Cooperation between competitive electric vehicle manufacturers: a strategic analysis of charging pile construction

Abstract

Electric vehicles (EVs) manufacturers building their own charging piles can enhance brand image and promote EVs sales, but which also induces huge investment. This paper studies the cooperation strategy of the charging pile construction between two competitive EV manufacturers. Assume that Manufacturer 1 (M1) has built its own branded charging stations. Considering whether Manufacturer 2 (M2) shares the construction costs of M1’s charging stations and whether M2 builds its own branded charging stations, this paper develops three corresponding models: non-cooperation, partial cooperation and fully cooperation, to explore the effect of cooperation on manufacturers’ decisions. Then we compare the results in the three models. Further, we extend our model to discuss the impacts of charging fee on the number of the charging piles and manufacturers’ profits. Finally, we present parameters’ sensitivity analysis. We find that (1) among the three strategies, M1’s and M2’s optimal strategies depend on the change of consumers’ sensitivity towards the number of the charging piles, the cost coefficient and cost sharing proportion; (2) there exists a dominant strategy that results in higher profits for both manufacturers compared to the other strategy; (3) M1 is more likely to gain more profits from non-cooperation and full cooperation and M2 is more likely to gain more profits from partial cooperation. (4) In non-cooperation and partial cooperation modes, an increase in charging fees will lead to an increase in the number of charging piles, while in full cooperation mode, an increase in charging fees may not necessarily lead to an increase in the number of charging piles.

Introduction

Promoting electric vehicles is widely regarded as a key strategy for reducing carbon emissions in the transportation sector^1,2,3. Encouraged by government policy incentives and growing consumer demand, an increasing number of automobile manufacturers are entering the EV market. As the number of EV producers continues to rise, competition among manufacturers has become increasingly intense.

In this increasingly competitive environment, many EV manufacturers have resorted to price wars to expand their market share. For instance, the price of the Tesla Model 3 was reduced by $4000 in 2023¹, making it a more attractive option for a broader range of consumers. Similarly, the BYD Leopard 5, which was officially launched on November 9, 2023, saw a price reduction of ¥50,000 across all models just eight months after its release². In addition to price competition, some manufacturers invest in building their own branded charging infrastructure to boost EV sales. For example, NIO, a Chinese EV manufacturer, has established its own network of branded charging and battery swapping stations since its inception. As a result, its annual EV sales have grown from 20,000 to 160,000 over the past five years³. Tesla, a global leader in the EV industry, has also provided users with exclusive access to its Supercharger network, having built over 59,000 charging piles worldwide⁴.

However, EV manufacturers face two major challenges in building charging infrastructure: high construction costs⁴ and low utilization rates of charging piles^5,6. In order to increase revenue, several EVs manufacturers that initially invested in proprietary charging networks have gradually begun to open them to other brands. A notable example is Tesla, which has expanded access to its Supercharger network beyond its own vehicles. Since 2021, Tesla has piloted charger-sharing programs in Europe, and more recently in North America, enabling EVs from automakers such as Ford, General Motors, and BYD to utilize its charging infrastructure.

In China, Xpeng (XPEV) and NIO have launched a collaborative charging network interconnection initiative, aiming to enhance the convenience of the charging experience for all EV users⁷. Meanwhile, IONITY—a joint venture among BMW, Ford, Mercedes-Benz, Volkswagen, Audi, and Porsche—is actively developing a pan-European charging network⁸.

Opening up their own branded charging networks among competing manufacturers can enhance overall charging convenience for EV users and potentially boost EV adoption. However, this strategy also carries the risk of consumer loss or brand dilution. For instance, many consumers choose Tesla in part due to the exclusive convenience offered by its Supercharger network. If Tesla were to make its Superchargers accessible to all EV users, it could lead to greater congestion and diminished convenience for Tesla’s own customers. This may prompt potential buyers to consider competitor brands, thereby weakening customer loyalty. Moreover, as the Supercharger network is a key differentiating feature of Tesla, opening it to competitors could potentially erode its brand premium. Therefore, EV manufacturers must carefully weigh the trade-offs between openness and exclusivity when deciding whether to engage in joint charging initiatives.

The above real-world examples motivate us to examine how EV manufacturers respond to increasing competition within the industry. Accordingly, this paper focuses on the following research questions:

• (1) Given the construction cost of charging piles and the potential increase in EV sales driven by proprietary charging infrastructure, should an EV manufacturer invest in building charging piles? If so, how many piles should be constructed, considering the competitive dynamics in the EV market?

• (2) For manufacturers that have built their own branded charging infrastructure, should they open and share their charging piles with the competitors? If so, under what conditions?

• (3) Among competing EV manufacturers, which cooperative strategy for developing charging infrastructure most effectively enhances both EV sales and profitability?

To answer these questions, considering the sensitivity of customer demand to the number of charging piles, we develop three competition models. We assume M1 has constructed own branded charging infrastructure, while M2 has not.

We first analyze the non-cooperative scenario, in which M1 does not share the charging piles with M2. Next, we examine the partial cooperation scenario, where M1 shares the charging piles with M2 (M2 produces EVs that could charge in the charging piles built by M1) and M2 shares a part of the construction cost of M1’s charging piles. Finally, we explore the full cooperation scenario, in which both manufacturers build their own charging piles and agree to mutual sharing (M2/M1 produces EVs that could charge in the charging piles built by M1/M2). For each scenario, we determine the optimal number of charging piles need to be built and the corresponding EV prices. We then compare the outcomes of the three scenarios across three key dimensions: EV price, the total number of charging piles, and manufacturers’ profits.

Our paper contributes to the literature in the following three key aspects. First, we examine the construction of self-owned charging stations by competing EV manufacturers, whereas most existing studies have primarily focused on the deployment and planning of charging infrastructure by a monopolistic EV manufacturer^9,10. Second, we explore various cooperation strategies among competing manufacturers in building charging infrastructure and analyze their strategic choices. In contrast, previous research has largely concentrated on collaborations between EV manufacturer and the government in the construction of charging stations^2,11. Third, we investigate how the sharing versus non-sharing of self-owned charging stations affects EV demand, and offer managerial insights on whether manufacturers should open their self-owned charging infrastructure to competitors.

The remainder of the paper is organized as follows. Section “Literature review” reviews existing literature. Section “Model” provides model descriptions and model solutions. Section “Comparison among three scenarios” compares the results obtained from the three models. Section “Extension” considers the model extension. Section “Numerical example” provides the numerical examples. Section “Conclusion” concludes the paper and points out future research directions. All proofs are given in the Appendix.

Literature review

This paper investigates the charging pile (station) construction cooperation strategy between two competitive EV manufacturers, hence this paper is closely related to two streams of literature: (1) charging pile (station) construction, (2) competition and cooperation in supply chain management.

Charging pile (station) construction

Charging infrastructure is an important obstacle affecting the large-scale diffusion of EVs^12,13. Zhang et al.¹⁴ investigated the charging station density’s marginal effect and proved that increasing the density of charging stations can expand the market share of EVs. Zhu et al.¹⁵ proposed a new charging pile planning model for plug-in electric vehicles, which deals with both the location of charging piles and how many charging piles are to be built at each charging station. Dorce et al.¹⁶ used a gamified survey to explore the willingness to pay for EV charging. Wang and Deng⁸ examined the charging station construction mode and analyzes which model is more conducive to the EVs rollout and which business mode can be more profitable. Kumar et al.¹¹ formulated four different models for charging infrastructure development and considered the impact of subsidy on EV manufacturer. Yu et al.¹⁰ analyzed the cost-sharing contract of charging station construction between retailers and manufacturers, and considers the scenario when the manufacturer and the government build charging stations at the same time. Shi et al.¹⁷ addressed spatial mismatch between supply and demand of charging piles through the design of charging pile construction and government subsidies. Yu et al.² examined the number of charging piles to be built by the government and EV manufacturers and compared the consumer subsidies and charging station subsidies. Huang et al.¹⁸ designed a novel agent-based evolutionary game model that incorporates consumers’ microscopic behavior into the dynamics of charging station diffusion to study the relationship between charging infrastructure and electric vehicle applications. Gulbahar et al.¹⁹ proposed a decision model for determining the best locations for EV charging stations along highways. Feng et al.¹⁶ studied the optimal charging station construction strategy of manufacturers by establishing a game theory model. Zhu et al.²⁰ used the Stackelberg game method to explore the installation optimization problem between the government and investors in the intelligent charging station market.

