution of maximization of $E_M^{}$ and $E_{}^{}$ is consistent, as shown in Fig. 3. It shows that ex-post service strategy can not only maximize the efficiency of MT, but also improve the overall efficiency of the rail-water multimodal transport system.

Fig. 3.

Distribution for maximum efficiency

Indirect effect analysis

Multi-modal operator’s service effort indirectly trigger market expansion effects and free-riding effects. This section focuses on the impact of indirect effects. First, we analyze the impact of the market expansion effect and compare the interaction mechanisms between optimal rates, traffic volume, and market expansion effect under different strategies. Second, it cuts from the perspective of the free-riding effect.

The impact of market expansion effect

Corollary 2

The market expansion effect not only drives higher transport rates and total traffic volume, but can also help MT capture OS market share.

According to the equilibrium solutions in different strategies, the relationship between the decision variables of each carrier and the market expansion effect is discussed respectively, as shown in Table 1. “$+$” in Table 1 indicates that the rate is positively related to the market expansion effect. Under the equilibrium state, the service effort of MT and the market expansion effect present a linkage reaction, and the service investment of MT expands the market demand. The market expansion effect, in turn, promoted the rise of service level. By comparison, it is found that the market expansion effect positively impacts the increase of transportation rate.

Table 1.

The effect of market expansion

	Cr	Cl	Dr	Dl
$\frac{\partial t^{\ast }}{\partial \theta }$	$+$	$+$	$0<\lambda <\frac{1}{4}\left(7-\sqrt{41}\right),-$ or $\frac{1}{4}\left(7-\sqrt{41}\right)<\lambda <1,+$	$+$
$\frac{\partial w_{\mathit{os}}^{\ast }}{\partial \theta }$	$+$	$+$	$+$	$+$
$\frac{\partial w_M^{\ast }}{\partial \theta }$	Null	Null	$+$	$+$
$\frac{\partial v^{\ast }}{\partial \theta }$	$+$	$+$	$+$	$+$
$\frac{\partial q_M^{\ast }}{\partial \theta }$	$+$	$+$	$+$	$+$
$\frac{\partial q_{\mathit{os}}^{\ast }}{\partial \theta }$	$-$	$-$	$-$	$-$
$\frac{\partial \left(q_M^{\ast }+q_{\mathit{os}}^{\ast }\right)}{\partial \theta }$	$+$	$+$	$+$	$+$

It is worth noting that the relationship between shipping rates and the market expansion effect of OS under strategy Dr is also affected by the free-riding effect. The announcement of service effort by MT in the final stage means that OS has less time to free ride. When the free-riding effect is large, OS gains more positive influence and increases its sales. At the same time, the decentralized procurement mode of MT also limits the profit realization of OS. Therefore, when the free-riding effect is large, OS has the incentive to increase the transportation rate to ensure its maximum profits. From the point of view of traffic volume, MT’s improvement of service level in any mode expands the potential transportation demand in the market, and the improvement of its own service level attracts more shippers to choose to buy, thus squeezing the traffic volume of OS and leading to the decline of the carrier volume of OS. Although the market expansion effect has different impacts on the two companies, the total traffic volume of the rail-water multimodal transportation market rises, which indicates that the increased traffic volume of MT is greater than the decreased traffic volume of OS.

The market expansion effect brought by the service effort of MT has a positive impact on the market volume and rate of MT in the four operational strategies, so what impact does it have on the profits of each carrier in the rail-water multimodal transport system? Next, we explore the influence of market expansion effect on profits through Table 2.

Table 2.

The effect of market expansion on income

	Cr	Cl	Dr	Dl
$\frac{\partial {\pi }_R^{\ast }}{\partial \theta }$	$+$	$+$	$+$	$+$
$\frac{\partial {\pi }_{\mathit{os}}^{\ast }}{\partial \theta }$	$\theta >\frac{-1-8\lambda -9{\lambda }^2+30{\lambda }^3}{-10-84\lambda -170{\lambda }^2+32{\lambda }^3},+$	$\theta >\frac{85+12\lambda -36{\lambda }^2+96{\lambda }^3-32{\lambda }^4}{1050-80\lambda +112{\lambda }^2-64{\lambda }^3+32{\lambda }^4},+$	$+$	$+$
$\frac{\partial {\pi }_M^{\ast }}{\partial \theta }$	$+$	$+$	$+$	$+$
$\frac{\partial {\pi }_{\prod }^{\ast }}{\partial \theta }$	$+$	$+$	$+$	$+$

Corollary 3

The market expansion effect can improve the profits of each carrier and the total profits of the system in the rail-water multimodal transport system.

Market expansion effect has a positive impact on rate and profits, which shows that the strategy of MT to improve service level, boost market demand and then increase profits is feasible. By comparison, it is found that the market expansion effect under the decentralized procurement mode can improve the income of all participants. This is because MT purchases from OS and R, respectively. At the same time, the market expansion effect improves the total volume and rate and indirectly improves the profits of upstream carriers. Combined with the volume analysis of corollary 2, although MT occupies the market share of OS under the centralized procurement mode, the transport business of MT still depends on OS to provide. Even if the market expansion effect reduces the market competition between MT and OS, the shipping volume of OS still decreases with the market expansion effect. Still, its profit will increase instead of decrease. When the market expansion effect is small, it means that OS and MT have to compete for limited traffic demand. MT enhances its degree of service effort and increases its competitiveness, crowding out the market share and resulting in a decline in the profits of liner companies. Although free-riding behavior caused by the service effort exists simultaneously, the income contribution brought by the market expansion effect far exceeds the service cost and other expenses, so the income of the rail-water multimodal transport system will increase accordingly.

The impact of free-riding effect

Through the analysis, it is found that the market expansion effect has a positive impact on the rate and volume increase. However, the service investment of MT not only brings the market expansion effect but also has a spillover effect, which generates the free-riding behavior of OS. Next, we discuss the influence of the free-riding effect on rates and calculate the derivatives of rates and service effort on free-riding effect. The results are shown in Table 3. $\left(+,-\right)$ in Table 3 indicates that within a specific range, the profit increases (or decreases) with the increase of $\lambda$.

Table 3.

The effect of free-riding behaviour

	Cr	Cl	Dr	Dl
$\frac{\partial t^{\ast }}{\partial \lambda }$	$-$	$-$	$0<\theta <\frac{11-16\lambda -28{\lambda }^2}{-74+40\lambda +8{\lambda }^2},-$	$-$
$\frac{\partial w_{\mathit{os}}^{\ast }}{\partial \lambda }$	$-$	$-$	$\frac{1}{2}\left(-1+\sqrt{6}\right)<\lambda <1,+$	$\frac{1}{6}\left(-68+19\sqrt{14}\right)<\lambda <1,-$
$\frac{\partial w_M^{\ast }}{\partial \lambda }$	Null	Null	$-$	$-$
$\frac{\partial v^{\ast }}{\partial \lambda }$	$-$	$-$	$\frac{1}{2}\left(-1+\sqrt{6}\right)<\lambda <1,+$	$-$
$\frac{\partial {\pi }_{\mathit{os}}^{\ast }}{\partial \lambda }$	$+,-$	$+,-$	$+,-$	$+,-$
$\frac{\partial {\pi }_M^{\ast }}{\partial \lambda }$	$+,-$	$-$	$+,-$	$-$

Under the centralized procurement mode, MT only buys rail-water multimodal transportation services from OS, thus increasing the market competition between OS and MT. Moreover, because the service costs are all borne by MT, when the free-riding effect is large, MT will reduce the degree of service effort to alleviate the market competition caused by the free-riding effect. However, when the free-riding effect is large $\left(\frac{1}{2}\left(-1+\sqrt{6}\right)<\lambda <1\right)$, the degree of service effort under the decentralized procurement mode increases with the free-riding effect, which is different from that under the centralized procurement mode. This is because MT can offset the competitive threat caused by the free-riding effect through the positive impact of the service under the decentralized procurement mode. In other words, MT can attract more shippers to rail-water multimodal transportation services by improving the service. Besides, MT has more bargaining power in negotiations R and OS after the increase of traffic volume. It also achieves the goal of increasing, resulting in a decline in the profits of OS by compressing the rates.

Corollary 4.

MT can counter the negative impact of the free-riding effect with ex-post services.

From the comparison in Table 3, it is found that when MT adopts the ex-post service strategy, its income can increase with the increase of the free-riding effect. This is because the ex-post service strategy reduces the chance of OS to free ride, i.e., the shipper enters the market immediately after MT publishes its degree of service effort. However, when MT chooses the advance service strategy, the profits decrease with the increase of the free-riding effect. It indicates that when MT needs to alleviate the spillover effect, it can make response plans through different service strategies. In a significant departure from the traditional findings, it is believed that free-riding behavior is beneficial to oneself, but in strategy Dr, the free-riding behavior of OS is not necessarily a positive influence on itself. This is because when the market expansion effect is large, it means that the service level of MT is higher. Although OS can obtain the benefits of the free-riding effect, more shippers will choose MT, which will drive increased profits for MT.

Strategy preference and conflict analysis

In the preliminary analysis, we discuss the service effort of MT and the influence of market expansion effect and free-riding effect brought by service effort on rates and profits. We find that the market expansion effect has a generally positive impact on the rate and profits. However, the spillover effect of services leads to the free-riding behavior of OS, and its impact is controversial in academic circles. The traditional view is that the free-riding behavior can intensify the market competition and reduce the company’s profits. However, it is proved by inference that free-riding effect can help the multi-modal operator to improve its profits under certain circumstances. The above analysis focuses on the four operational strategies and discusses the impacts of the market expansion effect and free-riding effect. But it is more important to improve the efficiency and provide strategy selection for carriers. Next, this paper analyses the practices and options of various carriers in different strategies.

Market share competition comparison

OS and MT compete in the terminal market of rail-water multimodal transportation, and the market share represented by the traffic volume has an important impact on the carrier’s profit, which further affects the strategic choice of multi-modal operator. Therefore, we first analyze the traffic volume in different operational strategies in this section.

Proposition 1.

The traffic volume of MT is higher than that of OS, and the difference in traffic volume increases with the increase of market expansion effect.

In the equilibrium result of competition, let $q_{\mathit{os}}^{\ast }=q_M^{\ast }$, we can get the boundary curve of two carriers’ traffic volume and draw it as shown in Fig. 4. It can be clearly found that MT has more advantages in traffic volume competition. When the market expansion effect $\theta$ is large, MT bears more traffic volume and occupies the main market share, which is due to the linkage reaction between the service effort and market expansion effect of MT, can promote each other to increase. Corollary 2 also strongly supports this result. It is worth noting that $q_{\mathit{os}}^{\ast }<q_M^{\ast }$ is permanently established in strategy Dr, which means that MT has a competitive advantage in traffic volume in this strategy, and the scale advantage of MT under the decentralized procurement mode is obvious. Combined with corollary 2, the degree of service effort rises in tandem with the market expansion effect. MT can obtain the quantity discount of upstream carriers by virtue of the scale traffic volume brought by services. Therefore, MT is motivated to strive for more market share.

Fig. 4.

Market share comparison between MT and OS

Selection preference of multi-modal operator

Multi-modal operator invades the rail-water multimodal market and ensures its competitive advantages through the choice of service strategy and procurement mode. We compare and select according to the balanced income of the multi-modal operator in four operational strategies. The value of service strategies under centralized and decentralized modes was discussed, and it was found that MT could maximize its profits by adopting an ex-post service strategy. This section continues to discuss the value of MT’s strategy selection. From Fig. 5, it is found that the procurement mode of MT is mainly affected by the market expansion effect. MT tends to choose the decentralized procurement mode when the market expansion effect is small. On the contrary, when the market expansion effect is large, it should choose the centralized procurement mode. This is because greater market expansion means that more shippers will enter the market. At this time, MT will invest more resources to ensure the high-quality transportation service and then choose to purchase the integrated transportation service of OS. When the market expansion effect is small, MT will adopt the decentralized procurement mode, enhancing its bargaining power and maintaining a particular profit margin.

Fig. 5.

Multi-modal operator’s profit comparison under different procurement modes

The above research discussed the decision-making of multi-modal operator after invading the market from two perspectives of service strategy and purchasing mode, but the choice of operational strategy should be compared from two perspectives. Therefore, we further discuss the choice of operational strategy of MT under the constraint of profit maximization.

Proposition 2.

Under the joint decision on service strategy and procurement mode, MT should choose strategy Dr or Cr, and the service effort can significantly improve the profit of MT.

Through analysis, we find that ${\pi }_M^{Dl\ast }-{\pi }_M^{Dr\ast }<0$ and ${\pi }_M^{Cl\ast }-{\pi }_M^{Cr\ast }<0$. Let ${\pi }_M^{Dr\ast }={\pi }_M^{Cr\ast }$, we can draw Fig. 6. The choice of procurement mode by MT depends on the size of the market expansion effect. When the market expansion effect is small, MT chooses strategy Dr, because there is less new demand from shippers. On the one hand, MT can attract more traffic volume by providing high-quality rail-water multimodal transportation services. On the other hand, MT can integrate railway and water transportation to enhance its profit margin. When the market expansion effect is large, MT should choose strategy Cr. After more shippers enter the market, MT still needs to provide high-quality services, and the cost of integrating water transportation and railway transportation by itself is higher. MT’s purchase of integrated multimodal services of OS can not only ensure that MT focuses on service competition but also negotiate with OS through large-scale transportation volume. Therefore, when there is a great new demand in the market, MT should consider purchasing integrated multimodal services of OS.

