Insights for Sustainable Urban Transport via Private Charging Pile Sharing in the Electric Vehicle Sector

Abstract

The growth of the electric vehicle (EV) market is significantly influenced by the development of EV charging infrastructure. In China, the surge in private charging piles has led to the promotion of the private charging pile sharing model (PCPSM) as a strategic solution to overcome infrastructure challenges. This research develops a tripartite evolutionary game model among pile owners, property companies, and EV users to explore the promotion of the sharing model. Innovatively, it integrates prospect theory to capture the decision-making psychology of the participants. Using system dynamics and numerical simulation, an in-depth analysis is conducted on the effects of 15 key factors influencing strategic decisions, culminating in the formulation of feasible incentive mechanisms. The research reveals that: 1) Exclusive reliance on private pile sharing between pile owners and EV users is unstable, highlighting the need for greater involvement from property companies; 2) Managing crucial factors, including property management costs, charging pile usage prices, and profit-sharing ratios, within appropriate limits is essential for the sustainable growth of PCPSM; 3) Enhancing players’ awareness of potential losses and decreasing their risk preference are effective in encouraging proactive strategy adoption; and 4) The practice of pile owners contributing a specific proportion of management fees to property companies, along with dynamic government incentives, considerably elevates the propensity of property companies to engage actively in the sharing model. This study provides novel insights into enhancing PCPSM, with wide-reaching implications for the sustainability of the EV sector and urban transportation systems.

Keywords

electric vehicle private charging pile sharing model prospect theory evolutionary game system dynamics

The increasing adoption of electric vehicles (EVs) plays a pivotal role in diminishing dependence on fossil fuels and mitigating greenhouse gas emissions. As public interest in sustainable, low-carbon transport grows, EVs are becoming a vital transport choice globally, effectively replacing conventional fossil-fuel-powered vehicles ( 1 ). Nevertheless, the growth of the EV market is hindered by inadequate electric vehicle charging infrastructure (EVCI) ( 2 ). Many regions and countries have not yet achieved the international target of a 1.5:1 vehicle-to-pile ratio ( 3 , 4 ). In China, despite a leading global ratio of approximately 2:1 as of June 2022, challenges such as charging difficulties and prolonged waiting times persist, deterring prospective EV buyers ( 5 ).

Addressing environmental concerns with EVs is juxtaposed against the lag in developing EVCI, which increasingly conflicts with the burgeoning EV sector and the escalating demand for charging solutions, underscoring the intricacies of advancing comprehensive development in this area. On the one hand, constructing public charging piles involves high investment barriers and significant initial costs. The expansion of charging infrastructure, particularly public stations, is constrained by limited land availability, thus creating a “chicken or egg” conundrum concerning the concurrent growth of EVs and their infrastructure ( 6 , 7 ). This issue is especially acute in densely populated urban areas with limited parking, where fulfilling the growing demand for charging piles is increasingly problematic. On the other hand, operationally, inefficiencies in management lead to the suboptimal use of EVCI. Public stations are plagued by poor planning and uneven distribution, and the numerous existing public charging platforms lack integrated mechanisms, failing to achieve standardized procedures in information sharing and payments ( 4 , 8 ). Combined with insufficient maintenance, this results in significant variations between congested and underutilized areas. Despite the higher number and rapid development of private charging piles, their use is inconsistent—often idle during the day and active at night ( 9 ).

The swift development of a comprehensive EVCI is pivotal for the growth of the EV industry ( 10 , 11 ). Globally, governments have enacted policies to expedite advancements in this domain. Notably, China has made significant strides, as evidenced by May 2023 data from the China Electric Vehicle Charging Infrastructure Promotion Alliance (EVCIPA), which shows the country houses 63.56 million charging piles, 67% of which are privately operated, exhibiting an impressive annual growth rate of 24.5% ( 12 ). This surge in private charging piles underscores their potential to mitigate the disparity between the availability of vehicles and charging stations ( 9 , 13 ). Private charging pile owners have the option to enlist their facilities on sharing platforms, detailing aspects such as charging protocols, site locations, accessibility, and pricing, thereby facilitating access for other EV users ( 9 ). This burgeoning model has notably enhanced usage efficiency, simultaneously generating income for owners and economizing charging costs for consumers. Nevertheless, the model’s widespread implementation is impeded by issues such as safety concerns, managerial complexities, and early developmental challenges. Despite China pioneering the concept of the private charging pile sharing model (PCPSM) in 2014, EVCIPA’s data indicates that, as of May 2023, a mere 2% of these stations have embraced this model ( 12 , 13 ).

The expansion of PCPSMs involves a broad range of stakeholders, encountering a notable challenge: the reluctance of property management entities ( 14 , 15 ). This hesitation is fueled by the increased complexity in management and heightened security concerns, stemming from external vehicles accessing residential communities and the installation of charging piles. This scenario often results in lukewarm responses from property managers, who are unmotivated because of a lack of direct financial benefits. As PCPSM becomes integral to the EV industry, the Chinese government, both at central and local levels, has initiated policies providing financial support for the establishment and operation of EVCI by homeowner associations and property service companies ( 16 ). Despite these efforts, policy support is still in the preliminary stages, leaving the development of concrete business models somewhat ambiguous. There is a noticeable gap in research focusing on the dynamics of property managers’ involvement in PCPSMs. Furthermore, the majority of studies utilizing evolutionary game theory to investigate decision-making behaviors concentrate predominantly on tangible gains and losses, often overlooking the critical influence of psychological factors ( 17 ). This research seeks to fill this gap by proposing novel business models grounded in game theory, offering strategic insights for market players, and guiding policymakers in crafting effective incentives. Such an approach underscores the viability of PCPSMs, furthering the sustainable growth of the EV sector and urban mobility solutions.

This study makes contributions in the following areas:

1) While prevailing studies on EVCI predominantly concentrate on public charging piles, our study delves into the underexplored realm of private charging piles, particularly addressing the intricacies of promoting PCPSM. We have developed an evolutionary game model tailored to the operational dynamics of PCPSM, incorporating the often-neglected perspective of property companies’ support.

2) Utilizing prospect theory, our research interprets the psychological cost-benefit perceptions relative to risk and uncertainty among participants in nascent charging models. This novel approach probes into how prospect theory’s psychological constructs influence the behavioral evolution of stakeholders.

3) Through detailed analysis, our study discerns the decision-making patterns of stakeholders under diverse influencing factors, identifying property companies as key players in surmounting the hurdles of private pile sharing promotion. We have innovatively formulated a targeted and effective dynamic incentive mechanism, contributing a novel solution to this complex issue.

The structure of this paper is organized as follows. The next section provides an extensive review of the literature, examining current operational models for charging infrastructure, the role of governmental policies in their evolution, and the utilization of evolutionary game theory within this context. The section after that develops an evolutionary game model for PCPSM, integrating the aspect of collaboration with property companies, grounded in prospect theory, and performs an in-depth stability analysis of the model. The penultimate section employs the system dynamics (SD) model to conduct a detailed numerical simulation analysis, probing into the key factors influencing the model, the implementation of prospect theory, and strategies for enhancing incentive mechanisms. The final section delivers conclusions derived from the research and explores the potential policy ramifications.

Literature Review

Sharing Model of Private Charging Piles

The disparity between the supply and demand of charging infrastructure represents a significant hurdle in the widespread adoption of EVs ( 18 ). Private charging piles, typically located in residential areas, offer enhanced convenience and competitive pricing compared with public options ( 19 ). The escalating conflict between public and private charging piles is garnering significant interest ( 20 ). Therefore, leveraging underutilized private charging piles is crucial in mitigating this disparity.

PCPSM, adopting a peer-to-peer (P2P) sharing economy format, facilitates the matching of resources between supply and demand through online platforms, allowing EV users to charge at lower costs while generating additional income for charging pile owners and enhancing utilization rates, thereby promoting resource sharing ( 21 ). Although PCPSM has the potential to foster collaborative consumption and improve resource efficiency, its widespread implementation faces numerous challenges, particularly because of the complexity arising from the involvement of multiple stakeholders.

