Game Analysis of Different Transportation Modes in a Corridor Considering Carbon Emission Costs

Abstract

This study aims to elucidate the implications of carbon peak and carbon neutrality targets on corridor logistics, with a focus on the burgeoning high-speed rail express sector. Acknowledging carbon emissions taxation as an integral component of transportation costs, it examines the competitive dynamics among four cargo transport modes: high-speed rail (HSR), conventional rail, aviation, and road transport. Within a strategic game framework, carriers are analyzed based on freight rates, speed, safety, reliability, and convenience to assess their utility. A dynamic game model and a solution approach are developed, aiming to optimize freight-related variables, maximize carriers’ generalized profits, and enhance shippers’ utility. Empirical validation is provided through case studies in Xi’an and Lanzhou, northwest China, affirming the model’s efficacy. The findings reveal the strong competitive edge of the high-speed rail express in the corridor, offering valuable insights for carrier pricing strategies, emission tax rate setting, and macro-policy adjustments in the transport sector.

1. Introduction

Entering the 21st century, a surge in economic activities and e-commerce expansion has led to a growing demand for small-batch, multi-batch, and time-sensitive goods in the logistics market. In response, logistics companies specializing in express freight are vigorously enhancing their service quality and speed. This effort challenges the traditional express freight models and unveils new prospects for the advancement of high-speed rail freight. Managed by China Railway Express, high-speed rail express signifies a significant transformation in railway freight transportation, leveraging daily high-speed rail EMU trains for the expedited delivery of goods. Launched initially on the Beijing–Shanghai high-speed railway in early 2014, this service expanded rapidly. By the end of 2019, it had achieved comprehensive coverage across major cities, including municipalities directly under the Central Government, provincial capitals, and key prefecture-level cities in the central and eastern regions, marking a significant shift in competitive dynamics among various transportation modes within the express freight corridor.

The innovation of this study lies in the introduction of a novel dynamic game model that analyzes the competitive dynamics among different transportation modes within the context of the “Belt and Road” initiative, particularly emphasizing the advantages of high-speed rail express under carbon emission tax conditions. The academic contributions include:

• A comprehensive utility function that integrates five key factors: transportation cost, speed, safety, reliability, and convenience.
• The introduction of carbon emission tax as a crucial component of transportation costs, with a quantitative analysis of its impact on transportation mode selection.
• Empirical validation of the model’s effectiveness, providing theoretical support for policymakers and logistics enterprises.

Numerous studies highlight the high-speed railway (HSR)’s significant contribution to emission reduction, surpassing other transportation modes. Fu et al. (2013) [1] delineated the carbon emission phases throughout HSR’s lifecycle—infrastructure construction, operation, and recovery—providing specific formulas for calculating carbon dioxide emissions at each phase. Despite higher emissions during the construction phase, HSR’s overall lifecycle emissions are lower than highways’, showing a decreasing trend with higher ridership rates, thus affirming HSR’s effective emission reduction capabilities. Alfonso et al. (2016) [2] discussed the balance between carbon emissions saved by shifting transport demand from air to HSR and those generated by increased HSR demand, considering the established rapid air transport framework. They argued that the environmental impact of air travel on local air pollution and climate change varies with flight duration, crew size, fuel usage, and the airport’s proximity to urban centers, especially for suburban airports. Conversely, HSR’s environmental impact largely hinges on the mix of power generation sources, track length, and energy consumption. Wang et al. (2022) [3] examined HSR’s effects on the economy and environment in China’s impoverished counties, assessing the environmental benefits difference between conventional railways and HSR. Utilizing a time-varying difference-in-difference approach, they quantified HSR’s impact on carbon dioxide emissions, revealing a 10.3% emission reduction in counties with railway access compared to those without, with HSR operations specifically showing greater emission reductions than conventional railways.

The pricing strategy of high-speed rail express plays a pivotal role in determining the variety and volume of goods it attracts. Analyzing this pricing issue requires a clear understanding of the relationships between various transportation modes and shippers within the express freight corridor. Research in the cargo transportation market predominantly centers on consignors’ transportation mode selection and the competition among different modes. Doyme et al. [4] (2019) utilized an n-player non-cooperative game to simulate the equilibrium outcomes of the Australian aviation market, estimating fares, frequencies, passenger numbers, and market shares for domestic competitors, thereby affirming the capability of game-theoretic models to mirror real-world transportation market dynamics. Talebian et al. [5] (2016) emphasized the significance of train technology in the competition between high-speed rail and airlines, proposing a three-stage game-theoretic model where the rail agency initially selects track technology and speed, followed by the determination of rail and airline frequencies, and ultimately, fare setting by each mode. Raturi et al. [6] (2019) explored competition along two Indian corridors, contrasting high-speed rail with bus and airline services through a three-stage entry game that assesses the high-speed rail entrant’s speed choice, fare and frequency settings by alternative modes, and high-speed rail’s market entry decisions. Arencibia [7] (2015) employed a discrete selection model to identify key determinants in goods transportation mode choice. As game theory matures, international scholars have increasingly applied its concepts to cargo transportation’s economic challenges. Gutiérrez-Hita and Ruiz-Rúa [8] (2019) introduced a quantity-setting duopoly model in a railway passenger market, examining the welfare implications of duopoly versus monopoly scenarios based on operational costs and access charges. Sun B and Xu Z [9] (2023) developed an evolutionary game model to refine decision-making in the interplay between civil aviation and high-speed rail, adjusting passenger ticket and carbon trading prices to foster the growth of the high-speed transportation system. Yang et al. [10] (2023) developed a comprehensive optimization model for various transportation tasks and modes considering carbon emissions, aiming at transportation distance, transportation time, and carbon emissions. They optimized the path, transportation mode, and carbon dioxide emissions, providing a reference for logistics enterprises to carry out multimodal transportation. Javad Akhlaghinia et al. [11] (2021) implemented a distributed architecture for an environmentally friendly traffic guidance system using an agent-based paradigm. The system utilizes real-time traffic data collection from multiple sources and provides multimodal transportation route recommendations to travelers through dedicated applications.

