Optimal Incentive Mechanism: Balancing the Complex Risk Preferences of Shared Battery Swapping Station Enterprises Under Dual Asymmetric Information

Abstract

This study investigates how the government can design optimal incentive mechanisms for Shared Battery Swapping Station (SBSS) construction enterprises under the conditions of dual information asymmetry (effort and operational efficiency) and heterogeneous corporate risk preferences. Employing game theory and principal–agent theory, this research constructs a mathematical model between the government and the enterprises to derive the optimal incentive coefficient and fixed subsidy, with theoretical results verified through numerical simulations. The findings reveal that risk-averse enterprises require higher incentives and subsidies. Compared to a single asymmetry scenario, the incentive coefficient is lower under dual information asymmetry. The government’s utility increases with an enterprise’s risk aversion beyond a critical threshold, while the enterprise’s utility remains at its reservation level. Our findings reveal a critical trade-off: while risk-averse enterprises require higher fixed subsidies and incentive coefficients, the presence of dual information asymmetry forces the government to paradoxically lower the incentive coefficient to prevent information rent extraction by less efficient firms. Furthermore, we identify a critical threshold where it becomes more beneficial for the government to contract with highly risk-averse firms due to their predictable behavior, even as the enterprise’s utility is held at its reservation level. These results provide a quantitative basis for policymakers to design a ’menu of contracts’—offering stable, high-subsidy options for risk-averse players and performance-based incentives for risk-neutral ones—rather than a one-size-fits-all policy.

1. Introduction

In the past decade, the electric vehicle (EV) industry has experienced exponential growth. In 2024, global sales of electric vehicles have increased by 25% year-on-year to more than 17 million. Many countries are promoting EV adoption through various policy measures. For instance, the United States provides tax incentives for electric vehicle buyers through the Inflation Reduction Act. According to the U.S. Internal Revenue Service (IRS), eligible new electric vehicle buyers can enjoy a tax credit of up to USD 7500, and second-hand electric vehicle buyers can obtain a tax credit equivalent to 30% of the purchase price and up to USD 4000 [1]. The promotion of electric vehicles (EVs) continues to face significant challenges, including high purchase costs, battery degradation, long charging times, and limited range [2]. While the Battery Charging Station (BCS) model has been introduced to alleviate some of these issues [3,4] its inherent limitations still pose significant obstacles to widespread adoption. To address these challenges, the Battery Swapping Station (BSS) model has been proposed as an innovative solution.

By addressing key barriers to EV adoption, the BSS model offers a promising pathway to accelerate the transition to sustainable transportation [2]. Chinese electric vehicle manufacturer NIO, which established its first BSS in May 2018, has significantly expanded its network, and as of June 2024 has installed 2432 BSSs across China. In addition, Aulton, a leading provider of BSS in China, has announced plans to establish over 10,000 BSSs by 2025, aiming to service more than 10 million new energy vehicles. The implementation of battery swapping mode in the Italian market has been demonstrated to effectively mitigate the issues associated with prolonged charging durations and inadequate infrastructure [5]. While companies like NIO and Aulton continue to expand their BSS networks, the growth of the industry, particularly towards developing widely accessible, shared infrastructure, is increasingly shaped by government policy. Hence, the government’s policy-driven measures also introduce new challenges, particularly in managing the interaction between governments and enterprises. For example, in Hainan Province, the local government rolled out financial subsidies for every battery swap station built, offering up to 500,000 RMB per station to incentivize companies. Similarly, in Beijing, financial compensation was introduced to support battery swapping infrastructure. Consequently, Shared Battery Swapping Stations (SBSS) largely rely on government funding and compensation to sustain these initiatives. This reliance is compounded by significant technological and economic hurdles. The viability of the SBSS model is fundamentally tied to the durability and lifespan of the batteries themselves, which constitute a massive capital investment. Therefore, technological advancements aimed at creating more durable and safer batteries, for instance, through advanced material science [6], are crucial for reducing the long-term operational costs and risks for station operators.

In response to the growing demand, governments around the world have begun taking proactive measures to promote SBSS development [7]. In China, the government has encouraged the integration of new energy sources with SBSS to enhance sustainability and energy efficiency. France and Germany are also providing corresponding policy support for electric vehicles, including financial subsidies and support for the construction of SBSS. That said, the implementation of government policy incentives is not without challenges, as it can give rise to moral hazards and adverse selection due to information asymmetries between enterprises and the government [8,9]. On the one hand, moral hazard can occur when companies that receive subsidies act in ways that are detrimental to long-term operational efficiency. For example, moral hazard in the SBSS industry can manifest when subsidized companies focus primarily on expanding the number of stations rather than improving service quality or operational efficiency. On the other hand, adverse selection in the context of SBSS means that enterprises with lower operational efficiency or poorer quality may be more likely to enter the market and claim subsidies. For instance, a company with subpar operational capacity may underreport the number of battery swaps performed or overstate its technological capabilities, such as the efficiency of its battery swapping equipment, in order to secure more financial support. Furthermore, enterprises’ risk preferences—whether risk-neutral or risk-averse—can significantly influence their approach to SBSS construction. Risk-neutral companies may take on greater operational risks, investing in long-term development but with a higher chance of financial losses if subsidies fall short of covering their initial investment. Conversely, risk-averse companies may be more inclined to secure stable government subsidies to minimize financial uncertainty, potentially avoiding investments in innovative or cost-saving technologies. This kind of behavior not only leads to inefficient allocation of public funds but also undermines the long-term development of the SBSS network by prioritizing short-term financial gains over sustainable growth.

While the principal–agent framework is well-established for analyzing incentive problems under asymmetric information [10,11,12], its application to the nascent field of Shared Battery Swapping Station (SBSS) infrastructure reveals significant research gaps. The existing literature often addresses moral hazard or adverse selection in isolation. Furthermore, while corporate risk preferences are acknowledged as important, few studies have systematically integrated heterogeneous risk preferences with the complexities of dual asymmetric information within a single, unified model. This oversight is critical, as government policies that ignore the interplay between these factors are likely to be inefficient or even counterproductive in the real world. Therefore, the primary contribution of this paper is to fill this gap by developing a comprehensive model that simultaneously addresses these intertwined challenges, providing nuanced insights for SBSS policy design.

Hence, the government should base its approach to SBSS construction on a principal–agent relationship with SBSS construction enterprises. This approach requires the government to consider the risk preferences of SBSS construction enterprises while regulating their methods for building and operating SBSS. By designing incentive mechanisms, the government can effectively guide enterprise behavior and align their actions with public objectives. This study, with a focus on incentive contracts, aims to investigate how these dual challenges influence the government’ s ability to design such mechanisms and explore their broader implications of these challenges and their mitigation through incentive design. Specifically, our analysis seeks to address the following research questions: (1) When SBSS construction enterprises are required by contract to reduce moral hazard, how does the heterogeneity of their risk preferences influence the design of government incentive mechanisms? (2) When the government crafts incentive contracts to simultaneously address moral hazard and adverse selection, how should the intensity of these incentives be calibrated in response to the risk aversion levels of enterprises? Does the impact of the heterogeneity in risk preferences among SBSS construction enterprises remain consistent, or does it fluctuate based on the specific terms and conditions of the contracts? (3) How does the government’s consideration of the risk preferences of SBSS construction enterprises influence the optimal utility for both parties involved in the game? Does the design of the two incentive mechanisms ultimately hinder the enterprise’s operations or drive a utility equilibrium between the two parties in the game?

To answer these questions, we construct a principal–agent model between the government and the SBSS construction enterprises from the perspective of game theory. By considering the risk preference of enterprises under the condition of asymmetric information, the design of government incentive contracts focuses on four scenarios: considering SBSS construction enterprises’ risk neutrality under single symmetric information, considering SBSS construction enterprises’ risk aversion under single asymmetric information, considering risk neutrality under dual asymmetric information, and considering risk aversion under dual asymmetric information. In this context, ’single asymmetric information’ refers specifically to scenarios where only moral hazard exists, while ’dual asymmetric information’ addresses scenarios where both moral hazard and adverse selection are present, reflecting the challenges previously discussed. The interests of both parties are balanced considering information asymmetry and risk preference. This study aims to develop a principal–agent model to illuminate the challenges of dual asymmetric information and heterogeneous risk preferences in the SBSS context. The primary contribution of this study is the development of a principal–agent model that systematically analyzes the interplay between dual information asymmetry and heterogeneous risk preferences in the SBSS context. By doing so, we provide analytical solutions for optimal incentive contracts, revealing how policymakers must balance incentive provision against the costs of risk and information asymmetry.

First, when SBSS construction enterprises are contractually obligated to mitigate moral hazard, the diversity in their risk preferences plays a crucial role in shaping the effectiveness of government incentive mechanisms. Risk-averse enterprises are inherently more cautious and likely to adhere strictly to contractual obligations to avoid potential losses, making them more responsive to structured incentives. Consequently, the government can design incentive mechanisms that offer higher financial rewards or stability to these risk-averse firms, encouraging them to invest in and maintain SBSS infrastructure diligently. On the other hand, firms with lower risk aversion may require different incentive structures. By recognizing and accommodating the varying degrees of risk aversion, the government can tailor its incentives to align more closely with the intrinsic motivations of each enterprise, thereby enhancing overall policy effectiveness and reducing instances of moral hazard.

Second, in scenarios where the government must address both moral hazard and adverse selection, the calibration of incentive intensity becomes pivotal, especially in the presence of heterogeneous risk preferences among SBSS construction enterprises. Enterprises with high levels of risk aversion benefit significantly from more substantial incentives, as these rewards help mitigate their concerns about potential losses and encourage them to engage proactively in SBSS deployment. The impact of risk preference heterogeneity remains consistent in that more risk-averse firms consistently require stronger incentives to ensure compliance and optimal performance. However, the specific terms and conditions of the contracts, such as the balance between fixed subsidies and variable incentives, can cause fluctuations in how these incentives are perceived and utilized by different enterprises. Therefore, the government must adopt a flexible and dynamic approach in designing incentive contracts, allowing adjustments based on ongoing assessments of enterprise behavior and risk profiles to maintain alignment with policy goals.

Third, the government’s consideration of SBSS construction enterprises’ risk preferences significantly influences the optimal utility for both parties involved, fostering a balanced operational equilibrium. By tailoring incentive mechanisms to accommodate varying risk aversion levels, the government ensures that enterprises receive appropriate rewards that align with their risk tolerance, thereby enhancing their utility and willingness to participate in SBSS initiatives. This strategic alignment between government incentives and enterprise preferences drives a mutually beneficial relationship, where enterprises operate efficiently without feeling financially overburdened. Consequently, the design of these incentive mechanisms does not hinder enterprise operations; instead, it promotes a utility equilibrium where both the government’s policy objectives and the enterprises’ operational and financial goals are harmoniously achieved. This equilibrium facilitates the sustainable and healthy development of the SBSS ecosystem, ensuring long-term success and widespread adoption of electric vehicles.

The rest of the paper is organized as follows. This will be followed by a literature review in Section 2, which contextualizes the study within existing frameworks. Section 3 presents the model, analyzing SBSS construction enterprises’ risk preferences under single and dual asymmetric information, focusing on both risk neutrality and risk aversion scenarios. Section 4 and Section 5 detail the optimal incentive mechanisms, specifically discussing the optimal incentive coefficient and the optimal fixed subsidy. Section 6 explores the trade-offs between utility and complexity, offering a balanced perspective on contract design. Finally, Section 7 concludes the paper by summarizing the results and providing management policy recommendations to promote the sustainable and effective development of SBSS construction enterprises.

2. Literature Review

In recent years, significant progress has been made in the research of SBSS construction enterprises, from the optimization of operation mode to the integration of management, from technological innovation to the exploration of commercialization path, which provides valuable theoretical basis and practical guidance for the promotion of shared battery swapping mode. This study is closely related to two major areas of literature: Influence of government policies on corporate behavior and dynamics of incentive-based government subsidies and corporate decision-making.