Fig. 6.

Strategy selection of MT

This paper assumes that service effort has an important impact on MT’s competition in the market, which not only improves the potential transportation demand but also brings free-riding behavior of OS. So, what is the impact of service effort on MT’s profit? The analysis shows that when $\theta >\frac{-1-5\lambda }{-2-14\lambda +4{\lambda }^2}$, MT has the highest degree of service effort in strategy Cr,While when $\theta <\frac{-1-5\lambda }{-2-14\lambda +4{\lambda }^2}$, the highest degree of service effort is in strategy Dr. Interestingly, the graphs of the highest degree of service effort, the highest efficiency, and the highest revenue are consistent, which shows that service effort has an important impact on the profit of MT. This is because the ex-post service strategy adopted by MT can not only cope with the profit decline caused by the free-riding behavior of OS. Besides, it can also make response measures from the perspective of service after more fully collecting competitors' information, market expectations, and so on.

Analysis of coping strategies of railway company and liner company

MT has the initiative in the market and can independently decide the operational strategy, so what are the changes in the profits of R and OS in different operational strategies? Next, we compare the income of three carriers in the rail-water multimodal transport system.

Occupying the first-mover advantage in the market is conducive to achieving the maximum overall system profit. By calculating the income of the three carriers under different operational strategies, we find that ${\pi }_M^{\ast }<{\pi }_{\mathit{os}}^{\ast }<{\pi }_R^{\ast }$ is always established, which indicates that the first-mover advantage is beneficial to the improvement of benefits. R and OS first determine the price of rail-water multimodal services purchased by MT. As a downstream company, MT can only give up part of its profit to meet the shipper’s demand, which results in higher profit for railway company and liner company. The order of profit of each carrier remains unchanged under different procurement modes, which indicates that the sequence of the event plays a decisive role in companies’ profit. Railway company and liner company ensure their maximum profit in the rail-water multimodal transport system by their first-mover advantage. Although other carriers cannot make decisions when MT chooses the operational strategy, MT can design its own coping strategies to maximize its profit. Therefore, we compare the maximum profit of each carrier under each of the four operating strategies to obtain Fig. 7 and Table 4.

Fig. 7.

Analysis of the consistency between system efficiency and carrier profit maximization

Table 4.

Comparison between efficiency and profit preference

	Optimal system efficiency	MT prefers	OS prefers	R prefers
Region 1	Dl	Cr	Cr	Dr
Region 2	Dl	Dr	Dr	Dr
Region 3	Dr	Dr	Dr	Dr
Region 4	Cl	Dr	Dr	Dr
Region 5	Cl	Dr	Cl	Dr
Region 6	Cl	Dr	Cr	Dr

Proposition 3.

Different carriers in the rail-water multimodal transport system can achieve win–win benefits.

Comparing the decision-making preferences of various carriers in the rail-water multimodal transport system, when MT chooses strategy Dr, three carriers can achieve a win–win situation and maximize their profits. This is because MT ensures the maximum of its own profit and traffic volume through the decision-making of ex-post service strategy. Since the separate purchase of water transportation services and railway transportation services improves the profit of OS and R, it realizes Pareto optimality in strategy Dr. When MT choose strategy Cr, there is a win–win situation between MT and OS, which indicates that OS should keep in line with the operational strategy of MT to maximize its own profits.

Railway company gains the most in strategy Dr, so it is obvious that R and MT can achieve a win–win state and even achieve Pareto optimization. Under the decentralized procurement mode, R has realized dual-channel sales, and the railway transportation services for OS and MT are single-source procurement. It means that railway company is in the upstream monopoly state, which is also similar to the current situation of China’s railway freight. Railway transportation is regulated and managed by China Railway. In addition, the analysis of $w_M^{Dr\ast }-w_{\mathit{os}}^{Dr\ast }$ shows that when the market expansion effect is large, the railway transportation rate ordered by MT is also large. Combined with Fig. 4, we can find that the traffic volume of MT occupies most of the market share. The railway company has achieved higher profits by selling railway transportation services to MT. Not only is the freight rate sold to MT higher than that sold to OS, but also the traffic volume sold to MT is greater than that sold to OS. Therefore, driven by the increase in traffic volume and rate, R prefers strategy Dr. However, when the market expansion effect is large, if MT chooses strategy Cr, the profit of the railway company will decline. Combined with the analysis of corollary 3, to deal with the threat caused by the decline of profits, R can appropriately increase $w_{\mathit{os}}^{\ast }$ and squeeze the profit space of downstream liner company through the price increase.

Analysis of system efficiency and social welfare

In the previous parts of our analysis, we discussed the selection preference of operational strategy and the value of service effort of MT, but does the market entry of MT help to improve the efficiency of the rail-water multimodal transport system? In this section, we analyze the multimodal operator's strategic preference and efficiency choice from the perspective of system efficiency. Based on the overall optimal income of the system, the optimal profit ${\pi }_Z$ can be obtained by the centralized decision-making of the rail-water multimodal transport system, and $\frac{{\pi }_{\prod }^{}}{{\pi }_Z}$ is defined as system efficiency. Figure 7 can be obtained by analyzing the consistency between the highest system efficiency and the highest profit of MT.

Proposition 4.

MT service efforts may harm system efficiency.

By comparison, it is found that the goal of optimal system efficiency and maximum profit is only to achieve a “win–win” in region 3. The free-riding effect in this region is high, but the market expansion effect is small. This is because MT’s selection of strategy Dr can mitigate the negative impacts caused by the free-riding effect; on the other hand, it can improve MT’s profit through the decentralized procurement mode. For the rail-water multimodal transport system, it can not only alleviate the market competition, but also improve the income of each carrier, to improve efficiency. But There is a deviation between the profit maximization of MT and the decision of the optimal system efficiency, which means that when MT chooses the operational strategy with the profit maximization, it will reduce the overall efficiency of the system. It also suggests that there may be over-investment in MT's service efforts, and while carrier revenues are optimized, intense terminal competition leads to a less efficient system.

We analyze the relationship between the strategic choice of MT and the maximization of social welfare. Shippers choose to purchase transportation services from OS or MT in the competitive market, and the expression of consumer surplus is:

$$U=\left(q_{\mathit{os}}^{}+q_M^{}\right)-\frac{1}{2}\left(q_{\mathit{os}}^2+q_M^2+2q_{\mathit{os}}^{}{q}_M^{}\right)-\left(p_{\mathit{os}}^{}{q}_{\mathit{os}}^{}+p_M^{}{q}_M^{}\right)$$ 17

According to the total profit of the system in different operational strategies, social welfare can be calculated (As shown in Fig. 8). In strategy Dr, the consumer surplus is the smallest, but social welfare can be maximized. It indicates that relevant have grabbed more consumer surplus to achieve their goal of maximizing their income. Moreover, social welfare is significant when MT chooses the ex-post service strategy. Compared with the previous consumer surplus, we find that the consumer surplus in the ex-post service strategy is small. This finding shows that all carriers' profits contribute more to social welfare and that MT can extract more consumer surplus through the ex-post service strategy.

Fig. 8.

Optimal consumer surplus and social welfare

Extension

Uncertainty of potential market demand

The potential market demand of this study is expanded from standardization to 1 to a, which is similar to the hypothesis of Li et al. (2021). The inverse demand functions are shown in (1) and (2). The change of potential demand does not affect the structure of the profit function, but only the inverse demand function. Therefore, the profit function is the same as the basic model, and the extended solution process is similar to the basic model and can be referred to the detailed process in appendix.

Proposition 5.

The potential market demand is extended to the uncertain situation, and the solution conclusion is basically consistent with the main model, indicating that the analysis of the basic model is robust.

Interestingly, the influence of market expansion effect on the change of rate, traffic volume, and balanced service effort in the expansion model is completely consistent with that in the main model. It directly shows that standardizing the potential market demand in the main model has no influence on the discussion of management enlightenment.

It is worth noting that although the potential market demand expands to an uncertain situation and the market scale may be enlarged, free-riding behavior is still a “double-edged sword,” which still poses a threat to the income of MT and OS. However, MT can deal with the negative impact of the free-riding effect through ex-post service strategy strategies, which is similarly explained in the main model. Surprisingly, when $\frac{\partial v}{\partial \lambda }>0$, $\theta <\frac{1+10\lambda +26{\lambda }^2+10{\lambda }^3+25{\lambda }^4}{18+156\lambda +300{\lambda }^2-180{\lambda }^3+10{\lambda }^4}$ can be obtained, which indicates that the optimal degree of service effort increases with the increase of free-riding effect in a certain range in the extended model. On the contrary, in the main model, the equilibrium service level of MT with the increase of free-riding effect. The difference in the conclusion is that the expansion of potential market demand can alleviate the market competition between OS and MT. Still, the uncertain market demand makes MT achieve the goal of increasing traffic volume by improving services.

By comparing the profits of each carrier, we find that ${\pi }_M^{}<{\pi }_{\mathit{os}}^{}<{\pi }_R^{}$ is still established even when the market demand is uncertain, which strongly proves the role of first-mover advantage. Comparing the strategic preferences of various carriers, we conclude that the win–win strategy is still strategy Dr. The explanation has been explained in Proposition 3, not tired in words here.

The impact of differentiated services

In the basic model, it is assumed that OS and MT provide homogeneous services. However, due to the difference in service providers’ business scope or market reputation in practice, the services are heterogeneous, and shippers also consider the service differences when making decisions. Therefore, we design a competition model under differentiated services, and the inverse demand function is:

$$p_{\mathit{os}}^{}=1+\lambda v-q_{\mathit{os}}^{}-b{q}_M^{}$$ 18

$$p_M^{}=1+\theta +v-b{q}_{\mathit{os}}^{}-q_M^{}$$ 19

Among them, $b\left(0<b<1\right)$ represents the cross-effect coefficient of OS and MT, which can also be understood as the differentiated service competition level between the two companies. Similar assumptions are adopted for competition between different carriers. Due to the complexity of the results, we show the conclusion through numerical simulation. To achieve generalization, we set $\theta =3$ and $b=0.5$ (see Fig. 9).

Fig. 9.

Profits of each carrier

Proposition 6.

Homogenized services of MT and OS intensify competition and weaken the coping role of ex-post service strategy on the free-riding effect.

Firstly, we analyze the strategic preference of each carrier in the differentiated strategy. MT and OS have the biggest profits in strategy Cr, while railway company has higher profits in strategy Dr, which shows that when there is differentiated service competition in the downstream of the rail-water multimodal transport system, the profits of the upstream railway company will be damaged, and the two downstream carriers have the motivation to form collusion to improve the profits jointly. Different from the main model, when the differentiation service competition level is moderate, the profit of multi-modal operator decreases with the increase of free-riding effect under the ex-post service strategy. Comparing the profit of each carrier, R’s profit is still the highest within the system. Secondly, we discuss the influence of MT’s decision to adopt strategy Cr on other carriers and further analyze the coping strategies of OS and R (see Fig. 10). When the differentiated service competition level is small ($b=0.2$), the profit of each carrier increases with the increase of free-riding effect, which is consistent with the conclusion of the main model, that is, ex-post service strategy can cope with the free-riding effect caused by service effort to a certain extent.

Fig. 10.

Changes in profits

The influence of free-riding behavior

In the basic model analysis, we assume that MT provides rail-water multimodal transportation service with the free-riding effect, which makes OS motivated to free ride. However, as MT enters the market later, it is difficult for MT to provide services that have a free-riding effect on OS. In other words, when the market influence of MT is low, the free-riding effect can be ignored. Due to the complexity of the results, we show the conclusion through numerical simulation. To achieve generalization, we set $\lambda =\left\{0,0.2,0.5,0.8\right\}$, and limit $\theta$ according to the value range. Figure 11 shows the profit changes of each carrier in the rail-water multimodal system under different operational strategies.

Fig. 11.

Changes in profit with free-riding effect

Proposition 7.

The analysis conclusion of the basic model is robust, and the ex-post service strategy can help MT cope with the free-riding behavior, which has different impacts on carriers.

From the comparison in Fig. 11, it is found that MT can obtain greater benefits by choosing the ex-post service strategy. Due to the heterogeneity of the response effect of different procurement modes on free-riding behavior, ${\pi }_M^{}$ increases with the increase of free-riding behavior in strategy Dr, which is consistent with the analysis conclusion of the basic model.

R and OS can also get better profits under the ex-post service strategy, which can achieve a win–win state. It is worth noting that in strategy Dr, ${\pi }_R^{}$ and ${\pi }_{\mathit{os}}^{}$ has a “U” relationship with free-riding effect. When the spillover degree is zero, R and OS can get the highest profit, which decreases with the increase of the free-riding effect. When the free-riding effect is high, their profit increases with the increase of the free-riding effect. This finding is similar to the main model. On the surface, the free-riding behavior of OS is conducive to the improvement of its own sales, but it actually damages its own profits to a certain extent.