Echoing the research methodologies of international scholars, Chinese experts have also extensively employed game theory tools to explore competitive dynamics. Yang and Zhang [12] (2012) devised a modified Hotelling–Bertrand model to examine product differentiation, positioning airlines as profit maximizers against high-speed rail (HSR) operators, who aim to optimize a blend of welfare and profit due to their public ownership. Their analysis of a 600 km corridor in China revealed that increasing welfare weights leads to reduced HSR ticket prices and airfares, a decrease in flight frequency, and an increase in HSR frequency, with both flight and HSR frequencies rising as the business passenger segment expands. Liu [13] (2008) formulated a Stackelberg game model for transportation mode selection, grounded in the fundamental principles of logistics transportation mode choice and the determinants influencing shippers’ preferences. Zhu et al. [14] (2021) explored the competitive effects of frequent flight delays on HSR and air transport, developing a game-theoretical model to deduce the equilibrium strategies of both parties. Their findings indicate a gradual decline in revenues for both HSR and air transport as flight delays escalate. Luo [15,16] (2012) analyzed cargo transportation mode selection through behavioral analysis, constructing a Stackelberg game model based on carrier incentive behavior, corroborated with practical examples. Ma et al. [17] (2021) addressed passengers’ service level sensitivity, price, and other vital factors, investigating if a revenue-sharing contract could harmonize the interests between air and high-speed rail transport, proposing an optimal revenue distribution scheme conducive to an ideal state of intermodal integration. Wang et al. [18] (2014) developed three game models to scrutinize the rivalry between two carriers in the container express market, further expanding the application of game theory in assessing transportation mode competition. Tian et al. [19] (2023) used Stackelberg game theory to establish a game model for air traffic control and the role played by airlines in the process of scheme selection. By obtaining the preferences of airlines for the initial allocation scheme, the range of options available for selection was narrowed, and the optimal allocation scheme was ultimately determined.

In summary, the scholarly investigation into the cargo transportation market primarily concentrates on analyzing pricing strategies across different transportation modes, variations in competition and market share within freight corridors, and the decision-making behaviors and preferences of consignors. Game theory has been extensively applied to dissect the competitive dynamics among these modes. However, despite the burgeoning express cargo transportation market and the advent of high-speed rail express transportation, there is a notable scarcity of literature addressing the competition that includes high-speed rail express within the express cargo transportation sphere. Similarly, few studies have considered carbon emissions as a criterion for examining the competitive landscape among express transportation options. Addressing this gap, the current study leverages game theory methodologies to explore the competitive interactions and market share distribution between high-speed rail express transportation and other modes, aiming to identify optimal freight pricing.

This study not only reveals the economic benefits of high-speed rail express but also emphasizes its importance in environmental protection. Compared to road and air transport, high-speed rail express has significantly lower carbon emissions, effectively reducing greenhouse gas emissions. Additionally, although conventional railways may be less competitive in fast freight services, their high-capacity transport capabilities offer advantages in long-distance transportation. Road transport, on the other hand, maintains a crucial position in short-distance and regional freight due to its flexibility. However, its high emission characteristics will pose greater environmental challenges in the future. By adopting reasonable pricing strategies and selecting appropriate transportation modes, logistics companies can meet market demands while achieving sustainable development goals. This research endeavors to offer a theoretical framework for transportation enterprises navigating market competition and to assist government bodies in refining emission tax rates and regulating the transportation market.

2. Problem

Express goods typically encompass small parcels, mail, and perishable food items. Given the nature of these goods, high-speed rail (HSR), conventional rail, air transport, and highways each present unique advantages and drawbacks within the same express freight corridor. This diversity leads to distinct service characteristics, guiding consignors to select transportation modes that align with their specific requirements. The choice among these modes is influenced by seven key factors: freight rates, timeliness, safety, reliability, capacity, accessibility, and convenience.

This study posits that companies and shippers across all transportation modes act rationally in the economic sense, striving to maximize their interests through decision making and competition. In the competitive arena, carriers set their freight rates based on inherent characteristics and operational costs, crafting a strategic freight rate framework and adapting their competitive tactics in response to market dynamics, all with the aim of capturing a larger market share. Regardless of the transportation mode, carriers’ ultimate objective is to minimize costs and maximize profits. They do not operate in isolation but are subject to external competition and market shifts. The interaction among transportation modes in the express freight market constitutes a strategic game, where enterprises reveal pricing strategies to attract shippers, who, in turn, share their requirements to select the most suitable transportation mode. Hence, the game is characterized by complete information and dynamic interaction. In the initial stage, various transportation modes develop freight rate strategies aimed at maximizing their revenue, thereby optimizing their overall profit. Subsequently, shippers assess the freight rates across different modes and choose the one that best suits their express goods’ needs, with the goal of maximizing their utility.

The foundational elements of game theory encompass rules, players, strategies, sequence, payoffs, and information. In the context of this study, the shipper acts as a game participant, making strategic decisions. Drawing from the preceding analysis, the utility function emerges as the most apt mathematical instrument for quantifying the consignor’s satisfaction. This research employs the freight rate, timeliness, safety, reliability, and convenience of express delivery as critical factors influencing the shippers’ choices. Consequently, a utility function for shippers is formulated, serving as the payoff for their involvement in the game.

3. Game Model

3.1. Shipper’s Payoff

The initial step involves quantifying the five characteristic indicators as follows

(1)

Freight rate: $p_k$ represents the freight rate of mode $k$. The freight rate is the actual cost that the shipper needs to pay when choosing the transportation service. In this paper, freight rate is used as a decision variable in the game between various modes of transportation, and in the game between the shipper and various modes of transportation.

(2)

Timeliness: $s_k$ represents the timeliness index of mode $k$. In this paper, the total transportation time refers to the time from the collection of express goods to the arrival of express goods at their destination. As shown in the diagram in Figure 1.

In Figure 1, O is the collection point of the goods, D is the destination of the goods, and A and B, respectively, represent two different freight stations of a certain mode of transportation separately. This method subdivides the transportation process of goods. Not only the collection from O to A but also the distribution from B to O is considered, which makes the quantification of the timeliness index more accurate. The timeliness $s_k$ is inversely proportional to the total transportation time. So,

$$s_{\mathrm{k}}={\displaystyle \frac{1}{(\frac{L_k}{v_k}+t_k^s+t_k^p){{\displaystyle \lambda }}_{OD}}}$$

(1)

where $s_k$ is the transportation timeliness index by mode $k$. $L_k$ is the transport distance by mode $k$. $v_k$ represents the average speed by mode $k$. $t_k^s$ represents the time required from O to A. $t_k^p$ represents the time required to reach point D. ${{\displaystyle \lambda }}_{OD}$ is the time value of express goods.

Many scholars have studied methods for determining the time value of goods [20,21]. According to the characteristics of the research question, this paper refers to the method in Construction Project Economic Evaluation Methods and Parameters (Second Edition) issued by the Former State Planning Commission and the Ministry of Construction in 1993, which gives the calculation formula for the time value of express cargo as

$${{\displaystyle \lambda }}_{OD}={\displaystyle \frac{{{\displaystyle \mathrm{Pr}}}_k\times {{\displaystyle R}}_{ }}{365\times 16}}$$

(2)

where ${{\displaystyle \mathrm{Pr}}}_k$ represents the average price of goods transported by mode $k$; $R$ is the social discount rate.