Above studies have explored the layout of charging stations, the number of charging stations need to build, and the impact of subsidies on the construction of charging stations. However, little research study the cooperation and sharing of charging station among competing manufacturers. Chen and Feng²¹ analyzed whether EV charging service providers should build their own stations or share infrastructure. They compare two scenarios: (1) the two manufacturers build their own piles but do not share with each other; (2) one manufacturer builds and shares charging piles and the other manufacturer pay a fixed fee to the manufacturer. Obviously, our paper is different from Chen and Feng²¹.

Competition and cooperation in supply chain management

Druehl et al.²² considered the impact of the cooperation of downstream monopoly manufacturers on the environmental innovation decisions of upstream suppliers. Jena et al.²³ studied two remanufacturing firms that cooperate on both price and service, and designed a quantity discount contract to coordinate the supply chains of a retailer and two manufacturers. Kraft et al.²⁴ examined alternative strategies for manufacturers in terms of cooperation or competition. Jamali et al.²⁵ discussed the optimal greenness level when the green supply chain competes with non-green supply chains and the manufacturer employs dual channels to sell product. Guo et al.²⁶ studied the impact of two competing retailers and consumers’ green decisions on the manufacturer’s green product development under wholesale contract. Zhou et al.²⁷ studied cooperative competitive equilibrium and coordination strategies under different conditions by constructing Stackelberg game and bargaining game. Cheng and Fan²⁸ explored the production competition strategy between fuel vehicle manufacturer and new energy vehicle manufacturer. Zhu et al.²⁹ developed a two-tier supply chain to examine the performance of government R&D subsidy by considering supply chain downstream competition. Wu et al.³⁰ concluded that critical sustainability management practices may be affected by cooperative-competitive relationships between manufacturers and suppliers.

The existing literature on the coopetition of manufacturers focus on product design and pricing, while this paper studies the EV price competition and charging service cooperation between manufacturers. Chen et al.⁷ studied the compatibility of charging stations when two manufacturers built stations simultaneously, while this article focuses on the cooperation strategies between the two manufacturers in charging station construction, and analyzes how they cooperate to maximize their profits.

Model

We consider that there are two types of EV manufacturers: M1 and M2. M1 produces EV1 and builds the charging pile for his own branded EV1, such as Tesla, and EV1 could charge at the piles built by M1. M2 only produces EV2 and may or may not build her own branded charging pile at present.

Considering the impacts of the own branded charging piles on EV sales, M2 needs to determine whether to build her own branded charging piles. Assume that M2 has three choices: (1) does not build charging piles and M2’s EVs does not use M1’s charging piles; (2) does not build charging pile and M2’s EVs use M1’s charging piles; (3) build charging piles and share the charging piles with M1.

Hence, we consider three scenarios for the two competitive manufacturers:

Scenario Non-Cooperation (NC): M1 builds his own branded charging piles and M2 does not build her own branded piles. And EV2 produced by M2 cannot be charged at the piles built by M1. This scenario reflects market fragmentation, where manufacturers act in pure competition.

Scenario Partial-Cooperation (PC): M1 builds his own branded charging infrastructure and M2 does not build her own branded piles. But M2 shares a certain percent of the construction cost with M1. Then the EV2 produced by M2 could use the charging stations built by M1. For example, in 2023, Ford and General Motors announced plans to adopt Tesla’s charging standard, with future vehicles designed to be compatible with Tesla charging stations. They also committed to enabling existing vehicles to access Tesla chargers via adapters. This collaboration allowed Tesla to expand its user base, while Ford and GM gained shared network access through investments, such as paying interface authorization fees or charging service fees.

Scenario Fully-Cooperation (FC): Both M1 and M2 build their own branded charging piles and each manufacturer is only responsible for the construction costs of their own charging stations. But EVs produced by M1 (M2) could use the charging piles built by M2 (M1). For example, both NIO and XPeng initially built their own charging stations exclusively for their own customers. In 2023, the two companies announced that they would grant mutual access to some of their charging stations, allowing vehicle owners to unlock and use each other’s charging infrastructure via their respective mobile apps.

We discuss the following three scenarios, i.e., Scenario NC, Scenario PC and Scenario FC, as shown in Fig. 1.

Fig. 1

Model structure.

Similar to Yu et al.¹⁰, consider the impacts of EV price and the number of the charging piles on consumers demand, we assume that the EV demand is as follows, $D = a - p + \gamma n$, where $a,p,\gamma ,n$ represent the potential market demand, the EV price, the sensitivity of the EV demand to the number of charging piles, and the number of charging piles, respectively. As the number of charging piles increases, the land cost and electricity storage cost will significantly increase, following Kumar et al.¹¹, we assume that the construction cost required to build n charging piles is $\frac{1}{2}{k}{n}^{2}$, where k is the cost coefficient.

Before modeling, we first give following assumptions.

Assumption 1

In Scenario NC, M1’s cost coefficient of charging pile construction $k_{1}$, the degree of substitutability $\theta$, and the sensitivity of the number of charging piles to the demand for EVs, $\gamma_{1}$ , satisfy: $\gamma_{1}^{2} < \frac{{k_{1} \left( {\theta^{2} - 4} \right)^{2} }}{8}$.

Assumption 2

In Scenario PC, M1’s cost coefficient of charging pile construction $k_{1}$, the degree of substitutability $\theta$, M1’s cost sharing rate in Scenario PC, $\mu$, and the sensitivity of the number of charging piles to EV demand, $\gamma_{2}$, satisfy: $\gamma_{2}^{2} < \frac{{k_{1} \mu \left( {\theta - 2} \right)^{2} }}{2}$.

Assumption 3

In Scenario FC, the cost coefficient of charging pile construction $k_{1}$ and $k_{2}$ , the degree of substitutability $\theta$, and the sensitivity of the number of charging piles to the demand $\gamma_{3}$, satisfy: $\gamma_{3}^{2} < \frac{{k_{1} k_{2} \left( {\theta - 2} \right)^{2} }}{{2\left( {k_{1} + k_{2} } \right)}}$.

Above assumptions in our model are primarily made to ensure the existence and uniqueness of the optimal solutions. Similar parameter assumptions are commonly adopted in the supply chain and game-theoretic modeling literature to ensure the existence of interior and unique optimal solutions (e.g.,^31,32,33). Future research may consider relaxing these assumptions, potentially using numerical methods to address more general or empirical settings.

All notations and parameters used throughout the paper are summarized in Table 1.

Table 1 Notations description.

Benchmark model (Scenario NC)

In this model, we assume that M1 produces EV1 and build his own branded charging piles. And M2 only produces EV2 and does not build charging piles. The EVs produced by M2 cannot be charged by the charging piles built by M1. In this scenario, two manufacturers compete completely, and M1 intends to attract more consumers by building his own branded charging stations. The game sequence is M1 first decides the number of charging piles need to build, and then M1 and M2 separately determine the prices of their EVs.