The impact of government policies on corporate behavior has become a significant issue in corporate operations management [13,14,15,16,17,18,19]. The government has not yet introduced specific financial support policies for the shared battery swapping model. Research by Comelli [20] indicates that while the battery swapping system is commercially viable, its success is contingent upon government support and policies. Furthermore, Jiang et al. [21] analyzed the collaboration between electric vehicle manufacturers and battery suppliers, recommending government support to achieve a win–win outcome.

Recent advancements in the measurement of risk preferences have provided new tools for understanding the behavior of firms. Deck et al. [22] propose a simple method to measure the high-order Arrow–Pratt risk aversion coefficient, which provides a new perspective for understanding the behavior of individuals at different risk preference levels. Pesenti et al. [23] proposed a novel risk budget allocation method, which is a risk-diversified portfolio strategy. This method ensures that any self-financing dynamic risk budgeting strategy with an initial wealth of 1 is a scaled version of the unique solution to these optimization problems. Hou and Lu [24] pointed out that it may be more effective to allocate more inventory risks to risk-averse retailers than to let risk-neutral manufacturers take more risks. Liu [25]’s research reveals how to achieve more effective risk management by optimizing risk allocation under heterogeneous beliefs. Smith [26] discussed the relationship between risk sharing, fiduciary duty, and corporate risk attitude.

The interaction between government incentive subsidies and firms has garnered considerable attention in the recent literature [27]. For instance, Liu et al. [28] presents a strategy for R&D cooperation among enterprises within the battery-swapping market, emphasizing the necessity for government-implemented subsidies and relevant policies. Similarly, Wu [2] developed a differential game model to explore the interaction strategies between the government and supply chain firms concerning technological innovation under government subsidies. Zhang et al. [29] demonstrates that, under certain conditions, combining subsidy policies with collaborative innovation contracts can achieve optimal economic efficiency. Furthermore, Zhu et al. [30] found that subsidies are the most effective means for maximizing social welfare. Benkhodja et al. [31] discovered that subsidies support green firms by reducing labor costs, decreasing pollution, and promoting growth in green sectors. In another study, He et al. [32] constructed a game model that accounts for the heterogeneous transformation potential of power companies and multiple policy goals, suggesting that the government should avoid fluctuations in subsidy budgets to maintain stability.

The most relevant literature for this study examines government incentive subsidies through the lenses of game theory and information asymmetry [10,33,34,35,36,37,38]. Research by He and Chen [39] highlights the critical role of government incentives and subsidies in encouraging companies to meet their environmental responsibilities. In this context, local governments are urged to take proactive measures to support green initiatives, including fostering a favorable regulatory environment and ensuring transparent information disclosure. Additionally, Cai et al. [40]’s research uniquely incorporates market value into the incentive contract framework, demonstrating that consumer preference for low-carbon products not only motivates manufacturers’ carbon reduction efforts but also complements government subsidies.

Our review of the existing literature reveals a critical, multifaceted research gap. While prior studies have provided valuable insights into incentive design, they tend to address the core challenges in a fragmented manner. For instance, a significant body of work [41,42,43,44] focuses on moral hazard, while another stream [45,46,47,48] examines adverse selection. However, few studies tackle the joint problem of dual asymmetric information, where a principal must design contracts to simultaneously mitigate both issues, a situation highly characteristic of infrastructure projects like SBSS.

Furthermore, while the importance of corporate risk aversion is acknowledged in finance and economics, its integration into principal–agent models of government subsidies is often oversimplified or ignored. Most models either assume risk neutrality or do not explore how heterogeneous risk preferences across firms should influence optimal contract design [49,50,51].

While the canonical principal–agent models of Laffont and Martimort [52] and Bolton and Dewatripont [53] provide the foundational tools for analyzing contracts under either moral hazard or adverse selection separately, a significant gap remains at the intersection of these complex issues. Our review reveals that few studies, particularly in the context of government subsidies for infrastructure, have systematically integrated the joint problem of dual asymmetric information with the complexities of heterogeneous corporate risk preferences.

Building on prior research, our study adopts a comprehensive approach to addressing the challenges of risk preferences and dual asymmetry information in SBSS construction enterprises. Specifically, we examine both effort levels and operational efficiency, developing a suite of incentive mechanism models to inform government policy design. These mechanisms aim to encourage SBSS construction enterprises to accurately report their operational levels and optimize effort decisions, while balancing the trade-offs between costs and incentive benefits. By integrating considerations of risk preferences and mitigating the complexities introduced by dual information asymmetry—namely, adverse selection and moral hazard—our approach enhances both operational efficiency and economic performance. Ultimately, our study not only tackles the theoretical and practical challenges posed by information uncertainty but also offers actionable strategies to promote the sustainable and healthy development of the SBSS construction enterprise ecosystem.

3. Model

Advancements in electric vehicle (EV) technology have significantly boosted their market popularity. However, the limited availability of shared battery swapping stations (SBSS) remains a primary obstacle to the widespread adoption of EVs. Our model seeks to develop incentive mechanisms that align government policies with the operational strategies of SBSS construction enterprises, thereby fostering both operational efficiency and economic performance. Furthermore, by incorporating considerations of risk preferences and addressing the complexities introduced by dual information asymmetry—specifically, adverse selection and moral hazard—our approach enhances the overall effectiveness of SBSS deployment.

We argue that the government will face some problems based on information asymmetry when concluding contracts with SBSS construction enterprises. At this time, the game timeline between the enterprises and the government is shown in Figure 1. At T1, the government formulates the contract through the information structure and risk preference of SBSS construction enterprises. During the T2–T3 period, the government and SBSS construction enterprises sign a contract. When the contract comes into effect at T4, SBSS construction enterprises will determine their level of effort. At T5, the SBSS construction enterprise delivers its output, and the government pays the corresponding remuneration according to the contract.

We assume that both the government and SBSS construction enterprises are rational, that is, both sides want to maximize their interests as much as possible in decision-making [54]. The government guides the behavior of SBSS construction enterprises by formulating policies and providing subsidies to achieve the goal of increasing the delivered production of SBSS. SBSS construction enterprises will look for the lowest cost and highest profit operation mode under the framework of these policies [55].

In the principal–agent relationship between the government and SBSS construction enterprises, the information observation ability shows significant differences. The government can observe the operational level $\theta$ of the enterprise through indirect indicators such as market data and financial reports. However, the specific effort level e of the enterprise is a decision variable, and its actual input level is unobservable. The SBSS construction enterprise’s effort can be quantified [56] and quantified as a variable e, then the SBSS construction enterprise’s construction project delivered production can be expressed as a linear function: $\pi (e, \theta )=ae+\theta ,$ where $\pi$ can be understood as the delivered production benefit of SBSS construction enterprises entrusted by the government, the parameter a represents the scaling factor for the marginal impact of effort, with $a>0$ and a typical range of $[0.01, 50]$ to accommodate both theoretical normalization and empirical calibration in SBSS contexts. In numerical simulations, $a=20$ corresponds to CNY $200,000$ per unit effort (e.g., aligned with provincial subsidies). When it evolves to dual information asymmetry (moral hazard and adverse selection), the government faces dual observation obstacles: it is impossible to monitor the efforts of enterprises in real time e, and it is impossible to accurately identify the real operational level $\theta$ of enterprises. $\theta$ is an index used by the government to evaluate the operation level of SBSS construction enterprises, general cognition: $\theta$ obeys continuous uniform distribution $U(0, \overline{\theta }).$ This simplification allows us to derive closed-form solutions and clearly illustrate the core trade-offs in the incentive design. While other distributions might better reflect the reality of many industries, the qualitative insights from our model are expected to be robust. We further address this as a limitation in the discussion section.

SBSS delivered production $\pi (e, \theta )$ for observable SBSS construction enterprises at the end of the contract period. The government will provide contract remuneration $S\left[\pi \right(e, \theta \left)\right]=\alpha +\beta \pi ,$ where $\alpha$ represents the fixed subsidy given by the government to the enterprises, $\beta$ represents the government’s incentive coefficient for corporate delivered production, which measures the marginal impact of delivered production levels, that is, the level of additional subsidies per unit of delivered production.

The cost function of SBSS construction enterprise’s efforts to produce is expressed as $C\left(e\right)={\displaystyle \frac{1}{2}}b{e}^2,b>0,$ where b denotes the cost coefficient of the SBSS construction enterprises $C\left(e\right)>0, C^{\prime }\left(e\right)>0, C^{″}\left(e\right)>0,$ it shows that the improvement of effort level leads to the increase of effort cost, and the marginal cost will gradually increase.

The main notations used throughout this paper are summarized in

Table 1. Summary of model notations.

Symbol	Description	Type/Note
Choice Variables
$\alpha , \beta$	Fixed subsidy and incentive coefficient.	$\alpha \left(\theta \right), \beta \left(\theta \right)$ in dual asymmetry.
e	Effort level exerted by the SBSS enterprise.	Agent’s choice, $e>0$.
Parameters
$\theta$	Operational efficiency level of the enterprise.	Agent’s type, $\theta \sim U(0, \overline{\theta })$.
a	Marginal productivity of effort.	Exogenous, $a>0$.
b	Marginal cost of effort.	Exogenous, $b>0$.
$\rho$	Coefficient of absolute risk aversion.	Agent’s risk preference, $\rho \ge 0$.
${\sigma }^2$	Variance of the random output shock.	Exogenous, ${\sigma }^2>0$.
$\overline{w}$	Reservation utility of the enterprise.	Exogenous.
Functions
$\pi (e, \theta )$	Delivered production output, $\pi =ae+\theta$.	Output function.
$C\left(e\right)$	Cost of effort, $C\left(e\right)={\displaystyle \frac{1}{2}}b{e}^2$.	Cost function.
$S\left(\pi \right)$	Total remuneration, $S\left(\pi \right)=\alpha +\beta \pi$.	Contract function.
$U_G, U_E$	Utility of the government and enterprise.	Objective functions.
$f\left(\theta \right), F\left(\theta \right)$	PDF and CDF of the type distribution.	Distribution functions.
$\mu \left(\theta \right)$	Hazard rate, $\mu \left(\theta \right)=f\left(\theta \right)/(1-F(\theta \left)\right)$.	Hazard rate function.

.

Based on the Arrow–Pratt theory [57,58] of microeconomics, by setting the government’s risk preference to be neutral, the expected utility function can be expressed as $E\left(U_G\right)$:

$$E\left(U_G\right)=E[\pi (e, \theta )-S\left(\pi \right)]=-\alpha +(ae+\theta )(1-\beta ).$$

(1)

The utility function of SBSS construction enterprises can be expressed as $U_N$:

$$\begin{array}{cc}\hfill & E\left(U_E\right)=E\left(w\right)=E[S\left(\pi \right)-C\left(e\right)]=E\left(\alpha +\beta ae+\beta \theta -{\displaystyle \frac{1}{2}}b{e}^2\right),\hfill \end{array}$$

(2)

where w is the real market income of SBSS construction enterprises, we considered the different information structures and risk preferences of the SBSS construction enterprises, $U_E$ will change according to different scenarios.