(3)

Safety: ${{\displaystyle d}}_k$ represents the safety index of the transportation mode $k$. The safe delivery of goods means minimizing the rate of damaged goods and the rate of transportation accidents. Therefore, the rate of intact goods and the rate of transportation accidents are used to quantify the safety index:

$${{\displaystyle d}}_k={{\displaystyle w}}_k\times ({{\displaystyle 1}}_{ }-{{\displaystyle gl}}_k)\times {{\displaystyle 100\%}}_{ }$$

(3)

where ${{\displaystyle w}}_k$ represents the rate of intact goods by mode $k$. And ${{\displaystyle gl}}_k$ represents the accident rate by mode $k$.

(4)

Reliability: ${{\displaystyle e}}_k$ represents the reliability index of the transportation mode $k$. According to the above analysis, weather is the factor that has the greatest influence on transportation modes from the outside world. This paper mainly discusses the influence of weather changes on the transportation process and uses the punctuality rate of the transportation mode to quantify the reliability index.

(5)

Convenience: ${{\displaystyle c}}_k$ represents the convenience index of the transportation mode $k$. The convenience index will be quantified according to the data obtained from the survey of shippers.

The above five utility measurement indicators, timeliness, safety, reliability and convenience, are all related to transportation services. Assuming that the carrier does not consider the incentive problem temporarily when formulating the freight rate strategy set, then the service quality is fixed. But if a consignor chooses a lower freight rate, they may have to sacrifice a certain service quality. If the goods mainly receive high-quality service, it is necessary to pay a correspondingly higher freight rate. For high-net-worth goods with strong timeliness requirements, sometimes the consignor will do whatever it takes to ensure the fastest delivery of the goods. Therefore, as shown in Figure 2, the freight rate and the consignor’s utility are similar to a normal correlation.

Generally, there are two principles of addition and multiplication used to calculate utility [22]. This paper upholds that reliability is independent of the other service quality indicators. In fact, when other characteristics and reliability of a certain mode are better at the same time, the service quality utility of this mode can be the best, so the reliability and other service characteristics should be multiplied. Then, the utility function of the shipper is established for the mode of transportation as follows:

$${{\displaystyle u}}_k({{\displaystyle p}}_k)={{\displaystyle \alpha }}_1\times {{\displaystyle e}}^{{\displaystyle \frac{-{({{\displaystyle p}}_k)}^2+{{\displaystyle \sigma }}_k\times {{\displaystyle p}}_k}{{{\displaystyle \rho }}_k}}}+({{\displaystyle \alpha }}_3\times {{\displaystyle d}}_k+{{\displaystyle \alpha }}_2\times {{\displaystyle s}}_k+{{\displaystyle \alpha }}_5\times {{\displaystyle c}}_k)\times {{\displaystyle e}}_k$$

(4)

where ${{\displaystyle u}}_k$ represents the utility of the consignor of cargo for transportation mode $k$; ${{\displaystyle \alpha }}_1$, ${{\displaystyle \alpha }}_2$, ${{\displaystyle \alpha }}_3$, ${{\displaystyle \alpha }}_4$, ${{\displaystyle \alpha }}_5$ represent the weights of the freight rate, timeliness, safety, reliability, and convenience characteristics of transportation mode $k$, respectively. In this paper, the traditional analytic hierarchy process (AHP) method will be used to determine the value of weight. ${{\displaystyle \rho }}_k$ and ${{\displaystyle \sigma }}_k$ are undetermined coefficients. In general, ${{\displaystyle \rho }}_k$ takes a relatively large number, and ${{\displaystyle \sigma }}_k$ takes the average value of the initial tariff strategy set in order to reduce the calculation error.

3.2. Payoff of Four Modes of Transportation

As participants in the game, the four modes of transportation will always pursue the maximization of their respective benefits. Let the set of transportation modes be $N=(\mathrm{1,2},\mathrm{3,4})$; $k=\mathrm{1,2},\mathrm{3,4}$ correspond to general-speed railway, HSR, air and highway, respectively. The set of freight rates given by the operator of the transportation mode is ${{\displaystyle P}}_k=\{{{\displaystyle p}}_k^1,{{\displaystyle p}}_k^2,\cdots ,{{\displaystyle p}}_k^{{{\displaystyle m}}_k}\}$. The initial freight rate set of the game can be given according to the actual freight rate in the current market and its reasonable floating range. ${{\displaystyle p}}_k^i$ is a specific selected freight rate in set ${{\displaystyle P}}_k$. Through the foregoing summary and analysis, it can be concluded that, in a broad sense, the revenue of transportation modes is equal to the revenue minus the cost, in which the cost includes transportation cost and risk cost.

At the same time, based on the strategic goal of green sustainable development, from a government perspective, in order to achieve carbon peaking and carbon neutrality goals, a carbon tax policy can be implemented to regulate the market. As far as transportation is concerned, the carbon emissions of different transportation modes are different, so the emission tax should be added to the cost of the transportation modes. Under the government’s macro control, the market should be the main body, and consignors should choose the most favorable mode of transportation for themselves. The revenue function of express freight mode $k$ is established as follows, $\forall k\in N$:

$${{\displaystyle f}}_k({{\displaystyle p}}_k^i)={{\displaystyle p}}_k^i\times {{\displaystyle q}}_k^i-{{\displaystyle b}}_k\times {{\displaystyle q}}_k^i\times {{\displaystyle L}}_k-({{\displaystyle g}}_k+{{\displaystyle w}}_k\times ({{\displaystyle t}}_k^{sh}-{{\displaystyle t}}_k^x)\times {{\displaystyle \lambda }}_{OD}+{{\displaystyle Tax}}_k) i=\mathrm{1,2},\cdots ,m_k$$

(5)

$${{\displaystyle {{\displaystyle 0}}_{ }<p}}_k^{\min }{{\displaystyle <}}_{ }{{\displaystyle p}}_k^i{{\displaystyle <}}_{ }{{\displaystyle p}}_k^{\max }$$

(6)

where ${{\displaystyle q}}_k^i$ represents the freight volume shared when the operator of the transportation mode $k$ chooses a pure strategy rate ${{\displaystyle p}}_k^i$. $L_k$ represents the distance of transportation mode $k$. ${{\displaystyle \mathrm{g}}}_{\mathrm{k}}$ represents the constant cost of the transportation mode $k$. ${{\displaystyle \mathrm{b}}}_{\mathrm{k}}$ represents the unit variable cost of the transportation mode $k$. ${{\displaystyle w}}_k$ represents the delay rate of transportation mode $k$. ${{\displaystyle t}}_k^{sh}$ represents the actual transport time of the transportation mode $k$. ${{\displaystyle t}}_k^x$ represents the stipulated transport time. ${{\displaystyle Tax}}_k$ represents emission tax. ${{\displaystyle p}}_k^{\min }$ represents the upper limit of freight rates and ${{\displaystyle p}}_k^{\max }$ represents the lower limit of freight rates. The corresponding equation is ${{\displaystyle f}}_k({{\displaystyle p}}_k^i){{\displaystyle {{\displaystyle \in }}_{ }F}}_k=({{\displaystyle p}}_k^i|{\displaystyle k}\in N,i=1,2,\cdots ,{{\displaystyle m}}_k)$.