Current research focuses on pricing strategies and benefit distribution mechanisms within PCPSM, exploring the interaction between the P2P sharing model and the Business-to-Consumer (B2C) public charging model. For instance, the impact of pricing strategies has been investigated using the dual matrix game model ( 13 ) and the Stackelberg game model ( 21 ). Considering EV users as key participants, Zhao et al. ( 9 ) have extended this to a bilateral bargaining game model, more accurately reflecting the interactions among pile owners, electricity retailers, and EV users during the pricing process. Wang et al. ( 15 ) innovatively studied the benefit distribution mechanism among pile owners, sharing platforms, and community managers within PCPSM. Moreover, with the inclusion of more participants, related studies have introduced various innovative attempts. Empirical investigations ( 2 ) and data-driven approaches ( 22 ) have been utilized to explore the potential of new business models with additional participants, such as community-shared charging stations ( 23 ) and the charging infrastructure-public-private partnership (EVCI-PPP) operation model considering real estate agencies ( 24 ).

Table 1 summarizes relevant research on PCPSM, including study regions, specific models, research contents, participating entities, and research methodologies. Existing studies have considered the involvement of various stakeholders within the sharing model, with a focus primarily on economic issues between these entities, thereby providing insights for the operational aspects of PCPSM. However, these studies often fail to fully address the practical challenges encountered during the proliferation of the sharing model, particularly the insufficient support from property companies because of management difficulties and security risks, which affects the market share of the sharing model. Consequently, this research incorporates property companies as a key stakeholder, exploring new strategies to promote the development of PCPSM from the perspectives of economic benefits and psychological risks of all involved parties.

Table 1.

Relevant Research on the Private Charging Pile Sharing Model

Publication	Region	Model	Research content	Stakeholders	Research methodology
Zhao et al. ( 13 )	China	P2P	Pricing strategies	① Private charging pile owners ② Public EVCI operators	Dual matrix game model
Hu et al. ( 21 )	China	P2P	Pricing strategies	① The electricity distributors ② Public EVCI operators ③ Consumers	Stackelberg game model
Zhao et al. ( 9 )	China	P2P	Pricing strategies	① Private charging pile owners ② Electricity retailers ③ EV users	Bilateral bargaining game model
Wang et al. ( 15 )	China	P2P	Benefit distribution mechanisms	① Private charging pile owners ② Sharing platforms ③ Community managers	Modified Shapley values and game model
Plenter et al. ( 2 )	Germany	P2P	Business model potential study	① Sharing platforms ② Consumers (potential peer-providers)	Empirical investigation
Yang et al. ( 22 )	China	P2P	Business model potential study	① EV users	Data-driven simulation
Azarova et al. ( 23 )	Austria	Community-shared charging stations	Business model potential study	① Consumers (potential EVowners and actual EV owners)	Empirical investigation
Huang et al. ( 24 )	China	EVCI-PPP	Business model potential study	① Real estate agencies ② Operators of EVCI ③ EV users	Evolutionary game model

Note: EV = electric vehicle; EVCI = electric vehicle charging infrastructure; P2P = peer-to-peer; PPP = public-private partnership.

Impact of Government Policies on the Development of EVCI

Government policies play a crucial role in promoting the development of EVCI, with existing studies primarily focusing on the impacts of subsidies, tax measures, and adjustments to electricity pricing on the construction and growth of EVCI ( 25 – 29 ). However, most research on additional policies has been conducted in conjunction with, or based on, subsidy policies. For instance, Fang et al. proposed a balanced policy of subsidies and taxes to encourage EVCI construction, while Song et al. found that a variable pricing policy for charging was more effective in increasing station revenue compared with standalone pricing reductions or subsidy policies ( 27 , 28 ). Therefore, economic subsidy policies remain a central focus of research.

The diffusion of EVCI is closely related to the adoption of EVs, prompting scholars to examine the differential impacts of subsidies on various stakeholders ( 30 ). Kumar et al. posited that, when the government subsidizes consumers who purchase EVs, the social welfare resulting from whether or not they subsidize EVCI is the same ( 31 ). However, other researchers argue that the effects of subsidizing EVCI or EVs require detailed analysis. For instance, certain studies highlight the unique advantages of EVCI subsidies in sustaining the promotion of EV adoption and addressing the spatial mismatch between EVCI supply and demand ( 32 , 33 ). Moreover, the effectiveness of subsidy approaches varies with market structures. Shao et al. demonstrated through data simulation that subsidizing EVs is more effective in the U.S. and German markets, whereas subsidizing EVCI yields greater effectiveness in the Chinese market ( 8 ). Nonetheless, as China’s EV industry gradually matures and EV subsidy policies are being phased out, some scholars have explored the development directions of the new-energy vehicle industry, suggesting that subsidizing EVCI remains a crucial supplementary strategy to ensure the widespread adoption of EVs ( 34 – 38 ). Thus, researching effective EVCI subsidy policies continues to be essential in practice.

Table 2 summarizes the research related to incentive policies for the diffusion of EVCI, highlighting important insights into the impact of various policies on the construction and operation of public charging piles. However, there is a notable scarcity of studies focusing on policy incentives for PCPSM, and among these, only a few have considered the declining subsidies for EVs. The focus has predominantly been on static subsidies with fixed amounts, without adequately considering the potential for dynamic adjustment of EVCI subsidy strategies throughout the diffusion cycle, which could lead to wasteful allocation of policy resources. Additionally, in the practice of PCPSM in China, some regions have implemented support policies involving the participation of property companies, but the theoretical foundations behind these measures remain relatively underexplored ( 39 ). There is a lack of policy-oriented research on the internal mechanisms driving differences in outcomes because of various parameters. This paper, based on the constructed PCPSM, proposes a specific incentive mechanism, including the methods of incentive and the internal mechanisms of related factors. It aims to understand the micro-level impact of government policies on the decisions of property companies, providing a better basis for decision-making to achieve efficient and stable development of the sharing model.

Table 2.

Relevant Research Related to Incentive Policies for the Diffusion of Electric Vehicle Charging Infrastructure (EVCI)

Publication	Policy				Consider EV/EVCI dynamic subsidies
Publication	Subsidize EV	Subsidize EVCI	Tax	Electricity	Consider EV/EVCI dynamic subsidies
Fetene et al. ( 25 )	×	√	×	√	×
Zhang and Xu ( 26 )	×	√	×	×	×
Fang et al. ( 27 )	×	√	√	×	×
Song et al. ( 28 )	×	√	×	√	×
Ma and Fan ( 29 )	√	√	×	√	×
Kumar et al. ( 31 )	√	√	×	×	×
Hu et al. ( 32 )	√	√	×	√	×
Zhang and Dou ( 33 )	√	√	×	×	×
Shao et al. ( 8 )	√	√	×	×	×
Ji et al. ( 34 )	√	×	×	×	EV
Kong et al. ( 35 )	√	√	×	×	EV
Wang et al. ( 36 )	√	×	×	×	EV
Wang et al. ( 37 )	√	√	×	×	EV
Song et al. ( 38 )	√	√	×	×	EV

Note: √ = the topic is included in the publication; × = the topic is not included in the publication; EV = electric vehicle.