The EV demand functions are $D_{1}^{NC} = \tilde{a} - p_{1}^{NC} + \theta p_{2}^{NC} + \gamma_{1} (n+n^{NC})$, $D_{2}^{NC} = \tilde{a} - p_{2}^{NC} + \theta p_{1}^{NC} { + }\gamma_{1} n$, where n represents the number of existing public charging piles $n^{NC}$ represents the number of the public charging piles built by M1, $\theta$ is the degree of product substitutability. For simplicity, we rewrite the demand functions as $D_{1}^{NC} = a - p_{1}^{NC} + \theta p_{2}^{NC} + \gamma_{1} n^{NC}$, $D_{2}^{NC} = a - p_{2}^{NC} + \theta p_{1}^{NC}$, where $a{ = }\tilde{a} + \gamma n$. The event sequence is as follows: (1) M1 decides the number of charging pile $n^{NC}$; (2) M1 and M2 determine the EV prices $p_{1}^{NC}$,$p_{2}^{NC}$. For simplicity, we omitted EV’s production cost.

Then the profit functions of M1 and M2 are,

$\pi_{1}^{NC} = D_{1}^{NC} p_{1}^{NC} - \frac{1}{2}k_{1} n^{NC2}$,

$\pi_{2}^{NC} = D_{2}^{NC} p_{2}^{NC}$.

Lemma 1

The optimal number of the charging piles built by M1 and EV prices are given as follows, $n^{NC*} = \frac{{4a\gamma_{1} \left( {2 + \theta } \right)}}{{k_{1} \left( {\theta^{2} - 4} \right)^{2} - 8\gamma_{1}^{2} }}$,$p_{1}^{NC*} = \frac{{ak_{1} \left( {2 - \theta } \right)\left( {\theta + 2} \right)^{2} }}{{k_{1} \left( {\theta^{2} - 4} \right)^{2} - 8\gamma_{1}^{2} }}$,$p_{2}^{NC*} = \frac{{a\left[ {k_{1} \left( {2 + \theta } \right)\left( {4 - \theta^{2} } \right) - 4\gamma_{1}^{2} } \right]}}{{k_{1} \left( {\theta^{2} - 4} \right)^{2} - 8\gamma_{1}^{2} }}$.

Lemma 1 provides the optimal solutions, revealing that the optimal number of charging piles and EV prices are influenced by the degree of product substitutability, the sensitivity of consumer demand to the number of charging piles, and the construction cost coefficient. To further examine the impacts of these key factors, we present the following propositions.

Proposition 1

As the degree of substitutability $\theta$ increases,$n^{NC*},$ $p_{1}^{NC*}$ and $p_{2}^{NC*}$ increase.

From Proposition 1, we can see that as the degree of substitutability increases, the number of charging piles and EV prices increase. This is because higher substitutability incentivizes M1 to invest in charging piles construction. A large number of the charging piles increase operational costs, which in turn could lead to higher prices. Thus, while competition typically leads to lower prices, the cost of differentiation and investment in infrastructure can push prices upward.

Proposition 2

As the sensitivity of the EV demand to the number of charging piles, ${\gamma }_{1}$, increases, $n^{NC*}$, $p_{1}^{NC*}$ and $p_{2}^{NC*}$ increase.

Proposition 2 shows that the optimal number of the charging piles and EV prices increase as consumer sensitivity to the number of charging piles increases. This is because the greater the customer preference for a brand’s own charging stations, the more of these proprietary charging piles need to be constructed. This not only increases costs, leading to higher EV prices, but also drives up competitors’ prices. Given this trend, manufacturers should proactively adjust their infrastructure strategies by increasing investment in branded charging networks to strengthen customer loyalty. At the same time, they must carefully control costs to prevent excessive price increases.

Proposition 3

As the EV charging pile construction cost coefficient $k_{1}$ increases, $n^{NC*}$, $p_{1}^{NC*}$ and $p_{2}^{NC*}$ decrease.

Proposition 3 demonstrates that both the optimal number of charging piles and EV prices decrease as the construction cost coefficient of charging infrastructure increases. This is because, as the cost of building charging piles rises, M1 is compelled to reduce the number of charging piles in order to control expenses. However, the reduction in charging infrastructure diminishes the convenience of EV usage, which negatively impacts consumer purchase intention. Consequently, both M1 and its competitor M2 are forced to lower EV prices to stimulate demand and maintain market share. Therefore, manufacturers should adopt flexible infrastructure planning strategies that align with cost fluctuations. This may include prioritizing high-impact locations for charging piles to maximize utility per unit cost or exploring modular, lower-cost construction technologies.

Partial cooperation scenario (PC)

As is well known cost-sharing contract is widely adopted in the real world, in this model, we use the cost-sharing contract to connect M1 and M2. We assume that the M1 produces EVs and build his charging piles. M2 produces EVs and M2 jointly build charging piles with M1 through a cost-sharing contract, and then the charging pile can also serve EVs produced by M1 and M2. In this scenario, manufacturers aim to reduce costs through joint deployment of charging stations while preserving competitive autonomy in EV production and pricing. This reflects a partial alliance where the infrastructures are shared but market strategies remain independent. The game sequence is M1 first decides the number of charging piles cooperatively constructed by both parties, and then M1 and M2 separately determine the prices of their EVs.

Similar to Scenario NC, the demand function for M1 and M2 are $D_{1}^{PC} = a - p_{1}^{PC} + \theta p_{2}^{PC} + \gamma_{2} n^{PC}$, $D_{2}^{PC} = a - p_{2}^{PC} + \theta p_{1}^{PC} + \gamma_{2} n^{PC}$. The event sequence is as follows: (1) M1 decides the number of charging pile $n^{NC}$; (2) M1 and M2 determine the EV prices,$p_{1}^{NC}$, $p_{2}^{NC}$. For simplicity, we omitted EV’s production cost.

The profit functions of M1, M2 are.

$\pi_{1}^{PC} = D_{1}^{PC} p_{1}^{PC} - \mu \frac{1}{2}k_{1} n^{PC2}$,

$\pi_{2}^{PC} = D_{2}^{PC} p_{2}^{PC} - \left( {1 - \mu } \right)\frac{1}{2}k_{1} n^{PC2}$.

Lemma 2

The optimal number of the charging piles built by M1 and EV prices are given as follows, $n^{PC*} = \frac{{2a\gamma_{2} }}{{k_{1} \mu \left( {\theta - 2} \right)^{2} - 2\gamma_{2}^{2} }}$,$p_{1}^{PC*} = p_{2}^{PC*} = \frac{{ak_{1} \mu \left( {2 - \theta } \right)}}{{k_{1} \mu \left( {\theta - 2} \right)^{2} - 2\gamma_{2}^{2} }}$.

Lemma 2 gives the optimal solutions. And we find that the optimal number of the charging piles and the EV prices are related to the degree of product substitutability, the sensitivity of the EV demand to the number of charging piles and cost coefficient. In order to explore the impacts of these factors, we present following Propositions.

Proposition 4

As the degree of substitutability $\theta$ increases, $n^{PC*}$, $p_{1}^{PC*}$ and $p_{2}^{PC*}$ increase.

Proposition 5

As the sensitivity of the EV demand to the number of charging piles, $\gamma_{2}$ increases, $n^{PC*}$, $p_{1}^{PC*}$ and $p_{2}^{PC*}$ increase.

Proposition 6

As the EV charging pile construction cost factor $k_{1}$ increases, $n^{PC*}$, $p_{1}^{PC*}$ and $p_{2}^{PC*}$ decrease.

Propositions 4–6 are similar to Propositions 1–3, so we will not explain them repeatedly.