In this paper, the Logit model is used to solve the freight volume ${{\displaystyle q}}_k^i$ of transportation mode $k$. The form of the Logit model when combined with the problem in this paper is as follows, $\forall k\in N$:

$${{\displaystyle q}}_k^i={{\displaystyle Q}}_{ }\times {\displaystyle \frac{\exp ({{\displaystyle u}}_k({{{\displaystyle p}}_k}^i))}{{\displaystyle \sum _k^{ }\exp ({{\displaystyle u}}_k({{{\displaystyle p}}_k}^i))}}} i=\mathrm{1,2},\cdots ,m_k$$

(7)

where $Q$ is the total amount of goods in all modes in the express freight corridor. ${{\displaystyle u}}_k({{{\displaystyle p}}_k}^i)$ is the utility of the consignor on mode $k$ when the price of mode $k$ is ${{\displaystyle p}}_k^i$.

3.3. Dynamic Game Model

The overarching goal of maximizing generalized revenue for each transportation mode forms the crux of their optimal response to the competitive landscape. Within this dynamic game framework, operators of transportation modes initiate the process by presenting a decision set. The consignor then evaluates this set, makes a choice, and communicates this decision back to the mode operator. Subsequently, the operator revises their optimal strategy, steering the game towards an equilibrium state. The freight rate acts as the critical link connecting the objective targets. This study zeroes in on the distribution of freight volume shares among transportation modes within the express freight corridor, aiming to enhance both the revenue of each mode and the utility for the consignor. To this end, the following assumptions are posited:

(1)

All participants, including modes of transport and consignors, are presumed to be rational in the economic sense, engaging in competition with the primary aim of maximizing their personal benefits.

(2)

Acting on rational grounds, the consignor selects the transportation mode that promises the highest utility, disregarding influences from non-utility factors, such as previous engagements with any other transport mode.

(3)

The transport capacity available from various modes within the express delivery corridor exceeds the total demand from consignors.

Each mode of transportation will investigate and integrate the information in the market before formulating the freight rate strategy and will also collect information about competitors via various sources. In today’s information era, it is not difficult for each mode of transportation to acquire information about competitors’ costs and benefits. Furthermore, the timing of the freight rate strategy set by the operators of various modes of transportation is relatively static, so a complete information static game is played among various modes of transportation at the initial stage of the game. When choosing a mode of transportation, the consignor will definitely consider the maximization of his own utility and then observe the freight rates of each mode of transportation. At this time, the owner’s utility will feed back to the operators of each mode of transportation, so that they can adjust the freight rate strategy. Therefore, the four modes of transportation and the consignor are regarded as five game participants and put in a game space, so the game between various modes of transportation and consignor are completely information dynamic. Tariff ${{\displaystyle P}}_k=\{{{\displaystyle p}}_k^1,{{\displaystyle p}}_k^2,\cdots ,{{\displaystyle p}}_k^{{{\displaystyle m}}_k}\}$ is the set of strategies owned by all players, and each objective function is exactly the payoff of each player. Set a game as

$${{\displaystyle G}}_{ }{{\displaystyle =}}_{ }\{({{\displaystyle P}}_k),({{\displaystyle F}}_k),({{\displaystyle U}}_k),{{\displaystyle N}}_{ }\}.$$

(8)

$\forall k\in N$, the operator of transportation mode $k$ will only choose a freight rate in ${{\displaystyle P}}_k$ each time it participates in the game. Therefore, any ${{\displaystyle p}}_k^i\in {{\displaystyle P}}_k$ is the pure tariff strategy of the operator of transportation mode $k$, and $i=\mathrm{1,2},\cdots ,m_k$. ${{\displaystyle F}}_k=({{\displaystyle p}}_k^i|k\in N,i=1,2,\cdots ,{{\displaystyle m}}_k)$ is the income function under the pure situation ${\displaystyle \prod _{k\in N}{{\displaystyle P}}_k}=\{{{\displaystyle P}}_k|k=1,2,\cdots ,n\}$. For any transportation mode, once the tariff strategy set is given, the consignor can determine a set of utility values according to the utility function. The consignor’s utility value set is ${{\displaystyle U}}_k=\{{{\displaystyle u}}_k({{\displaystyle p}}_k^1),{{\displaystyle u}}_k({{\displaystyle p}}_k^2),\cdots ,{{\displaystyle u}}_k({{\displaystyle p}}_k^{{{\displaystyle m}}_k})\}$ where the maximum value in ${{\displaystyle U}}_k$ is the maximum utility of the consignor to the transportation mode $k$. So,

$${{\displaystyle u}}_k({{{\displaystyle p}}_k}^i)={{\displaystyle \alpha }}_1\times {{\displaystyle e}}^{{\displaystyle \frac{-{({{{\displaystyle p}}_k}^i)}^2+{{\displaystyle \sigma }}_k\times {{{\displaystyle p}}_k}^i}{{{\displaystyle \rho }}_k}}}+({{\displaystyle \alpha }}_3\times {{\displaystyle d}}_k+{{\displaystyle \alpha }}_2\times {{\displaystyle s}}_k+{{\displaystyle \alpha }}_5\times {{\displaystyle c}}_k)\times {{\displaystyle e}}_k.$$

(9)

Transposing Equation (9) leads to Equation (10):

$${{\displaystyle u}}_k({{{\displaystyle p}}_k}^i)-{{\displaystyle \Delta }}_k^{hf}={{\displaystyle \alpha }}_1\times {{\displaystyle e}}^{{\displaystyle \frac{-{({{{\displaystyle p}}_k}^i)}^2+{{\displaystyle \sigma }}_k\times {{{\displaystyle p}}_k}^i}{{{\displaystyle \rho }}_k}}}.$$

(10)