Analysis of environmental impact

This section discusses the impact of environmental costs on social welfare. The carbon intensity per unit of cargo transported through the rail-water multimodal transport system is $\tau$ (He et al., 2019). Therefore, the environmental cost is $S=\phi \tau \left(q_{\mathit{os}}^{}+q_M^{}\right)$ (Yenipazarli, 2016). The $\phi$ translates a unit of emissions into a monetary unit (Cachon, 2014). Let $S{W}_e^{}=SW-S$ denotes sustainable social welfare. The relationship between $S{W}_e^{}$ and key factors is discussed by numerical simulation. For the sake of illustration, we set it to $\tau =1$ and assume that the carbon emission intensity increases linearly (Levi & Nault, 2004). The findings of this study can easily be extended to the case of $\tau >0$. Figure 12 illustrates this case. We find that $S{W}_e^{\mathit{Dl}}$ is minimal. Comparison with Fig. 8 reveals that the range at the maximum of $S{W}^{\mathit{Dl}}$ is extremely small, reducing the sustainable social welfare due to the consideration of the negative environmental costs.

Fig. 12.

Sustainable social welfare when $\phi =0.4$

For government management, the optimal choice is the Cr strategy, which achieves maximum sustainable social welfare. Moreover, $S{W}^{\mathit{Cr}}$ also achieves the maximum, indicating the robustness of the model results. While MT and OS can realize the maximum profit under the Cr strategy, the profit of R is compressed. Therefore, policy development should consider targeting carriers for subsidies to achieve better emission reduction targets. It is alarming to note that $S{W}_e^{\mathit{Cr}}$ and $S{W}_e^{\mathit{Dr}}$ decrease as $\theta$ increases. Interestingly, the effect of $\theta$ on $S{W}_e^{}$ is heterogeneous. It shows that emission reduction policies cannot only be considered in terms of the carriage volume but should also focus on the interaction between the carrier's operating strategy and the market size.

Conclusion

Rail-water multimodal transportation has been widely used in transportation organizations, and it is of great significance for the transportation industry to realize emission peak. Based on this background, this paper focuses on the operational strategies of the multi-modal operator after it invades the market and analyzes the direct effect and indirect effects caused by the multi-modal operator’s service effort. We concentrate on the impact of service effort on system efficiency and analyze the interaction mechanism between the market expansion effect and the free-riding effect triggered by service effort. This paper summarizes four operational strategies of the rail-water multimodal transport system according to the different service strategies and procurement modes. We construct a multi-agent co-opetition game model including a railway company, a liner company, and a multi-modal operator to describe the multi-stage decision-making problem of each carrier in the rail-water multimodal transport system. We obtain the equilibrium decision-making of each carrier with the constraint of profit maximization. Subsequently, conflicts between the choices of each carrier in the rail-water multimodal transport system is analyzed from the perspective of maximizing the company’s profit, and the strategic preference is further discussed from the perspective of optimal system efficiency. Comparisons are made under the two dimensions of profit and efficiency to achieve a win–win situation. The analysis shows that: (1) multi-modal operators choose ex-post service strategy, which is beneficial to themselves. Ex-post service strategy can achieve their optimal service input efficiency and deal with the free-riding behavior of liner companies. Counter-intuitively, the liner companies’ free-riding behavior may harm their own revenue. (2) The choice of procurement mode of multi-modal operators is highly related to the market expansion effect brought by their service effort. When the market expansion effect is small, multi-modal operators should choose the centralized procurement mode. On the contrary, choosing the decentralized procurement mode can ensure profit maximization. (3) Although the service effort of multi-modal operators forms a competitive threat to liner companies, it can still enhance the profit of each carrier in the rail-water multimodal transport system to achieve the Pareto optimality of “win–win” for all three parties. (4) The service effort of multi-modal operators can not only improve their own operational efficiency, but also optimize the efficiency of the rail-water multimodal transport system. However, wariness should be exercised about the negative effects of over-investment. Moreover, the strategy choice under the constraint of profit maximization is consistent with the strategy choice of efficiency optimization (when multimodal companies choose strategy Dr), both the profits of each carrier and the system efficiency can achieve a win–win situation. However, only in strategy Dr, the efficiency of each carrier is consistent with the optimal efficiency under the centralized procurement mode.

Therefore, this paper has the following implications for the management decision of rail-water multimodal transportation: (1) From the perspective of industry governance, the decision-making departments should formulate support policies for professional multi-modal operators, improve the income and social operation efficiency of the rail-water multimodal transport system by incubating new multi-modal operators, and cultivate the operators of the rail-water multimodal transportation, which is consistent with the “3—year action plan” issued by the Ministry of Transport of the People’s Republic of China. In comparison with the development of multimodal transport in the United States, the cultivation of leading operators has a positive effect on the development of the market. (2) The government should pay attention to the late entrants in the management of rail-water multimodal transport systems and guide the carriers to cooperate from the upstream and downstream perspectives to achieve the best system efficiency by formulating a package of guiding policies. Not only that, but policy guidance can also lead to better emission reduction targets. (3) From the decision-making of multi-modal operators, the procurement mode should be determined according to the market transportation demand, and the uncertainty brought by the fluctuation of traffic volume shall be handled flexibly. While service effort incurs costs, choosing the ex-post service strategy can reduce the negative impact of competitors’ free-riding behavior and ensure that their service investment is accurately transmitted to shippers. Therefore, the direction of MT's service efforts is Digital critical empowerment in conjunction with industry development could be an essential driver. Shipper 360™ by J.B. Hunt simplifies business processes and enables consignments to be shipped via smartphone. The service experience for shippers is greatly enhanced. (4) From the perspective of railway companies and liner companies as market followers, such carriers should focus on the decision-making and selection of the multi-modal operators, dynamically track the purchasing modes of the multi-modal operators, timely update their sales strategies and ensure consistency with the multi-modal operators, so as to achieve the goal of maximizing profit.

This paper still has the following expansion directions. First, this paper assumes that the shipper can only book space from a liner company or a multi-modal operator. There may be different management implications if competition among multiple liner companies or multimodal operators is considered. Secondly, this paper assumes that the shipper makes bookings according to the freight rate. In fact, the carrier’s social responsibility and brand effect will have a comprehensive impact on the shipper’s decision. Therefore, new studies can be carried out from the perspective of the utility of the shipper’s purchase of rail-water multimodal transportation services.

Appendix

The appendices consist of two parts: Appendix A lists all the notations used in the paper, and Appendix B provides the proofs for all the results in the main paper.

A Model notations

See Table 5.

Table 5.

Summary of Model Notations

Notation	Explanation
$t_{}^i$	OS’s transport service rates under $i$ procurement mode, $i\in \left\{C,D\right\}$,where C means centralized procurement mode, D means decentralized procurement mode
$w_k^i$	The rate at which the railway company offers carriage to carrier $k$ under $i$ procurement mode, $k\in \left\{OS,M\right\}$, where OS means liner company, M means multi-modal operator
$p_k^i$	Carrier $k$’s rates for transport services under $i$ procurement mode
$q_k^i$	Volume of Carrier $k$ under $i$ procurement mode
$a$	The potential market traffic volume of OS
$v$	The degree of service effort of the MT
$\lambda$	The positive spillover degree of service effort to OS, $0<\lambda <1$
$\theta$	The MT’s market expansion effect
${\pi }_{}^i$	Profits for each member of the rail-water multimodal transport system under $i$ procurement mode
${\pi }_{\prod }^{}$	The total profits in the rail-water multimodal transport system under $i$ procurement mode.
$E_M$	The service input efficiency of MT
$E_{}^{}$	The influence of service input of MT on the system
${\pi }_Z$	The optimal profit of centralized decision-making of railway and water transport system
$S{W}^i$	Social welfare under $i$ procurement mode
$S{W}_e$	Sustainable social welfare
$U^i$	Consumer surplus under $i$ procurement mode

B Proofs of Results in section 4 and Section 5

B1 Equilibrium results and constraints

1. Strategy Cr.

According to the order of backward induction, the equilibrium in phase 3 is discussed first. The Hesse matrix of ${\pi }_M^{\mathit{Cr}}$ with respect to $q_M^{\mathit{Cr}}$ and $v$ is:

$$H=\left(\left(\begin{array}{cc}\frac{{\partial }^2{\pi }_M^{\mathit{Cr}}}{\partial {(q_M^{\mathit{Cr}})}^2}& \frac{{\partial }^2{\pi }_M^{\mathit{Cr}}}{\partial q_M^{\mathit{Cr}}\partial v}\\ \frac{{\partial }^2{\pi }_M^{\mathit{Cr}}}{\partial v\partial q_M^{\mathit{Cr}}}& \frac{{\partial }^2{\pi }_M^{\mathit{Cr}}}{\partial v_{}^2}\end{array}\right)\right)=\left(\left(\begin{array}{cc}-2& 1\\ 1& -1\end{array}\right)\right)$$ 1.1

According to the calculation results of the Hesse matrix, ${\pi }_M^{\mathit{Cr}}$ is a strictly concave function about $q_M^{\mathit{Cr}}$ and $v$. Therefore, the equilibrium traffic volume and the degree of service effort of MT satisfy the following conditions:

$$-v+q_M^{\mathit{Cr}}=0$$ 1.2

$$1-q_{\mathit{os}}^{\mathit{Cr}}-2q_M^{\mathit{Cr}}-t^{\mathit{Cr}}+v+\theta =0$$ 1.3

The maximization condition of ${\pi }_{\mathit{os}}^{\mathit{Cr}}$ in phase 2 with respect to $q_{\mathit{os}}^C$ is combined with the above equation, and it can be calculated that:

$$q_M^{\mathit{Cr}}=v=\frac{1-2t^{\mathit{Cr}}+w_{\mathit{os}}^{\mathit{Cr}}+2\theta }{1+\lambda }$$ 1.4

$$q_{\mathit{os}}^{\mathit{Cr}}=\frac{t_{}^{\mathit{Cr}}-w_{\mathit{os}}^{\mathit{Cr}}+\theta \left(\lambda -1\right)+\lambda -t_{}^{\mathit{Cr}}\lambda }{1+\lambda }$$ 1.5

Bringing $q_M^{\mathit{Cr}}$ and $q_{\mathit{os}}^{\mathit{Cr}}$ into ${\pi }_{\mathit{os}}^{\mathit{Cr}}$, one can get $\frac{{\partial }^2{\pi }_{\mathit{os}}^{\mathit{Cr}}}{\partial {(t^{\mathit{Cr}})}^2}=\frac{2\left[-1+\lambda \left(-4+\lambda \right)\right]}{{\left(1+\lambda \right)}^2}<0$. By solving $\frac{\partial {\pi }_{\mathit{os}}^{\mathit{Cr}}}{\partial t^{\mathit{Cr}}}=0$, one can get $t_{}^{\mathit{Cr}}=\frac{1+w_{\mathit{os}}^{\mathit{Cr}}+\lambda \left(3+5w_{\mathit{os}}^{\mathit{Cr}}+6\theta \right)-2{\lambda }^2\left(1+\theta \right)}{2\left[1-\left(-4+\lambda \right)\lambda \right]}$.

Discuss the equilibrium traffic volume of R in phase 1, Bring $q_M^{\mathit{Cr}}$, $q_{\mathit{os}}^{\mathit{Cr}}$, and $t_{}^{\mathit{Cr}}$ into the profit function of R, and let $\frac{\partial {\pi }_R^{\mathit{Cr}}}{\partial w_{\mathit{os}}^{\mathit{Cr}}}=0$, the equilibrium freight rate of R is: $w_{\mathit{os}}^{Cr\ast }=\frac{1+5\lambda +2\theta \left(1+\lambda \right)}{2+10\lambda }$.

It can be concluded that the equilibrium traffic volume, the degree of service effort, and profit of MT are:

$$q_M^{Cr\ast }=v^{\ast }=\frac{\theta }{1+5\lambda }+\frac{2\theta +\lambda }{2-2\lambda \left(-4+\lambda \right)},{\pi }_M^{Cr\ast }=\frac{{\left[4\theta +\lambda -2\theta \lambda \left(-9+\lambda \right)+5{\lambda }^2\right]}^2}{8{\left[1+\lambda \left(9+19\lambda -5{\lambda }^2\right)\right]}^2}.$$

The equilibrium traffic volume, rate, and profit of OS are:

$$q_{\mathit{os}}^{Cr\ast }=\frac{1+2\theta \left(\lambda -1\right)+3\lambda }{4-4\lambda \left(-4+\lambda \right)}-\frac{\theta }{1+5\lambda },t_{}^{Cr\ast }=1+\theta +\frac{1+5\lambda +2\theta \left(1+\lambda \right)}{-4+4\lambda \left(-4+\lambda \right)}$$

$${\pi }_{\mathit{os}}^{Cr\ast }=\frac{1}{80}\left[\frac{16{\theta }^2}{{\left(1+5\lambda \right)}^2}+\frac{4\theta \left(-5+16\theta \right)}{1+5\lambda }-\frac{5\left[1+4\lambda +4\theta \left(\theta +\lambda \right)\right]}{-1+\lambda \left(-4+\lambda \right)}\right]$$

The equilibrium rate and profit of R are:

$$w_R^{Cr\ast }=\frac{1+5\lambda +2\theta \left(1+\lambda \right)}{2+10\lambda },{\pi }_R^{Cr\ast }=\frac{{\left[-1-5\lambda -2\theta \left(1+\lambda \right)\right]}^2}{8\left(1+5\lambda \right)\left[-1+\lambda \left(-4+\lambda \right)\right]}$$

The total profit of the system under strategy Cr is:

$${\pi }_{\prod }^{Cr\ast }=\frac{\begin{array}{c}-{\left(1+5\lambda \right)}^2\left\{\left[-3+\lambda \left(-26-55\lambda +14\lambda {}^2\right)+4\theta +\lambda \left(22+154\lambda +398{\lambda }^2+249{\lambda }^3-80{\lambda }^4\right)\right]\right\}\\ +4{\theta }^2\left\{15+\lambda \left[156+\lambda \left(478+330\lambda -129{\lambda }^2+6{\lambda }^3\right)\right]\right\}\\ \end{array}}{16{\left(1+5\lambda \right)}^2{\left[-1+\lambda \left(-4+\lambda \right)\right]}^2}$$

• (2)Strategy Cl.