Fully cooperation scenario (FC)

In this subsection, both M1 and M2 build their own branded charging piles and the two EVs be charged at each other’s charging piles. Each of them bears the construction cost of their own branded charging stations. The charge piles built by M1 (M2) could charge EVs produced by M1 and M2. In other word, the two manufacturers open their own branded charging stations to each other. In this case, the two manufacturers cooperate in the usage of the own branded charging piles, but competes in EV prices. Similar to the cooperation between the NIO and Xpeng Motors in China. In this game, at first M1 and M2 determine the number of the charging piles need to built separately, then the two manufacturers decide the EV prices simultaneously.

Similar to Scenario NC, the demand function for M1 and M2 are $D_{1}^{FC} = a - p_{1}^{FC} + \theta p_{2}^{FC} + \gamma_{3} (n_{1}^{FC} + n_{2}^{FC} )$, $D_{2}^{FC} = a - p_{2}^{FC} + \theta p_{1}^{FC} + \gamma_{3} (n_{1}^{FC} + n_{2}^{FC} )$.

The profit functions of M1, M2 are

$\pi_{1}^{FC} = D_{1}^{FC} p_{1}^{FC} - \frac{1}{2}k_{1} n_{1}^{FC2}$,

$\pi_{2}^{FC} = D_{2}^{FC} p_{2}^{FC} - \frac{1}{2}k_{2} n_{2}^{FC2}$.

Lemma 3

The optimal number of the charging piles and EV prices are given as follows, $n_{1}^{FC*} = \frac{{2a\gamma_{3} k_{2} }}{{k_{1} k_{2} \left( {\theta - 2} \right)^{2} - 2\gamma_{3}^{2} \left( {k_{1} + k_{2} } \right)}}$,$n_{2}^{FC*} = \frac{{2a\gamma_{3} k_{1} }}{{k_{1} k_{2} \left( {\theta - 2} \right)^{2} - 2\gamma_{3}^{2} \left( {k_{1} + k_{2} } \right)}}$, $p_{1}^{FC*} = p_{2}^{FC} = \frac{{ak_{1} k_{2} \left( {2 - \theta } \right)}}{{k_{1} k_{2} \left( {\theta - 2} \right)^{2} - 2\gamma_{3}^{2} \left( {k_{1} + k_{2} } \right)}}$.

Lemma 3 gives the optimal solutions in the fully cooperation scenario. And we find that the optimal number of the charging piles and the EV prices are related to the degree of substitutability, the sensitivity of the EV demand to the number of charging piles and cost coefficient. In order to explore the impacts of these factors, we present following Propositions.

Proposition 7

As the degree of substitutability $\theta$ increases, $n^{FC*}$, $p_{1}^{FC*}$ and $p_{2}^{FC*}$ increase.

Proposition 8

As the sensitivity of the EV demand to the number of charging piles, $\gamma_{3}$, increases,$n^{FC*}$, $p_{1}^{FC*}$ and $p_{2}^{FC*}$ increase.

Proposition 9

As the EV charging pile construction cost factor $k_{i}$ increases, $n^{FC*}$, $p_{1}^{FC*}$ and $p_{2}^{FC*}$ decrease.

Propositions 7–9 are similar to Propositions 1–3, so we will not explain repeatedly.

Comparison among three scenarios

In this section, we will compare the equilibriums among the three scenarios and figure out the optimal choice in different conditions.

Corollary 1

• (1) If $\gamma_{1} > \frac{{ - A + \sqrt {A^{2} + 32\gamma^{2} k_{1} \left( {\theta^{2} - 4} \right)^{2} } }}{{16\gamma_{2} }}$, then $n^{NC*} > n^{PC*}$; if $\gamma_{1} < \frac{{ - A + \sqrt {A^{2} + 32\gamma^{2} k_{1} \left( {\theta^{2} - 4} \right)^{2} } }}{{16\gamma_{2} }}$, then $n^{NC*} < n^{PC*}$ ; where $A = 2\left( {\theta + 2} \right)\left[ {k_{1} \mu \left( {\theta - 2} \right)^{2} - 2\gamma_{2}^{2} } \right]$;

• (2) If $\gamma_{1} > \frac{{ - B + \sqrt {B^{2} + 32\gamma_{3}^{2} k_{1}^{2} \left( {k_{1} + k_{2} } \right)^{2} \left( {\theta^{2} - 4} \right)^{2} } }}{{16\left( {k_{1} + k_{2} } \right)\gamma_{3} }}$, then $n^{NC*} > n^{FC*}$; if $\gamma_{1} < \frac{{ - B + \sqrt {B^{2} + 32\gamma_{3}^{2} k_{1}^{2} \left( {k_{1} + k_{2} } \right)^{2} \left( {\theta^{2} - 4} \right)^{2} } }}{{16\left( {k_{1} + k_{2} } \right)\gamma_{3} }}$, then $n^{NC*} < n^{FC*}$ ; where $B = 2\left( {2 + \theta } \right)\left[ {k_{1} k_{2} \left( {\theta - 2} \right)^{2} - 2\gamma_{3}^{2} \left( {k_{1} + k_{2} } \right)} \right]$;

• (3) If $\gamma_{2} > \frac{{ - C + \sqrt {C^{2} + 8\gamma_{3}^{2} k_{1} \mu \left( {k_{1} + k_{2} } \right)^{2} \left( {\theta - 2} \right)^{2} } }}{{4\left( {k_{1} + k_{2} } \right)\gamma_{3} }}$, then $n^{PC*} > n^{FC*}$; if $\gamma_{2} < \frac{{ - C + \sqrt {C^{2} + 8\gamma_{3}^{2} k_{1} \mu \left( {k_{1} + k_{2} } \right)^{2} \left( {\theta - 2} \right)^{2} } }}{{4\left( {k_{1} + k_{2} } \right)\gamma_{3} }}$, then $n^{PC*} < n^{FC*}$; where $C = k_{1} k_{2} \left( {\theta - 2} \right)^{2} - 2\gamma_{3}^{2} \left( {k_{1} + k_{2} } \right)$.

Corollary 1 compares the number of charging piles built under different cooperation modes. Corollary 1 (1) presents the comparison between non-cooperation and partial cooperation: if the sensitivity of consumer demand to the number of the charging piles is very large in scenario NC, then the number of the charging piles need to be built in scenario NC is larger than that in scenario PC. Otherwise, M1 need to build more piles in scenario PC than in scenario NC.

Similar to Corollary 1 (1), Corollary 1 (2) compares the scenarios between non-cooperation and full cooperation: if the sensitivity of consumer demand to the number of the charging piles is very large in Scenario NC, then the number of the charging piles need to be built in Scenario NC is larger than that in Scenario FC. Otherwise, M1 need to build more piles in FC scenario than in Scenario NC.

Corollary 1 (3) compares the situations of partial cooperation and full cooperation.

In order to present the results of Corollary 1 more visually, we present Figs. 2, 3, and 4, which correspond to the results of Corollary 1 (1)—(3), respectively.

Fig. 2

Comparison of $n^{NC*}$ and $n^{PC*}$.

Fig. 3

Comparison of $n^{NC*}$ and $n^{FC*}$.

Fig. 4

Comparison of $n^{PC*}$ and $n^{FC*}$.

Additionally, from Fig. 2, we can see that if M2 bears a relatively high proportion of costs, there is a greater likelihood that the number of charging piles built under partial cooperation will be higher than those built without cooperation. Figure 3 illustrates that if M2’s cost coefficient of building charging piles is very high, the likelihood of a higher total number of charging piles being built in the fully cooperative mode compared to the non-cooperative mode is relatively low; Fig. 4 illustrates that if the cost proportion shared by M2 is relatively small during partial cooperation, there is a higher possibility that the total number of charging piles in full cooperation mode will be larger than that in partial cooperation.

Overall, Corollary 1 demonstrates that cooperation does not necessarily lead to a greater number of charging piles compared to non-cooperation, and that full cooperation does not always result in more charging piles than partial cooperation. This is mainly related to consumers’ sensitivity to charging piles.