Supposing that when the consignor’s utility to the mode of transport $k$ reaches the maximum, the optimal solution of the consignor’s utility function freight rate is ${{\displaystyle p}}_k^{h\ast }$. From the concept of pure strategy Nash equilibrium [23], for any transportation mode $\forall k\in N$, if ${{\displaystyle p}}_k^{y\ast }$ exists, it is right that all other modes of transport have a pure strategy tariff combination ${{\displaystyle p}}_{-k}^{*}=\left\{{{\displaystyle p}}_j^{*}\right|j\in N,j\ne k\}$, and the optimal response pure strategy satisfies the formula (11)

$${{\displaystyle F}}_k({{\displaystyle p}}_k^{y\ast }){{\displaystyle \ge }}_{ }{{\displaystyle F}}_k({{\displaystyle p}}_k^i,{{\displaystyle p}}_{-k}^{*}) i=\mathrm{1,2},\cdots ,m_k$$

(11)

$\left\{{{\displaystyle p}}_k^{y\ast }{{\displaystyle |}}_{ }{{\displaystyle \mathrm{k}}}_{ }{{\displaystyle \in }}_{ }{{\displaystyle N}}_{ }\right\}$ is called the pure strategy Nash equilibrium of the game between transportation modes. ${{\displaystyle p}}_{-k}^{*}=\{{{\displaystyle p}}_j^{*}{{\displaystyle |}}_{ }{{\displaystyle j}}_{ }\in {{\displaystyle N}}_{ }{{\displaystyle ,j}}_{ }{{\displaystyle \ne }}_{ }{{\displaystyle k}}_{ }\}$ is the optimal response strategy for all transportation modes except transportation mode $k$. Obviously, the game problem studied in this paper is a finite game, so according to the existence theorem of the Nash equilibrium, if there is no pure strategy Nash equilibrium, then there must be a mixed strategy Nash equilibrium. This is assuming that operators of various modes of transport have a probability distribution in their respective tariff strategies ${{\displaystyle \omega }}_k=({{\displaystyle \omega }}_k^1,{{\displaystyle \omega }}_k^2,\cdots {{\displaystyle \omega }}_k^i\cdots ,{{\displaystyle \omega }}_k^{{{\displaystyle m}}_k})$ [24]. ${{\displaystyle \omega }}_k^i$ means for all $i=\mathrm{1,2},\cdots ,m_k$, the probability that the operator of transportation mode $k$ chooses the freight rate ${{\displaystyle p}}_k^i$. And for any probability distribution, ${{\displaystyle \omega }}_k$ should satisfy the following constraint, $\forall k\in N$:

$${{\displaystyle \varpi }}_k=\{{{\displaystyle \omega }}_k{{\displaystyle |}}_{ }{{\displaystyle 0}}_{ }{{\displaystyle \le }}_{ }{{\displaystyle \omega }}_k^i{{\displaystyle \le }}_{ }{{\displaystyle 1}}_{ },{\displaystyle \sum _{i=1}^{{{\displaystyle m}}_k}{{\displaystyle \omega }}_k^i={{\displaystyle 1}}_{ }}\} i=\mathrm{1,2},\cdots ,m_k$$

(12)

Define the set of income ${{\displaystyle F}}_k={{\displaystyle (p}}_k^1,{{\displaystyle p}}_k^2,\cdots ,{{\displaystyle p}}_k^{{{\displaystyle m}}_k})$ is in a mixed strategy combination ${{\displaystyle \omega }}_k=({{\displaystyle \omega }}_k^1,{{\displaystyle \omega }}_k^2,\cdots {{\displaystyle \omega }}_k^i\cdots ,{{\displaystyle \omega }}_k^{{{\displaystyle m}}_k})$$\forall k\in N$, and the expectation is

$${{\displaystyle E}}_k({{\displaystyle \omega }}_k)={{\displaystyle E}}_k({{\displaystyle \omega }}_k^1,{{\displaystyle \omega }}_k^2,\cdots {{\displaystyle \omega }}_k^i\cdots ,{{\displaystyle \omega }}_k^{{{\displaystyle m}}_k})={{\displaystyle \omega }}_k•{{\displaystyle F}}_k={({{\displaystyle p}}_k^1,{{\displaystyle p}}_k^2,\cdots ,{{\displaystyle p}}_k^{{{\displaystyle m}}_k})}^T={\displaystyle \sum _{i=1}^{{{\displaystyle m}}_k}{{\displaystyle \omega }}_k^i{{\displaystyle f}}_k{{\displaystyle (p}}_k^i)}$$

(13)

If there is a ${{\displaystyle \omega }}_k^{*}$, its combination of mixed tariff strategy for all other modes of transport ${{\displaystyle \omega }}_{-k}^{*}=\{{{\displaystyle \omega }}_j^{*}{{\displaystyle |}}_{ }{{\displaystyle j}}_{ }\in {{\displaystyle N}}_{ }{{\displaystyle ,j}}_{ }{{\displaystyle \ne }}_{ }{{\displaystyle k}}_{ }\}$ is the optimal response mixed tariff strategy, and it satisfies the following formula

$${{\displaystyle E}}_k({{\displaystyle \omega }}_k^{*}){{\displaystyle \ge }}_{ }{{\displaystyle E}}_k({{\displaystyle \omega }}_k,{{\displaystyle \omega }}_{-k}^{*}){{\displaystyle \forall }}_{ }{{\displaystyle \omega }}_k{{\displaystyle \in }}_{ }{{\displaystyle \varpi }}_k$$

(14)

So, we can say that $\left\{{{\displaystyle \omega }}_k^{*}{{\displaystyle |}}_{ }{{\displaystyle \mathrm{k}}}_{ }{{\displaystyle \in }}_{ }{{\displaystyle N}}_{ }\right\}$ is the mixed strategy Nash equilibrium of the game.