Application of Evolutionary Game Theory in the Field of EVCI

The evolutionary game model, diverging from traditional game models, acknowledges that participants operate with bounded rationality, are not fully rational, and make decisions through trial and error, influenced by the game’s complexity and their perceptual limitations. This approach to game theory focuses on dynamic equilibrium, and in the realm of EVCI research, it is primarily applied to assess the viability of operational models and provide policy direction. Huang et al., utilizing this model, introduced real estate developers as a new evolutionary driving force in charging infrastructure development ( 24 ). In a similar vein, Li et al. created a multi-agent stochastic evolutionary game model to evaluate the developmental trajectories of EVCI strategies from the perspectives of governments, businesses, and consumers ( 40 ). Their empirical analysis in Shanghai indicated a need for a paradigm shift in government subsidies from consumer-oriented to support for private charging operators. Cao et al.’s study, using a similar model to examine interactions between EV users and charging operators, highlighted how various pricing strategies of charging piles influence user behavior, emphasizing the significance of regional pricing strategies for the stable growth of local EVCI ( 41 ). Furthermore, various researchers have amalgamated the evolutionary game model with SD, providing a more vivid illustration of the entire system’s evolutionary patterns and tendencies, thereby increasing the method’s adaptability and practical relevance ( 42 , 43 ).

In contrast to traditional evolutionary game models, which predominantly focus on tangible gains and losses, recent approaches have started to emphasize the paramount role of psychological attributes in decision-making processes. This shift has led to the increasing incorporation of prospect theory into evolutionary game theory, a trend particularly pertinent in tackling environmental and energy issues, such as trans-jurisdictional water pollution control and energy structure transformation ( 43 , 44 ). In the realm of EVCI, however, theoretical applications integrating prospect theory remain in their infancy. Notably, a study by Wang et al. analyzes the issue of private charging pile installation through the lens of prospect theory ( 19 ). Their work considers the effects of loss aversion and risk preference, providing a depiction of the behavioral evolution of bounded rational agents. Our study differs from Wang et al. primarily in that it focuses on the sharing process of private charging piles and takes into account the decisions of EV users as demand-side participants, with property companies playing the role of a third party beyond the supply and demand sides ( 19 ).

In summary, the integration of evolutionary game theory with SD models has proven to be a highly effective framework for delineating the evolutionary processes of EVCI operational models. In practical settings, the choices and behaviors of decision-makers are in a state of constant flux, influenced by external changes and prevailing uncertainties. This dynamic is especially pronounced in emerging operational models such as PCPSM, where stakeholders’ decisions are more driven by their perceived values rather than solely by tangible benefits—a nuance that is often not adequately addressed in conventional EVCI research. By synthesizing prospect theory with SD models and evolutionary game theory, this study investigates the impact of different perceptions of gains and losses on decision-making, offering new perspectives on the intricacies of EVCI operational models.

Evolutionary Game Model

Model Description

The realization of PCPSM necessitates a confluence of efforts, transcending the capabilities of a solitary entity and demanding cooperative actions from all stakeholders within the shared service management framework. Throughout the utilization of PCPSM, seamless communication is indispensable among EV users, charging pile owners, and property companies. There are three key stages involved in using a sharing private charge pile by EV users: 1) exploring available nearby sharing private charging pile details via the operational platform; 2) rationalizing charging decisions, influenced by owner-provided pile information and individual requisites; and 3) coordinating with the property company for requisite access to residential zones for charging purposes. In this complex process of using PCPSM, significant obstacles largely arise from property companies which frequently dismiss requests from non-resident vehicles to access sharing private charging piles, attributing their refusal to augmented safety risks and managerial expenditures ( 14 , 15 ).

In the proposed model, EV users symbolize the demand facet, predominantly focusing on the advantages and economic implications of opting for sharing private charging piles versus public piles. Conversely, pile owners constitute the supply facet, concentrating mainly on the merits and financial considerations inherent to their participation in PCPSM. Property companies operate as an influential third entity, molding both supply and demand dynamics, with their principal interests revolving around the potential gains and fiscal consequences of endorsing or opposing the scheme. The interactive strategies of these three factions are continuously refined, reacting dynamically to each other’s actions and decisions, an attribute consistent with the principles of evolutionary game theory. Moreover, the operational platforms play a pivotal intermediary role in PCPSM, facilitating interactions and service connections among pile owners, property companies, and EV users. However, their role is essentially that of facilitators rather than principal decision-makers. Therefore, our evolutionary game model considers these platforms as contextual factors, with a primary focus on the direct interactions and decision-making dynamics among pile owners, property companies, and EV users.

Model Assumptions

1) The game encompasses three key stakeholders: charging pile owners, property companies, and EV users. These stakeholders make decisions based on optimizing their benefits, influenced by variables such as available information and inherent preferences. Currently, the integration of property collaboration with PCPSM emerges as an avant-garde charging model in the niche market segment. Charging pile owners, before entering the sharing model, must evaluate perceived benefits and challenges such as the complexities of liaising with property companies, potential inconveniences stemming from occupied charging piles, and diminished sharing revenues because of potential non-cooperation by property companies. Property companies weigh the managerial costs linked with supporting such sharing, alongside the unforeseen safety risks posed by consumers accessing residential premises. EV users, on the other hand, evaluate the convenience, cost-effectiveness, and potential security implications associated with utilizing the sharing private charging piles. The participants exhibit marked disparities in information assimilation and risk perception, manifesting traits of bounded rationality. Consequently, the decision-making strategies adopted by these entities are largely influenced by their psychological expectations of the value derived from potential gains and losses, rather than the direct outcomes of the strategies themselves ( 45 ).

2) Traditional evolutionary game models typically focus the decisions of agents squarely on tangible gains and losses, often omitting the intricate psychological aspects of decision-makers. However, prospect theory encapsulates the psychological perception influences experienced by decision-makers amidst uncertain returns, utilizing “bounded rationality” as an underpinning premise and reflecting subjective risk preferences, thereby making it germane to this study ( 46 ). Within the confines of prospect theory, agents’ decision-making unfurls in two primary stages. Initially, decision-makers determine a reference point, utilized to compute relative utility and assess the gains and losses attributed to decisions; gains are perceived when outcomes surpass the reference point, and losses are acknowledged when they do not. Subsequently, leveraging anticipated gains, a subjective probability weight function, denoted $π (p_{i})$ , transforms the objective probability $p_{i}$ of the event $i$ materializing into a decision weight through which the decision-maker subjectively evaluates the event $i$ . The value function $v (Δ x_{i})$ supplants the utility model to express the subjective value perceived by the decision-maker, thereby establishing the perceived value $V$ as:

V = \sum_{i = 1}^{m} π (p_{i}) v (Δ x_{i})

(1)

The formulation for the weight function $π (p_{i})$ is expressed as:

π (p_{i}) = \frac{{p_{i}}^{γ}}{{[{p_{i}}^{γ} + {(1 - {p_{i}}^{γ})}^{γ}]}^{\frac{1}{γ}}}

(2)

where

$γ$ = the adjustment parameter, symbolizing the gradient of the decision function curve, with smaller $γ$ values yielding a more pronouncedly curved function graph.

Generally, individuals attribute a weight of 1 to events of high probability and a weight of 0 to those with minimal probabilities, leading to $π (1) = 1$ and $π (0) = 0$ . Events of moderate-to-high probability are typically undervalued, while those with low probabilities are often overvalued; thus, for larger $p_{i}$ values, $π (p_{i}) > p_{i}$ and, for smaller $p_{i}$ values, $π (p_{i}) < p_{i}$ .

The formulation for the value function $v (Δ x_{i})$ is expressed as:

v (Δ x_{i}) = {\begin{matrix} Δ {x_{i}}^{β} \\ - λ {(- Δ x_{i})}^{β} \end{matrix} \begin{matrix}  \end{matrix} \begin{matrix} Δ x_{i} \geq 0 \\ Δ x_{i} < 0 \end{matrix}

(3)

where

$Δ x_{i} = x - x_{i}$ = the deviation between the gain $x$ and the reference point $x_{0}$ ,

$Δ x_{i} \geq 0$ = relative gains,

$Δ x_{i} < 0$ = relative losses,

within $[0, 1]$ , $β$ = the risk preference coefficient, indicating the sensitivity of decision-makers to gains and losses—the larger their values, the greater sensitivity, and

$λ$ = the loss aversion coefficient, with larger values indicating a heightened sensitivity to losses. Typically, $λ > 1$ , suggesting that decision-makers exhibit greater sensitivity to losses than gains, aligning with the impact the adoption of the novel technology of PCPSM exerts on decision-makers.