Corollary 2

• (1) If $\gamma_{1} < \sqrt {\frac{{k_{1} \mu \left( {\theta^{2} - 4} \right)^{2} - k_{1} \mu \left( {\theta - 2} \right)^{2} + 2\gamma_{2}^{2} }}{8\mu }}$, then $p_{1}^{PC*} = p_{2}^{PC*} > p_{1}^{NC*} > p_{2}^{NC*}$; If $\sqrt {\frac{{k_{1} \mu \left( {\theta^{2} - 4} \right)^{2} - k_{1} \mu \left( {\theta - 2} \right)^{2} + 2\gamma_{2}^{2} }}{8\mu }} < \gamma_{1} < \sqrt {\frac{{k_{1} \left( {4 - \theta^{2} } \right)\left[ {k_{1} \mu \left( {2 - \theta } \right)\left( {4 - \theta^{2} } \right) - \left( {2 + \theta } \right)D} \right]}}{{4\left[ {2k_{1} \mu \left( {2 - \theta } \right) - D} \right]}}}$, then $p_{1}^{NC*} > p_{1}^{PC*} = p_{2}^{PC*} > p_{2}^{NC*}$; If $\gamma_{1} > \sqrt {\frac{{k_{1} \left( {4 - \theta^{2} } \right)\left[ {k_{1} \mu \left( {2 - \theta } \right)\left( {4 - \theta^{2} } \right) - \left( {2 + \theta } \right)D} \right]}}{{4\left[ {2k_{1} \mu \left( {2 - \theta } \right) - D} \right]}}}$, then $p_{1}^{NC*} > p_{2}^{NC*} > p_{1}^{PC*} = p_{2}^{PC*}$; where $D = k_{1} \mu \left( {\theta - 2} \right)^{2} - 2\gamma_{2}^{2}$;

• (2) If $\gamma_{1} < \sqrt {\frac{{k_{1} \left[ {k_{1} k_{2} \left( {\theta^{2} - 4} \right)^{2} - E} \right]}}{{8k_{1} k_{2} }}}$, then $p_{1}^{FC*} = p_{2}^{FC*} > p_{1}^{NC*} > p_{2}^{NC*}$; If $\sqrt {\frac{{k_{1} \left[ {k_{1} k_{2} \left( {\theta^{2} - 4} \right)^{2} - E} \right]}}{{8k_{1} k_{2} }} < } \gamma_{1} < \sqrt {\frac{{k_{1} \left( {4 - \theta^{2} } \right)\left[ {k_{1} k_{2} \left( {2 - \theta } \right)\left( {4 - \theta^{2} } \right) - \left( {2 + \theta } \right)C} \right]}}{{8k_{1} k_{2} \left( {2 - \theta } \right) - 4C}}}$, then $p_{1}^{NC*} > p_{1}^{FC*} = p_{2}^{FC*} > p_{2}^{NC*}$; If $\gamma_{1} > \sqrt {\frac{{k_{1} \left( {4 - \theta^{2} } \right)\left[ {k_{1} k_{2} \left( {2 - \theta } \right)\left( {4 - \theta^{2} } \right) - \left( {2 + \theta } \right)C} \right]}}{{8k_{1} k_{2} \left( {2 - \theta } \right) - 4C}}}$, then $p_{1}^{NC*} > p_{2}^{NC*} > p_{1}^{FC*} = p_{2}^{FC*}$ ; where $E = k_{1} \left( {2 + \theta } \right)^{2} \left[ {k_{1} k_{2} \left( {\theta - 2} \right)^{2} - 2\gamma_{3}^{2} \left( {k_{1} + k_{2} } \right)} \right]$;

• (3) If $\gamma_{2} > \sqrt {\frac{{k_{1} \mu \left[ {k_{1} k_{2} \left( {\theta - 2} \right)^{2} - C} \right]}}{{2k_{1} k_{2} }}}$, then $p_{1}^{PC*} = p_{2}^{PC*} > p_{1}^{FC*} = p_{2}^{FC*}$; If $\gamma_{2} < \sqrt {\frac{{k_{1} \mu \left[ {k_{1} k_{2} \left( {\theta - 2} \right)^{2} - C} \right]}}{{2k_{1} k_{2} }}}$, then $p_{1}^{PC*} = p_{2}^{PC*} < p_{1}^{FC*} = p_{2}^{FC*}$.

Corollary 2 compares EVs prices under different cooperation models. Corollary 2 (1) compares the scenarios of non-cooperation and partial cooperation. There are two threshold points: if the sensitivity of consumer demand to charging pile in Scenario NC is very large, $\gamma_{1} > \sqrt {\frac{{k_{1} \left( {4 - \theta^{2} } \right)\left[ {k_{1} \mu \left( {2 - \theta } \right)\left( {4 - \theta^{2} } \right) - \left( {2 + \theta } \right)D} \right]}}{{4\left[ {2k_{1} \mu \left( {2 - \theta } \right) - D} \right]}}}$ , then the EV prices in Scenario PC are less than those in Scenario NC; If the sensitivity of consumer demand to charging pile in Scenario NC is at a moderate level, that is, $\sqrt {\frac{{k_{1} \mu \left( {\theta^{2} - 4} \right)^{2} - k_{1} \mu \left( {\theta - 2} \right)^{2} + 2\gamma_{2}^{2} }}{8\mu }} < \gamma_{1} < \sqrt {\frac{{k_{1} \left( {4 - \theta^{2} } \right)\left[ {k_{1} \mu \left( {2 - \theta } \right)\left( {4 - \theta^{2} } \right) - \left( {2 + \theta } \right)D} \right]}}{{4\left[ {2k_{1} \mu \left( {2 - \theta } \right) - D} \right]}}}$, then EV1’s (EV2’s) price in Scenario PC is smaller (bigger) than that in Scenario NC; if the sensitivity of consumer demand to charging pile in Scenario NC is very small $\left( {\gamma_{1} < \sqrt {\frac{{k_{1} \mu \left( {\theta^{2} - 4} \right)^{2} - k_{1} \mu \left( {\theta - 2} \right)^{2} + 2\gamma_{2}^{2} }}{8\mu }} } \right)$, then EVs prices in Scenario PC is bigger than those in Scenario NC.

Corollary 2 (2) compares the situations of non-cooperation and full cooperation. Corollary 2 (3) compares the scenarios of partial cooperation and full cooperation. The results are similar to Corollary 2(1) and (2). Full cooperation may not necessarily lead to an increase in prices for both products, as it also depends on the relationship between $\gamma_{2}$ and $\gamma_{3}$.

To illustrate Corollary 2 more clearly, we present the following example. Tesla has already built charging piles, while Xpeng Motors currently does not (Scenario NC). If both parties reach a cooperation agreement, XPENG will start building charging piles and share their own charging piles with each other (Scenario FC). At this case, if Tesla potential consumers are very sensitive to the use of Tesla charging piles by XPENG’s consumers $\left( {\gamma_{1} > \sqrt {\frac{{k_{1} \left( {4 - \theta^{2} } \right)\left[ {k_{1} k_{2} \left( {2 - \theta } \right)\left( {4 - \theta^{2} } \right) - \left( {2 + \theta } \right)C} \right]}}{{8k_{1} k_{2} \left( {2 - \theta } \right) - 4C}}} } \right)$, leading to a decrease in demand for Tesla, then both Tesla and XPENG manufacturers are likely to need to lower product prices ($p_{1}^{NC*} > p_{2}^{NC*} > p_{1}^{FC*} = p_{2}^{FC*}$); On the contrary, if consumers are not very sensitive to the number of the charging piles, the two manufacturers may increase product prices, depending on the relationship between r₁ and $\gamma_{3}$.