According to the order of backward induction, the equilibrium in phase 4 is discussed first. Calculate the first derivatives of ${\pi }_M^{\mathit{Cl}}$ and ${\pi }_{\mathit{os}}^{\mathit{Cl}}$ with respect to $q_M^{\mathit{Cl}}$ and $q_{\mathit{os}}^{\mathit{Cl}}$, respectively, and one can get:

$$\frac{\partial {\pi }_M^{\mathit{Cl}}}{\partial q_M^{\mathit{Cl}}}=1-q_{\mathit{os}}^{\mathit{Cl}}-2q_M^{\mathit{Cl}}-t_{}^{\mathit{Cl}}+v+\theta$$ 1.6

$$\frac{\partial {\pi }_{\mathit{os}}^{\mathit{Cl}}}{\partial q_{\mathit{os}}^{\mathit{Cl}}}=1-2q_{\mathit{os}}^{\mathit{Cl}}-q_M^{\mathit{Cl}}-w_{\mathit{os}}^{\mathit{Cl}}+v\lambda$$ 1.7

By solving the above two equations simultaneously, one can obtain:

$$q_M^{\mathit{Cl}}=\frac{1}{3}\left(1-2t_{}^{\mathit{Cl}}+2v+w_{\mathit{os}}^{\mathit{Cl}}+2\theta -v\lambda \right)$$ 1.8

$$q_{\mathit{os}}^{\mathit{Cl}}=\frac{1}{3}\left(1+t_{}^{\mathit{Cl}}-v-2w_{\mathit{os}}^{\mathit{Cl}}-\theta +2v\lambda \right)$$ 1.9

Bringing $q_M^{\mathit{Cl}}$ and $q_{\mathit{os}}^{\mathit{Cl}}$ into the profit function of OS in phase 3, solving $\frac{\partial {\pi }_{\mathit{os}}^{\mathit{Cl}}}{\partial t_{}^{\mathit{Cl}}}=\frac{1}{9}\left[5-10t_{}^{\mathit{Cl}}+5w_{\mathit{os}}^{\mathit{Cl}}+4\theta +v\left(4+\lambda \right)\right]=0$, one can get the best rate of OS is:

$$t_{}^{\mathit{Cl}}=\frac{1}{10}\left(5+4v+5w_{\mathit{os}}^{\mathit{Tl}}+4\theta +\lambda v\right).$$

Substituting the optimal response obtained above in the profit function of R and calculating in phase 2 according to the profit maximization conditions of R, one can obtain:$w_{\mathit{os}}^{\mathit{Cl}}=\frac{1}{10}\left(5+2v+2\theta +3\lambda v\right)$.

As MT adopts prior decision-making, it decides the degree of service effort in phase 1. Substituting the optimal response obtained above in ${\pi }_M^{\mathit{Cl}}$ and calculating according to the profit maximization conditions of MT, one can get: $v^{\ast }=\frac{8\theta \left(\lambda -1\right)}{-17-16\lambda +8{\lambda }^2}$.

Sorting and simplification, the traffic volume, the degree of service effort, and profit of MT under the equilibrium state are:

$$q_M^{Cl\ast }=\frac{10\theta }{17-8\lambda \left(\lambda -2\right)},v_{}^{\ast }=\frac{1}{2}\left[1+\theta +\frac{\theta \left(7+8\lambda \right)}{-17+8\lambda \left(\lambda -2\right)}\right],{\pi }_M^{Cl\ast }=\frac{4{\theta }^2}{17-8\lambda \left(\lambda -2\right)}$$

The equilibrium traffic volume, rate, and profit of OS are:

$$q_{\mathit{os}}^{Cl\ast }=\frac{1}{4}\left[1+\theta +\frac{\theta \left(47+8\lambda \right)}{-17+8\lambda \left(\lambda -2\right)}\right],t_{}^{Cl\ast }=\frac{1}{4}\left[3\left(1+\theta \right)+\frac{\theta +24\theta \lambda }{-17+8\lambda \left(\lambda -2\right)}\right]$$

$${\pi }_{\mathit{os}}^{Cl\ast }=\frac{\begin{array}{c}{\left[17-8\lambda \left(\lambda -2\right)\right]}^2+4\theta \left[-17+8\lambda \left(\lambda -2\right)\right]\left[5+4\lambda \left(\lambda -1\right)\right]\\ +4{\theta }^2\left[525+8\lambda \left(\lambda -1\right)\left(5-2\lambda +2{\lambda }^2\right)\right]\\ \end{array}}{16{\left[17-8\lambda \left(\lambda -2\right)\right]}^2}$$

The equilibrium traffic volume, rate and profit of R are:

$$w_{\mathit{os}}^{Cl\ast }=\frac{1}{2}\left[1+\theta +\frac{\theta \left(7+8\lambda \right)}{-17+8\lambda \left(\lambda -2\right)}\right],{\pi }_R^{Cl\ast }=\frac{{\left[-17+8\lambda \left(\lambda -2\right)-10\theta +8\lambda \theta \left(\lambda -1\right)\right]}^2}{8{\left[17-8\lambda \left(\lambda -2\right)\right]}^2}$$

The total profit of the system under strategy Cl is:

$${\pi }_{\prod }^{Cl\ast }=\frac{\begin{array}{c}3{\left[17-8\lambda \left(\lambda -2\right)\right]}^2+4\theta \left[-17+8\lambda \left(\lambda -2\right)\right]\left[-5+12\lambda \left(\lambda -1\right)\right]\\ +4{\theta }^2\left\{847+8\lambda \left[37-15\lambda +6{\lambda }^2\left(\lambda -2\right)\right]\right\}\\ \end{array}}{16{\left[17-8\lambda \left(\lambda -2\right)\right]}^2}$$

• (3)Strategy Dr.

According to the order of backward induction, the equilibrium in phase 3 is discussed first. The Hesse matrix of ${\pi }_M^{\mathit{Dr}}$ with respect to $q_M^{\mathit{Dr}}$ and $v$ is:

$$H=\left(\left(\begin{array}{cc}\frac{{\partial }^2{\pi }_M^{\mathit{Dr}}}{\partial {(q_M^{\mathit{Dr}})}^2}& \frac{{\partial }^2{\pi }_M^{\mathit{Dr}}}{\partial q_M^{\mathit{Dr}}\partial v}\\ \frac{{\partial }^2{\pi }_M^{\mathit{Dr}}}{\partial v\partial q_M^{\mathit{Dr}}}& \frac{{\partial }^2{\pi }_M^{\mathit{Dr}}}{\partial v_{}^2}\end{array}\right)\right)=\left(\left(\begin{array}{cc}-2& 1\\ 1& -1\end{array}\right)\right)$$ 1.10

According to the calculation results of the Hesse matrix, ${\pi }_M^{\mathit{Dr}}$ is a strictly concave function about $q_M^{\mathit{Dr}}$ and $v$. Therefore, the equilibrium traffic volume and the degree of service effort of MT satisfy the following conditions:

$$-v+q_M^{\mathit{Dr}}=0$$ 1.11

$$1-q_{\mathit{os}}^{\mathit{Dr}}-2q_M^{\mathit{Dr}}-t^{\mathit{Dr}}+v-w_M^{\mathit{Dr}}+\theta =0$$ 1.12

The maximization condition of ${\pi }_{\mathit{os}}^{\mathit{Dr}}$ in phase 2 with respect to $q_{\mathit{os}}^{\mathit{Dr}}$ is combined with the above equation, and it can be calculated that:

$$q_M^{\mathit{Dr}}=v^{\mathit{Dr}}=\frac{1-2t^{\mathit{Dr}}+w_{\mathit{os}}^{\mathit{Dr}}-2w_M^{\mathit{Dr}}+2\theta }{1+\lambda }$$ 1.13

$$q_{\mathit{os}}^{\mathit{Dr}}=\frac{t_{}^{\mathit{Dr}}-w_{\mathit{os}}^{\mathit{Dr}}+w_M^{\mathit{Dr}}-\theta +\lambda -\lambda \left(t_{}^{\mathit{Dr}}+w_M^{\mathit{Dr}}-\theta \right)}{1+\lambda }$$ 1.14

Bringing $q_{\mathit{MT}}^{\mathit{Dr}}$ and $q_{\mathit{os}}^{\mathit{Dr}}$ into ${\pi }_{\mathit{os}}^{\mathit{Dr}}$, one can get $\frac{{\partial }^2{\pi }_{\mathit{os}}^{\mathit{Dr}}}{\partial {(t^{\mathit{Dr}})}^2}<0$.By solving $\frac{\partial {\pi }_{\mathit{os}}^{\mathit{Dr}}}{\partial t^{\mathit{Dr}}}=0$, one can get $t_{}^{\mathit{Dr}}=\frac{-1+w_{\mathit{os}}^{\mathit{Dr}}-3\lambda \left(1+w_{\mathit{os}}^{\mathit{Dr}}-2w_M^{\mathit{Dr}}+2\theta \right)+2{\lambda }^2\left(1-w_M^{\mathit{Dr}}+\theta \right)}{2\left[-1+\left(-4+\lambda \right)\lambda \right]}$.

Discuss the optimal sales volume of R in stage 1, and substitute $q_M^{\mathit{Dr}}$、 $q_{\mathit{os}}^{\mathit{Dr}}$ and $t_{}^{\mathit{Dr}}$ into the profit function of R. We let $\frac{\partial {\pi }_R^{\mathit{Dr}}}{\partial w_{\mathit{os}}^{\mathit{Dr}}}=0$ and $\frac{\partial {\pi }_R^{\mathit{Dr}}}{\partial w_M^{\mathit{Dr}}}=0$ to obtain the equilibrium freight price of R.

Sorting and simplification, the traffic volume, the degree of service effort, and profit of MT under the equilibrium state are:

$$q_M^{Dr\ast }=v^{\ast }=\frac{-1-4\theta -2\lambda }{-3-16\lambda +4{\lambda }^2},{\pi }_M^{Dr\ast }=\frac{{\left(1+4\theta +2\lambda \right)}^2}{2{\left(-3-16\lambda +4{\lambda }^2\right)}^2}$$

The equilibrium traffic volume, rate, and profit of OS are:

$$q_{\mathit{os}}^{Dr\ast }=\frac{3\theta -3\lambda -2\theta \lambda }{-3-16\lambda +4{\lambda }^2},t_{}^{Dr\ast }=\frac{1-2\theta +9\lambda +14\theta \lambda -4{\lambda }^2-4\theta {\lambda }^2}{6+32\lambda -8{\lambda }^2}$$

$${\pi }_{\mathit{os}}^{Dr\ast }=\frac{1+2\theta +10{\theta }^2+11\lambda +10\theta \lambda +32{\theta }^2\lambda +32{\lambda }^2+32\theta {\lambda }^2-8{\theta }^2{\lambda }^2-8{\lambda }^3-8\theta {\lambda }^3}{2{\left(-3-16\lambda +4{\lambda }^2\right)}^2}$$

The equilibrium rate and profit of R are:

$$w_{\mathit{os}}^{Dr\ast }=\frac{-2-2\theta -9\lambda +2{\lambda }^2}{-3-16\lambda +4{\lambda }^2},w_M^{Dr\ast }=\frac{3+6\theta +13\lambda +14\theta \lambda -4{\lambda }^2-4\theta {\lambda }^2}{6+32\lambda -8{\lambda }^2}$$

$${\pi }_R^{Dr\ast }=\frac{-1-2\theta -4{\theta }^2-5\lambda -4\theta \lambda }{2\left(-3-16\lambda +4{\lambda }^2\right)}$$

The total profit of the system under strategy Dr is:

$${\pi }_{\prod }^{Dr\ast }=\frac{5+2\theta \left(8+19\theta \right)+46\lambda +2\theta \lambda \left(35+48\theta \right)-8{\lambda }^2\left(1+\theta \right)\left(-14+3\theta \right)-4{\lambda }^3\left(7+6\theta \right)}{2{\left[3-4\lambda \left(-4+\lambda \right)\right]}^2}$$

• (4)Strategy Dl.