We will discuss the problem of constructing a game model between the government and the SBSS construction enterprises of the SBSS in the continuous system. As shown in Figure 2, the government needs to consider the information structure and risk preference of SBSS construction enterprises in the contract design. In the case of single asymmetric information, we assume that the government can grasp the operational level range of SBSS construction enterprises, but cannot directly observe the specific level of effort paid by the company to the contract. The rationality of this hypothesis is that the government, as a policy maker and resource allocator, can roughly evaluate the overall operation of the enterprises through market data, financial statements, and operation indicators. However, due to the complexity of internal management and the implicit characteristics of the level of effort, it is difficult for the government to accurately grasp the actual payment of the enterprises in the process of contract implementation. Direct monitoring and evaluating the effort level of SBSS construction enterprises not only requires a lot of information acquisition costs but also may lead to information distortion and agency problems. Therefore, this hypothesis is more in line with the actual operating conditions than fully grasping the efforts of SBSS construction enterprises, which helps to simplify the model construction, reduce transaction costs, and reflect the regulatory difficulties faced by the government under information asymmetry. Under the background of dual information asymmetry, due to the government’s efforts to the operation level of SBSS construction enterprises, it is particularly important to design dual incentive mechanism to promote information transparency and sharing. In the contract design, fully considering the degree of risk preference of SBSS construction enterprises can better measure the impact of risks on the utility of all parties, and avoid waste of resources or failure of cooperation due to improper risk allocation. Through reasonable incentive allocation on the basis of risk preference, we can strive to achieve a better Pareto of the system, so that all parties can not further improve their own utility under the existing resources and information structure, so as to promote the efficient operation and sustainable development of the SBSS mode.

4. Single Incentive Mechanism

4.1. The Risk Neutrality of SBSS Construction Enterprises Under Single Asymmetric Information

This part studies the contract under the background of single asymmetric information in the traditional principal–agent theory. In this type of scenario, the government has the ability to obtain the real operation level of SBSS construction enterprises but cannot know the effort level in advance.

The optimization model of the traditional principal–agent theory is established, and the objective function is established to maximize its expected utility [52]. The principal–agent model is established considering participation constraints (PC), incentive compatibility constraints (IC), and limited liability constraints (LL). Set the objective function to maximize the expected utility of the government: ${max}_{(\alpha ,\beta )}E\left(U_G^{SN}\right)={max}_{(\alpha ,\beta )}E[\pi (e, \theta )-S\left(\pi \right)].$

Set participation constraints (PC) to ensure that SBSS construction enterprises can obtain retained profits when accepting contracts: $E\left({U^{SN}}_E\right)={U^{SN}}_E=S\left(\pi (e, \theta )\right)-C\left(e\right)\ge \overline{w}.$

The incentive compatibility constraint (IC) is set to encourage SBSS construction enterprises to obtain the maximum profit when using the optimal effort level: ${max}_e\left[S\left(\pi (e, \theta )\right)-C\left(e\right)\right].$

Set a limited liability constraint (LL) to ensure that the fixed subsidy and incentive coefficient in the government’s incentive contract are not negative: $\alpha \ge 0,1\ge \beta \ge 0.$

Lemma 1.

The expression of the principal–agent model can be converted to scenario SN:

$$\begin{array}{cc}\hfill & \underset{(\alpha ,\beta )}{max}[-\alpha +(ae+\theta )(1-\beta )]\hfill \\ & s.t.\hfill \\ & \left(PC\right)\alpha +\beta ae+\beta \theta -{\displaystyle \frac{1}{2}}b{e}^2\ge \overline{w},\hfill \\ & \left(IC\right)\underset{e}{max}\left(\alpha +\beta ae+\beta \theta -{\displaystyle \frac{1}{2}}b{e}^2\right),\hfill \\ & \left(LL\right)\alpha \ge 0,1\ge \beta \ge 0.\hfill \end{array}$$

(3)

(a) In the case of single asymmetric information, the optimal contract $({\alpha }^{\ast SN}, {\beta }^{\ast SN})$ provided by the government and the optimal effort level $e^{\ast SN}$ of SBSS construction enterprises can be expressed as

$${\alpha }^{\ast SN}=\overline{w}-\theta -{\displaystyle \frac{a^2}{2b}},{\beta }^{\ast SN}=1,e^{\ast SN}={\displaystyle \frac{a}{b}}.$$

(4)

(b) The optimal utility of the government and SBSS construction enterprises can be obtained:

$$U_G^{\ast SN}=-\overline{w}+\theta +{\displaystyle \frac{a^2}{2b}}.$$

(5)

$$U_E^{\ast SN}=\overline{w}.$$

(6)

All the proofs of Lemmas, Corollaries, and Propositions are put into Appendix A.

4.2. The Risk Aversion of SBSS Construction Enterprises Under Single Asymmetric Information

Under single asymmetric information, this subsection examines the scenario of risk aversion among SBSS construction enterprises. The government aims to maximize its expected utility by incorporating participation constraints, incentive compatibility constraints, and limited liability constraints into its model.

According to principal–agent theory [11,12,52], since the SBSS construction enterprises (agents) are risk-averse, their objective is to maximize their expected utility. We assume the enterprises exhibit Constant Absolute Risk Aversion (CARA), with a utility function of the form $U\left(w\right)=-exp(-\rho w)$, where w is the enterprise’s income and $\rho >0$ is the Arrow–Pratt coefficient of absolute risk aversion. The final output $\pi$ is subject to external random shocks, $ϵ$, which are assumed to be normally distributed with $N(0, {\sigma }^2)$.

The enterprise’s income is $w=S\left(\pi \right)-C\left(e\right)=\alpha +\beta \pi -C\left(e\right)$. Under the CARA-Normal framework, maximizing the expected utility $E\left[U\right(w\left)\right]$ is equivalent to maximizing the agent’s certainty equivalent, given by $E\left[w\right]-(1/2)\rho Var\left(w\right)$. Here, the expected income is $E\left[w\right]=\alpha +\beta E\left[\pi \right]-C\left(e\right)=\alpha +\beta (ae+\theta )-(1/2)b{e}^2$, and its variance is $Var\left(w\right)=Var\left(\beta \pi \right)={\beta }^{2}Var\left(\pi \right)={\beta }^2{\sigma }^2$. Thus, the enterprise’s objective function is to maximize: $\alpha +\beta ae+\beta \theta -(1/2)b{e}^2-(1/2)\rho {\beta }^2{\sigma }^2$. The term $(1/2)\rho {\beta }^2{\sigma }^2$ represents the risk premium.

Lemma 2.

The principal–agent model expression can be written as scenario SA:

$$\begin{array}{cc}& \underset{(\alpha ,\beta )}{max}[-\alpha +(ae+\theta )(1-\beta )]\hfill \\ & s.t.\hfill \\ & \left(PC\right)\alpha +\beta ae+\beta \theta -{\displaystyle \frac{1}{2}}b{e}^2-{\displaystyle \frac{1}{2}}\rho {\beta }^2{\sigma }^2\ge \overline{w},\hfill \\ & \left(IC\right)\underset{e}{max}\left(\alpha +\beta ae+\beta \theta -{\displaystyle \frac{1}{2}}b{e}^2-{\displaystyle \frac{1}{2}}\rho {\beta }^2{\sigma }^2\right),\hfill \\ & \left(LL\right)\alpha \ge 0,1\ge \beta \ge 0.\hfill \end{array}$$

(7)

(a) In the case of single asymmetric information, the optimal contract $\left({\alpha }^{\ast SA},{\beta }^{\ast SA}\right)$ provided by the government considering the risk aversion of SBSS construction enterprises and the optimal effort level of SBSS construction enterprises $e^{\ast SA}$ can be expressed as

$$\begin{array}{cc}\hfill & {\alpha }^{\ast SA}=\overline{w}-{\displaystyle \frac{a^6}{b{(a^2+b\rho {\sigma }^2)}^2}}-{\displaystyle \frac{\theta a^2}{a^2+b\rho {\sigma }^2}}+{\displaystyle \frac{b{a}^6}{2{(b{a}^2+\rho b^2{\sigma }^2)}^2}}+{\displaystyle \frac{\rho {\sigma }^2{a}^4}{2{(a^2+b\rho {\sigma }^2)}^2}},\hfill \\ \hfill & {\beta }^{\ast SA}={\displaystyle \frac{a^2}{a^2+b\rho {\sigma }^2}},e^{\ast SA}={\displaystyle \frac{a^3}{b{a}^2+\rho b^2{\sigma }^2}}.\hfill \end{array}$$

(8)

(b) The optimal utility of the government and SBSS construction enterprises can be obtained:

$$\begin{array}{cc}\hfill & U_G^{\ast SA}=\theta -\overline{w}-{\displaystyle \frac{b{a}^6}{2{(b{a}^2+\rho b^2{\sigma }^2)}^2}}-{\displaystyle \frac{\rho {\sigma }^2{a}^4}{2{(a^2+b\rho {\sigma }^2)}^2}}+{\displaystyle \frac{a^4}{b{a}^2+\rho b^2{\sigma }^2}}.\hfill \end{array}$$

(9)

$$U_E^{\ast SA}=\overline{w}.$$

(10)

(c) The optimal effort level of SBSS construction enterprises $e^{\ast SA}$ has the following properties: $e^{\ast SA}$ decreases with the increase of effort cost coefficient b, $e^{\ast SA}$ increases with the increase of marginal influence degree of effort level a, and $e^{\ast SA}$ decreases with the increase of risk aversion degree $\rho .$

Lemma 2 constitutes a theoretical extension of the canonical principal–agent paradigm by formally incorporating enterprises’ risk aversion properties. Through rigorous mathematical derivation, this proposition systematically examines how heterogeneous risk preferences influence the architecture of incentive-compatible contracts.

4.3. Optimal Incentive Coefficient Under Single Asymmetric Information

Designing effective incentive mechanisms under single asymmetric information requires a deep understanding of the diverse risk preferences exhibited by SBSS construction enterprises. Risk-averse firms, in particular, respond uniquely to government incentives compared to their risk-neutral counterparts. By accounting for these varying risk attitudes, the government can tailor incentive coefficients to better align with the operational behaviors and stability of SBSS construction enterprises. The following Proposition 1 elucidates how risk aversion influences the optimal allocation of incentive coefficient in scenarios characterized by moral hazard.

Proposition 1.

Risk-averse enterprises receive higher incentive coefficients in moral hazard situations:

$$\begin{array}{cc}\hfill & {\beta }^{\ast SN}<{\beta }^{\ast SA}.\hfill \end{array}$$

(11)

Proposition 1 underscores the pivotal role that heterogeneous risk preferences play in shaping the government’s optimal incentive coefficients under single asymmetric informational contexts. Specifically, the government assigns higher incentive coefficients (${\beta }^{\ast SN}<{\beta }^{\ast SA}$) to risk-averse SBSS construction enterprises in scenarios characterized by single asymmetric information (moral hazard). This strategic allocation is motivated by the fact that risk-averse firms demonstrate more stable and predictable behaviors, thereby reducing the uncertainty associated with achieving policy objectives.

Corollary 1.

Under moral hazard, the optimal incentive coefficient increases with enterprise risk preference ρ: i.e., ${\displaystyle \frac{\partial }{\partial \rho }}({\beta }^{\ast SN}-{\beta }^{\ast SA})={\displaystyle \frac{a^{2}b{\sigma }^2}{{(a^2+b\rho {\sigma }^2)}^2}}\ge 0.$

This positive relationship implies that the government adjusts the incentive coefficients to accommodate enterprises’ varying risk preferences, thereby enhancing the effectiveness of subsidy allocations. By recognizing and responding to the heterogeneity in risk preferences, the government ensures that its incentive mechanisms remain robust and aligned with enterprise behaviors and policy objectives, even in the presence of informational asymmetries.

4.4. Optimal Fixed Subsidy Under Single Asymmetric Information

Proposition 2.

Risk-averse enterprises receive higher fixed subsidies in moral hazard situations:

$$\begin{array}{cc}\hfill & {\alpha }^{\ast SN}<{\alpha }^{\ast SA}.\hfill \end{array}$$

(12)

Proposition 2 demonstrates that the government allocates higher fixed subsidies (${\alpha }^{\ast SN}<{\alpha }^{\ast SA}$) to risk-averse enterprises under single asymmetric information scenarios (moral hazard). This allocation reflects the government’s confidence in the predictable and stable behavior of risk-averse firms, which better aligns with long-term public objectives and reduces uncertainty in achieving policy goals. Risk-averse enterprises are less likely to misuse subsidies—for example, by avoiding excessive station expansion for short-term gains—and instead prioritize sustainable improvements in service quality and operational efficiency. This strategic alignment ensures that subsidy funds are effectively utilized to foster desired outcomes. Consequently, the heterogeneity in risk preferences informs the government’s approach to designing fixed subsidies, enabling it to guide enterprise behavior and ensure the successful implementation of contracts.