Due to the complexity of the profit function, we use numerical experiments to compare two manufacturers’ profits under above three modes.

Figures 5, 6, and 7 partition the space into three distinct regions, labeled as I, II and III. In our numerical study, we also drew partitions under other combinations, and they are qualitatively the same except for minor differences in size for the three regions.

Fig. 5

Comparison of $\pi_{i}^{NC}$ and $\pi_{i}^{PC}$ with different $\mu$.

Fig. 6

Comparison of $\pi_{i}^{NC}$ and $\pi_{i}^{FC}$ with different $k_{2}$.

Fig. 7

Comparison of $\pi_{i}^{PC}$ and $\pi_{i}^{FC}$ with different $\mu$.

Since ${\gamma }_{1}$ represents the increase in EV1 sales resulting from adding one charging station when M1 does not charging piles with M2, and $\gamma_{2}$ represents the increase in EV1 and EV2 sales from adding one charging station when M1 and M2 partially cooperate. Note that ${{\gamma }_{1}>\gamma }_{2}$ in most cases. This is because, under Scenario PC the increase in EV1 sales from adding one charging station is less than that under Scenario NC: sharing M1’s charging stations with EV2 users may lead to longer charging wait times for EV1 users, which in turn reduces EV1 sales.

Therefore, when ${\gamma }_{1}$ keeps a constant, the difference between ${\gamma }_{1}$ and $\gamma_{2}$ can to some extent reflect EV1 users’ negative reactions to sharing M1’s charging infrastructure with EV2. From this perspective, we analyze following Figs. 5, 6, and 7.

Figure 5 visually shows the comparison of M1’s and M2’s profits between non-cooperation and partial cooperation, where colored curves Ai and Bi correspond to the thresholds for distinguishing $\pi_{1}$ and $\pi_{2}$ in different scenarios. In Fig. 5, we can see that if sharing charging pile has a significant negative impact on EV1’s demand, partial cooperation will lead to a decrease in two manufacturers’ profits (Region I); And if the impact of sharing charging piles on the demand for EV1 is very small, then partial cooperation will lead to an increase in two manufacturers’ profits (Region III). In Region II, M2 could obtain more profit with partial cooperation than with non-cooperation. And M2 is more likely to gain more profits from partial cooperation than M1 $\left( {II + III > I} \right)$. By comparing the two graphs in Fig. 5, we can see that the larger the proportion of cost sharing by M2, the greater the likelihood that M2 will gain more profits from partial cooperation.

Similar to Figs. 5 and 6 presents the comparison of M1’s and M2’s profits between non-cooperation and full cooperation. Figure 6 shows that non-cooperation strategy consistently dominates the full cooperation strategy in Regions I, while full cooperation strategy always dominates the non-cooperation scenario in Region III. Region II is a conflict zone, that is, M1 could obtain more profit in non-cooperation strategy while M2 obtains more profit in full cooperation strategy. By comparing the two graphs in Fig. 6, we find that the higher the charging pile cost coefficient of M2, the less likely it is to obtain greater profits from full cooperation.

Figure 7 shows that partial cooperation scenario consistently dominates the fully cooperation scenario in Regions I, while fully cooperation scenario always dominates the partial cooperation scenario in Region III. Similar to Figs. 5 and 6, for M1’ profitability, partial cooperation scenario is more favorable in region II, but for M2 the fully cooperation scenario is more effective. In addition, by comparing the two graphs in Fig. 7, we can see that the larger the proportion of cost sharing by M2, the greater the likelihood that M2 will gain more profits from partial cooperation than from full competition.

In sum, in above figurers there are main two types of regions: congruence and conflict regions, following Wang et al.³⁴ and Zhang et al.³⁵. A congruence region means that one scenario of cooperation dominates the other under two manufacturers’ profits. In Figs. 5 and 6, we can see that regions I and III are two congruence regions, where Scenario NC consistently dominates Scenarios PC and FC in regions I, while Scenarios PC and FC always dominate Scenario NC in regions III. Regions II are conflict regions. For M1’ profitability, Scenario NC is more favorable in regions II, but for M2 the cooperation scenarios are more effective. This illustrates the need for manufacturers to choose between a cooperative or independent operation strategy based on the characteristics of the market area in order to maximize profits.

Figure 7 shows that partial cooperation scenario consistently dominates the fully cooperation scenario in regions I, while Scenario FC always dominates the Scenario PC in region III. Similar to Figs. 5 and 6, for M1’ profitability, Scenario PC is more favorable in region II, but for M2 Scenario FC is more effective. Therefore, manufacturers need to assess market conditions, cost-effectiveness, and potential advantages of cooperation in order to develop an optimal cooperation model.

Extension

In the previous sections, we assume that the charging revenue is zero, since the revenue obtained from providing charging service is relatively low compared to the charging pile construction cost. With the continuous increase in EV sales, charging service fees for electric vehicles have also become a significant income source. Therefore, in this section, we explore the impact of charging fee on the decision-making of EV manufacturers. Similar to Shi et al.¹⁷, we assume that the average charging revenue per electric vehicle brings to the manufacturer is f. In both non-cooperative and partially cooperative models, the profit function of M1 becomes

$\widetilde{\pi}_{1}^{NC} (\widetilde{n}^{NC} ,\widetilde{p_{1}}^{NC} ) = \widetilde{D_{1}}^{NC}\widetilde{ p_{1}}^{NC} - \frac{1}{2}k_{1} \widetilde{n}^{NC2} + f\widetilde{D_{1}}^{NC}$,

$\widetilde{\pi}_{1}^{PC} (\widetilde{n}^{PC} ,\widetilde{p}_{1}^{PC} ) = \widetilde{D}_{1}^{PC}\widetilde{ p}_{1}^{PC} - \mu \frac{1}{2}k_{1} \widetilde{n}^{PC2} + f(\widetilde{D}_{1}^{PC} +\widetilde{D}_{2}^{PC} )$.

Since the profit functions of M2 in the two modes remains unchanged, we will not present them here.

In the full cooperative mode, we assume that EV owners with $\alpha$ ratio charge at the charging pile built by M1, and $1 - \alpha$ ratio EV consumers charge at the station built by M2. Therefore, the profit functions of the two manufacturers are

$$\widetilde\pi_{1}^{FC} \left( {\widetilde n_{1}^{FC} ,\widetilde p_{1}^{FC} } \right) = \widetilde D_{1}^{FC} \widetilde p_{1}^{FC} - \frac{1}{2}k_{1} \widetilde n_{1}^{FC2} + \alpha f\left( {\widetilde D_{1}^{FC} + \widetilde D_{2}^{FC} } \right),$$

$$\widetilde \pi_{2}^{FC} \left( {\widetilde n_{2}^{FC} ,\widetilde p_{2}^{FC} } \right) = \widetilde D_{1}^{FC} \widetilde p_{2}^{FC} - \frac{1}{2}k_{2} \widetilde n_{2}^{FC2} + (1 - \alpha )f\left( {\widetilde D_{1}^{FC} + \widetilde D_{2}^{FC} } \right).$$