3) Charging pile owners are presented with a binary decision: to participate or abstain from PCPSM, symbolized by a set of behavioral strategies, $S_{H} = {P, NP}$ (where $P$ signifies participation and $NP$ denotes non-participation), with associated adoption probabilities $x$ and $1 - x$ , respectively. Property companies also navigate a binary choice, namely to support or resist PCPSM, characterized by a behavioral strategy set $S_{P} = {S, NS}$ (where $S$ indicates support and $NS$ signifies non-support), accompanied by respective adoption probabilities $y$ and $1 - y$ . Within the framework of this model, EV users and charging pile owners are defined as distinct entities; therefore, scenarios wherein consumers construct private charging piles independently are disregarded. Decision-making exclusively hinges on the choice between utilizing sharing private charging piles and public charging piles. Consequently, the set of behavioral strategies for EV users is articulated as $S_{C} = {SC, PC}$ (where $SC$ represents sharing charging and $PC$ denotes public charging), and the respective adoption probabilities for each strategy are delineated as $z$ and $1 - z$ .

4) Before electing participation in PCPSM, charging pile owners incur a perceived cost, denoted as $V (- C_{1})$ , associated with the time and effort invested in assimilating pertinent information. Additionally, the cost attributed to equipment upgrades and maintenance on joining the sharing mode is represented as $M$ . Conversely, charging pile owners opting against participation in PCPSM sidestep the associated inconvenience cost, thereby realizing a perceived benefit symbolized as $V (C_{1})$ . In instances where only one party among the property companies and charging pile owners chooses PCPSM, the non-participating entity bears a negotiation cost, $C_{2}$ , associated with liaising with the sharing platform. If a cooperative consensus to share is attained between both parties, the negotiation cost, $C_{2}$ , is proportionally distributed based on the parameter $m$ .

5) Within the model, EV users, characterized by bounded rationality and heterogeneity, meticulously weigh the costs and benefits derived from various charging methods, basing their selection between sharing private charging piles and public charging piles on personal needs and external conditions. Given the current practical circumstances of PCPSM, pile owners, enjoying the residential electricity rate—which, even when supplemented by a third-party platform service fee, remains lower than public charging pile rates—reap tangible benefits on adopting the sharing model. While public charging piles offer a swifter charging rate, albeit at a higher cost, sharing private charging piles offers relatively slower charging but at a reduced expense. It is posited that users utilizing public charging piles incur a cost of $R_{0}$ , whereas those employing sharing private charging piles expend $R_{1}$ for electricity and associated services. Consequently, when users opt for charging via sharing private piles, pile owners accrue actual benefits of $k R_{1}$ , wherein $k$ denotes the pricing benefit ratio for the pile owners, with $k$ residing in the interval $[0, 1]$ . At this juncture, if property companies support sharing, the benefit will be distributed proportionally with pile owners at a ratio $m$ . Moreover, EV users also derive subjective benefits based on the timeliness and convenience of the charging services. Given that consumer valuations of charging piles predominantly hinge on charging speed rather than location, the model assumes the perceived benefit of choosing public charging piles is $V (U_{0})$ and $V (U_{1})$ when selecting sharing private charging piles, with $U_{0} > U_{1}$ ( 21 ).

6) Regardless of whether pile owners choose to participate in sharing, property companies actively support PCPSM, accruing perceived benefits, $V (R_{2})$ , in the form of favorable resident sentiment and enhanced company image. Conversely, when pile owners apply for participation and property companies do not lend their support, a perceived loss, $V (- R_{2})$ , is incurred because of resultant damages to the company’s reputation and image. If a pile owner within the community participates in PCPSM, the ingress of external vehicles introduces uncertain security risks, invariably escalating the management challenges faced by property companies. It is assumed that, if property companies proactively support sharing, they need to bear the management cost, $C_{3}$ ; if they do not support it, a perceived risk cost, $V (- L_{0})$ , arises. When both property companies and pile owners support sharing, it attracts more EV users to opt for sharing private charging piles, thereby earning corresponding value-added benefits, $Δ R$ , which are distributed between the pile owner and the property company at a ratio $m$ . Additionally, EV users, opting for sharing private charging piles when property companies do not support PCPSM, will need to pay a parking fee, $C_{4}$ , to the property companies.

7) Inherently a private initiative, PCPSM operates under regulatory conditions that, at present, lack clear delineation of accident resolution. Should property companies withhold support for PCPSM, EV users, navigating through the uncertain security risks involved in opting for the sharing private charging piles, are likely to incur a perceived cost, denoted by $V (- L_{1})$ . In contrast, when property companies do lend their support, conducting regular inspections and maintenance within the community, users of private charging piles stand to gain a more secure sharing experience, thus acquiring a perceived benefit, expressed as $V (L_{1})$ .

8) Prevailing research substantiates that a predominant number of charging pile owners, motivated by prospective benefits, manifest a propensity to participate in PCPSM ( 15 ). The principal deterrent for EV users and property companies in opting for PCPSM is chiefly ascribable to the ambivalence surrounding both benefits and risks ( 47 ). Viewed from an economic perspective, this demonstrates an absence of incentive ( 19 ). Consequently, this study, integrating practical aspects, propounds corresponding incentive mechanisms. The first assumption states that pile owners, who opt for PCPSM, are required to pay a management fee of $s_{1} C_{3}$ to property companies, where $s_{1}$ is the coefficient of management fee collection, such that $s_{1} \in [0, 1]$ ( 19 ). Property companies, in turn, will periodically inspect and maintain charging piles for which the management fee has been paid. The second assumption asserts that the government will provide subsidies to property companies that support PCPSM to offset management fees, with the companies receiving a subsidy of $s_{2} C_{3}$ , where $s_{2}$ is the coefficient of government subsidy intensity, such that $s_{2} \in [0, 1]$ ( 16 , 39 ).

Grounded on the above assumptions, the related symbols and definitions are elaborated on in Table 3.

Table 3.

Description of Parameter Symbols

Symbol	Description
$M$	Equipment upgrade and maintenance costs for pile owners participating in PCPSM
$C_{1}$	Pile owners gaining insight into the costs of time and effort spent on PCPSM
$C_{2}$	Negotiation costs of engaging with sharing platforms required to participate in PCPSM
$C_{3}$	Management costs incurred by property companies when they support PCPSM
$C_{4}$	Parking fees for EV users choosing to use sharing private charging piles when the property company does not support
$R_{0}$	Electricity and service fees for EV users using public charging piles
$R_{1}$	Electricity and service fees for EV users using shared private charging piles
$R_{2}$	Gains in resident goodwill and improved company image that property companies bring when they support PCPSM
$Δ R$	Value-added benefits when pile owners partner with property companies to participate in PCPSM
$U_{0}$	The value of public charging piles to EV users
$U_{1}$	The value of sharing private charging piles to EV users
$L_{0}$	Cost of risk to be borne when property companies do not support PCPSM
$L_{1}$	Risk costs to EV users choosing to use sharing private charging piles when property companies do not support PCPSM
$k$	Pricing proportion of revenue for charging pile owners
$m$	Proportion of costs and benefits when pile owners partner with property companies to participate in PCPSM
$s_{1}$	Coefficient of the management fee paid by the charging pile owners to the property companies
$s_{2}$	Coefficient of government subsidy to property companies
$γ$	Adjustment parameter symbolizing the gradient of the decision function curve
$λ$	The loss aversion coefficient, $λ > 1$
$β$	The risk preference coefficient, $0 < β < 1$
$Δ x_{i}$	The deviation between the gain $x$ and the reference point $x_{0}$
$p_{i}$	The objective probability of the event $i$
$x$	The probability of charging pile owners choosing to participate in PCPSM
$y$	The probability of property companies choosing to support PCPSM
$z$	The probability of EV users choosing to utilize sharing private charging piles
$π ()$	Weight function
$v ()$	Value function
$V ()$	Perceived value function

Note: EV = electric vehicle; PCPSM = private charging pile sharing model.