According to the order of backward induction, the equilibrium in phase 4 is discussed first. Calculate the first derivatives of ${\pi }_M^{\mathit{Dl}}$ and ${\pi }_{\mathit{os}}^{\mathit{Dl}}$ with respect to $q_M^{\mathit{Dl}}$ and $q_{\mathit{os}}^{\mathit{Dl}}$, respectively, and one can get:

$$\frac{\partial {\pi }_M^{\mathit{Dl}}}{\partial q_M^{\mathit{Dl}}}=1-q_{\mathit{os}}^{\mathit{Dl}}-2q_M^{\mathit{Dl}}-t_{}^{\mathit{Dl}}+v-w_M^{\mathit{Dl}}+\theta$$ 1.15

$$\frac{\partial {\pi }_{\mathit{os}}^{\mathit{Dl}}}{\partial q_{\mathit{os}}^{\mathit{Dl}}}=1-2q_{\mathit{os}}^{\mathit{Dl}}-q_M^{\mathit{Dl}}-w_{\mathit{os}}^{\mathit{Dl}}+v\lambda$$ 1.16

By solving the above two equations simultaneously, one can obtain:

$$q_M^{\mathit{Dl}}=\frac{1}{3}\left(1-2t_{}^{\mathit{Dl}}+2v+w_{\mathit{os}}^{\mathit{Dl}}-2w_M^{\mathit{Dl}}+2\theta -v\lambda \right)$$ 1.17

$$q_{\mathit{os}}^{\mathit{Dl}}=\frac{1}{3}\left(1+t_{}^{\mathit{Dl}}-v-2w_{\mathit{os}}^{\mathit{Dl}}+w_M^{\mathit{Dl}}-\theta +2v\lambda \right)$$ 1.18

Bringing $q_M^{\mathit{Dl}}$ and $q_{\mathit{os}}^{\mathit{Dl}}$ into the profit function of OS in phase 3, solving $\frac{\partial {\pi }_{\mathit{os}}^{\mathit{Dl}}}{\partial t_{}^{\mathit{Dl}}}==0$, one can get the best rate of OS is:

$$t_{}^{\mathit{Dl}}=\frac{1}{10}\left(5+4v-w_{\mathit{os}}^{\mathit{Dl}}-4w_M^{\mathit{Dl}}+4\theta +v\lambda \right).$$

Substituting the optimal response obtained above in the profit function of R and calculating in phase 2 according to the profit maximization conditions of R, one can obtain:$w_M^{\mathit{Dl}}=\frac{1}{38}\left(15+22v+22\theta -7v\lambda \right)$,$w_{\mathit{os}}^{\mathit{Dl}}=\frac{2}{19}\left(5+v+\theta +4v\lambda \right)$.

Sorting and simplification, the traffic volume, the degree of service effort and profit of MT under the equilibrium state are:

$$q_M^{Dl\ast }=\frac{19+76\theta }{329+6\left(8-3\lambda \right)\lambda },v_{}^{\ast }=\frac{2\left(1+4\theta \right)\left(-4+3\lambda \right)}{-329+6\lambda \left(-8+3\lambda \right)},{\pi }_M^{Dl\ast }=\frac{{\left(1+4\theta \right)}^2}{329-6\lambda \left(-8+3\lambda \right)}$$

The equilibrium traffic volume, rate and profit of OS are:

$$q_{\mathit{os}}^{Dl\ast }=\frac{-68+57\theta -2\left(7+4\theta \right)\lambda +6\left(1+\theta \right){\lambda }^2}{-329+6\lambda \left(-8+3\lambda \right)},$$

$$t_{}^{Dl\ast }=\frac{-193-114\theta -4\left(7+4\theta \right)\lambda +12\left(1+\theta \right){\lambda }^2}{-658+12\lambda \left(-8+3\lambda \right)}$$

$${\pi }_{\mathit{os}}^{Dl\ast }=\frac{\begin{array}{c}7\left(1845+38\theta \left(5+57\theta \right)\right)-4\left(7+4\theta \right)\left(-155+38\theta \right)\lambda +4\left(-367+\theta \left(-239+146\theta \right)\right){\lambda }^2\\ -48\left(1+\theta \right)\left(7+4\theta \right){\lambda }^3+72{\left(1+\theta \right)}^2{\lambda }^4\\ \end{array}}{2{\left(329+6\left(8-3\lambda \right)\lambda \right)}^2}$$

The equilibrium rate and profit of R are:

$$w_M^{Dl\ast }=\frac{-269-418\theta -4\left(7+4\theta \right)\lambda +12\left(1+\theta \right){\lambda }^2}{-658+12\lambda \left(-8+3\lambda \right)}$$

$$w_{\mathit{os}}^{Dl\ast }=\frac{2\left(-87-19\theta -2\left(7+4\theta \right)\lambda +6\left(1+\theta \right){\lambda }^2\right)}{-329+6\lambda \left(-8+3\lambda \right)},$$

$${\pi }_R^{Dl\ast }=\frac{\begin{array}{c}28775+13718\theta \left(1+2\theta \right)+9212\lambda +5264\theta \lambda +4\left(-791+\theta \left(-763+64\theta \right)\right){\lambda }^2\\ -96\left(1+\theta \right)\left(7+4\theta \right){\lambda }^3+144{\left(1+\theta \right)}^2{\lambda }^4\\ \end{array}}{2{\left(329+6\left(8-3\lambda \right)\lambda \right)}^2}$$

The total profit of the system under strategy Dl is:

$${\pi }_{\prod }^{Dl\ast }=\frac{\begin{array}{c}21174+\theta \left(10156+26563\theta \right)+6824\lambda +4\theta \left(931+116\theta \right)\lambda +6\left(-389-358\theta +22{\theta }^2\right){\lambda }^2\\ -72\left(1+\theta \right)\left(7+4\theta \right){\lambda }^3+108{\left(1+\theta \right)}^2{\lambda }^4\\ \end{array}}{{\left(329+6\left(8-3\lambda \right)\lambda \right)}^2}$$

In order to ensure that the equilibrium results under the four operating modes are meaningful, that is, to ensure that the rates, traffic volumes, and profits are greater than 0, the constraints of this study can be obtained, and the feasible region of the results can be expressed as:

$$\left\{\begin{array}{c}0\le \theta \le -\frac{3\lambda }{-3+2\lambda },if0\le \lambda \le \frac{1}{3}\\ 0\le \theta \le \frac{-1-8\lambda -15{\lambda }^2}{-6-24\lambda +14{\lambda }^2},if\frac{1}{3}<\lambda \le 0.688\\ 0\le \theta \le \frac{17+16\lambda -8{\lambda }^2}{30-8\lambda +8{\lambda }^2},if0.688<\lambda <1\\ \end{array}\right)$$ 1.19

The proof is over.

B2 The impact of free riding behavior on profit

First, one can get the derivative of the profit of MT and OS on the spillover effect under strategy Cr:

$$\frac{\partial {\pi }_M^{Cr\ast }}{\partial \lambda }=-\frac{\begin{array}{c}\left[-\lambda \left(1+5\lambda \right)-4\theta +2\theta \lambda \left(-9+\lambda \right)\right]\\ \left\{-{\left(1+5\lambda \right)}^2\left(1+{\lambda }^2\right)+18\theta +2\theta \lambda \left[78+150\lambda +5{\lambda }^2\left(-18+\lambda \right)\right]\right\}\\ \end{array}}{4{\left(1+5\lambda \right)}^3{\left[-1+\lambda \left(-4+\lambda \right)\right]}^3}$$

$$\frac{\partial {\pi }_{\mathit{os}}^{Cr\ast }}{\partial \lambda }=\frac{1}{16}\left\{-\frac{32{\theta }^2}{{\left(1+5\lambda \right)}^3}+\frac{4\theta \left(5-16\theta \right)}{{\left(1+5\lambda \right)}^2}-\frac{4\left(1+\theta \right)}{-1+\lambda \left(-4+\lambda \right)}+\frac{2\left(-2+\lambda \right)\left[1+4\lambda +4\theta \left(\theta +\lambda \right)\right]}{{\left[-1+\lambda \left(-4+\lambda \right)\right]}^2}\right\}$$

Under restricted conditions, when $\theta <\frac{1+10\lambda +26{\lambda }^2+10{\lambda }^3+25{\lambda }^4}{18+156\lambda +300{\lambda }^2-180{\lambda }^3+10{\lambda }^4}$,$\frac{\partial {\pi }_M^{Cr\ast }}{\partial \lambda }>0$. Similarly, one can get $\frac{\partial {\pi }_{\mathit{OS}}^{Cr\ast }}{\partial \lambda }>0$ when $\theta <\frac{1}{2}\sqrt{\frac{{\left[1+\lambda \left(9+19\lambda -5{\lambda }^2\right)\right]}^2\left(9+A_2\right)}{{\left(14+A_1\lambda \right)}^2}}+\frac{\left(1+5\lambda \right)\left(3+\lambda \left(25+3\lambda \left(16+5\left(-1+\lambda \right)\lambda \right)\right)\right)}{28+2\lambda A_1}$, where $A_1=165+\lambda \left(623+\lambda \left(639+\lambda \left(-433+40\lambda \right)\right)\right),A_2=\lambda \left(92+\lambda \left(314+125\lambda \left(4+5\lambda \right)\right)\right)$.

The above calculation range represents that the profit of MT and OS increases with the spillover degree $\lambda$. In the opposite range, the income decreases with the increase of the free riding behavior, so it is expressed as $\left(+,-\right)$ in Table 2. Then, we discuss the changes of $\frac{\partial {\pi }_M^{Cl\ast }}{\partial \lambda }$ and $\frac{\partial {\pi }_{\mathit{OS}}^{Cl\ast }}{\partial \lambda }$ in strategy Cl and get $\frac{\partial {\pi }_M^{Cl\ast }}{\partial \lambda }<0$, which means the profit of MT decreases with the increase of spillover effect. When $\theta >\frac{-629+326\lambda +1296{\lambda }^2-304{\lambda }^3-64{\lambda }^4}{-4370+4836\lambda -648{\lambda }^2+368{\lambda }^3+64{\lambda }^4}$,$\frac{\partial {\pi }_{\mathit{OS}}^{Cl\ast }}{\partial \lambda }<0$. It shows that the profit of OS decreases with the increase of spillover degree $\lambda$. But in other range, it rises with the increase of $\lambda$. Similarly, in strategy Dr, when $\theta <\frac{5-4\lambda -4{\lambda }^2}{-32+16\lambda }$, the profit of MT increases with the increase of spillover degree. The profit of OS increases with the increase of spillover degree, and the following conditions need to be met:

$$\theta <\frac{-17+32\lambda +24{\lambda }^2+64{\lambda }^3-16{\lambda }^4}{16\left(14+25\lambda -24{\lambda }^2+4{\lambda }^3\right)}+\frac{1}{16}\sqrt{\frac{\begin{array}{c}513+6480\lambda +26064{\lambda }^2+39808{\lambda }^3\\ +28256{\lambda }^4-56064{\lambda }^5+23808{\lambda }^6-4096{\lambda }^7+256{\lambda }^8\\ \end{array}}{{\left(14+25\lambda -24{\lambda }^2+4{\lambda }^3\right)}^2}}$$

In strategy Dl, the profit of MT decreases with the increase of spillover degree, and the condition for the profit of OS decreases with the increases of spillover degree is:

$$\theta >\frac{-47005+61096\lambda +24318{\lambda }^2-6444{\lambda }^3-216{\lambda }^4}{-103474+94070\lambda -13896{\lambda }^2+6660{\lambda }^3+216{\lambda }^4}.$$

The proof is over.

B3 System efficiency analysis

The rail-water multimodal transport system should not only maximize the carriers' profits, but also be examined in terms of efficiency. At the policy level, governments promote the increase of rail-water multimodal transportation volume and promote the development of this mode of transport organization. Can the system efficiency be improved with the participation of multiple entities in the rail-water multimodal transport system? Centralized decision-making completely eliminates the double marginalization effect between R, OS, and MT. It enables the system to operate at optimal efficiency. Therefore, this study benchmarks the centralized decision-making of the rail-water multimodal transport system, calculates the profits of the system under the benchmark model, and compares the profits under different operation strategies with the benchmark revenue to further explore the change of efficiency.

First, we discuss the profit of the rail-water multimodal transport system. As the double marginalization effect disappears, the overall system profit is derived from the multimodal services provided by OS and MT. The cost of downstream to upstream booking has disappeared, but the service effort cost of MT still needs to be considered. Therefore, the profit function of the rail-water multimodal transport system is:

$$\underset{}{max}{\pi }_Z=p_M{q}_M^{}+p_{\mathit{os}}{q}_{\mathit{os}}^{}-\frac{v^2}{2}$$ 3.1

Bringing the inverse demand function into Eq. (3.1), we can obtain the equilibrium traffic volume and the degree of service effort, respectively. The second-order condition determines the equilibrium solution of the profit function ${\pi }_Z$. Bringing the equilibrium capacity and service effort level back into the revenue function, we can get:

$${\pi }_Z^{}=\frac{1-2{\theta }^2-2\lambda \left(1+\theta \right)+{\left(1+\theta \right)}^2{\lambda }^2}{4{\left(-1+\lambda \right)}^2}$$ 3.2

In order to compare the total revenue of the rail-water multimodal transport system under different operational strategies, we analyze it by ratio. The system efficiency of strategy Dr is $\frac{{\pi }_{\prod }^{Dr\ast }}{{\pi }_Z^{}}$. Similarly, the system efficiencies under other operational strategies are $\frac{{\pi }_{\prod }^{Dl\ast }}{{\pi }_Z^{}}$, $\frac{{\pi }_{\prod }^{Cr\ast }}{{\pi }_Z^{}}$, and $\frac{{\pi }_{\prod }^{Dl\ast }}{{\pi }_Z^{}}$. We start the discussion with the objective of optimal efficiency, so that the system efficiency is maximized and compared. In the range of Eq. (1.19), the range of maximum efficiency in strategy Cr is discussed, satisfying the following constraint:

$$\frac{{\pi }_{\prod }^{Cr\ast }}{{\pi }_Z^{}}>\frac{{\pi }_{\prod }^{Dr\ast }}{{\pi }_Z^{}},\frac{{\pi }_{\prod }^{Cr\ast }}{{\pi }_Z^{}}>\frac{{\pi }_{\prod }^{Dl\ast }}{{\pi }_Z^{}},\frac{{\pi }_{\prod }^{Cr\ast }}{{\pi }_Z^{}}>\frac{{\pi }_{\prod }^{Dl\ast }}{{\pi }_Z^{}}$$ 3.3

The simplification shows that Eq. (3.3) does not hold. We discuss the system efficiency under the other three models in turn to obtain Fig. 13. This shows that the ex-post service strategies under the centralized procurement mode are not conducive to the system efficiency. This is because MT orders rail-water multimodal services from OS, which makes OS profit, intensifies the double marginalization effect upstream and downstream of the supply chain, and makes the overall efficiency of the system the lowest. We superimpose the efficiency maximization scope diagram with the benefit maximization scope diagram, which can analyze the decision conflict under the dual objective constraints of system efficiency and profit. The results of the analysis are described in Proposition 4.