Corollary 2.

Under moral hazard, the government’s optimal fixed subsidy difference (${\alpha }^{\ast SN}-{\alpha }^{\ast SA}$) increases with enterprise risk preference ρ: i.e., ${\displaystyle \frac{\partial }{\partial \rho }}({\alpha }^{\ast SN}-{\alpha }^{\ast SA})\le 0$.

Corollary 2 indicates that under single asymmetric information scenarios (moral hazard), the enterprise risk preference parameter ($\rho$) has a non-positive effect on the government’s optimal fixed subsidy. Specifically, as the risk preference parameter $\rho$ increases, the difference in optimal fixed subsidies (${\alpha }^{\ast SN}-{\alpha }^{\ast SA}$) either decreases or remains unchanged.

Corollary 3.

Enhanced policy support is provided for risk-averse enterprises via integrated incentive coefficients and fixed subsidies under moral hazard.

Corollary 3 synthesizes the results of Propositions 1 and 2 to demonstrate that under single asymmetric information scenarios (moral hazard), the government provides both higher incentive coefficients (${\beta }^{\ast SN}<{\beta }^{\ast SA}$, as stated in Proposition 1) and higher fixed subsidies (${\alpha }^{\ast SN}<{\alpha }^{\ast SA}$) to risk-averse enterprises. This result implies that when contracting with firms known for their conservative financial strategies, the government must offer a larger share of the revenue to induce the same level of effort. For instance, a government tender for SBSS construction could include clauses that offer more favorable revenue-sharing terms to state-owned enterprises or established infrastructure players, who are typically more risk-averse, compared to more agile but risk-tolerant startups.

5. Dual Incentive Mechanism

This section discusses the incentive interaction model between government and enterprises under the dual asymmetric information situation (including moral hazard and adverse selection). By comprehensively analyzing the risk preference and operation level of enterprises, the government can design more accurate and efficient incentive schemes to promote the stable development of enterprises and the realization of policy objectives.

5.1. The Risk Neutrality of SBSS Construction Enterprises Under Dual Asymmetric Information

Starting from this section, we consider that in the real scenario, the government cannot obtain the effort level e and operation level of SBSS construction enterprises in advance $\theta$ (i.e., dual information asymmetry). When discussing this kind of dual asymmetric information, we assume that the fixed subsidy and incentive coefficient provided by the government are nonlinear functions of the operation level of SBSS construction enterprises, which can be expressed as $\alpha (·)$ and $\beta (·)$, respectively. Following the standard approach in the principal–agent literature, we focus on direct revelation mechanisms where the government offers a menu of contracts, and the enterprise’s choice reveals its private information $\theta$. Specifically, we consider a class of contracts that are linear in the final output $\pi$ but allow the coefficients to be type-dependent, i.e., the remuneration is $S(\pi , \theta )=\alpha \left(\theta \right)+\beta \left(\theta \right)\pi$. This linear contract form is a common and powerful simplification, widely used since the seminal work of Holmström and Milgrom [10], as it captures the fundamental trade-off between incentives and risk-sharing while maintaining analytical tractability. While the fully optimal contract might be nonlinear in $\pi$, the linear form provides a close approximation and yields clear, intuitive policy insights.

The hypothesis has a priori knowledge: $\mu \left(\theta \right)=f\left(\theta \right)/1-F\left(\theta \right)$ is a monotonically increasing risk rate, where $f\left(\theta \right)$ is the probability density function and $F\left(\theta \right)$ is the cumulative probability distribution function, such that $f\left(\theta \right)=\mathrm{d}F\left(\theta \right)/\mathrm{d}\theta$.

The objective function is set to maximize the expected utility of the government: ${max}_{(\alpha ,\beta )}E\left(U_G^{DN}\right)={max}_{(\alpha ,\beta )}[\pi (e, \theta )-S\left(\pi (e, \theta )\right)].$

Set participation constraints (PC) to ensure that SBSS construction enterprises can obtain retained utility when accepting contracts: $E\left[U_E^{DN}(\theta , \theta )\right]=U_E^{DN}(\theta , \theta )=S\left(\pi (e, \theta )\right)-C\left(e\right)\ge \overline{w}.$

Set incentive compatibility constraint 1 (IC1) to ensure that SBSS construction enterprises can do their best to complete the contract: ${max}_e\left[U_E^{DN}(\theta , \theta )\right]={max}_e[S\left(\pi (e, \theta )\right)-C\left(e\right)].$

Set incentive compatibility constraint 2 (IC2) to ensure that SBSS construction enterprises are more efficient than false reports when reporting real operational levels: $E\left[U_E^{DN}(\theta , \theta )\right]\ge E\left[U_E^{DN}({\theta }^{\prime },\theta )\right],$ where $E\left[U_E^D({\theta }^{\prime },\theta )\right]$ represents the contract obtained $[\alpha \left({\theta }^{\prime }\right),\beta \left({\theta }^{\prime }\right)]$ due to false reporting of operational levels ${\theta }^{\prime },$ thereby determining the expected utility ${\int }_0^{\overline{\theta }}\theta \left\{S[\pi (e,{\theta }^{\prime })\right]-C\left(e\right)\}f\left(x\right)\mathrm{d}x.$

Set up limited liability constraints (LL) to ensure that the fixed subsidy and incentive coefficient of the contract is not negative: $\alpha \ge 0, 1\ge \beta \ge 0.$

Lemma 3.

The principal–agent model expression can be converted to scenario DN:

$$\begin{array}{cc}\hfill & \underset{(\alpha ,\beta )}{max}{\int }_0^{\overline{\theta }}\left[-\alpha \left(\theta \right)+\left(ae\right(\theta )+\theta )(1-\beta (\theta \left)\right)\right]f\left(\theta \right)d\theta \hfill \\ \hfill & s.t.\hfill \\ \hfill & \left(PC\right)\alpha \left(\theta \right)+\beta \left(\theta \right)ae\left(\theta \right)+\beta \left(\theta \right)\theta -{\displaystyle \frac{1}{2}}be{\left(\theta \right)}^2\ge \overline{w},\hfill \\ \hfill & \left(IC\mathit{1}\right)\underset{e}{max}\left[\alpha \left(\theta \right)+\beta \left(\theta \right)ae+\beta \left(\theta \right)\theta -{\displaystyle \frac{1}{2}}b{e}^2\right],\hfill \\ \hfill & \left(IC\mathit{2}\right)U_E^{DN}(\theta , \theta )\ge U_E^{DN}({\theta }^{\prime },\theta ),\hfill \\ \hfill & \left(LL\right)\alpha \left(\theta \right)\ge 0,1\ge \beta \left(\theta \right)\ge 0.\hfill \end{array}$$

(13)

(a) In the case of dual asymmetric information, the optimal contract $({\alpha }^{\ast DN}\left(\theta \right),{\beta }^{\ast DN}\left(\theta \right))$ provided by the government and the optimal effort level $e^{\ast DN}\left(\theta \right)$ of SBSS construction enterprises can be expressed as

$$\begin{array}{cc}\hfill & \alpha {\left(\theta \right)}^{\ast DN}=\overline{w}+{\int }_0^{\overline{\theta }}\beta {\left(x\right)}^{\ast DN}dx-{\displaystyle \frac{{\left(\beta {\left(\theta \right)}^{\ast DN}\right)}^2{a}^2}{2b}},\hfill \\ \hfill & \beta {\left(\theta \right)}^{\ast DN}=1-{\displaystyle \frac{b\left(1-F\left(\theta \right)\right)}{a^{2}f\left(\theta \right)}},e^{\ast DN}\left(\theta \right)={\displaystyle \frac{a}{b}}-{\displaystyle \frac{1-F\left(\theta \right)}{af\left(\theta \right)}}.\hfill \end{array}$$

(14)

(b) The optimal utility of the government and SBSS construction enterprises can be obtained:

$$\begin{array}{cc}\hfill & U_G^{\ast DN}={\int }_0^{\overline{\theta }}\left({\displaystyle \frac{a^2}{2b}}\beta {\left(x\right)}^{\ast DN}+x\right)dF\left(x\right)-{\int }_0^{\overline{\theta }}\left(1-F\left(x\right)\right)\beta {\left(x\right)}^{\ast DN}dx-\overline{w}.\hfill \end{array}$$

(15)

$$U_E^{\ast DN}\left(\theta \right)=\overline{w}+{\int }_0^{\overline{\theta }}\beta {\left(x\right)}^{\ast DN}dx+x-{\displaystyle \frac{xb\left(1-F\left(x\right)\right)}{a^{2}f\left(x\right)}}.$$

(16)

(c) The optimal effort level of SBSS construction enterprises $e^{\ast DN}\left(\theta \right)$ under scenario DN has the following properties: $e^{\ast DN}\left(\theta \right)$ decreases as the effort cost coefficient b, $e^{\ast DN}\left(\theta \right)$ increases with the increase of the marginal effect of effort a, $e^{\ast DN}\left(\theta \right)$ increases with the increase of operation level $\theta .$

Lemma 3 demonstrates how the government adopts the optimal contract under dual information asymmetry. It balances the following objectives: encouraging SBSS construction enterprises to exert their best effort while also motivating them to report their business level honestly.

5.2. The Risk Aversion of SBSS Construction Enterprises Under Dual Asymmetric Information

Based on the dual information asymmetry of the contract information structure discussed in the previous section, this section considers the contract of risk aversion of SBSS construction enterprises. The principal–agent model is established to maximize its expected utility and consider the participation constraint, incentive compatibility constraint 1 (IC1), incentive compatibility constraint 2 (IC2), and limited liability constraint (LL).

Lemma 4.

Then the principal–agent model expression can be written as Scenario DA:

$$\begin{array}{cc}\hfill & \underset{(\alpha ,\beta )}{max}{\int }_0^{\overline{\theta }}[-\alpha \left(\theta \right)+(ae\left(\theta \right)+\theta )(1-\beta \left(\theta \right))]f\left(\theta \right)d\theta \hfill \\ \hfill & s.t.\hfill \\ \hfill & \left(PC\right)\alpha \left(\theta \right)+\beta \left(\theta \right)ae\left(\theta \right)+\beta \left(\theta \right)\theta -{\displaystyle \frac{1}{2}}be{\left(\theta \right)}^2-{\displaystyle \frac{1}{2}}\rho \beta {\left(\theta \right)}^2{\sigma }^2\ge \overline{w},\hfill \\ \hfill & \left(IC\mathit{1}\right)\underset{e}{max}\left(\alpha \left(\theta \right)+\beta \left(\theta \right)ae+\beta \left(\theta \right)\theta -{\displaystyle \frac{1}{2}}b{e}^2-{\displaystyle \frac{1}{2}}\rho \beta {\left(\theta \right)}^2{\sigma }^2\right),\hfill \\ \hfill & \left(IC\mathit{2}\right)U_E^{DA}(\theta ,\theta )\ge U_E^{DA}({\theta }^{\prime },\theta ),\hfill \\ \hfill & \left(LL\right)\alpha \left(\theta \right)\ge 0,1\ge \beta \left(\theta \right)\ge 0.\hfill \end{array}$$

(17)