Construction of the Payoff Matrix Based on Prospect Theory

Considering the preceding assumptions, the payoff matrix—reflecting the interactions among property companies, charging pile owners, and EV users within PCPSM, as guided by prospect theory—is delineated in Table 4.

Table 4.

Payoff Matrix of Tripartite Evolutionary Game Based on Prospect Theory

Electric vehicle user	Pile owner	Property company
Electric vehicle user	Pile owner	$S (y)$	$NS (1 - y)$
$SC (z)$	$P (x)$	$[\begin{matrix} m (k R_{1} + Δ R - C_{2}) - M + V (- C_{1}) - s_{1} C_{3} \\ V (R_{2}) + (1 - m) (k R_{1} + Δ R - C_{2}) + s_{1} C_{3} + s_{2} C_{3} - C_{3} \\ V (U_{1}) - R_{1} + V (L_{1}) \end{matrix}]$	$[\begin{matrix} k R_{1} - M + V (- C_{1}) - C_{2} \\ V (- R_{2}) + V (L_{0}) + C_{4} \\ V (U_{1}) - R_{1} + V (- L_{1}) - C_{4} \end{matrix}]$
$SC (z)$	$\begin{matrix} NP \\ (1 - x) \end{matrix}$	$[\begin{matrix} V (C_{1}) \\ V (R_{2}) + s_{2} C_{3} - C_{2} - C_{3} \\ 0 \end{matrix}]$	$[\begin{matrix} V (C_{1}) \\ V (- L_{0}) \\ 0 \end{matrix}]$
$\begin{matrix} PC \\ (1 - z) \end{matrix}$	$P (x)$	$[\begin{matrix} - M + V (- C_{1}) - m C_{2} - s_{1} C_{3} \\ V (R_{2}) + s_{1} C_{3} + s_{2} C_{3} - (1 - m) C_{2} - C_{3} \\ V (U_{0}) - R_{0} \end{matrix}]$	$[\begin{matrix} - M + V (- C_{1}) - C_{2} \\ V (- R_{2}) + V (- L_{0}) \\ V (U_{0}) - R_{0} \end{matrix}]$
$\begin{matrix} PC \\ (1 - z) \end{matrix}$	$\begin{matrix} NP \\ (1 - x) \end{matrix}$	$[\begin{matrix} V (C_{1}) \\ V (R_{2}) + s_{2} C_{3} - C_{2} - C_{3} \\ V (U_{0}) - R_{0} \end{matrix}]$	$[\begin{matrix} V (C_{1}) \\ 0 \\ V (U_{0}) - R_{0} \end{matrix}]$

Note: $V (- C_{1}) = π (1) v (- C_{1}) + π (0) v (- C_{1}) = - λ C_{1}^{β}$ ; $V (C_{1}) = C_{1}^{β}$ ; $V (- R_{2}) = - λ R_{2}^{β}$ ; $V (R_{2}) = R_{2}^{β}$ ; $V (- L_{0}) = - λ L_{0}^{β}$ ; $V (L_{0}) = L_{0}^{β}$ ; $V (- L_{1}) = - λ L_{1}^{β}$ ; $V (L_{1}) = L_{1}^{β}$ ; $V (U_{0}) = U_{0}^{β}$ ; $V (U_{1}) = U_{1}^{β}$ .

The elements of the payoff matrix in Table 4 correspond to the payoff values under specific strategy combinations. However, this static perspective is insufficient to reveal the evolutionary trajectories of strategies over time and their ultimate impacts. To delve into this dynamic process, this paper introduces the replicator dynamics equation to dynamically describe the evolution of strategy proportions within a population. Specifically, if the average payoff of a strategy exceeds the population’s average, the proportion of the population adopting this strategy will increase over time; conversely, it will decrease. The general form is expressed as follows:

\frac{d x_{i}}{dt} = x_{i} (t) [U_{i} (x) - \bar{U}]

(4)

where

$d x_{i} / dt$ = the rate of change over time of the proportion of strategy $i$ ,

$x_{i} (t)$ = the proportion of the population adopting strategy $i$ at time $t$ ,

$U_{i} (x)$ = the payoff of strategy $i$ , and

$\bar{U}$ = the average payoff of all strategies.

Further, incorporating the model’s assumptions, the probabilities of pile owners, property companies, and EV users adopting strategies $P$ , $S$ , and $SC$ , denoted as $x$ , $y$ , and $z$ , respectively, dynamically evolve over time $t$ . Thus, the replicator dynamics equations for their respective strategies can be represented as follows (see Appendix A for a detailed derivation process):

F (x) = x (x - 1) {\begin{matrix} C_{2} + M + V (C_{1}) - V (- C_{1}) + y [s_{1} C_{3} + (m - 1) C_{2}] \\ - zk R_{1} + yz [- m Δ R + (1 - m) k R_{1}] \end{matrix}}

(5)

F (y) = y (y - 1) {\begin{matrix} C_{2} + C_{3} - V (R_{2}) - s_{2} C_{3} + x [V (- L_{0}) + V (- R_{2}) - s_{1} C_{3} - m C_{2}] \\ + zV (- L_{0}) + xz [- V (- L_{0}) + C_{4} + (m - 1) (k R_{1} + Δ R)] \end{matrix}}

(6)

F (z) = - z (z - 1) {\begin{matrix} R_{0} - V (U_{0}) + x [V (U_{1}) - R_{1} + V (- L_{1}) - C_{4}] \\ + xy [V (L_{1}) - V (- L_{1}) + C_{4}] \end{matrix}}

(7)

Stability Analysis of the Tripartite Game

Equilibrium Point Analysis in the Evolutionary Game Model

In pursuit of establishing stable strategies within a dynamic system, the replicator dynamic equations for the three participating parties in the game are equated to 0, denoted as:

F (x) = 0, F (y) = 0, F (z) = 0

(8)

On resolving the replicator dynamic equations via MATLAB R2022b, 15 equilibrium points are ascertained. Friedman’s study asserts that evolutionarily stable strategies (ESS) reside within pure strategies ( 48 ). Given that the direct solutions for mixed strategy equilibrium points in tripartite evolutionary game theory are complex, mixed strategies are initially precluded, focusing the analysis solely on the stability of eight pure strategy equilibrium points; conditions for implementing mixed strategies will be analytically addressed in subsequent local equilibrium point analyses.

Within the context of this study, stability of equilibrium points can be obtained through the analysis of the Jacobian matrix ( 49 ). For the tripartite evolutionary game model, an equilibrium point is deemed ESS when the eigenvalues $θ_{i}^{j}$ (indicating the $j - th$ eigenvalue of the $i - th$ equilibrium point) of the Jacobian matrix $J$ (see Equation 9) are uniformly negative. It is considered an unstable point if all eigenvalues are positive, and a saddle point when one or two eigenvalues are positive. According to Lyapunov’s stability analysis, the eigenvalues and stability analyses of the pure strategy equilibrium points can be derived, as illustrated in Tables 5 and 6 ( 50 ).