In summary, the dynamic game model of the express freight corridor is established as the following:

$${{\displaystyle G}}_{ }=[{{\displaystyle N}}_{ },({{\displaystyle P}}_k),({{\displaystyle F}}_k),({{\displaystyle U}}_k)]$$

(15)

First of all, various transportation modes conduct a complete information static game with the goal of maximizing their own generalized benefits:

$$\begin{array}{l}{{\displaystyle \max }}_{ }{{\displaystyle f}}_k({{\displaystyle p}}_k^{y\ast },{{\displaystyle p}}_{-k}^{*})\\ \left\{\begin{array}{l}{{\displaystyle s.t{.}^{ }p}}_k^i\in {{\displaystyle P}}_k\\ {{\displaystyle \begin{array}{cc} & \end{array}0<p}}_k^{\min }{{\displaystyle <}}_{ }{{\displaystyle p}}_k^i{{\displaystyle <}}_{ }{{\displaystyle p}}_k^{\max }\\ {{\displaystyle \begin{array}{cc} & \end{array}0}}_{ }{{\displaystyle <}}_{ }{{\displaystyle q}}_k^i{{\displaystyle <}}_{ }{{\displaystyle Q}}_{ }\\ i=1,2,\cdots ,m_k,\forall k\in N\end{array}\right.\end{array}$$

(16)

Then, the consignor observes the strategies of various modes of transport with the goal of maximizing his own utility and participates in the game.

$$\begin{array}{l}{{\displaystyle \max }}_{ }{{\displaystyle u}}_k({{\displaystyle p}}_k^{h\ast },{{\displaystyle p}}_{-k}^{*})\\ \left\{\begin{array}{l}s.t.{{\displaystyle { }^{ }p}}_k^i\in {{\displaystyle P}}_k\\ \begin{array}{cc} & \end{array}i=1,2,\cdots ,m_k,\forall k\in N\end{array}\right.\end{array}$$

(17)

In Formula (17),

$${{\displaystyle f}}_k({{\displaystyle p}}_k^i)\ge {{\displaystyle p}}_k^i\times {{\displaystyle q}}_k^i-{{\displaystyle b}}_k\times {{\displaystyle q}}_k^i\times {{\displaystyle L}}_k-({{\displaystyle g}}_k+{{\displaystyle w}}_k\times ({{\displaystyle t}}_k^{sh}-{{\displaystyle t}}_k^x)\times {{\displaystyle \lambda }}_{OD}+{{\displaystyle Tax}}_k)$$

(18)

$${{\displaystyle f}}_k({{\displaystyle p}}_k^i)\le {{\displaystyle p}}_k^i\times {{\displaystyle q}}_k^i-{{\displaystyle b}}_k\times {{\displaystyle q}}_k^i\times {{\displaystyle L}}_k-({{\displaystyle g}}_k+{{\displaystyle w}}_k\times ({{\displaystyle t}}_k^{sh}-{{\displaystyle t}}_k^x)\times {{\displaystyle \lambda }}_{OD}+{{\displaystyle Tax}}_k)$$

$${{\displaystyle u}}_k({{{\displaystyle p}}_k}^i)\ge {{\displaystyle \alpha }}_1\times {{\displaystyle e}}^{{\displaystyle \frac{-{({{{\displaystyle p}}_k}^i)}^2+{{\displaystyle \sigma }}_k\times {{{\displaystyle p}}_k}^i}{{{\displaystyle \rho }}_k}}}+{{\displaystyle \Delta }}_k^{hf}$$

(19)

$${{\displaystyle u}}_k({{{\displaystyle p}}_k}^i)\le {{\displaystyle \alpha }}_1\times {{\displaystyle e}}^{{\displaystyle \frac{-{({{{\displaystyle p}}_k}^i)}^2+{{\displaystyle \sigma }}_k\times {{{\displaystyle p}}_k}^i}{{{\displaystyle \rho }}_k}}}+{{\displaystyle \Delta }}_k^{hf}$$

$$i=\mathrm{1,2},\cdots ,m_k, \forall k\in N$$

(20)

In theory, when ${{\displaystyle p}}_k^{h\ast }$ and ${{\displaystyle p}}_k^{y\ast }$ are equal, it is considered that the game reaches equilibrium and the equilibrium solution is ${{\displaystyle p}}_k^{*}={{\displaystyle p}}_k^{h\ast }={{\displaystyle p}}_k^{y\ast }$.

In that case, we suppose that the optimal sharing rate of transportation mode $k$ when the game reaches equilibrium is ${{\displaystyle r}}_k^{\ast }$, and the transportation volume shared by transportation mode $k$ is ${{\displaystyle q}}_k^{\ast }$, when the game is in equilibrium:

$${{\displaystyle r}}_k^{\ast }={{\displaystyle q}}_k^{\ast }{{\displaystyle /}}_{ }{{\displaystyle Q}}_{ }\times {{\displaystyle 100\%}}_{ }$$

(21)

The solution to the game is the combination of the optimal strategies of all participants, that is, the state when ${{\displaystyle p}}_k^{y\ast }$, the optimal freight rate of the various modes of transportation games is theoretically equal to ${{\displaystyle p}}_k^{h\ast }$, the freight rate with the maximum utility of the owner. Nevertheless, equality is difficult to realize. Hence, the difference between ${{\displaystyle p}}_k^{y\ast }$ and ${{\displaystyle p}}_k^{h\ast }$ is close to the infinitesimal value instead of equal to zero. In the equilibrium state of the game, no player has the motivation to change his strategy.

According to the previous analysis and the actual situation, when consignors choose the mode of transportation for the purpose of maximizing their own utility, because they want to obtain high-quality service, the utility will increase with the increase in freight charges. But there must be a cap ${{\displaystyle p}}_k^{h\ast }$ for the freight charges. Beyond this cap, the satisfaction of the consignor will decrease with the increase in freight charges. Therefore, various modes will not make the freight rate rise indefinitely, and they will not reduce the freight rate indefinitely because of the cost. In the game, the owner must choose one of the modes of transportation to transport the goods. Therefore, there is an optimal solution to the game problem in this paper, and the game can reach equilibrium.

In this model, the constraint is non-convex, so it is difficult to solve. According to the characteristics of the model, this paper takes the proportion of freight rate in the utility of the consignor as weight. According to the characteristics of the problem studied by the model, the advantage is that the appropriate iteration step can be determined. The process of solution is the process of constantly adjusting the freight rate strategy of the transportation mode to gradually approach the game equilibrium, which is also the process of the dynamic game. The solving steps are as follows:

Step 1: Set the iteration number $x=0$, give the initial tariff policy set ${{\displaystyle P}}_k^0$ of transportation mode $k$, and assign a quite small positive value to the elements in the vector ${{\displaystyle \epsilon }}_{ }^{ }=\left\{{{\displaystyle \epsilon }}_{\mathrm{k}}\right|{{\displaystyle k}}_{ }\in {{\displaystyle N}}_{ }\}$.

Step 2: For a given set of tariff strategies ${{\displaystyle P}}_k^x$, use Equation (9) to obtain the utility value ${{\displaystyle u}}_k^x$ and the optimal solution ${{\displaystyle p}}_k^{hx}$ of the freight owner to the mode $k$, then calculate the freight volume ${{\displaystyle q}}_k^{x,i}$ of the mode $k$ according to the logit model and finally, use Equation (5) to obtain the generalized revenue value ${{\displaystyle f}}_k^x$ and the optimal solution ${{\displaystyle p}}_k^{y\ast }$.