Similar to 3.2 and 4.2, we can obtain the optimal number of charging piles and EV prices under non-cooperative mode, partial cooperative mode and full cooperative mode are respectively,

$$\begin{gathered} \tilde{n}^{NC*} = \frac{{4\gamma_{1} \left[ {a\left( {2 + \theta } \right) - f\left( {2 - \theta^{2} } \right)} \right]}}{{k_{1} \left( {4 - \theta^{2} } \right)^{2} - 8\gamma_{1}^{2} }},\;\;\tilde{p}_{1}^{NC*} = \frac{{ak_{1} \left( {2 - \theta } \right)\left( {\theta + 2} \right)^{2} + 2fk_{1} \left( {4\gamma_{1}^{2} - 4 + \theta^{2} } \right)}}{{k_{1} \left( {\theta^{2} - 4} \right)^{2} - 8\gamma_{1}^{2} }}, \hfill \\ \tilde{p}_{2}^{NC*} = \frac{{a\left[ {k_{1} \left( {2 + \theta } \right)\left( {4 - \theta^{2} } \right) - 4\gamma_{1}^{2} } \right] + fk_{1} \theta \left( {4\gamma_{1}^{2} - 4 + \theta^{2} } \right)}}{{k_{1} \left( {\theta^{2} - 4} \right)^{2} - 8\gamma_{1}^{2} }}; \hfill \\ \end{gathered}$$

$$\tilde{n}^{PC*} = \frac{{2a\gamma_{2} \left( {\theta + 2} \right) + f\gamma_{2} \left( {\theta^{3} - 3\theta^{2} + 8} \right)}}{{\left( {\theta + 2} \right)\left[ {k_{1} \mu \left( {\theta - 2} \right)^{2} - 2\gamma_{2}^{2} } \right]}},$$

$$\tilde{p}_{1}^{PC*} = \frac{{ak_{1} \mu \left( {\theta + 2} \right)\left( {2 - \theta } \right) + f\gamma_{2} \left( {6 + \theta - \theta^{2} } \right) - 2fk_{1} \mu \left( {2 - 3\theta + 2\theta^{2} } \right)}}{{\left( {\theta + 2} \right)\left[ {k_{1} \mu \left( {\theta - 2} \right)^{2} - 2\gamma_{2}^{2} } \right]}},$$

$$\tilde{p}_{2}^{PC*} = \frac{{ak_{1} \mu \left( {\theta + 2} \right)\left( {2 - \theta } \right) + f\gamma_{2} \left( {4 + 3\theta - \theta^{2} } \right) - fk_{1} \mu \theta \left( {2 - 3\theta + \theta^{2} } \right)}}{{\left( {\theta + 2} \right)\left[ {k_{1} \mu \left( {\theta - 2} \right)^{2} - 2\gamma_{2}^{2} } \right]}};$$

$$\tilde{n}_{1}^{FC*} = \frac{{2a\gamma_{3} k_{2} \left( {4 - \theta^{2} } \right) + 2f\gamma_{3}^{2} \left( {2\alpha - 1} \right)\left( {4 + 3\theta - \theta^{2} } \right) + 2fk_{2} \theta \left( {2 - 3\theta + \theta^{2} } \right) - \alpha fk_{2} \left( {16 - 4\theta - 12\theta^{2} + 7\theta^{3} - \theta^{4} } \right)}}{{\left( {4 - \theta^{2} } \right)\left[ {k_{1} k_{2} \left( {\theta - 2} \right)^{2} - 2\gamma_{3}^{2} \left( {k_{1} + k_{2} } \right)} \right]}},$$

$$\tilde{n}_{2}^{FC*} = \frac{{2a\gamma_{3} k_{1} \left( {4 - \theta^{2} } \right) + 2f\gamma_{3}^{2} \left( {2\alpha - 1} \right)\left( {4 + 3\theta - \theta^{2} } \right) + fk_{1} \left( {16 - 8\theta - 6\theta^{2} + 5\theta^{3} - \theta^{4} } \right) - \alpha fk_{1} \left( {16 - 4\theta - 12\theta^{2} + 7\theta^{3} - \theta^{4} } \right)}}{{\left( {4 - \theta^{2} } \right)\left[ {k_{1} k_{2} \left( {\theta - 2} \right)^{2} - 2\gamma_{3}^{2} \left( {k_{1} + k_{2} } \right)} \right]}},$$

$$\tilde{p}_{1}^{{FC*}} = \frac{\begin{gathered} ak_{1} k_{2} \left( {4 - \theta ^{2} } \right) + f\gamma _{3}^{2} k_{1} \left( {4 + 3\theta - \theta ^{2} } \right) - fk_{1} k_{2} \theta \left( {2 - 3\theta + \theta ^{2} } \right) - \alpha f\gamma _{3}^{2} k_{1} \left( {2 + 5\theta - \theta ^{2} } \right) \hfill \\ \quad \quad + \alpha f\gamma _{3}^{2} k_{2} \left( {6 + 8\theta - \theta ^{2} } \right) - \alpha fk_{1} k_{2} \left( {4 - 8\theta + 5\theta ^{2} - \theta ^{3} } \right) \hfill \\ \end{gathered} }{{\left( {2 + \theta } \right)\left( {k_{1} k_{2} \left( {\theta - 2} \right)^{2} - 2\gamma _{3}^{2} \left( {k_{1} + k_{2} } \right)} \right)}},$$

$$\tilde{p}_{2}^{{FC*}} = \frac{\begin{gathered} ak_{1} k_{2} \left( {4 - \theta ^{2} } \right) + f\gamma _{3}^{2} k_{1} \left( {6 + \theta - \theta ^{2} } \right) + 2fk_{1} k_{2} \left( {2 + 3\theta + \theta ^{2} } \right) - \alpha f\gamma _{3}^{2} k_{1} \left( {6 + \theta - \theta ^{2} } \right) \hfill \\ \quad \quad + \alpha f\gamma _{3}^{2} k_{2} \left( {2 + 5\theta - \theta ^{2} } \right) - \alpha fk_{1} k_{2} \left( {4 + 8\theta - 5\theta ^{2} + \theta ^{3} } \right) + 2f\gamma _{3}^{2} k_{2} \left( {1 - \theta } \right) \hfill \\ \end{gathered} }{{\left( {2 + \theta } \right)\left( {k_{1} k_{2} \left( {\theta - 2} \right)^{2} - 2\gamma _{3}^{2} \left( {k_{1} + k_{2} } \right)} \right)}};$$

By solving the partial derivative of the optimal number of charging piles with respect to charging fees, we can conclude that in both non-cooperative and partial cooperative modes, as the charging fee increases, the number of charging piles constructed by M1 also increases. In the fully cooperation mode, as the charging service fee increases, the number of charging piles built by the two manufacturers does not necessarily increase, and the trend is closely related to $\alpha ,\theta ,k_{i} ,\gamma_{3}$.

In the three cooperation modes, by solving the partial derivative of the optimal EV price with respect to charging fees, we can see that if consumers are sensitive to the number of charging piles, i.e. when $\gamma_{i}$ is very large, then the EV price increases with the charging fee. On the contrary, the EV price decreases with charging fee when $\gamma_{i}$ is very small.

Numerical example

We conduct several numerical examples to complement our analytical findings. These examples illustrate how the optimal solution changes, including the optimal number of charging piles, the prices and demands of EVs, and the manufacturers’ profits with the key parameters.

To validate the robustness of our analytical results, we perform simulations using real-world data, following the assumptions of the model and consistent with Yu et al.¹⁰, we set the basic parameters as follow: $\theta = 0.6$, $\gamma_{1} = \gamma_{2} = \gamma_{3} = 22$, $k_{1} = k_{2} = k = 2000$, $\mu = 0.6$. These parameter choices fully satisfy the assumptions of our model.

Example 1

In this example, we will explore the impacts of the degree of substitutability on the number of the charging piles, the EVs prices and demands, manufacturers’ profit. Let $\theta \in \left[ {0,1} \right]$, and the other parameters are the same as above, then we can get Figs. 8, 9, 10, and 11.

Fig. 8

Impact of $\theta$ on the number of charging piles.

Fig. 9

Impact of $\theta$ on the prices of EVs.

Fig. 10

Impact of $\theta$ on the demands of EVs.

Fig. 11

Impact of $\theta$ on the profits of manufacturers.