Fig. 13.

Optimal efficiency in different ranges

The proof is over.

B4 Figure composition and condition analysis

Based on the conditions in Eq. (1.19), Fig. 14 can be formed as follows:

Fig. 14.

Feasible region

Analysis of Fig. 2. Under Mode C, comparing the profits of MT, OS and R. We can obtain that ${\pi }_M^{Cr\ast }>{\pi }_M^{Cl\ast }$ and ${\pi }_R^{Cr\ast }>{\pi }_R^{Cl\ast }$ are constant.

Let ${\pi }_{\mathit{os}}^{Cl\ast }={\pi }_{\mathit{os}}^{Cr\ast }$. The thresholds can be calculated as follows:

$$f\left({\theta }_1\right)=\frac{-1-8\lambda -9{\lambda }^2+30{\lambda }^3}{2\left(-5-42\lambda -85{\lambda }^2+16{\lambda }^3\right)}-\frac{1}{2}\sqrt{\frac{1+16\lambda +77{\lambda }^2-8{\lambda }^3-1029{\lambda }^4-2424{\lambda }^5-1065{\lambda }^6+400{\lambda }^7}{{\left(-5-42\lambda -85{\lambda }^2+16{\lambda }^3\right)}^2}}$$

$$f\left({\theta }_2\right)=\frac{-1-8\lambda -9{\lambda }^2+30{\lambda }^3}{2\left(-5-42\lambda -85{\lambda }^2+16{\lambda }^3\right)}+\frac{1}{2}\sqrt{\frac{1+16\lambda +77{\lambda }^2-8{\lambda }^3-1029{\lambda }^4-2424{\lambda }^5-1065{\lambda }^6+400{\lambda }^7}{{\left(-5-42\lambda -85{\lambda }^2+16{\lambda }^3\right)}^2}}$$

$f\left({\theta }_1\right)$ and $f\left({\theta }_2\right)$ are shown in the Fig. 15.

Fig. 15.

OS revenue distribution

Superimpose Fig. 16 to form the figure below:

Fig. 16.

Preference distribution of service strategy of each carrier (Centralized procurement mode)

Under Mode D, comparing the profits of MT, OS and R. We compare equilibrium profits and find that the following conditions are satisfied

$${\pi }_M^{Dr\ast }=\frac{{\left(1+4\theta +2\lambda \right)}^2}{2{\left(-3-16\lambda +4{\lambda }^2\right)}^2}>{\pi }_M^{Dl\ast }=\frac{{\left(1+4\theta \right)}^2}{329-6\lambda \left(-8+3\lambda \right)}$$

$\begin{array}{c}{\pi }_R^{Dr\ast }=\frac{-1-2\theta -4{\theta }^2-5\lambda -4\theta \lambda }{2\left(-3-16\lambda +4{\lambda }^2\right)}\\ >{\pi }_R^{Dl\ast }=\frac{\begin{array}{c}28775+13718\theta \left(1+2\theta \right)+9212\lambda +5264\theta \lambda \\ +4\left(-791+\theta \left(-763+64\theta \right)\right){\lambda }^2\\ -96\left(1+\theta \right)\left(7+4\theta \right){\lambda }^3+144{\left(1+\theta \right)}^2{\lambda }^4\\ \end{array}}{2{\left(329+6\left(8-3\lambda \right)\lambda \right)}^2}\\ \end{array}$ For the OS profit comparison, we let ${\pi }_{\mathit{os}}^{Dr\ast }={\pi }_{\mathit{os}}^{Dl\ast }$ and calculate that we get:

$f\left({\theta }_3\right)=\frac{\begin{array}{c}\frac{\begin{array}{c}102256+502577\lambda +1662290{\lambda }^2-7920{\lambda }^3\\ -71716{\lambda }^4-6044{\lambda }^5-30752{\lambda }^6+12144{\lambda }^7-1152{\lambda }^8\\ \end{array}}{\begin{array}{c}8\left(-118244-291184\lambda +426889{\lambda }^2-181528{\lambda }^3\right)\\ \left(+51732{\lambda }^4-18288{\lambda }^5+6652{\lambda }^6-1536{\lambda }^7+144{\lambda }^8\right)\\ \end{array}}\\ +\frac{1}{8}\surd ((18018229824+175007196672\lambda \\ +317074261585{\lambda }^2-232374284708{\lambda }^3+2125968539876{\lambda }^4\\ +1939651856056{\lambda }^5-398060008360{\lambda }^6-511652732336{\lambda }^7\\ +42530825584{\lambda }^8+52724584224{\lambda }^9-4284739440{\lambda }^{10}\\ -2378723328{\lambda }^{11}+244964736{\lambda }^{12}+35707392{\lambda }^{13}-4292352{\lambda }^{14})\\ \end{array}}{\begin{array}{c}\left(-118244-291184\lambda +426889{\lambda }^2-181528{\lambda }^3+51732{\lambda }^4\right)\\ {\left(-18288{\lambda }^5+6652{\lambda }^6-1536{\lambda }^7+144{\lambda }^8\right)}^2\\ \end{array}}$ The range of $f\left({\theta }_3\right)$ is shown in Fig. 17.

Fig. 17.

OS revenue distribution

Superimpose Fig. 18 to form the figure below:

Fig. 18.

Preference distribution of service strategy of each carrier (Decentralized procurement mode)

Analysis of Fig. 3. We take the equilibrium results into $E_M^{}=\frac{2{\pi }_M^{}}{v^2}$ and calculate that we get:

$$E_M^{\mathit{Cr}}=\frac{{\left(4\theta +\lambda -2\theta \left(-9+\lambda \right)\lambda +5{\lambda }^2\right)}^2}{4{\left(1+\lambda \left(9+\left(19-5\lambda \right)\lambda \right)\right)}^2},$$

$$E_M^{\mathit{Cl}}=-\frac{8{\theta }^2}{-17+8\left(-2+\lambda \right)\lambda },$$

$$E_M^{\mathit{Dr}}=\frac{{\left(1+4\theta +2\lambda \right)}^2}{{\left(3-4\left(-4+\lambda \right)\lambda \right)}^2},$$

$$E_M^{\mathit{Dl}}=-\frac{2{\left(1+4\theta \right)}^2}{-329+6\lambda \left(-8+3\lambda \right)}.$$

The comparison reveals that $E_M^{\mathit{Cr}}>E_M^{\mathit{Cl}}$ and $E_M^{\mathit{Dr}}>E_M^{\mathit{Dl}}$ hold constant. Comparing the magnitudes of $E_M^{\mathit{Cr}}$ and $E_M^{\mathit{Dr}}$, so that $E_M^{\mathit{Cr}}-E_M^{\mathit{Dr}}=0$, the calculation gives a threshold value of $\frac{-1-5\lambda }{-2-14\lambda +4{\lambda }^2}$.

Similar to the above process, the calculation gives:

$$E_{}^{\mathit{Cr}}=\frac{\begin{array}{c}-{\left(1+5\lambda \right)}^2\left(-3+\lambda \left(-26+\lambda \left(-55+14\lambda \right)\right)\right)\\ +4\theta \left(1+\lambda \left(22+\lambda \left(154+\lambda \left(398+\left(249-80\lambda \right)\lambda \right)\right)\right)\right)\\ +4{\theta }^2\left(15+\lambda \left(156+\lambda \left(478+3\lambda \left(110+\lambda \left(-43+2\lambda \right)\right)\right)\right)\right)\\ \end{array}}{8{\left(1+5\lambda \right)}^2{\left(-1+\left(-4+\lambda \right)\lambda \right)}^2},$$

$$E_{}^{\mathit{Cl}}=\frac{\begin{array}{c}3{\left(17-8\left(-2+\lambda \right)\lambda \right)}^2\\ +4\theta \left(-17+8\left(-2+\lambda \right)\lambda \right)\left(-5+12\left(-1+\lambda \right)\lambda \right)\\ +4{\theta }^2\left(847+8\lambda \left(37+3\lambda \left(-5+2\left(-2+\lambda \right)\lambda \right)\right)\right)\\ \end{array}}{8{\left(17-8\left(-2+\lambda \right)\lambda \right)}^2}$$

$$E_{}^{\mathit{Dr}}=\frac{\begin{array}{c}5+2\theta \left(8+19\theta \right)+46\lambda +2\theta \left(35+48\theta \right)\lambda \\ -8\left(1+\theta \right)\left(-14+3\theta \right){\lambda }^2-4\left(7+6\theta \right){\lambda }^3\\ \end{array}}{{\left(3-4\left(-4+\lambda \right)\lambda \right)}^2}$$

$$E_{}^{\mathit{Dl}}=\frac{2\left(\begin{array}{c}21174+\theta \left(10156+26563\theta \right)\\ +6824\lambda +4\theta \left(931+116\theta \right)\lambda \\ +6\left(-389-358\theta +22{\theta }^2\right){\lambda }^2\\ -72\left(1+\theta \right)\left(7+4\theta \right){\lambda }^3+108{\left(1+\theta \right)}^2{\lambda }^4\\ \end{array}\right)}{{\left(329+6\left(8-3\lambda \right)\lambda \right)}^2}$$

The comparison reveals that $E_{}^{\mathit{Cr}}>E_{}^{\mathit{Cl}}$ and $E_{}^{\mathit{Dr}}>E_{}^{\mathit{Dl}}$ hold constant. Let $E_{}^{\mathit{Cr}}-E_{}^{\mathit{Dr}}=0$. The calculation gives a threshold value of $\frac{-1-5\lambda }{-2-14\lambda +4{\lambda }^2}$.

Interestingly, the calculation found the thresholds to be the same. The ranges are therefore shown in Fig. 19.

Fig. 19.

Distribution for maximum efficiency

Analysis of Fig. 4. First, under the Cr strategy, let $q_M^{Cr\ast }>q_{\mathit{os}}^{Cr\ast }$. The calculation gives:

$$\left\{\begin{array}{c}\frac{-1-6\lambda -5{\lambda }^2}{-14-60\lambda +18{\lambda }^2}<\theta <-\frac{3\lambda }{-3+2\lambda },if0.0764<\lambda <\frac{1}{3}\\ \frac{-1-6\lambda -5{\lambda }^2}{-14-60\lambda +18{\lambda }^2}<\theta <\frac{-1-8\lambda -15{\lambda }^2}{-6-24\lambda +14{\lambda }^2},if\frac{1}{3}<\lambda \le 0.688\\ \frac{-1-6\lambda -5{\lambda }^2}{-14-60\lambda +18{\lambda }^2}<\theta <\frac{17+16\lambda -8{\lambda }^2}{30-8\lambda +8{\lambda }^2},if0.688<\lambda <1\\ \end{array}\right)$$ 4.1

Comparing Eq. (4.1) with Eq. (1.19), the threshold for $q_M^{Cr\ast }>q_{\mathit{os}}^{Cr\ast }$ is $\frac{-1-6\lambda -5{\lambda }^2}{-14-60\lambda +18{\lambda }^2}$. As shown in Fig. 20.

Fig. 20.

Market share comparison between MT and OS (Cr strategy)

Secondly, under the Cl strategy, let $q_M^{Cl\ast }>q_{\mathit{os}}^{Cl\ast }$. The calculation gives:

$$\left\{\begin{array}{c}\frac{17+16\lambda -8{\lambda }^2}{70-8\lambda +8{\lambda }^2}<\theta <-\frac{3\lambda }{-3+2\lambda },if0.249<\lambda <\frac{1}{3}\\ \frac{17+16\lambda -8{\lambda }^2}{70-8\lambda +8{\lambda }^2}<\theta <\frac{-1-8\lambda -15{\lambda }^2}{-6-24\lambda +14{\lambda }^2},if\frac{1}{3}<\lambda \le 0.688\\ \frac{17+16\lambda -8{\lambda }^2}{70-8\lambda +8{\lambda }^2}<\theta <\frac{17+16\lambda -8{\lambda }^2}{30-8\lambda +8{\lambda }^2},if0.688<\lambda <1\\ \end{array}\right)$$ 4.2

Therefore, the threshold for $q_M^{Cl\ast }>q_{\mathit{os}}^{Cl\ast }$ is $\frac{17+16\lambda -8{\lambda }^2}{70-8\lambda +8{\lambda }^2}$. As shown in Fig. 21.

Fig. 21.