(a) In the case of dual information asymmetry, the optimal contract provided by the government considering the risk aversion of SBSS construction enterprises $({\alpha }^{\ast DA}\left(\theta \right),{\beta }^{\ast DA}\left(\theta \right))$ and the optimal effort level of SBSS construction enterprises $e^{\ast DA}\left(\theta \right)$ can be expressed as

$$\begin{array}{cc}\hfill & \alpha {\left(\theta \right)}^{\ast DA}=\overline{w}+{\int }_0^{\overline{\theta }}\beta {\left(x\right)}^{\ast DA}dx-{\displaystyle \frac{{\left(\beta {\left(\theta \right)}^{\ast DA}\right)}^2{a}^2}{2b}},\hfill \\ \hfill & \beta {\left(\theta \right)}^{\ast DA}={\displaystyle \frac{a^2-\frac{b\left(1-F\left(\theta \right)\right)}{f\left(\theta \right)}}{a^2+b\rho {\sigma }^2}},e^{\ast DA}\left(\theta \right)={\displaystyle \frac{a^3-\frac{ab\left(1-F\left(\theta \right)\right)}{f\left(\theta \right)}}{b{a}^2+\rho b^2{\sigma }^2}}.\hfill \end{array}$$

(18)

(b) The optimal utility function of the government and SBSS construction enterprises can be obtained:

$$\begin{array}{cc}\hfill & U_G^{\ast DA}={\int }_0^{\overline{\theta }}\left[{\displaystyle \frac{a^2}{2b}}\beta {\left(x\right)}^{\ast DA}+x-{\displaystyle \frac{1}{2}}\rho {\left(\beta {\left(x\right)}^{\ast DA}\right)}^2{\sigma }^2\right]dF\left(x\right)\hfill \\ \hfill & -{\int }_0^{\overline{\theta }}\left(1-F\left(x\right)\right)\beta {\left(x\right)}^{\ast DA}dx-\overline{w}.\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & U_E^{\ast DA}\left(\theta \right)=\overline{w}+{\int }_0^{\overline{\theta }}\beta {\left(x\right)}^{\ast DA}dx+{\displaystyle \frac{x{a}^2-\frac{xb\left(1-F\left(x\right)\right)}{f\left(x\right)}}{a^2+b\rho {\sigma }^2}}-{\displaystyle \frac{1}{2}}\rho {\left(\beta {\left(\theta \right)}^{\ast DA}\right)}^2{\sigma }^2.\hfill \end{array}$$

(20)

(c) In Scenario DA, the optimal effort level $e^{\ast DA}\left(\theta \right)$ of SBSS construction enterprises has the following properties: $e^{\ast DA}\left(\theta \right)$ increases with the increase of risk aversion degree ρ, $e^{\ast DA}\left(\theta \right)$ decreases with the increase of effort cost coefficient b, $e^{\ast DA}\left(\theta \right)$ increases with the increase of marginal influence a of effort level, and $e^{\ast DA}\left(\theta \right)$ increases with the increase of operation level $\theta .$

Lemma 4 outlines the parameters and results of the optimal incentive mechanism designed by the government under dual asymmetric information (Scenario DA). This result provides a theoretical foundation for the government in formulating policies to support SBSS construction enterprises, highlighting the significant impact of information asymmetry and risk aversion on the design of the incentive mechanism.

5.3. Optimal Incentive Coefficient Under Dual Asymmetric Information

Before delving into the details of Proposition 3, it is essential to explore how the government’s incentive mechanisms are influenced by the interplay of risk preferences and asymmetric information. By examining the relationships between risk-neutral and risk-averse SBSS construction enterprises under both single and dual asymmetric information, we uncover a consistent pattern in the design of optimal incentive coefficients. These coefficients are adjusted to balance the trade-offs between enterprise effort and government cost efficiency, ensuring alignment with policy objectives. The following proposition formalizes this consistency and highlights the government’s strategic use of incentives to address the challenges posed by diverse risk preferences and informational complexities.

Proposition 3.

Risk-averse enterprises receive higher incentive coefficients under dual asymmetric information:

$$\begin{array}{cc}\hfill & {\beta }^{\ast DN}<{\beta }^{\ast DA}.\hfill \end{array}$$

(21)

Proposition 3 underscores under dual asymmetric information scenarios, which encompass both moral hazard and adverse selection, the government assigns higher incentive coefficients (${\beta }^{\ast DN}<{\beta }^{\ast DA}$) to risk-averse enterprises. This adjustment reflects the government’s recognition of the enhanced predictability and stability that risk-averse enterprises bring. Consequently, the heterogeneity in enterprise risk preferences informs the government’s strategy to design incentive coefficients that effectively guide enterprise behavior and ensure the successful implementation of contracts under dual asymmetric information conditions.

Corollary 4.

A positive influence exists on enterprise risk preference on the government’ s incentive coefficient difference under dual asymmetric information:i.e.,$\left\{\begin{array}{cc}{\displaystyle \frac{\partial }{\partial \rho }}({\beta }^{\ast DN}-{\beta }^{\ast DA})>0\hfill & if\mu \left(\theta \right)>{\displaystyle \frac{b}{a^2}},\hfill \\ {\displaystyle \frac{\partial }{\partial \rho }}({\beta }^{\ast DN}-{\beta }^{\ast DA})\le 0\hfill & if{\displaystyle \frac{b}{a^2}}\ge \mu \left(\theta \right)>0.\hfill \end{array}\right.$

Corollary 4 highlights the impact of enterprise risk aversion on the government’ s incentive coefficient under dual asymmetric information. It provides valuable insights for the government in designing incentive contracts, emphasizing the need to adjust incentive strategies based on the type of enterprise, particularly when risk aversion and information asymmetry coexist.

5.4. Optimal Fixed Subsidy Under Dual Asymmetric Information

Proposition 4 shows that the relationship between fixed subsidies depends on the threshold $\rho {\sigma }^2$ under dual asymmetric information (a combination of risk aversion, uncertainty, and operational parameters): 1. If $\rho {\sigma }^2<{\displaystyle \frac{a^2(a^2-2b)}{2b^2}}$, the government assigns higher fixed subsidies under risk neutrality (${\alpha }^{\ast DN}\left(\theta \right)>{\alpha }^{\ast DA}\left(\theta \right)$); 2. If $\rho {\sigma }^2>{\displaystyle \frac{a^2(a^2-2b)}{2b^2}}$, the government assigns lower fixed subsidies under risk neutrality (i.e., higher under risk aversion) (${\alpha }^{\ast DN}\left(\theta \right)<{\alpha }^{\ast DA}\left(\theta \right)$). This shift reflects the government’s strategy to balance incentives and manage the additional risks. Proposition 4 reveals a critical threshold for $\rho {\sigma }^2$ that determines the government’s optimal fixed subsidy strategy. It is important to note that this threshold is derived based on an approximation detailed in Appendix A, where the incentive coefficient under risk aversion is simplified for high values of risk $\rho {\sigma }^2$. This approximation is robust under our model’s parameterization, where the marginal impact of effort a is significantly larger than the effort cost coefficient b (e.g., $a=20$, $b=2$ in our numerical examples). A brief sensitivity analysis confirms that as long as this condition holds, the approximated threshold remains a reliable indicator of the policy switch point. The key insight—that the government’s preference for contracting with risk-neutral or risk-averse firms depends on a threshold of risk—is therefore a robust finding of our model.

Proposition 4.

The threshold effect of risk preferences on government fixed subsidy allocation exists under dual asymmetric information:

$$\begin{array}{cc}\hfill & if\rho {\sigma }^2<{\displaystyle \frac{a^2(a^2-2b)}{2b^2}},then{\alpha }^{\ast DN}\left(\theta \right)>{\alpha }^{\ast DA}\left(\theta \right),\hfill \\ \hfill & if\rho {\sigma }^2>{\displaystyle \frac{a^2(a^2-2b)}{2b^2}},then{\alpha }^{\ast DN}\left(\theta \right)<{\alpha }^{\ast DA}\left(\theta \right).\hfill \end{array}$$

(22)

Due to Proposition 4’s complexity, we employ numerical analysis to interpret further and expand its implications. Figure 3 illustrate the relationship between the operational level ($\theta$), the risk aversion level ($\rho$), and the fixed subsidy differences ($\mathsf{\Delta }\alpha$). The specific values referenced in the reference: $a=20,$ $b=2,$ ${\sigma }^2=0.4.$

At the Figure 3, in the upper region of the graph, where $\rho$ is higher, ${\alpha }^{\ast DN}<{\alpha }^{\ast DA}$. It is assumed that the operation level $\theta$ of the SBSS construction enterprises obeys a continuous uniform distribution $U(0,1000),$ then there is a probability density function $f\left(\theta \right)=1/1000$, a cumulative probability distribution function $F\left(\theta \right)=\theta /1000$, and a risk rate $\mu \left(\theta \right)=1/(1000-\theta )$. From the perspective of the government providing fixed subsidies, risk-averse SBSS construction enterprises receive higher subsidies under dual asymmetric information to address the compounded risks of moral hazard and adverse selection. To counteract these risks, the government strategically offers higher fixed subsidies to risk-averse enterprises, whose preference for stability and aversion to high-risk strategies align better with long-term operational and policy goals, ensuring a more efficient allocation of resources and sustainable outcomes.

Corollary 5.

There exists a positive influence of the enterprise’s risk preference on the government’s fixed subsidy difference under dual asymmetric information: i.e., ${\displaystyle \frac{\partial }{\partial \rho }}({\alpha }^{\ast DN}-{\alpha }^{\ast DA})={\displaystyle \frac{b{\sigma }^2}{{\left(a^2+b\rho {\sigma }^2\right)}^2}}{\int }_0^{\overline{\theta }}\left[a^2-{\displaystyle \frac{b\left(1-F\left(x\right)\right)}{f\left(x\right)}}\right]dx-{\displaystyle \frac{a^2}{b}}{\beta }^{\ast DA}\left(\theta \right)\left[a^2-{\displaystyle \frac{b\left(1-F\left(x\right)\right)}{f\left(x\right)}}\right][{\displaystyle \frac{b{\sigma }^2}{{\left(a^2+b\rho {\sigma }^2\right)}^2}}{]}^2\ge 0.$

Corollary 5 establishes a non-negative relationship between corporate risk aversion and government fixed subsidies under dual asymmetric information. The mathematical framework demonstrates that the optimal subsidy level either increases or remains constant as corporate risk aversion intensifies. This finding provides critical insights for policymakers: Enterprises with heightened risk aversion require proportionally higher subsidies to offset perceived uncertainties and maintain operational engagement. Conversely, for risk-neutral firms exhibiting greater risk tolerance, subsidy adjustments become less imperative, as their strategic behavior aligns more naturally with policy objectives under incentive mechanisms.

5.5. Balancing Risk Preference and Asymmetry Information Through Incentive Mechanisms

5.5.1. The Impact of Asymmetry Information on Incentive Coefficient

Before delving into the details of Proposition 5, it is essential to explore how the government’s incentive mechanisms are influenced by the interplay of risk preferences and asymmetric information. By examining the relationships between risk-neutral and risk-averse SBSS enterprises under both single and dual asymmetric information, we uncover a consistent pattern in the design of optimal incentive coefficients. These coefficients are adjusted to balance the trade-offs between enterprise effort and government cost efficiency, ensuring alignment with policy objectives. The following proposition formalizes this consistency and highlights the government’s strategic use of incentives to address the challenges posed by diverse risk preferences and informational complexities.

Proposition 5.

A consistent reduction in incentive coefficients occurs under dual asymmetric information:

$$\begin{array}{cc}\hfill & {\beta }^{\ast SN}>{\beta }^{\ast DN}.\hfill \\ \hfill & {\beta }^{\ast SA}>{\beta }^{\ast DA}.\hfill \end{array}$$

(23)

Proposition 5 demonstrates that the government provides higher incentive coefficients (${\beta }^{\ast SN}>{\beta }^{\ast DN}$) and (${\beta }^{\ast SA}>{\beta }^{\ast DA}$) to enterprises under single asymmetric information compared to those under dual asymmetric information. The rationale for this consistency lies in the additional complexity introduced by dual asymmetric information (specifically, adverse selection regarding $\theta$). To mitigate the risk of overpaying less efficient types (low $\theta$) who might mimic high types, the optimal contract under dual asymmetry typically reduces the incentive power ($\beta$) compared to the single asymmetry (moral hazard only) case. This seemingly counter-intuitive result has a critical practical implication: when the government cannot easily verify a firm’s true operational efficiency (adverse selection), it must deliberately reduce the performance-based incentive to prevent inefficient firms from mimicking efficient ones and extracting excessive ‘information rents’.