J = {\begin{matrix} \frac{\partial F (x)}{\partial x} \frac{\partial F (x)}{\partial y} \frac{\partial F (x)}{\partial z} \\ \frac{\partial F (y)}{\partial x} \frac{\partial F (y)}{\partial y} \frac{\partial F (y)}{\partial z} \\ \frac{\partial F (z)}{\partial x} \frac{\partial F (z)}{\partial y} \frac{\partial F (z)}{\partial z} \end{matrix}

(9)

Based on the analysis of equilibrium points, it is evident that ESS will not emerge at $E_{5}$ or $E_{7}$ because of the presence of eigenvalues greater than 0 at these points. ESS may manifest at six equilibrium points — $E_{1}$ , $E_{2}$ , $E_{3}$ , $E_{4}$ , $E_{6}$ , and $E_{8}$ —only when conditions ①②③④⑤⑥ are satisfied, respectively. However, these conditions exhibit interdependence and can cause mutual disruptions. Notably, $E_{8}$ , depicting the ideal state of PCPSM with property company collaboration, cannot simultaneously be an ESS with $E_{4}$ and $E_{6}$ , as evidenced by $θ_{8}^{1} = - θ_{4}^{1}$ , $θ_{8}^{2} = - θ_{6}^{2}$ . This indicates that, in the tripartite evolutionary game model of PCPSM, while certain equilibrium points can locally achieve ESS status under specific conditions, these conditions often conflict with each other. This conflict results in a lack of a wholly stable center for the behavior strategies of the three parties involved in the game ( 51 ).

Table 5.

Eigenvalues of Jacobian Matrices $J$

Equilibrium point	Eigenvalue 1	Eigenvalue 2	Eigenvalue 3
$E_{1} (0, 0, 0)$	$V (- C_{1}) - M - V (C_{1}) - C_{2}$	$V (R_{2}) - C_{2} - (1 - s_{2}) C_{3}$	$R_{0} - V (U_{0})$
$E_{2} (0, 0, 1)$	$\begin{matrix} k R_{1} - M - V (C_{1}) + V (- C_{1}) - C_{2} \end{matrix}$	$\begin{matrix} V (R_{2}) - C_{2} - V (- L_{0}) - (1 - s_{2}) C_{3} \end{matrix}$	$V (U_{0}) - R_{0}$
$E_{3} (0, 1, 0)$	$\begin{matrix} V (- C_{1}) - M - V (C_{1}) - m C_{2} - s_{1} C_{3} \end{matrix}$	$C_{2} + (1 - s_{2}) C_{3} - V (R_{2})$	$R_{0} - V (U_{0})$
$E_{4} (0, 1, 1)$	$\begin{matrix} V (- C_{1}) - M - V (C_{1}) - s_{1} C_{3} \\ + m (k R_{1} + Δ R - C_{2}) \end{matrix}$	$\begin{matrix} C_{2} + (1 - s_{2}) C_{3} \\ - V (R_{2}) + V (- L_{0}) \end{matrix}$	$V (U_{0}) - R_{0}$
$E_{5} (1, 0, 0)$	$C_{2} + M + V (C_{1}) - V (- C_{1})$	$\begin{matrix} V (R_{2}) - V (- R_{2}) - (1 - m) C_{2} \\ - (1 - s_{1} - s_{2}) C_{3} - V (- L_{0}) \end{matrix}$	$\begin{matrix} V (U_{1}) - V (U_{0}) - R_{1} \\ + R_{0} + V (- L_{1}) - C_{4} \end{matrix}$
$E_{6} (1, 0, 1)$	$\begin{matrix} C_{2} + M + V (C_{1}) - V (- C_{1}) - k R_{1} \end{matrix}$	$\begin{matrix} (1 - m) (Δ R + k R_{1} - C_{2}) \\ - (1 - s_{1} - s_{2}) C_{3} - C_{4} \\ + V (R_{2}) - V (- R_{2}) - V (- L_{0}) \end{matrix}$	$\begin{matrix} V (U_{0}) - V (U_{1}) + R_{1} \\ - R_{0} - V (- L_{1}) + C_{4} \end{matrix}$
$E_{7} (1, 1, 0)$	$\begin{matrix} m C_{2} + M + s_{1} C_{3} + V (C_{1}) - V (- C_{1}) \end{matrix}$	$\begin{matrix} (1 - m) C_{2} + (1 - s_{1} - s_{2}) C_{3} \\ - V (R_{2}) + V (- R_{2}) + V (- L_{0}) \end{matrix}$	$\begin{matrix} V (U_{1}) - V (U_{0}) - R_{1} \\ + R_{0} + V (L_{1}) \end{matrix}$
$E_{8} (1, 1, 1)$	$\begin{matrix} M + V (C_{1}) - V (- C_{1}) \\ + m (C_{2} - Δ R - k R_{1}) + s_{1} C_{3} \end{matrix}$	$\begin{matrix} (1 - m) (C_{2} - Δ R - k R_{1}) \\ + (1 - s_{1} - s_{2}) C_{3} + C_{4} \\ + V (- L_{0}) + V (- R_{2}) - V (R_{2}) \end{matrix}$	$\begin{matrix} V (U_{0}) - V (U_{1}) \\ + R_{1} - R_{0} - V (L_{1}) \end{matrix}$

Table 6.

The Stability Analysis of the Equilibrium Points

Equilibrium point	Condition	Symbol	Stability
$E_{1} (0, 0, 0)$	① $θ_{1}^{1} < 0, θ_{1}^{2} < 0, θ_{1}^{3} < 0$	$-, unsure$ , unsure	ESS/saddle point
$E_{2} (0, 0, 1)$	② $θ_{2}^{1} < 0, θ_{2}^{2} < 0, θ_{2}^{3} < 0$	unsure, unsure, unsure	ESS/saddle/unstable point
$E_{3} (0, 1, 0)$	③ $θ_{3}^{1} < 0, θ_{3}^{2} < 0, θ_{3}^{3} < 0$	$-, unsure$ , unsure	ESS/saddle point
$E_{4} (0, 1, 1)$	④ $θ_{4}^{1} < 0, θ_{4}^{2} < 0, θ_{4}^{3} < 0$	unsure, unsure, unsure	ESS/saddle/unstable point
$E_{5} (1, 0, 0)$	⑤ $θ_{5}^{1} = - θ_{1}^{1} > 0, θ_{5}^{2} < 0, θ_{5}^{3} < 0$	$+, unsure$ , unsure	Saddle/unstable point
$E_{6} (1, 0, 1)$	⑥ $θ_{6}^{1} < 0, θ_{6}^{2} < 0, θ_{6}^{3} < 0$	unsure, unsure, unsure	ESS/saddle/unstable point
$E_{7} (1, 1, 0)$	⑦ $θ_{7}^{1} = - θ_{3}^{1} > 0, θ_{7}^{2} < 0, θ_{7}^{3} < 0$	$+, unsure$ , unsure	Saddle/unstable point
$E_{8} (1, 1, 1)$	⑧ $θ_{8}^{1} < 0, θ_{8}^{2} < 0, θ_{8}^{3} < 0$	unsure, unsure, unsure	ESS/saddle/unstable point

Note: ESS = evolutionarily stable strategies.

Incremental Stability Analysis of Unilateral Behavior Strategy

In accordance with the stability theorem of differential equations and the inherent properties of ESS, an ESS must exhibit robustness to minor perturbations ( 51 ). Specifically, when $F (x) = 0$ and $F' (x) < 0$ , $x$ is identified as ESS; analogous conditions apply for $F (y)$ and $F (z)$ . Based on this theory, this study undertakes an analysis of the behavioral strategy stability among the three parties involved in the game. The proofs of all the following propositions are shown in Appendix B.

Strategy Stability Analysis of Charging Pile Owners

By calculating $F (x) = 0$ , we can obtain Proposition 1.

Proposition 1(a): if $z = z_{0}$ , then for any $x$ , $F (x) = 0$ , all behavior strategies are stable.

Proposition 1(b): if $z \neq z_{0}$ , only $x = 0$ and $x = 1$ are stable points. And for $z > z_{0}$ , $x$ tends toward 1.

z_{o} = \frac{C_{2} + M + V (C_{1}) - V (- C_{1}) + y [s_{1} C_{3} + (m - 1) C_{2}]}{k R_{1} + y [m Δ R + (m - 1) k R_{1}]}

(10)

This indicates that, in determining participation in PCPSM, pile owners are more inclined to participate when there are reduced costs for equipment upgrades and maintenance ( $M$ ), understanding the intricacies of PCPSM ( $C_{1}$ ), liaising with the sharing platform ( $C_{2}$ ), and paying management fees to property companies ( $s_{1} C_{3}$ ). Additionally, greater pricing benefits for pile owners ( $k R_{1}$ ), increased value from property company involvement ( $Δ R$ ), and larger benefit distribution coefficient for pile owners ( $m$ ) further enhance this inclination.