Market share comparison between MT and OS (Cl strategy)

Thirdly, let $q_M^{Dl\ast }>q_{\mathit{os}}^{Dl\ast }$. The calculation gives:

$$\left\{\begin{array}{c}\frac{49+14\lambda -6{\lambda }^2}{133-8\lambda +6{\lambda }^2}<\theta <-\frac{3\lambda }{-3+2\lambda },if0.318<\lambda <\frac{1}{3}\\ \frac{49+14\lambda -6{\lambda }^2}{133-8\lambda +6{\lambda }^2}<\theta <\frac{-1-8\lambda -15{\lambda }^2}{-6-24\lambda +14{\lambda }^2},if\frac{1}{3}<\lambda \le 0.688\\ \frac{49+14\lambda -6{\lambda }^2}{133-8\lambda +6{\lambda }^2}<\theta <\frac{17+16\lambda -8{\lambda }^2}{30-8\lambda +8{\lambda }^2},if0.688<\lambda <1\\ \end{array}\right)$$ 4.3

Therefore, the threshold for $q_M^{Dl\ast }>q_{\mathit{os}}^{Dl\ast }$ is $\frac{49+14\lambda -6{\lambda }^2}{133-8\lambda +6{\lambda }^2}$. As shown in Fig. 22.

Fig. 22.

Market share comparison between MT and OS (Dl 模式) Dl strategy

Finally, under the Dr strategy, $q_M^{Dr\ast }>q_{\mathit{os}}^{Dr\ast }$ holds constant.

Analysis of Fig. 5. Let ${\pi }_M^{Cl\ast }={\pi }_M^{Dl\ast }$ the calculation gives the threshold is:

$$\begin{array}{cc}\hfill f\left({\theta }_4\right)& =\frac{17+16\lambda -8{\lambda }^2}{261-16\lambda +14{\lambda }^2}\hfill \\ \hfill & +\frac{1}{2}\sqrt{\frac{5593+6080\lambda -2170{\lambda }^2-672{\lambda }^3+144{\lambda }^4}{{\left(261-16\lambda +14{\lambda }^2\right)}^2}}\hfill \\ \hfill \end{array}$$

Similarly, so that ${\pi }_M^{Cr\ast }={\pi }_M^{Dr\ast }$ the calculation yields a threshold value of $\frac{-1-5\lambda }{-2-14\lambda +4{\lambda }^2}$. A new graphic is formed, as shown in Fig. 23.

Fig. 23.

Multi-modal operator’s profit comparison under different procurement modes

Analysis of Fig. 6. We find that ${\pi }_M^{Dl\ast }-{\pi }_M^{Dr\ast }<0$ and ${\pi }_M^{Cl\ast }-{\pi }_M^{Cr\ast }<0$. Let ${\pi }_M^{Dl\ast }={\pi }_M^{Cr\ast }$ the calculation gives the threshold is $\frac{-1-5\lambda }{-2-14\lambda +4{\lambda }^2}$. As shown in Fig. 24.

Fig. 24.

Strategy selection of MT

Analysis of Fig. 7. First, compare the profits of railway companies under different strategies. We calculate and find that ${\pi }_R^{Dr\ast }$ is the largest. Secondly, compare the profit of OS under different strategies. After comparison it is concluded that ${\pi }_{\mathit{os}}^{Dl\ast }$ is the smallest. We calculate the thresholds for ${\pi }_{\mathit{os}}^{Cl\ast }={\pi }_{\mathit{os}}^{Dr\ast }$, ${\pi }_{\mathit{os}}^{Cl\ast }={\pi }_{\mathit{os}}^{Cr\ast }$ and ${\pi }_{\mathit{os}}^{Dr\ast }={\pi }_{\mathit{os}}^{Cr\ast }$as:

$$\begin{array}{cc}& f\left({\theta }_5\right)=\frac{\begin{array}{c}1921+16224\lambda +49860{\lambda }^2+18160{\lambda }^3-14192{\lambda }^4\\ +9152{\lambda }^5-12096{\lambda }^6+4608{\lambda }^7-512{\lambda }^8\\ \end{array}}{\begin{array}{c}2\left(-1055+20664\lambda +88592{\lambda }^2-56544{\lambda }^3+38432{\lambda }^4\right)\\ \left(-21888{\lambda }^5+9728{\lambda }^6-2560{\lambda }^7+256{\lambda }^8\right)\\ \end{array}}+\hfill \\ \hfill & -\frac{2\sqrt{2}\sqrt{\begin{array}{cc}& 124848+1855431\lambda +11481807{\lambda }^2+45968262{\lambda }^3\hfill \\ \hfill & +125870518{\lambda }^4+144957304{\lambda }^5+382432{\lambda }^6-74796896{\lambda }^7\hfill \\ \hfill & -5170592{\lambda }^8+7850880{\lambda }^9+483328{\lambda }^{10}+2772992{\lambda }^{11}\hfill \\ \hfill & -2041856{\lambda }^{12}+450560{\lambda }^{13}-32768{\lambda }^{14}\hfill \\ \hfill \end{array}}}{\begin{array}{cc}& \left(-1055+20664\lambda +88592{\lambda }^2-56544{\lambda }^3+38432{\lambda }^4\right)\hfill \\ \hfill & {\left(-21888{\lambda }^5+9728{\lambda }^6-2560{\lambda }^7+256{\lambda }^8\right)}^2\hfill \\ \hfill \end{array}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill f\left({\theta }_6\right)=& \frac{\begin{array}{cc}& 204+1654\lambda +1365{\lambda }^2-12168{\lambda }^3-16411{\lambda }^4\hfill \\ \hfill & -3468{\lambda }^5+764{\lambda }^6+3360{\lambda }^7-800{\lambda }^8\hfill \\ \hfill \end{array}}{\begin{array}{cc}& 2\left(920+7548\lambda +14237{\lambda }^2-5778{\lambda }^3-6923{\lambda }^4\right)\hfill \\ \hfill & \left(-23032{\lambda }^5+12792{\lambda }^6-3264{\lambda }^7+400{\lambda }^8\right)\hfill \\ \hfill \end{array}}\hfill \\ \hfill & -\frac{\sqrt{\begin{array}{cc}& 10404+168708\lambda +756689{\lambda }^2-1447444{\lambda }^3-21688903{\lambda }^4\hfill \\ \hfill & -60936280{\lambda }^5-40748041{\lambda }^6+84195500{\lambda }^7\hfill \\ \hfill & +172242095{\lambda }^8+123643948{\lambda }^9-5042644{\lambda }^{10}-80545392{\lambda }^{11}\hfill \\ \hfill & -10758144{\lambda }^{12}+26622464{\lambda }^{13}-7010560{\lambda }^{14}+537600{\lambda }^{15}\hfill \\ \hfill \end{array}}}{\begin{array}{cc}& \left(920+7548\lambda +14237{\lambda }^2-5778{\lambda }^3-6923{\lambda }^4\right)\hfill \\ \hfill & {\left(-23032{\lambda }^5+12792{\lambda }^6-3264{\lambda }^7+400{\lambda }^8\right)}^2\hfill \\ \hfill \end{array}}\hfill \end{array}$$

$$f\left({\theta }_7\right)=\frac{1+17\lambda +76{\lambda }^2+8{\lambda }^3-344{\lambda }^4+80{\lambda }^5}{50+678\lambda +2948{\lambda }^2+3616{\lambda }^3-2112{\lambda }^4+256{\lambda }^5}$$

Finally, MT's strategy selection preferences are discussed. We calculate that ${\pi }_M^{Dl\ast }<{\pi }_M^{Dr\ast }$ and ${\pi }_M^{Cl\ast }<{\pi }_M^{Cr\ast }$ hold constant. Let ${\pi }_M^{Dr\ast }={\pi }_M^{Cr\ast }$ to a threshold value of $\frac{-1-5\lambda }{-2-14\lambda +4{\lambda }^2}$. All the thresholds discussed above are combined to form a new graphic, as shown in Fig. 25. The illustration represents a combination of the “MT-OS-R” strategy preferences.

Fig. 25.

Distribution of operational strategy preference of each carrier

Superimpose Figs. 13 and 25 to obtain Fig. 26.

Fig. 26.

Consistency analysis of system efficiency maximization and MT’s profit maximization

Table 6 explains the specific meaning of each area of Fig. 26.

Table 6.

Comparison between efficiency and profit preference

	Optimal system efficiency	MT prefers	OS prefers	R prefers
Region 1	Dl	Cr	Cr	Dr
Region 2	Dl	Dr	Dr	Dr
Region 3	Dr	Dr	Dr	Dr
Region 4	Cl	Dr	Dr	Dr
Region 5	Cl	Dr	Cl	Dr
Region 6	Cl	Dr	Cr	Dr

Analysis of Fig. 8. Calculated from Eq. (17) we get:

$$U^{Dl\ast }=\frac{\begin{array}{c}7265+1576\lambda -256{\lambda }^2+96{\lambda }^3-36{\lambda }^4\\ +{\theta }^2\left(-54511+304\lambda -676{\lambda }^2+672{\lambda }^3-252{\lambda }^4\right)\\ -4\theta \left(2907+379\lambda -298{\lambda }^2-246{\lambda }^3+72{\lambda }^4\right)\\ \end{array}}{2{\left(329+48\lambda -18{\lambda }^2\right)}^2}$$

$$U^{Dr\ast }=\frac{\begin{array}{c}-1+2\lambda +11{\lambda }^2-12{\lambda }^3+5{\theta }^2\left(-11-20\lambda +4{\lambda }^2\right)\\ +4\theta \left(-5-14\lambda -13{\lambda }^2+2{\lambda }^3\right)\\ \end{array}}{2{\left(3+16\lambda -4{\lambda }^2\right)}^2}$$

$$U_{}^{Cl\ast }=\frac{\begin{array}{c}{\left(17+16\lambda -8{\lambda }^2\right)}^2-4\theta \left(-85+124\lambda +28{\lambda }^2-288{\lambda }^3+96{\lambda }^4\right)\\ -4{\theta }^2\left(1975+120\lambda -8{\lambda }^2-224{\lambda }^3+112{\lambda }^4\right)\\ \end{array}}{32{\left(17+16\lambda -8{\lambda }^2\right)}^2}$$

$$U_{}^{Cr\ast }=-\frac{\begin{array}{c}{\left(1+5\lambda \right)}^2\left(-1-10\lambda -13{\lambda }^2+12{\lambda }^3\right)\\ +4{\theta }^2\left(47+468\lambda +1310{\lambda }^2+648{\lambda }^3-229{\lambda }^4+12{\lambda }^5\right)\\ -4\theta \left(1-8\lambda -162{\lambda }^2-556{\lambda }^3-343{\lambda }^4+60{\lambda }^5\right)\\ \end{array}}{32{\left(1+5\lambda \right)}^2{\left(-1-4\lambda +{\lambda }^2\right)}^2}$$

Comparing $U_{}^{Dl\ast }$, $U_{}^{Dr\ast }$, $U_{}^{Cl\ast }$ and $U_{}^{Cr\ast }$, it is concluded that $U_{}^{Dr\ast }$ is the smallest. First, assuming that the conditions $U_{}^{Cr\ast }>U_{}^{Dl\ast }$ and $U_{}^{Cr\ast }>U_{}^{Cl\ast }$ are satisfied, simplification gives a threshold as:

$$\begin{array}{c}\hfill f\left({\theta }_8\right)=\frac{\begin{array}{c}-154753-8936\lambda +12157942{\lambda }^2+49150172{\lambda }^3+39146827{\lambda }^4\\ +7981828{\lambda }^5-3624388{\lambda }^6-213488{\lambda }^7-726020{\lambda }^8+297840{\lambda }^9-28800{\lambda }^{10}\\ \end{array}}{\begin{array}{c}2\left(-4869283-48217660\lambda -130200590{\lambda }^2-34677384{\lambda }^3+76572245{\lambda }^4\right)\\ \left(-26934084{\lambda }^5+3774096{\lambda }^6-1429136{\lambda }^7+946420{\lambda }^8-262608{\lambda }^9+25200{\lambda }^{10}\right)\\ \end{array}}\\ \hfill +\frac{\sqrt{\begin{array}{cc}& -3750225927+275823730\lambda +1304194530733{\lambda }^2+17051673941972{\lambda }^3\hfill \\ \hfill & +97526326280322{\lambda }^4+288195373213868{\lambda }^5+441621691517162{\lambda }^6\hfill \\ \hfill & +365536759274256{\lambda }^7+295719789559381{\lambda }^8+235389709941490{\lambda }^9\hfill \\ \hfill & +76792357420481{\lambda }^{10}-105844303386836{\lambda }^{11}-15350831597888{\lambda }^{12}\hfill \\ \hfill & +13425555868032{\lambda }^{13}+721083810136{\lambda }^{14}-616300849344{\lambda }^{15}\hfill \\ \hfill & -92679220944{\lambda }^{16}+26345202912{\lambda }^{17}+5985893520{\lambda }^{18}-1799755200{\lambda }^{19}\hfill \\ \hfill & +116640000{\lambda }^{20}\hfill \\ \hfill \end{array}}}{\begin{array}{cc}& \left(-4869283-48217660\lambda -130200590{\lambda }^2-34677384{\lambda }^3+76572245{\lambda }^4\right)\hfill \\ \hfill & {\left(-26934084{\lambda }^5+3774096{\lambda }^6-1429136{\lambda }^7+946420{\lambda }^8-262608{\lambda }^9+25200{\lambda }^{10}\right)}^2\hfill \\ \hfill \end{array}}\end{array}$$