Corollary 6.

The nature of the optimal incentive coefficient difference provided by the government is determined by various factors:

(a) Negative impact of enterprise operational levels on government incentive coefficient difference under risk neutrality: ${\displaystyle \frac{\partial }{\partial \theta }}({\beta }^{\ast SN}-{\beta }^{\ast DN})=-{\displaystyle \frac{b}{a^2}}{\displaystyle \frac{{\mu }^{\prime }\left(\theta \right)}{{\left[\mu \left(\theta \right)\right]}^2}}\le 0.$

(b) Negative impact of enterprise operational levels on government incentive coefficient difference under risk aversion: ${\displaystyle \frac{\partial }{\partial \theta }}({\beta }^{\ast SA}-{\beta }^{\ast DA})=-{\displaystyle \frac{b}{a^2+b\rho {\sigma }^2}}{\displaystyle \frac{{\mu }^{\prime }\left(\theta \right)}{{\left[\mu \left(\theta \right)\right]}^2}}\le 0.$

This negative relationship (the difference decreases as $\theta$ increases) arises from the following reasons:

Information rent effect diminishes for higher types: Construction enterprises with higher operational levels are typically more efficient, enabling them to achieve policy objectives—such as constructing shared battery swapping stations—with fewer external incentives. As a result, the government can lower the incentive coefficient to optimize resource allocation.

Increased efficiency needs less distortion: Higher operational levels ($\theta$) mean the agent is naturally more productive. The distortion introduced by dual asymmetry (lowering $\beta$) has less impact or is less necessary at higher efficiency levels.

5.5.2. The Impact of Asymmetry Information on Fixed Subsidies

As governments aim to incentivize SBSS construction enterprises under conditions of asymmetric information, the allocation of fixed subsidies becomes a critical tool for addressing the diverse risk preferences and operational efficiencies of enterprises. Fixed subsidies are particularly important for encouraging risk-averse enterprises to participate in contracts and enhance their operational performance. By examining the interplay between risk aversion and operational efficiency, the following proposition demonstrates the government’s tendency to offer higher fixed subsidies to enterprises with greater risk aversion or higher operational capabilities. This approach reflects the strategic balancing of subsidy levels to ensure both enterprise participation and the achievement of broader policy goals, such as improved operational efficiency and economic sustainability.

Proposition 6.

There exist threshold effects of moral hazard and adverse selection on optimal fixed subsidies:

(a) Comparison of two optimal fixed subsidies under risk neutrality:

There exist $\overline{\theta }\left({\displaystyle \frac{4a^2}{b}}-\overline{\theta }\right)>0,$ such that

$$\begin{array}{cc}\hfill & if\theta <\overline{\theta }-\sqrt{\overline{\theta }\left({\displaystyle \frac{4a^2}{b}}-\overline{\theta }\right)},then{\alpha }^{\ast SN}>{\alpha }^{\ast DN},\hfill \\ \hfill & if\theta >\overline{\theta }-\sqrt{\overline{\theta }\left({\displaystyle \frac{4a^2}{b}}-\overline{\theta }\right)},then{\alpha }^{\ast SN}<{\alpha }^{\ast DN};\hfill \end{array}$$

(24)

There exist $\overline{\theta }\left({\displaystyle \frac{4a^2}{b}}-\overline{\theta }\right)=0,$ such that ${\alpha }^{\ast SN}<{\alpha }^{\ast DN}.$ (b) Comparison of two optimal fixed subsidies under risk aversion:

There exist $\mathsf{\Delta }={(4a^4+4a^{2}b\rho {\sigma }^2)}^2-4(b{a}^2+b^2\rho {\sigma }^2)(\rho {\sigma }^2{a}^4-4a^4\overline{\theta }+2a^{2}b{\overline{\theta }}^2)>0,$ such that

$$\begin{array}{cc}\hfill & if\theta <{\displaystyle \frac{4a^4+4a^{2}b\rho {\sigma }^2-\sqrt{\mathsf{\Delta }}}{2(b{a}^2+b^2\rho {\sigma }^2)}},or\theta >{\displaystyle \frac{4a^4+4a^{2}b\rho {\sigma }^2+\sqrt{\mathsf{\Delta }}}{2(b{a}^2+b^2\rho {\sigma }^2)}},then{\alpha }^{\ast SA}>{\alpha }^{\ast DA},\hfill \\ \hfill & if{\displaystyle \frac{4a^4+4a^{2}b\rho {\sigma }^2-\sqrt{\mathsf{\Delta }}}{2(b{a}^2+b^2\rho {\sigma }^2)}}<\theta <{\displaystyle \frac{4a^4+4a^{2}b\rho {\sigma }^2+\sqrt{\mathsf{\Delta }}}{2(b{a}^2+b^2\rho {\sigma }^2)}},then{\alpha }^{\ast SA}<{\alpha }^{\ast DA};\hfill \end{array}$$

(25)

There exists $\mathsf{\Delta }=0,$ such that: ${\alpha }^{\ast SA}>{\alpha }^{\ast DA}.$

Proposition 6 shows that as $\theta$ increases and surpasses the first threshold ($\overline{\theta }-\sqrt{\overline{\theta }\left({\displaystyle \frac{4a^2}{b}}-\overline{\theta }\right)}$), ${\alpha }^{\ast SN}<{\alpha }^{\ast DN}$, meaning DN becomes the preferred scenario for higher subsidies due to its ability to manage moral hazard effectively at higher efficiency levels, Figure 4 effectively captures the essence of Proposition 6(a), demonstrating how operational efficiency influences the optimal allocation of fixed subsidies under risk neutrality. The figure demonstrates a linear decrease in the subsidy difference as the operational level increases, highlighting how enhanced operational efficiency influences government subsidy allocation in the presence of single asymmetric information.

Under risk aversion, Figure 4 shows that the fixed subsidy difference exhibits a non-linear decreasing trend. At low operational levels ($\theta$), the subsidy difference is positive, meaning the fixed subsidy under single asymmetric information (${\alpha }^{\ast SA}$) is higher than that under dual asymmetric information (${\alpha }^{\ast DA}$). As $\theta$ increases, the subsidy difference gradually turns negative. Figure 4 shows a clear trend: the difference in optimal fixed subsidies between single and dual asymmetry scenarios decreases as operational efficiency increases, eventually turning negative. Based on this finding, we argue that a differentiated subsidy strategy would be more effective. Specifically, our results suggest that for less-efficient firms, high fixed subsidies are crucial for encouraging market entry and survival. Conversely, for established, high-efficiency firms, the government should shift towards performance-based contracts to maximize output and avoid over-subsidization.

The concept of threshold effects, as presented in Proposition 6, illustrates how the government’s subsidy allocation strategy shifts as the operational level ($\theta$) of SBSS construction enterprises crosses specific thresholds. As $\theta$ increases, indicating higher operational efficiency, the government allocates higher fixed subsidies to mitigate the increased risks associated with moral hazard and adverse selection.

Corollary 7.

The nature of the optimal fixed subsidy difference comparing single vs dual asymmetry depends on risk preference:

(a) The degree of influence of the fixed subsidy provided by the government on the operation level of the enterprises under risk neutrality exists with a negative impact: i.e., ${\displaystyle \frac{\partial }{\partial \theta }}({\alpha }^{\ast SN}-{\alpha }^{\ast DN})={\displaystyle \frac{b(\theta -\overline{\theta })}{a^2}}\le 0.$

(b) The degree of influence of the fixed subsidy provided by the government on the operation level of the enterprises under risk aversion exists with a positive impact: i.e., ${\displaystyle \frac{\partial }{\partial \theta }}({\alpha }^{\ast SA}-{\alpha }^{\ast DA})={\int }_0^{\overline{\theta }}{\displaystyle \frac{b{\sigma }^2}{{\left(a^2+b\rho {\sigma }^2\right)}^2}}\left[a^2-{\displaystyle \frac{b\left(1-F\left(x\right)\right)}{f\left(x\right)}}\right]dx+{\displaystyle \frac{{\left[a^2-\frac{b\left(1-F\left(x\right)\right)}{f\left(x\right)}\right]}^2{a}^2{\sigma }^2+a^6{\sigma }^2-a^4{\sigma }^{4}b\rho }{{\left(a^2+b\rho {\sigma }^2\right)}^3}}+{\displaystyle \frac{\theta a^{2}b{\sigma }^2}{{\left(a^2+b\rho {\sigma }^2\right)}^2}}+{\displaystyle \frac{a^4{\sigma }^2}{2{\left(a^2+b\rho {\sigma }^2\right)}^2}}\ge 0.$

Corollary 7 reveals that the effect of a fixed subsidy on enterprise operation levels depends on risk attitudes:

Under risk neutrality, subsidies don’t improve operations and may even harm them.

Under risk aversion, subsidies boost operations by reducing risk.

6. Balancing Utility and Complexity

Understanding how risk preferences and operational efficiency influence the utility outcomes for both the government and SBSS construction enterprises is key to designing effective incentive mechanisms. Under conditions of single and dual asymmetric information, these factors shape the trade-offs in utility allocation, highlighting the complex interplay between enterprise risk behavior, informational constraints, and government policy design. The following proposition explores how risk aversion and operational efficiency affect the optimal utility of both parties and the effectiveness of contract execution.

6.1. Risk Preferences and Utility Trade-Offs

Understanding the role of risk preferences is essential for evaluating how both the government and SBSS construction enterprises derive utility from their interactions. In the context of promoting shared battery swapping station construction, this principal–agent relationship is complex, involving significant investments, operational risks, and asymmetric information. These preferences influence the way each party values different outcomes, leading to trade-offs that impact overall utility. By analyzing the varying levels of risk aversion among SBSS construction enterprises, we can better comprehend how the optimal incentive mechanisms can be designed to improve utility for both the government and the enterprises, by effectively balancing risk-sharing and motivational effects.

Proposition 7.

The impact of risk aversion on the optimal utility of the parties to the contract under single asymmetric information is as follows:

(a) Comparison of two optimal utilities of government under single asymmetric information:

$$U_G^{\ast SN}<U_G^{\ast SA}.$$

(26)

(b) Comparison of two optimal utilities of SBSS construction enterprises under single asymmetric information:

$$U_E^{\ast SN}=U_E^{\ast SA}=\overline{w}.$$

(27)

In Proposition 7, we analyze the comparison of optimal utilities under asymmetric information scenarios. For the government under single asymmetric information, the optimal utility $U_G^{\ast SN}$ is less than $U_G^{\ast SA}$ (implying the government benefits from the agent being risk-averse, likely due to reduced incentive costs), while for SBSS construction enterprises, their optimal utility remains constant at their reservation level ($\overline{w}$) ($U_E^{\ast SN}=U_E^{\ast SA}$). Figure 5 provides a clear visualization of how operational efficiency and risk aversion influence government utility.

Proposition 8.