Strategy Stability Analysis of Property Companies

By calculating $F (y) = 0$ , we can obtain Proposition 2.

Proposition 2(a): if $x = x_{0}$ , then for any $y$ , $F (y) = 0$ , all behavior strategies are stable.

Proposition 2(b): if $x \neq x_{0}$ , only $y = 0$ and $y = 1$ are stable points. And for $x > x_{0}$ , $y$ approaches 1.

x_{0} = \frac{- C_{2} - (1 - s_{2}) C_{3} + V (R_{2}) - zV (- L_{0})}{[V (- L_{0}) + V (- R_{2}) - s_{1} C_{3} - m C_{2}] + z [- V (- L_{0}) + C_{4} + (m - 1) (k R_{1} + Δ R)]}

(11)

This suggests that the likelihood of property companies supporting PCPSM intensifies under several conditions: an elevation in reputational benefits from endorsing the sharing ( $R_{2}$ ), amplified risks associated with non-participation ( $L_{0}$ ), augmented economic stimuli from both pile owners ( $s_{1}$ ) and governmental bodies ( $s_{2}$ ), smaller benefit distribution coefficient for pile owners ( $m$ ), and an increase in the profit shares allocated between property companies and pile owners when joining the sharing model ( $k R_{1} + Δ R$ ). Conversely, when EV users are required to pay higher parking fees ( $C_{4}$ ) to property companies in scenarios where the latter does not support sharing, the propensity of property companies to join diminishes.

Strategy Stability Analysis of EV Users

By calculating $F (z) = 0$ , we can obtain Proposition 3.

Proposition 3(a): if $y = y_{0}$ , then for any $z$ , $F (z) = 0$ , all behavior strategies are stable.

Proposition 3(b): if $y \neq y_{0}$ , only $z = 0$ and $z = 1$ are stable points. And for $y > y_{0}$ , $z$ converges to 1.

y_{0} = - \frac{R_{0} - V (U_{0}) + x [V (U_{1}) - R_{1} + V (- L_{1}) - C_{4}]}{x [V (L_{1}) - V (- L_{1}) + C_{4}]}

(12)

This suggests that, when EV users face increasing costs ( $R_{0}$ ) and perceive reduced value ( $U_{0}$ ) at public charging piles, while simultaneously perceiving higher value ( $U_{1}$ ) and incurring lower costs ( $R_{1}$ ) with sharing private charging piles, there is a pronounced inclination toward utilizing sharing private charging piles. Moreover, when the parking fees levied by non-participating property companies ( $C_{4}$ ) are lowered, EV users display an even stronger preference for the sharing private charging option.

Based on unilateral game behavior strategy analysis, stakeholders including pile owners, property companies, and EV users consistently prioritize the maximization of their respective benefits. For fostering a collective adoption of proactive stances among these participants, it is essential to amplify the advantages of integrating into PCPSM. Nevertheless, the specific parameter, such as $m$ , introduces divergent interests among the entities. Consequently, pinpointing an optimal milieu for the dissemination of the sharing model and implementing mechanisms such as subsidies and incentives becomes indispensable in alleviating these inherent conflicts of interest.

Numerical Simulation of Tripartite Evolutionary Game

Evolutionary Game Model Based on System Dynamics (SD)

Evolutionary game models serve to analyze enduring game outcomes among charging pile owners, property companies, and EV users. However, these models inherently lack the capacity to portray the dynamic evolutionary trajectory of the game. Consequently, founded on the analyses presented in the section of “Evolutionary Game Model” this section introduces an SD model using Vensim_PLE, which explores the influence of varied parameter values on the strategic combinations of the three entities, and intuitively unveils the evolution path and tendencies of the entire dynamic system.

The SD model constructed by Vensim_PLE is shown in Figure 1. Within the model, the level variables represent the probabilities of players choosing a sharing strategy, denoted as $x$ , $y$ , and $z$ . The corresponding rate variables, representing the rate of change in probabilities, are denoted as $dx / dt$ , $dy / dt$ , and $dz / dt$ , which correspond to the replicator dynamics equations $F (x)$ , $F (y)$ , $F (z)$ , respectively. Additionally, the model includes 19 initial external variables and six intermediary variables, which correspond to the variables in the payoff matrix presented in Table 4.

Figure 1.

The system dynamics model of the evolutionary game.

Simulation Design and Path Evolution

In alignment with the replicator dynamics equation delineated in the section of “Evolutionary Game Model”, the probabilities of strategies $x$ , $y$ , and $z$ inherently function over time $t$ . The replicator dynamics equations, specifically Equations 5, 6, and 7, are discretized to facilitate an efficient simulation analysis, adopting an evolutionary time step, $Δ t$ , to trace asymptotic stability in their trajectories. Subsequent to decision-making within respective strategy sets, each entity among the three gaming parties accrues corresponding payoffs. Strategies are updated iteratively based on preceding outcomes until stabilization. Given the parameter values in the simulation model, simulation through Vensim_PLE yields the ESS of the three gaming parties.

To procure initial values requisite for the evolutionary simulation and to further explore the critical success factors inherent in PCPSM under cooperative property companies, the present study utilizes relevant empirical data from China to determine values for various exogenous variables, as detailed in Table 7.

Table 7.

Exogenous Variables in the Case of China

Exogenous variable	Value (CNY)	Residual lifetime (year)
Upgrade cost for sharing a private charging pile	5,000	5
Operation cost for sharing a private charging pile by pile owner	1,000	1
Operation cost for managing a private charging pile by property company	8,893	5
Power of private charging pile (/kW)	7	na
Power of public charging pile (/kW)	110	na
Unit electricity rate for residents (1 kWh)	0.4983	na
Unit electricity rate for commercial (1 kWh)	0.7683	na

Note: CNY = Chinese yuan renminbi; na = not applicable.

Source: Wang et al., Hu et al., and Zhang et al. ( 19 , 21 , 52 ).

Considering the extant research deficit addressing the diffusion of PCPSMs from the strategic perspectives of charging pile owners, property companies, and EV users, this research integrates actual scenarios with online surveys to derive initial values for strategy evolution. Data on PCPSM were retrieved from sources including relevant research and websites and were adjusted based on their average levels to formulate the simulation data; the resulting parameter values are compiled as shown in the corresponding Table 8. It is crucial to acknowledge that the data do not represent actual values but are simulation parameters, proportionally downsized based on particular average values to depict the relative magnitude of each parameter. The simulation is configured to run for 50 months with a time step of 1 month.

Table 8.

Parameters related to the Private Charging Pile Sharing Model

Parameter	Estimate	Source	Parameter	Estimate	Source
$M$	0.56	Wang et al. and Hu et al. ( 19 , 21 )	$U_{1}$	7.0	Hu et al. ( 21 ) and Sina Technology ( 53 )
$C_{1}$	0.12	Wang et al. and Hu et al. ( 19 , 21 )	$L_{0}$	1.0	Wang et al. ( 19 )
$C_{2}$	0.32	Wang et al. and Hu et al. ( 19 , 21 )	$L_{1}$	0.8	Wang et al. ( 19 )
$C_{3}$	1.0	na	$k$	0.5	Motor Home ( 54 )
$C_{4}$	2.0	Motor Home ( 54 )	$m$	0.7	na
$R_{0}$	8.0	Motor Home ( 54 )	$s_{1}$	0.5	na
$R_{1}$	5.0	Motor Home ( 54 )	$s_{2}$	0.5	na
$R_{2}$	0.2	Wang et al. ( 19 )	$λ$	1.2	Xu et al. ( 51 )
$Δ R$	5.0	na	$β$	0.88	Xu et al. ( 51 )
$U_{0}$	10.0	Hu et al. and Sina Technology ( 21 , 53 )

Note: na = not applicable.