Step 3: Judge whether the convergence condition $\left|{{\displaystyle p}}_k^{h\ast }-\left.{{\displaystyle p}}_k^{y\ast }\right|\right.\le {{\displaystyle \epsilon }}_k$ is satisfied or not. If it is satisfied, the optimal solution ${{\displaystyle p}}_k^{y\ast }$ or ${{\displaystyle p}}_k^{h\ast }$, the freight volume ${{\displaystyle q}}_k^x$ and freight volume sharing rate ${{\displaystyle r}}_k^x$ are all output and algorithm stops. Otherwise, go to step 4.

Step 4: Weight the price in the set ${{\displaystyle P}}_k^x$ with the reciprocal of the number of elements ${{\displaystyle m}}_{ }$ in the price set ${{\displaystyle p}}_k^{x+1,i}={{\displaystyle ({{\displaystyle 1}}_{ }{{\displaystyle -}}_{ }}}_{ }{{\displaystyle 1}}_{ }{{\displaystyle /}}_{ }{{\displaystyle m}}_{ }{{\displaystyle )}}_{ }\times {{\displaystyle p}}_k^{x,i}+{{\displaystyle (}}_{ }{{\displaystyle 1}}_{ }{{\displaystyle /}}_{ }{{\displaystyle m}}_{ }{{\displaystyle )}}_{ }\times {{\displaystyle p}}_k^{yx}$, $i=\mathrm{1,2},\cdots ,m_{k}x=x+1$, go to step 2.

3.4. Discussion of Equilibrium

After the four modes have realized the non-cooperative game, it can be considered whether two modes of transportation, HSR and general-speed railway, can enter the market in cooperation to compete with aviation and highways. For the railway, there is a problem of benefit distribution after cooperation between HSR and the general-speed railway. A reasonable distribution of benefits can allow the railway to improve its total income. The SHAP value is one of the methods used to solve the cooperative game problem for the distribution of cooperative benefits among various partners.

The SHAP value defines the benefit distribution vector of a multiplayer cooperative game:

$$\begin{gathered} {\varphi }_i(v)={\displaystyle \sum _{i\in S}}{\displaystyle \frac{(n-\left|S\right|)!(\left|S\right|-1)!}{n!}}\left[v\left(S\right)-v(S-\left\{i\right\})\right],S\in I \\ \begin{array}{cccc}& & & \end{array}i=\mathrm{1,2},3,...,n. \end{gathered}$$

(22)

In Formula (22), $S$ is the set of players $i$. $\left|S\right|$ represents the number of elements in the set $S$. $S-\left\{i\right\}$ represents the set removing $i$ from $S$. ${\varphi }_i\left(v\right)$ is the SHAP value.

4. Model Application

4.1. Case Description

This paper employs the express freight transportation between Xi’an and Lanzhou as an example. Assuming that 200 tons of express freight will be transported from Xi’an to Lanzhou by SF Express Xi’an Branch (Xi’an, China) on a certain day, the transportation modes $k=\mathrm{1,2},\mathrm{3,4}$ in the example successively represent the general-speed railway, HSR, air and highway. Basic data from Xi’an to Lanzhou were found through network analysis and actual investigation and are shown in

Table 1. Basic data.

	General-Speed Railway	HSR	Air	Highway
Transportation mileage (km)	676	570	550	650
Average operating speed (km/h)	100	200	800	70
Distance from cargo distribution center to urban area (km)	Lanzhou 0 Xi’an 0	Lanzhou 0 Xi’an 0	Lanzhou 70 Xi’an 30	Lanzhou 0 Xi’an 0
$t_k^s$	0	0	0.6	0
$t_k^p$	0	0	1.1	0
${{\displaystyle gl}}_k$	99.7%	99.7%	99.54%	91.9%
${{\displaystyle e}}_k$	85%	93%	75%	50%
${{\displaystyle c}}_k$	0.5	0.8	0.9	0.6

below.

The social discount rate is 8% and the average transportation price of goods is CNY1400/ton according to the former Ministry of Construction and National Development and Reform Commission of the People’s Republic of China in 2006. The value of ${{\displaystyle \lambda }}_{OD}$ is 0.0192, and the calculation results of the timeliness index are shown in

Table 2. Calculation results of timeliness index.

	1	2	3	4
${{\displaystyle s}}_k$	7.705	18.282	21.834	5.609

.

Based on the experts’ scoring of the utility index and the nature of express goods, the judgment matrix of the consignor of cargo on the utility index of express freight mode is established. The criteria level ${{\displaystyle B}}_1$, ${{\displaystyle B}}_2$, ${{\displaystyle B}}_3$, ${{\displaystyle B}}_4$, ${{\displaystyle B}}_5$ corresponds to the indexes of freight rate, timeliness, safety, reliability and convenience, see

Table 3. Utility index evaluation matrix.

	${{\displaystyle \mathit{B}}}_1$	${{\displaystyle \mathit{B}}}_2$	${{\displaystyle \mathit{B}}}_3$	${{\displaystyle \mathit{B}}}_4$	${{\displaystyle \mathit{B}}}_5$
${{\displaystyle B}}_1$	1	2	4	3	5
${{\displaystyle B}}_2$	1/2	1	1	2	4
${{\displaystyle B}}_3$	1/4	1	1	2	4
${{\displaystyle B}}_4$	1/3	1/2	1/2	1	5
${{\displaystyle B}}_5$	1/5	1/4	1/4	1/5	1

.

The weight ${{\displaystyle \alpha }}_j$ values of each index obtained by calculation are shown in

Table 4. Calculation results of utility index weight.

$\mathit{j}$	1	2	3	4	5
${{\displaystyle \alpha }}_j$	0.434	0.204	0.183	0.134	0.045

, in which $j=\mathrm{1,2},\mathrm{3,4},5$ correspond to the five indexes of freight rate, timeliness, safety, reliability and convenience, respectively.

Then, the values of ${\Delta }_k^{hf}$ can be calculated, respectively, as shown in

Table 5. Calculation results of ${\Delta }_k^{hf}$.

k	1	2	3	4
${\Delta }_k^{hf}$	0.510	3.672	3.508	0.670

.

4.2. Index Calculation of Generalized Income of Four Transportation Modes

The fixed cost and variable cost of the four transportation modes can be obtained by consulting the network data. Among them, the personnel management and salary expenses in the fixed costs are not much different, so they are not calculated. Among them, the fixed costs of the common railway and the HSR include the depreciation of vehicles and maintenance of vehicles and lines; the fixed cost of air transportation includes depreciation and the maintenance of aircraft; the fixed cost of the highway includes depreciation, maintenance and the insurance of vehicles and maintenance of roads. See

Table 6. Parameter values.