The impact of risk aversion on the optimal utility of the parties to the contract under dual asymmetric information is as follows:

(a) Comparison of two optimal utilities of government under dual asymmetric information:

$$\begin{array}{cc}\hfill if& \rho {\sigma }^2<{\displaystyle \frac{a^2(4b-a^2)}{b(a^2-2b)}},then{U}_G^{\ast DN}<U_G^{\ast DA};\hfill \\ \hfill if& \rho {\sigma }^2>{\displaystyle \frac{a^2(4b-a^2)}{b(a^2-2b)}},then{U}_G^{\ast DN}>U_G^{\ast DA}.\hfill \end{array}$$

(28)

(b) Comparison of two optimal utilities of SBSS construction enterprises under dual asymmetric information:

$$\begin{array}{cc}\hfill & if\rho {\sigma }^2<{\displaystyle \frac{2\theta a^2}{a^2-2\theta b}},then{U}_E^{\ast DN}>U_E^{\ast DA};\hfill \\ \hfill & if\rho {\sigma }^2>{\displaystyle \frac{2\theta a^2}{a^2-2\theta b}},then{U}_E^{\ast DN}<U_E^{\ast DA}.\hfill \end{array}$$

(29)

In Proposition 8, under dual asymmetric information, the government’s optimal utility $U_G^{\ast DN}$ surpasses $U_G^{\ast DA}$ when $\rho {\sigma }^2>{\displaystyle \frac{a^2(4b-1)}{b(1-2b)}}$, but the relationship reverses when $\rho {\sigma }^2<{\displaystyle \frac{a^2(4b-1)}{b(1-2b)}}.$ Figure 6 shows when $\theta$ or $\rho$ is low, the government achieves higher utility under scenario DA. Figure 6 illustrates the critical threshold at which the government’s preference shifts from risk-averse to risk-neutral enterprises under dual asymmetric information. Risk-neutral enterprises are more favorable under scenario DN because they can handle the complexities of moral hazard and adverse selection more efficiently, utilize subsidies more effectively, and align with government objectives at a lower cost. Figure 6 tells a policymaker that there is no universally ’better’ type of firm; the optimal partner choice is contingent on the specific operational context. Inside the mid-range $\theta$: The practical insight here is that for the majority of moderately efficient firms, predictability is more valuable than high potential. A risk-averse firm provides stable, albeit not stellar, performance, making them a reliable partner for achieving core policy objectives like network coverage. For very inefficient firms, a risk-neutral partner is preferred because their risk-taking nature might lead them to innovate out of their low-efficiency state. For very efficient firms, a risk-neutral partner is desirable because they can fully leverage their capabilities without requiring a costly risk premium, maximizing output for the government.

Similarly, for SBSS construction enterprises, $U_E^{\ast DN}$ is less than $U_E^{\ast DA}$ when $\rho {\sigma }^2>{\displaystyle \frac{2\theta a^2}{a^2-2\theta b}},$ but exceeds $U_E^{\ast DA}$ when $\rho {\sigma }^2<{\displaystyle \frac{2\theta a^2}{a^2-2\theta b}}.$ This highlights the dependency of optimal utilities on the degree of risk aversion. It reveals that in the context of asymmetric information, it is particularly critical to analyze the impact of specific risk preference parameters on incentive contract design. Figure 6 shows in the mid-range of $\theta$, $\mathsf{\Delta }<0$, indicating that enterprises perform better under scenario DA. Risk-averse enterprises thrive here because their cautious behavior aligns with stable operational improvements, incentivizes by higher fixed subsidies ($\alpha$) and incentive coefficients ($\beta$). The two regions, where $\mathsf{\Delta }>0$ represent scenarios where risk-neutral enterprises (DN) outperform risk-averse enterprises (DA) in terms of utility. Region 1: $\mathsf{\Delta }>0$ for low operational levels, risk-neutral enterprises, with their willingness to take risks and aggressively pursue operational improvements, are better equipped to navigate these uncertainties; Region 2: $\mathsf{\Delta }>0$ for high operational levels, SBSS construction enterprises are better positioned to leverage their capabilities, and risk-neutral enterprises excel by maximizing utility through risk-taking and aggressive strategies.

Considering that the impact of the risk preferences of SBSS construction enterprises under single asymmetric information is consistent across scenarios and that the optimal utility function of SBSS construction enterprises forms a straight line relative to their operational capacity, we now shift its focus to the more complex environment of dual information asymmetry.

6.2. Operational Efficiency and Utility Trade-Off

Asymmetric information within the SBSS market significantly influences the utility outcomes for both the government and enterprises. These informational disparities can lead to suboptimal decision-making and resource allocation. Operational efficiency becomes a critical factor in this context, determining how effectively enterprises can mitigate the adverse effects of information asymmetry. Enhancing operational efficiency allows enterprises to better align their actions with policy objectives, thereby improving utility for both parties and fostering a more balanced and sustainable SBSS ecosystem.

Proposition 9.

The impact of asymmetric information on the optimal utility of the parties to the contract under risk neutrality is as follows:

(a) Comparison of two optimal utilities of government under risk neutrality:

$$\begin{array}{cc}\hfill & if\theta >{\displaystyle \frac{(1-2b)a^2}{4b^2}}-{\displaystyle \frac{\overline{\theta }}{4}}+{\displaystyle \frac{b{\overline{\theta }}^2}{3a^2}},then{U}_G^{\ast SN}>U_G^{\ast DN};\hfill \\ \hfill & if\theta <{\displaystyle \frac{(1-2b)a^2}{4b^2}}-{\displaystyle \frac{\overline{\theta }}{4}}+{\displaystyle \frac{b{\overline{\theta }}^2}{3a^2}},then{U}_G^{\ast SN}<U_G^{\ast DN}.\hfill \end{array}$$

(30)

(b) Comparison of two optimal utilities of SBSS construction enterprises under risk neutrality:

$There$$exist$$\mathsf{\Delta }={\left({\displaystyle \frac{b\overline{\theta }}{a^2}}-1\right)}^2+{\displaystyle \frac{4b}{a^2}}\left(-\overline{\theta }+{\displaystyle \frac{b{\overline{\theta }}^2}{2a^2}}\right)>0,$$such$$that$:

$$\begin{array}{cc}\hfill & if\theta <{\displaystyle \frac{b\overline{\theta }-a^2-a^2\sqrt{\mathsf{\Delta }}}{-2b}}or\theta >{\displaystyle \frac{b\overline{\theta }-a^2+a^2\sqrt{\mathsf{\Delta }}}{-2b}},then{U}_E^{\ast SN}<U_E^{\ast DN},\hfill \\ \hfill & if{\displaystyle \frac{b\overline{\theta }-a^2-a^2\sqrt{\mathsf{\Delta }}}{-2b}}<\theta <{\displaystyle \frac{b\overline{\theta }-a^2+a^2\sqrt{\mathsf{\Delta }}}{-2b}},then{U}_E^{\ast SN}>U_E^{\ast DN};\hfill \end{array}$$

(31)

$There$$exsits$$\mathsf{\Delta }=0,$$such$$that$:

$$\begin{array}{cc}\hfill & U_E^{\ast SN}<U_E^{\ast DN}.\hfill \end{array}$$

(32)

Proposition 9 shows that when $\theta >{\displaystyle \frac{(1-2b)a^2}{4b^2}}-{\displaystyle \frac{\overline{\theta }}{4}}+{\displaystyle \frac{b{\overline{\theta }}^2}{3a^2}}$, the government achieves higher utility under the SN scenario ($U_G^{\ast SN}>U_G^{\ast DN}$). Conversely, when $\theta <{\displaystyle \frac{(1-2b)a^2}{4b^2}}-{\displaystyle \frac{\overline{\theta }}{4}}+{\displaystyle \frac{b{\overline{\theta }}^2}{3a^2}}$, the government’s utility under the DN scenario exceeds that under SN scenario ($U_G^{\ast DN}>U_G^{\ast SN}$). For SBSS construction enterprises, Figure 7 shows that the threshold $\theta$ can be used to distinguish between high and low operational levels of enterprises. At lower operational levels, enterprises tend to benefit more from SN scenario. This is because the additional fixed subsidies and higher incentive coefficients provided under SN scenario help to counteract the inefficiencies of these enterprises. The adverse selection and moral hazard issues are mitigated by these subsidies and incentives. Moreover, the simpler contracts characteristic of SN scenario enable these less efficient enterprises to achieve higher utility. As operational efficiency decreases, the complexity of incentives and the need for monitoring also decrease, making the simpler SN scenario contracts more suitable and advantageous for less efficient firms. At high operational levels, enterprises again benefit more under Dual Asymmetric Information (DN), as their high efficiency enables them to navigate the complexity of dual asymmetric information and maximize utility.

An interesting discovery is that at the threshold equilibrium, the utilities of the government and SBSS construction enterprises align, creating a balance point where neither single asymmetric information (SN) nor dual asymmetric information (DN) holds a clear advantage. This equilibrium suggests an optimal range of $\theta$ where the government can allocate resources flexibly between SN and DN scenarios without significant utility loss. The government can leverage this equilibrium as a guideline to tailor contracts and subsidies, optimizing resource allocation and policy outcomes. For SBSS construction enterprises, the equilibrium represents a level of operational efficiency where the choice of informational structure (SN or DN) has minimal impact on utility. Enterprises operating near this threshold can adapt effectively to either scenario with comparable utility outcomes, offering flexibility in strategic decision-making and contract negotiations.

Building upon the findings under risk neutrality in Proposition 9, it becomes evident that the interplay between asymmetric information and operational efficiency significantly influences the utility outcomes for both the government and SBSS construction enterprises. However, when risk aversion is introduced, the dynamics shift as enterprises prioritize minimizing risk over maximizing returns, altering their responsiveness to incentive mechanisms. This shift necessitates a closer examination of how asymmetric information affects the optimal utility of both parties under conditions of risk aversion, as explored in Proposition 10.

Proposition 10.

The impact of asymmetric information on the optimal utility of the parties to the contract under risk aversion is as follows:

(a) Comparison of two optimal utilities of government under risk aversion:

$$\begin{array}{cc}\hfill & if\theta \le -{\displaystyle \frac{\overline{\theta }}{4}}+{\displaystyle \frac{b{\overline{\theta }}^2}{3a^2}},then{U}_G^{\ast SA}\le U_G^{\ast DA};\hfill \\ \hfill & if\theta >-{\displaystyle \frac{\overline{\theta }}{4}}+{\displaystyle \frac{b{\overline{\theta }}^2}{3a^2}},then{U}_G^{\ast SA}>U_G^{\ast DA}.\hfill \end{array}$$

(33)

(b) Comparison of two optimal utilities of SBSS construction enterprises under risk aversion:

There exist

$$\begin{array}{cc}& \mathsf{\Delta }={\left({\displaystyle \frac{\rho {\sigma }^{2}b(a^2-b\overline{\theta })}{{(a^2+b\rho {\sigma }^2)}^2}}-{\displaystyle \frac{a^2-b\overline{\theta }}{a^2+b\rho {\sigma }^2}}\right)}^2-\hfill \\ & 4\left({\displaystyle \frac{\rho {\sigma }^2{b}^2}{2{(a^2+b\rho {\sigma }^2)}^2}}-{\displaystyle \frac{b}{a^2+b\rho {\sigma }^2}}\right)\left({\displaystyle \frac{\rho {\sigma }^2{(a^2-b\overline{\theta })}^2}{2{(a^2+b\rho {\sigma }^2)}^2}}-{\displaystyle \frac{(a^2-b\overline{\theta })\overline{\theta }+\frac{b}{2}{\overline{\theta }}^2}{a^2+b\rho {\sigma }^2}}\right)>0,\hfill \\ & suchthat\hfill \end{array}$$

$$\begin{array}{cc}\hfill & if\theta <{\displaystyle \frac{-\left(\frac{\rho {\sigma }^{2}b(a^2-b\overline{\theta })}{{(a^2+b\rho {\sigma }^2)}^2}-\frac{a^2-b\overline{\theta }}{a^2+b\rho {\sigma }^2}\right)-\sqrt{\mathsf{\Delta }}}{2\left(\frac{\rho {\sigma }^2{b}^2}{2{(a^2+b\rho {\sigma }^2)}^2}-\frac{b}{a^2+b\rho {\sigma }^2}\right)}}or\theta >{\displaystyle \frac{-\left(\frac{\rho {\sigma }^{2}b(a^2-b\overline{\theta })}{{(a^2+b\rho {\sigma }^2)}^2}-\frac{a^2-b\overline{\theta }}{a^2+b\rho {\sigma }^2}\right)+\sqrt{\mathsf{\Delta }}}{2\left(\frac{\rho {\sigma }^2{b}^2}{2{(a^2+b\rho {\sigma }^2)}^2}-\frac{b}{a^2+b\rho {\sigma }^2}\right)}},\hfill \\ \hfill & then{U}_E^{\ast SA}<U_E^{\ast DA},\hfill \\ \hfill & if{\displaystyle \frac{-\left(\frac{\rho {\sigma }^{2}b(a^2-b\overline{\theta })}{{(a^2+b\rho {\sigma }^2)}^2}-\frac{a^2-b\overline{\theta }}{a^2+b\rho {\sigma }^2}\right)-\sqrt{\mathsf{\Delta }}}{2\left(\frac{\rho {\sigma }^2{b}^2}{2{(a^2+b\rho {\sigma }^2)}^2}-\frac{b}{a^2+b\rho {\sigma }^2}\right)}}<\theta <{\displaystyle \frac{-\left(\frac{\rho {\sigma }^{2}b(a^2-b\overline{\theta })}{{(a^2+b\rho {\sigma }^2)}^2}-\frac{a^2-b\overline{\theta }}{a^2+b\rho {\sigma }^2}\right)+\sqrt{\mathsf{\Delta }}}{2\left(\frac{\rho {\sigma }^2{b}^2}{2{(a^2+b\rho {\sigma }^2)}^2}-\frac{b}{a^2+b\rho {\sigma }^2}\right)}},\hfill \\ \hfill & then{U}_E^{\ast SA}>U_E^{\ast DA};\hfill \end{array}$$