Secondly, assuming that the conditions $U_{}^{Cl\ast }>U_{}^{Dl\ast }$ and $U_{}^{Cl\ast }>U_{}^{Cr\ast }$ are satisfied, simplification gives a threshold as:

$$\begin{array}{cc}& f\left({\theta }_9\right)=\frac{\begin{array}{c}-22642453-16317780\lambda +6581392{\lambda }^2-15589744{\lambda }^3+1093596{\lambda }^4\\ +3473008{\lambda }^5-1093136{\lambda }^6+287616{\lambda }^7-42624{\lambda }^8\\ \end{array}}{\begin{array}{c}2\left(-150761259+42897192\lambda +12548596{\lambda }^2-26049408{\lambda }^3\right)\\ \left(+7494852{\lambda }^4-5905760{\lambda }^5+1529056{\lambda }^6-163968{\lambda }^7+28224{\lambda }^8\right)\\ \end{array}}\hfill \\ \hfill & -\frac{\sqrt{\begin{array}{cc}& 41041054263265+114884460881064\lambda +102331534807928{\lambda }^2\hfill \\ \hfill & +20135922886384{\lambda }^3-6279073334376{\lambda }^4+47644385925408{\lambda }^5\hfill \\ \hfill & +32927037217248{\lambda }^6-36147281875648{\lambda }^7-8636728029680{\lambda }^8\hfill \\ \hfill & +6740784700160{\lambda }^9+837853093888{\lambda }^{10}-540680411136{\lambda }^{11}-41126307840{\lambda }^{12}\hfill \\ \hfill +& 24598757376{\lambda }^{13}+615776256{\lambda }^{14}-668860416{\lambda }^{15}+47775744{\lambda }^{16}\hfill \\ \hfill \end{array}}}{\begin{array}{cc}& \left(-150761259+42897192\lambda +12548596{\lambda }^2-26049408{\lambda }^3\right)\hfill \\ \hfill & {\left(+7494852{\lambda }^4-5905760{\lambda }^5+1529056{\lambda }^6-163968{\lambda }^7+28224{\lambda }^8\right)}^2\hfill \\ \hfill \end{array}}\hfill \end{array}$$

$$\begin{array}{cc}\hfill f\left({\theta }_{10}\right)& =\frac{\begin{array}{c}-22642453-16317780\lambda +6581392{\lambda }^2-15589744{\lambda }^3+1093596{\lambda }^4\\ +3473008{\lambda }^5-1093136{\lambda }^6+287616{\lambda }^7-42624{\lambda }^8\\ \end{array}}{\begin{array}{c}2\left(-150761259+42897192\lambda +12548596{\lambda }^2-26049408{\lambda }^3+7494852{\lambda }^4\right)\\ \left(-5905760{\lambda }^5+1529056{\lambda }^6-163968{\lambda }^7+28224{\lambda }^8\right)\\ \end{array}}\hfill \\ & +\frac{\sqrt{\begin{array}{c}41041054263265+114884460881064\lambda +102331534807928{\lambda }^2\\ +20135922886384{\lambda }^3-6279073334376{\lambda }^4+47644385925408{\lambda }^5\\ +32927037217248{\lambda }^6-36147281875648{\lambda }^7-8636728029680{\lambda }^8\\ +6740784700160{\lambda }^9+837853093888{\lambda }^{10}-540680411136{\lambda }^{11}-41126307840{\lambda }^{12}\\ +24598757376{\lambda }^{13}+615776256{\lambda }^{14}-668860416{\lambda }^{15}+47775744{\lambda }^{16}\\ \end{array}}}{\begin{array}{c}\left(-150761259+42897192\lambda +12548596{\lambda }^2-26049408{\lambda }^3+7494852{\lambda }^4\right)\\ {\left(-5905760{\lambda }^5+1529056{\lambda }^6-163968{\lambda }^7+28224{\lambda }^8\right)}^2\\ \end{array}}\hfill \end{array}$$

Finally, the threshold is assumed to be the same as $f\left({\theta }_8\right)$, $f\left({\theta }_9\right)$ and $f\left({\theta }_{10}\right)$ when $U_{}^{Dl\ast }$ is at its maximum. We plot the calculated thresholds, as shown in Fig. 27.

Fig. 27.

Optimal consumer surplus

Analyse the relationship between the magnitude of social welfare under different strategies. First, when the conditions $S{W}_{}^{Cr\ast }>S{W}_{}^{Cl\ast }$, $S{W}_{}^{Cr\ast }>S{W}_{}^{Dr\ast }$ and $S{W}_{}^{Cr\ast }>S{W}_{}^{Dl\ast }$ are satisfied, the threshold is calculated as:

$$\begin{array}{cc}& \begin{array}{c}\hfill f\left({\theta }_{11}\right)=\frac{\begin{array}{cc}& -289987-3342452\lambda -12363318{\lambda }^2-16110384{\lambda }^3-7132415{\lambda }^4\hfill \\ \hfill & +3891908{\lambda }^5-2728396{\lambda }^6+1729152{\lambda }^7+78676{\lambda }^8-137040{\lambda }^9+14400{\lambda }^{10}\hfill \\ \hfill \end{array}}{\begin{array}{cc}& 2\left(-1834557-17327732\lambda -42510034{\lambda }^2-7082100{\lambda }^3+935827{\lambda }^4\right)\hfill \\ \hfill & \left(-2353304{\lambda }^5+1552304{\lambda }^6-438576{\lambda }^7+143788{\lambda }^8-36960{\lambda }^9+3600{\lambda }^{10}\right)\hfill \\ \hfill \end{array}}\end{array}\hfill \\ \hfill & -\frac{\sqrt{\begin{array}{cc}& 4456606693+317506789670\lambda +6256091367497{\lambda }^2+58270425988936{\lambda }^3\hfill \\ \hfill & +296948147124762{\lambda }^4+849990419139956{\lambda }^5+1279612785461698{\lambda }^6+\hfill \\ \hfill & 780537611702040{\lambda }^7-48419874889871{\lambda }^8-123861312496778{\lambda }^9\hfill \\ \hfill & +1209698972141{\lambda }^{10}-22188933658992{\lambda }^{11}+6492384738272{\lambda }^{12}\hfill \\ \hfill & +753857115584{\lambda }^{13}-225664411784{\lambda }^{14}+423270000672{\lambda }^{15}-185429031696{\lambda }^{16}\hfill \\ \hfill & -1597972320{\lambda }^{17}+12372555600{\lambda }^{18}-2181168000{\lambda }^{19}+116640000{\lambda }^{20}\hfill \\ \hfill \end{array}}}{\begin{array}{cc}& \left(-1834557-17327732\lambda -42510034{\lambda }^2-7082100{\lambda }^3+935827{\lambda }^4\right)\hfill \\ \hfill & {\left(-2353304{\lambda }^5+1552304{\lambda }^6-438576{\lambda }^7+143788{\lambda }^8-36960{\lambda }^9+3600{\lambda }^{10}\right)}^2\hfill \\ \hfill \end{array}}\hfill \end{array}$$

$$f\left({\theta }_{12}\right)=\frac{-1-55\lambda -575{\lambda }^2-2033{\lambda }^3-1520{\lambda }^4+2504{\lambda }^5-480{\lambda }^6}{-170-2402\lambda -10946{\lambda }^2-14430{\lambda }^3+7324{\lambda }^4-1040{\lambda }^5+64{\lambda }^6}$$

$$\begin{array}{c}\hfill f\left({\theta }_{13}\right)=\frac{\begin{array}{c}-612-7002\lambda -23525{\lambda }^2-12608{\lambda }^3+31722{\lambda }^4-21002{\lambda }^5\\ -88973{\lambda }^6+12724{\lambda }^7+41916{\lambda }^8-19040{\lambda }^9+2400{\lambda }^{10}\\ \end{array}}{\begin{array}{c}2\left(-4632-49746\lambda -161723{\lambda }^2-140992{\lambda }^3-10846{\lambda }^4-43126{\lambda }^5\right)\\ \left(+125593{\lambda }^6-51048{\lambda }^7+14360{\lambda }^8-3840{\lambda }^9+400{\lambda }^{10}\right)\\ \end{array}}\\ \hfill +\frac{2\sqrt{\begin{array}{c}23409+1037646\lambda +18311631{\lambda }^2+173768874{\lambda }^3+986791647{\lambda }^4\\ +3474603006{\lambda }^5+7491898755{\lambda }^6+9140315332{\lambda }^7+4503531747{\lambda }^8\\ -2397275654{\lambda }^9-4658146319{\lambda }^{10}-2084415782{\lambda }^{11}+1441564957{\lambda }^{12}\\ +1384726394{\lambda }^{13}-754570963{\lambda }^{14}-140766856{\lambda }^{15}+313774976{\lambda }^{16}\\ -191667200{\lambda }^{17}+61902400{\lambda }^{18}-10048000{\lambda }^{19}+640000{\lambda }^{20}\\ \end{array}}}{\begin{array}{c}\left(-4632-49746\lambda -161723{\lambda }^2-140992{\lambda }^3-10846{\lambda }^4-43126{\lambda }^5\right)\\ {\left(+125593{\lambda }^6-51048{\lambda }^7+14360{\lambda }^8-3840{\lambda }^9+400{\lambda }^{10}\right)}^2\\ \end{array}}\end{array}$$

$$f\left({\theta }_{14}\right)=\frac{-1-5\lambda }{-2-14\lambda +4{\lambda }^2}$$

Second, assuming that when $S{W}_{}^{Cl\ast }$ is maximum, the thresholds can be obtained as $f\left({\theta }_{13}\right)$ and $f\left({\theta }_{15}\right)$, respectively, and the calculation leads to:

$$\begin{array}{cc}& f\left({\theta }_{15}\right)=\frac{\begin{array}{c}6919+20996\lambda +28260{\lambda }^2-7936{\lambda }^3-57680{\lambda }^4\\ -1344{\lambda }^5+31424{\lambda }^6-12800{\lambda }^7+1536{\lambda }^8\\ \end{array}}{\begin{array}{c}2\left(-17123-18888\lambda +9728{\lambda }^2-114528{\lambda }^3+115808{\lambda }^4\right)\\ \left(-40064{\lambda }^5+10496{\lambda }^6-2560{\lambda }^7+256{\lambda }^8\right)\\ \end{array}}\hfill \\ \hfill & +\frac{\sqrt{\begin{array}{c}13205277+195094176\lambda +989911000{\lambda }^2+2034783984{\lambda }^3\\ +1903277008{\lambda }^4+1906200064{\lambda }^5+1553695296{\lambda }^6\\ -3013520128{\lambda }^7-2779885056{\lambda }^8+2850256896{\lambda }^9\\ +165004288{\lambda }^{10}-1167515648{\lambda }^{11}+880508928{\lambda }^{12}-\\ 409468928{\lambda }^{13}+115408896{\lambda }^{14}-17301504{\lambda }^{15}+1048576{\lambda }^{16}\\ \end{array}}}{\begin{array}{c}\left(-17123-18888\lambda +9728{\lambda }^2-114528{\lambda }^3+115808{\lambda }^4\right)\\ {\left(-40064{\lambda }^5+10496{\lambda }^6-2560{\lambda }^7+256{\lambda }^8\right)}^2\\ \end{array}}\hfill \end{array}$$

Finally, when the conditions $S{W}_{}^{Dl\ast }>S{W}_{}^{Dr\ast }$, $S{W}_{}^{Dl\ast }>S{W}_{}^{Cl\ast }$ and $S{W}_{}^{Dl\ast }>S{W}_{}^{Cr\ast }$ are satisfied, the threshold is calculated as

$$\begin{array}{cc}& f\left({\theta }_{16}\right)=\frac{\begin{array}{c}255560-250993\lambda -910394{\lambda }^2+341384{\lambda }^3-261256{\lambda }^4\\ +135940{\lambda }^5+10616{\lambda }^6-11424{\lambda }^7+1152{\lambda }^8\\ \end{array}}{\begin{array}{c}4\left(-456908-212005\lambda -47591{\lambda }^2-120804{\lambda }^3+76528{\lambda }^4\right)\\ \left(-21412{\lambda }^5+6484{\lambda }^6-1536{\lambda }^7+144{\lambda }^8\right)\\ \end{array}}\hfill \\ \hfill & +\frac{\sqrt{\begin{array}{c}40541017104+631553052060\lambda +3359904177429{\lambda }^2+6653705308272{\lambda }^3\\ +2866217167384{\lambda }^4-214899692560{\lambda }^5+758071383592{\lambda }^6-1109648704480{\lambda }^7\\ +130466172992{\lambda }^8+61985827968{\lambda }^9-19821723440{\lambda }^{10}+13833306240{\lambda }^{11}\\ -4322392704{\lambda }^{12}-351046656{\lambda }^{13}+364393728{\lambda }^{14}-58226688{\lambda }^{15}+2985984{\lambda }^{16}\\ \end{array}}}{\begin{array}{c}4\left(-456908-212005\lambda -47591{\lambda }^2-120804{\lambda }^3+76528{\lambda }^4\right)\\ {\left(-21412{\lambda }^5+6484{\lambda }^6-1536{\lambda }^7+144{\lambda }^8\right)}^2\\ \end{array}}\hfill \end{array}$$

We plot the calculated thresholds, as shown in Fig. 28.

Fig. 28.

Optimal social welfare

The proof is over.