Figures 8, 9 and 10 illustrate the impact of EV degree of substitutability on the number of charging piles and the EVs prices and demands. As the degree of substitutability increases, manufacturers may increases the number of charging piles and adjust their pricing strategies to enhance their market competitiveness. The number of charging piles and the prices of EVs are highest when M1 and M2 are fully cooperation (Scenario FC). This could be attributed to the economies of scale that collaboration bring about. This is because when the two manufacturers fully cooperate, the total number of charging piles and the overall charging convenience reach their maximum level, thereby stimulating consumer demand for EVs and ultimately driving an increase in EV prices.

Figure 11 illustrates the impact of degree of substitutability on the profits of M1 and M2. As θ increases, M1’s and M2’s profits increase. And the profits of M1 and M2 are highest when M1 and M2 are fully cooperation (Scenario FC). This is due to the increase of the EVs’ price and demand.

Example 2

In this example, we will explore the impacts of the sensitivity of the EV demand to the number of charging pilesfor EVs on the number of the charging piles, the EV price and demands, manufacturer’s profit. Let $\gamma_{i} \in \left[ {10,30} \right]$, and the other parameters are the same as above, then we can get Figs. 12, 13, 14, and 15.

Fig. 12

Impact of $\gamma$ on the number of charging piles.

Fig. 13

Impact of $\gamma$ on the price of EVs.

Fig. 14

Impact of $\gamma$ on the demand of EVs.

Fig. 15

Impact of $\gamma$ on the profit of manufacturers.

Figures 12, 13, 14, and 15 show that the number of charging piles and the price of EVs increase as the sensitivity of the EV demand to the number of charging pilesfor EVs increases, as predicted in Section “Model”. We find that EV1’s and EV2’s demand and M1’s and M2’s profits increase with $\gamma$. An increase in consumer sensitivity of EV demand to the number of charging piles leads to the installation of more charging piles, which in turn stimulates EV demand through the indirect network effect. Consequently, higher EV prices generate greater revenue per unit sold for both manufacturers, and, combined with the growth in EV demand, the profits of M1 and M2 increase accordingly.

Example 3

In this example, we will explore the impacts of the proportion of cost sharing by M1 on the number of the charging piles, the EV price and demands, manufacturer’s profit. Let $\mu \in \left[ {0.4,1} \right]$, and the other parameters are the same as above, then we can get Figs. 16, 17, 18, and 19.

Fig. 16

Impact of $\mu$ on the number of charging piles.

Fig. 17

Impact of $\mu$ on the price of EVs.

Fig. 18

Impact of $\mu$ on the demand of EVs.

Fig. 19

Impact of $\mu$ on the profit of manufacturers.

Figures 16, 17, 18, and 19 show that in Scenario PC, as the proportion of cost sharing by M1 increases, the number of charging piles, the prices and demands for EVs, and the profits of M1 and M2 decrease. In the partial cooperation scenario, M1 reduces the number of charging piles to cut costs, which lowers consumer demand and EV prices. As a result, the profits of both manufacturers decline due to fewer charging piles and reduced EV demand.

Example 4

In this example, we will explore the impacts of the EV charging pile construction cost factor on the number of the charging piles, EVs prices and demands, manufacturers’ profits. Let $k_{i} \in \left[ {2000,3000} \right]$, and the other parameters are the same as above, then we can get Figs. 20, 21, 22, 23, 24, and 25.

Fig. 20

Impact of $k_{1}$ on the number of charging piles.

Fig. 21

Impact of $k_{1}$ on the price of EVs.

Fig. 22

Impact of $k_{1}$ on the demand of EVs.

Fig. 23

Impact of $k_{1}$ on the profit of manufacturers.

Fig. 24

Impact of $k_{2}$ on SPs’ number and EV’ price.

Fig. 25

Impact of $k_{2}$ on the EV’ demand and M_i profit.

Figures 20, 21, 22, 23, 24, and 25 demonstrates that the optimal number of the charging piles and EV prices decrease as the EV charging pile construction cost coefficient increases, as predicted in Section “Model”. We also find that EV1’s and EV2’s demands and M1’s and M2’s profits decrease with the cost coefficient. The rising cost of charging pile construction increases the financial burden on M1 and M2, potentially leading them to scale back investment in charging infrastructure. This reduction diminishes the convenience and attractiveness of EVs, thereby lowering consumer demand. At the same time, to remain competitive, manufacturers may be forced to either reduce EV prices or absorb higher operating costs, both of which can erode profitability. Consequently, higher charging pile construction costs exert a direct negative impact on EV demand and manufacturers’ profits in the EV market.

Conclusion

To attract more consumers to purchase electric vehicles (EVs), some EV manufacturers have begun investing in the construction of charging infrastructure. However, building charging piles requires significant capital investment. This paper examines how competing manufacturers can collaborate on charging pile deployment to reduce costs and enhance profitability.

We consider a scenario where manufacturer M1 has its own branded charging piles, while M2 does not. Three cooperation modes are analyzed: non-cooperation, partial cooperation, and full cooperation. Under non-cooperation, M1’s charging piles are exclusively available to its own EV customers (EV1), and are not accessible to M2’s customers (EV2). In partial cooperation, M2 contributes a portion of the construction costs for M1’s charging piles, and EV2 customers are allowed to use M1’s charging infrastructure. Under full cooperation, M2 also builds its own branded charging piles, and both manufacturers share access to each other’s charging infrastructure. By analyzing and comparing the equilibrium outcomes under each mode, we derive several key managerial insights.

First, as consumer sensitivity to the number of available charging piles increases, both manufacturers are incentivized to build more charging piles under all three modes. This leads to higher EV prices, increased demand, and improved profits for both manufacturers, with the most pronounced effect observed in the full cooperation scenario. Similarly, the degree of product substitutability between EV1 and EV2 influences strategic decisions: greater substitutability intensifies competition and motivates both manufacturers to invest more in charging infrastructure. As a result, EV prices, demand, and profits all increase.

Second, the optimal strategy for M1 and M2 varies depending on three factors: consumer sensitivity to charging pile availability, the charging pile construction cost coefficient, and the cost-sharing ratio. For M1, a critical factor is the potential negative impact of opening its branded charging piles to M2’s customers. If this impact is significant—e.g., due to increased wait times—M1 may reduce its investment in charging infrastructure, leading to lower profits. However, if the impact is minimal, cooperation can encourage M1 to build more charging piles. For M2, its strategic choice is primarily driven by the cost-sharing ratio and its own construction cost coefficient. A higher cost-sharing proportion makes partial cooperation more attractive than non-cooperation or full cooperation. Conversely, a higher construction cost coefficient makes full cooperation less profitable for M2.

Third, there often exists a dominant strategy that yields higher profits for both manufacturers compared to alternative strategies, and the likelihood of this mutually beneficial outcome is high. However, there are also potential conflict scenarios in which one manufacturer’s profit increases at the expense of the other’s. For example, M1 may benefit more from non-cooperation or full cooperation, while M2 may gain more from partial cooperation.

Finally, increasing the charging fee has different effects under each cooperation mode. In the non-cooperation and partial cooperation modes, a higher charging fee typically leads to more charging piles being constructed. In contrast, under full cooperation, an increase in charging fees does not necessarily result in a higher number of charging piles. Additionally, across all three modes, if consumers are highly sensitive to the number of charging piles, EV prices will rise alongside charging fees.

Future research could explore several directions. First, this paper assumes that each manufacturer produces only one type of EV, while in reality, a manufacturer may offer multiple EV models. For example, Tesla’s charging infrastructure is increasingly being opened to selected models from other manufacturers. Future work could investigate a scenario where a competing manufacturer offers two types of EVs and examine which models are granted access to shared charging infrastructure. Second, the sharing of self-branded charging piles may influence customer loyalty, which in turn affects EV sales. This dimension also merits further exploration.