(34)

$There$$exsits$$\mathsf{\Delta }=0,$$such$$that$

$$\begin{array}{cc}\hfill & U_E^{\ast SA}<U_E^{\ast DA}.\hfill \end{array}$$

(35)

According to Proposition 10(a), the government’s utility under SA ($U_G^{\ast SA}$) surpasses that under DA ($U_G^{\ast DA}$) when $\theta >-{\displaystyle \frac{\overline{\theta }}{4}}+{\displaystyle \frac{b{\overline{\theta }}^2}{3a^2}}$. This is shown by the region where the red line rises above zero. Conversely, when $\theta \le -{\displaystyle \frac{\overline{\theta }}{4}}+{\displaystyle \frac{b{\overline{\theta }}^2}{3a^2}}$, the government’s utility under SA exceeds that under DA, as indicated by the negative region of the red line. From the SBSS construction enterprise’s perspective, Figure 8 highlights how operational efficiency ($\theta$) affects the utility difference between single asymmetric information (SA) and dual asymmetric information (DA) scenarios under risk aversion. The blue curve ($U_E^{\ast SA}-U_E^{\ast DA}$) represents the utility difference for SBSS construction enterprises. A key focus is the fan-shaped region formed by the intersection of the two lines above the zero line, which reveals critical insights into how enterprises can optimize utility.

The fan-shaped region in Figure 8 represents a critical range where single asymmetric information offers enterprises the greatest utility advantage. Within this zone, the simplified structure of SA reduces the burden of risk costs, allowing enterprises to operate more flexibly and efficiently. However, outside this range, enterprises may benefit more from the tailored incentives of dual asymmetric information, depending on their operational efficiency. By aligning their strategies with these insights, enterprises can maximize their utility while effectively engaging with government incentive mechanisms.

Propositions 9 and 10 explore how asymmetric information impacts the optimal utilities of the government and SBSS construction enterprises under different risk conditions: risk neutrality (Proposition 9) and risk aversion (Proposition 10). The findings highlight the critical role of operational efficiency ($\theta$) and risk aversion level ($\rho$) in shaping the relative advantages of single asymmetric information (SA) and dual asymmetric information (DA).

6.3. Sensitivity Analysis and Robustness

While our main analysis reveals critical thresholds that guide the government’s contracting strategy, it is important to understand how these results depend on the underlying economic environment. Therefore, this section investigates the sensitivity of our key propositions to variations in the model’s core parameters. We focus on how changes in risk aversion, productivity, and external uncertainty affect the government’s preference between risk-averse and risk-neutral firms.

As illustrated in Figure 9 (the pseudo-code for generating the numerical results in Figure 9 is provided in Appendix B, ensuring the reproducibility of our sensitivity analysis), the sensitivity analysis reveals a consistent pattern across all scenarios: the government’s utility difference starts negative for low values of $\theta$ and becomes positive as $\theta$ increases, crossing the zero line at a critical threshold. The government’s utility difference starts negative for low values of $\theta$ and becomes positive as $\theta$ increases, crossing the zero line at a critical threshold. This indicates that the government’s optimal contracting partner depends on the enterprise’s operational efficiency. Specifically, risk-averse firms are preferred partners when efficiency is low, while risk-neutral firms are favored when efficiency is high, pointing to a strategic choice between promoting operational predictability and incentivizing maximum performance.

Delving deeper into the specific parametric effects provides further policy-relevant insights. First, higher enterprise risk aversion (Panel (b)) systematically expands the government’s preference for DA contracts. As $\rho$ increases, the crossover threshold for $\theta$ shifts to the right, because the cost of compensating a more risk-averse agent for bearing risk under a DN-style contract becomes prohibitively high. Second, this effect is counteracted by enterprise productivity (Panel (c)). When a is high, the government is more eager to unlock the agent’s potential, making it switch its preference to the high-powered DN contract at a much lower efficiency threshold. Finally, a more uncertain external environment (Panel (d)) reinforces the government’s desire for stability, mirroring the effect of higher risk aversion. The increased volatility makes the predictability of a DA contract more attractive, thus enlarging the range of firms for which it is the optimal choice.

7. Conclusions

Our findings demonstrate that treating moral hazard, adverse selection, and risk aversion as interconnected, rather than separate, challenges leads to fundamentally different and more effective policy prescriptions.

7.1. Summary of Findings

In the context of SBSS construction, this study explores how the government, acting as the principal, can design optimal incentive mechanisms to encourage SBSS construction enterprises (agents) to improve SBSS output under conditions of both single and dual information asymmetry regarding the enterprises. We analyze models incorporating enterprise risk preferences under single asymmetric information (moral hazard only) and subsequently under dual asymmetric information (moral hazard and adverse selection). Finally, a comparative analysis of the incentive contracts derived under four distinct scenarios highlights their differences and reveals implications for policy design.

In our study, the optimal incentive mechanism and its corresponding properties under four scenarios are compared in depth, and the decision-making and utility changes of the government and the SBSS construction enterprises under different scenarios are analyzed. Through proposition, model comparison, and numerical analysis, the following conclusions are obtained: (1) Under single asymmetric information (moral hazard), while the government’s incentive contract successfully encourages enterprises to increase their output levels, a key challenge arises: enterprises only achieve their reservation utility (the minimum required for participation), which represents merely the baseline for their willingness to cooperate. (2) Compared with single information asymmetry, when considering the asymmetry of the operation level of SBSS construction enterprises (i.e., dual asymmetric information), the change in the operation level of SBSS construction enterprises with different risk preferences will cause a loss in government utility, potentially harming the government’s interests. However, at this time, the optimal utility function of the SBSS construction enterprises presents a convex function, indicating that when the operation level of the SBSS construction enterprises reaches a certain level, the optimal utility of the enterprises will increase with the increase of the real operation level. Of course, this structure may also increase the information rent the government must pay to mitigate moral hazard.

In the study of the optimal incentive mechanism of shared battery stations, although the principal–agent theory framework is adopted, and the risk preference and information asymmetry of SBSS construction enterprises are considered, there are still some limitations, including simplified assumptions of the actual situation, limited data sources, insufficient universality of the model, insufficient consideration of the diversity of risk preferences, the impact of changes in policy and market environment, and the lack of full integration of social and environmental factors. In addition, in this study, we also lack the design of risk-sharing contract, which makes the contract unable to effectively disperse and avoid risks between the parties, limiting the possibility of the system to achieve Pareto optimality. Reasonable risk-sharing mechanisms can prevent the risk from being too concentrated, reduce the occurrence of moral hazard and adverse selection, and improve the stability and operation efficiency of the system. Therefore, these limitations may affect the accuracy and practicability of the model, and provide space for further discussion and improvement for future research, including improving the design of risk-sharing contracts to better balance the interests of principals and agents and promote the optimization of incentive mechanisms for shared battery stations.

7.2. Management Policy Recommendations

Based on the above conclusions, we give two suggestions: (1) Implement a “Menu of Contracts” to screen firms. Our model demonstrates that firms’ responses to incentives depend critically on their unobservable efficiency and risk preferences. Instead of a one-size-fits-all policy, the government should offer a menu of contracts. A central finding of our study is that a one-size-fits-all policy is suboptimal. Propositions 1 and 2 analytically demonstrate that under single asymmetry, risk-averse firms require both higher incentive coefficients and higher fixed subsidies to be properly incentivized. Proposition 3 extends this insight to the dual-asymmetry case, confirming that risk preferences consistently demand differentiated contracts.

(2) Our analysis of utility thresholds reveals that the optimal contract type depends on the firm’s operational efficiency level. This directly supports the design of threshold-based policies. For instance, the government can define a clear Key Performance Indicator (KPI) for efficiency, such as the “average number of battery swaps per station per day” or “station operational up time. For firms below the benchmark (e.g., <50 swaps/day), they could be offered a “standard support package” with higher fixed grants to ensure their survival and growth. For firms exceeding the benchmark (e.g., >50 swaps/day), they become eligible for a “high-performance incentive scheme” with a greater share of revenue, designed to maximize their output and leverage their proven efficiency.

7.3. Limitations and Future Research

While this study provides a comprehensive theoretical framework for designing incentive mechanisms under dual asymmetry and heterogeneous risk preferences, we acknowledge several limitations that stem from our simplifying assumptions. These limitations, however, pave the way for valuable future research directions.

First, as noted in our model setup, we assume that the firms’ operational efficiency levels follow a uniform distribution. This assumption is crucial for analytical tractability, as it allows us to derive closed-form solutions that clearly reveal the fundamental trade-offs. Future research could explore alternative and potentially more empirically grounded distributions, such as normal, log-normal, or beta distributions. While such an extension would likely necessitate numerical simulations rather than analytical solutions, it could yield deeper insights into how the density of high- and low-efficiency firms in a market impacts optimal policy design.

Second, our model assumes that firms have a constant risk aversion coefficient. This allows us to isolate the direct effect of the level of risk aversion on the optimal contract. A natural extension would be to consider state-dependent risk preferences, where a firm’s appetite for risk might change with its accumulated capital or recent performance. This would transform the problem into a more complex dynamic setting but could better capture the behavior of firms in a fluctuating market environment.

Third, the current framework is a static, one-shot model. It does not capture the long-term, repeated interactions between the government and SBSS enterprises. A significant extension would be to develop a dynamic contract model. In such a setting, the government could learn about a firm’s type over time from its performance, and contracts could be designed to adapt based on this evolving information, incorporating elements like reputation and long-term investment incentives.

Fourth, our analysis is restricted to linear incentive contracts. While this structure is standard and yields tractable solutions, a fully optimal incentive scheme could potentially be nonlinear in performance outcomes. Future research could explore the design and implications of such nonlinear contracts, although this would likely require more complex numerical methods and might sacrifice the clarity of the closed-form solutions we derive.

Finally, our model assumes that performance outcomes are clearly observed. Future work could incorporate stochastic performance signals, where the government receives a noisy or imperfect measure of the firm’s true output. This would introduce an additional layer of risk that must be optimally shared between the principal and the agent, further complicating the contract design but bringing the model closer to real-world monitoring challenges.

By addressing these avenues, future research can build upon our foundational model to develop even more nuanced and robust policy recommendations for fostering the sustainable development of SBSS infrastructure.