Parameter	1	2	3	4
${{\displaystyle \mathrm{g}}}_{\mathrm{k}}$	15,000,000	25,000,000	45,450,000	30,000
${{\displaystyle \mathrm{b}}}_{\mathrm{k}}$	30	42	95	210
${{\displaystyle Tax}}_k$ [25]	0.217	0.150	14.336	1.218

for ${{\displaystyle \mathrm{g}}}_{\mathrm{k}}$, ${{\displaystyle \mathrm{b}}}_{\mathrm{k}}$, and ${{\displaystyle Tax}}_k$.

4.3. Calculation Model Equilibrium Solution

According to the survey, the current freight rates in the express freight corridor from Xi’an to Lanzhou are shown in

Table 7. Market freight rate.

$\mathit{k}$	1	2	3	4
Freight rate (CNY/ton)	2000	1200	2800	250

.

As only the maximum and minimum freight rates are regulated without a mandatory unified pricing policy in China, the initial freight strategy set for the four transportation modes is established around the current freight rate. This set uses a 40% fluctuation range from the upper and lower price limits, with a 10% increment from the current rate as the differential. Consequently, the initial tariff strategy set is defined as detailed in

Table 8. Initial strategy set of four transportation modes.

Transportation Modes	Initial Tariff Strategy Set (CNY/Ton)
HSR	[1200,1400,1600,1800,2000,2200,2400,2600,2800]
Highway	[720,840,960,1080,1200,1320,1440,1560,1680]
Air	[1680,1960,2240,2520,2800,3080,3360,3640,3920]
General-speed railway	[150,175,200,225,250,275,300,325,350]

.

Utilizing the collected data, the equilibrium solution is computed through a weighted iterative method, executed via MATLAB 7.0 programming. The results are depicted in Figure 3.

The Figure 4 demonstrates that the algorithm exhibits significant convergence. After 60 iterations, the final freight rates and sharing rates for various transportation modes at game equilibrium are determined. These calculation results are presented in

Table 9. Equilibrium calculation results.

Optimal Solution	${{\displaystyle \mathit{p}}}_{\mathit{k}}^{\mathit{y}\ast }$ (CNY)	${{\displaystyle \mathit{p}}}_{\mathit{k}}^{\mathit{h}\ast }$ (CNY)	${{\displaystyle \mathit{r}}}_{\mathit{k}}^{\ast }$ (%)	${{\displaystyle \mathit{f}}}_{\mathit{k}}^{\ast }$
High-speed railway	1964.6	1964.1	48	21,307
Expressway	1395	1394.3	6	1707
Transport aviation	2591.5	2590.7	42	24,225
General railway	234.76	234.66	4	190

.

4.4. Equilibrium Solution Analysis

The analysis of the calculation results presented in Table 9 reveals that the game equilibrium freight rates for the conventional railway, high-speed rail (HSR), air transport, and highways are set at CNY235, 1965, 2591, and 1395, respectively. These rates ensure the highest utility for the consignor. Correspondingly, the equilibrium distribution of market shares among these four transportation modes is established at 4%, 48%, 42%, and 6%. To validate the equilibrium solution, the range of fluctuation for the initial pricing strategy is widened to 50% of the current actual rates, as detailed in

Table 10. Adjusted initial tariff strategy set.

Transportation Modes	Initial Tariff Strategy Set (CNY/ton)
High-speed railway	[1000,1200,1400,1600,1800,2000,2200, 2400,2600,2800,3000]
Expressway	[600,720,840,960,1080,1200,1320,1440,1560,1680,1800]
Transport aviation	[1400,1680,1960,2240,2520,2800,3080,3360,3640,3920,4200]
General railway	[125,150,175,200,225,250,275,300,325,350,375]

.

In Figure 5, after 63 iterations, the equilibrium solution was achieved, yielding a calculation result identical to the one before the adjustment of the initial tariff strategy. This outcome signifies that no participant can gain additional benefits by altering their tariff strategy, confirming that the game has indeed reached equilibrium. Compared to current pricing, the attributes of high speed, punctuality, stability, and low emissions of HSR offer it a significantly strong competitive edge. The study suggests reducing the freight rate from CNY2000/ton to CNY1965/ton to capture more traffic volume and generate increased revenue. Historically, air transport accounted for over 80% of freight volume. However, faced with HSR’s robust competition, airlines may minimize freight source loss by lowering prices. Meanwhile, conventional railways, though less competitive in fast freight services, can leverage their high-volume capacity by reducing prices during peak periods (such as around the Double Eleven Global Carnival) or for long distances. Despite its lower share rate, the highway’s flexibility remains unmatched by other modes. The findings indicate that highways could boost revenue from the current rate of CNY1200/ton to CNY1395/ton. Transport mode operators can use these game equilibrium results as a theoretical basis to efficiently allocate resources, engage in healthy competition, better fulfill consignors’ needs, increase their income, and contribute to economic growth.

5. Conclusions

This study employs game theory methodologies and concepts to examine the competitive dynamics between high-speed rail (HSR) and other transportation modes within the express freight corridor. By imposing a carbon emission tax on different transportation methods, the pursuit of low-carbon objectives is translated into a cost factor considered by shippers. The results of the game indicate that, with strategic pricing, HSR emerges as an ideal choice for rapid freight services, capturing a significant share of the goods market and securing notable benefits. Furthermore, enhanced cooperation between rail transport enterprises to synchronize HSR with conventional rail services could amplify revenues. However, the case study presented here is confined to distances within 1000 km, where HSR exhibits peak competitiveness.

This study not only reveals the economic benefits of high-speed rail express but also emphasizes its importance in environmental protection. Compared to road and air transport, high-speed rail express has significantly lower carbon emissions, effectively reducing greenhouse gas emissions. Additionally, although conventional railways may be less competitive in fast freight services, their high-capacity transport capabilities offer advantages in long-distance transportation. Road transport, on the other hand, maintains a crucial position in short-distance and regional freight due to its flexibility. However, its high emission characteristics pose greater environmental challenges to the future. By adopting reasonable pricing strategies and selecting appropriate transportation modes, logistics companies can meet market demands while achieving sustainable development goals.

Future research should account for how deepening integration between transport services, efficient logistics, and the development of a comprehensive transportation system and network might influence game equilibrium with varying transport distances. Additionally, exploring revenue distribution and pricing strategies for intermodal transportation through cooperative game theory could bridge the gap between theoretical findings and their practical applications. Moreover, this paper simplifies the carbon emission tax as a constant factor, highlighting the necessity for more comprehensive carbon tax standards to elucidate their impact on transportation competitiveness.