Since the initial probabilities of the three parties are randomly distributed within $[0, 1]$ , the evolutionary trajectories of the gaming system are analyzed. As evidenced in Figure 2a, variances in these initial probabilities lead to distinct final strategy selections among the game’s stakeholders. This observation further corroborates the conclusions drawn in the section of “Evolutionary Game Model”, indicating an absence of a wholly stable equilibrium in the gaming dynamics involving pile owners, property companies, and EV users.

Figure 2.

The evolution path diagram of the tripartite game system under different initial probabilities: (a) two-dimensional diagram and (b) three-dimensional diagram.

Figure 2b shows that, despite differing initial probabilities, many trajectories converge to two equilibrium points: (0,0,0) and (1,1,1). Lower initial probabilities tend to drive the system toward ESS (0,0,0), while higher probabilities push it toward ESS (1,1,1). In practical EV charging contexts, property companies confront multifaceted challenges, such as revenue distribution, managerial expenditures, and risk-related costs. Consequently, equilibrating expenses, anchored only on sharing revenues, proves arduous, leading some companies to eschew the sharing framework. Confronted by the compounded challenges of high costs and risks, both pile owners and EV users could potentially abstain from joining the sharing mechanism.

Ideally, the envisioned model entails property companies and pile owners adjusting their respective decisions to actualize collaborative sharing and mutual benefits. Concurrently, consumers should actively engage with this sharing paradigm, optimizing the utilization of idle resources. Given this backdrop, forthcoming analyses will focus on gauging the impact of parameter fluctuations on the evolutionary game’s stability, seeking configurations that bolster the progression of PCPSM. This exploration is intended to proffer actionable recommendations for the efficacious deployment of PCPSM.

Parameter Sensitivity Analysis

To identify the factors that significantly influence the decisions of stakeholders in the game, a sensitivity analysis was conducted. This involved adjusting each factor within its actual range of values while keeping other factors constant, and setting the simulation observation period to 50 months. Furthermore, considering that PCPSM is still in its early stages of development, all initial probabilities in this section were set to 0.1, that is, $x (0) = 0.1$ , $y (0) = 0.1$ , and $z (0) = 0.1$ .

Risk Preference Coefficient ( $β$ )

To investigate the sensitivity of parameters within the context of the prospect theory, we assigned $β$ values of 0.77, 0.88, and 0.99, corresponding to low-, medium-, and high-risk preferences, respectively. The outcomes are presented in Figure 3. This demonstrates that, for $β$ values of 0.77 and 0.88, $x$ and $y$ exhibit an initial decrease followed by an increase, while $z$ increases monotonically, with all converging to 1. From this, one can deduce that, at lower $β$ , the tripartite agents are predisposed to participate in PCPSM, but this inclination wanes as $β$ rises. For $β = 0.99$ , $x$ , $y$ , and $z$ consistently decrease, stabilizing at 0.

Figure 3.

The evolution of tripartite behavior under different risk preference coefficient ( $β$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

In prospect theory, the risk preference coefficient signifies the marginal decline in the value function, reflecting decision-makers’ psychological expectations of gains and losses. A higher $β$ value indicates that decision-makers may face smaller profits or larger losses when selecting strategies. Simulation results demonstrate that $β$ has a significant impact on the decision-making behavior of participants in PCPSM. When pile owners, property companies, and EV users perceive strong uncertainty in the losses and gains associated with participating in the sharing model, they exhibit a negative attitude toward PCPSM, and this psychological effect has a more pronounced impact on EV users, which increases the challenges in promoting PCPSM. This implies that enhancing incentives and publicity to reduce public perception of uncertainty in gains is crucial in promoting the sharing model.

Loss Aversion Coefficient ( $λ$ )

Considering $λ > 1$ , we assigned $λ$ values of 1.1, 1.2, 1.5, and 2.0 in our simulations to discern the effects of the game participants’ loss aversion on the evolutionary trajectory of the system. Results are depicted in Figure 4. The data suggest that, with increasing $λ$ , the convergence speeds of $x$ , $y$ , and $z$ toward stability of 1 intensify, with no deviations in $λ$ influencing the eventual evolutionary outcome. This underscores that game participants exhibiting pronounced loss aversion show a greater propensity toward PCPSM. Significantly, $y$ is the most influenced by variations in $λ$ . This is largely because of property companies serving as an intermediary. In comparison with pile owners and EV users, property companies demonstrate a heightened awareness of potential losses. Their involvement in PCPSM mitigates apprehensions related to unforeseen risks and probable losses, precluding potentially more severe actual losses from non-participation.

Figure 4.

The evolution of tripartite behavior under different loss aversion coefficient ( $λ$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

From a strategic standpoint, to facilitate the widespread adoption of PCPSM, measures can be undertaken to amplify the tripartite game participants’ perceived value of losses, thereby accelerating their alignment with the sharing model.

Equipment Upgrade and Maintenance Costs ( $M$ )

In this section, $M$ is set to 0.36, 0.46, 0.56, 0.66, and 0.76, to examine its impact on the evolutionary game, as illustrated in Figure 5. With the increase in $M$ , the initial growth rate of $x$ slows down, followed by a significant rise in the evolutionary curve between time = 10 and 20. Overall, the rate at which $x$ , $y$ , and $z$ approach a stable state of ESS (1,1,1) accelerates initially but then slows down, with the system reaching stability fastest when $M = 0.56$ .

Figure 5.

The evolution of tripartite behavior under different equipment upgrade and maintenance costs ( $M$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

Thus, it is observed that pile owners, directly affected by $M$ , are initially attracted by a lower $M$ but, ultimately, an appropriate $M$ value facilitates their quicker participation in the sharing model. Increasing investment in equipment upgrades and maintenance can effectively enhance equipment performance and lifespan, thereby attracting property companies and EV users to opt for PCPSM. However, beyond a certain threshold, further investment does not lead to proportional improvements in the sharing system’s performance, resulting in a decline in investment efficiency. This phenomenon leads to diminishing returns for pile owners joining the sharing model, consequently slowing the rate at which the system approaches an ESS (1,1,1). In summary, both too low and too high values of $M$ can decelerate the rate at which the three parties join the sharing model, while a reasonable $M$ value can promote faster and more stable development of PCPSM.

Negotiation Costs of Engaging with Sharing Platforms (C₂)

The parameter $C_{2}$ was set at 0.12, 0.32, 0.52, 0.72, and 0.92 to observe its impact on the system’s evolutionary speed and stability, as shown in Figure 6. As $C_{2}$ increases, the rate at which $x$ , $y$ , and $z$ approach a stable state of 1 slows down. Notably, when $C_{2} = 0.92$ , significant fluctuations in the values of $x$ and $z$ were observed, along with a marked decrease in the rate at which $y$ approaches 1.

Figure 6.

The evolution of tripartite behavior under different negotiation costs of engaging with sharing platforms ( $C_{2}$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

This observation underscores the profound influence of negotiation costs on strategic decision-making within PCPSM. These costs, shared by pile owners and property companies, disproportionately affect property companies because of their typically lower profit margins in the sharing arrangement. Elevated negotiation costs may reduce property companies’ profitability or, in some cases, lead to financial losses, prompting a more cautious approach toward supporting the sharing model. This hesitancy, in turn, affects the likelihood of EV users opting for private charging piles, thereby emphasizing the pivotal role of property company support in influencing EV user behavior. A lack of collaboration from property companies may lead to instability in the market. Therefore, to foster the development of PCPSM, strategies to negotiation contact costs or provide other incentives to stakeholders are necessary.

Management Costs Incurred by Property Companies (C₃)

The parameter $C_{3}$ was set to 0.5, 1.0, 1.5, 2.0, and 2.5 to examine the effects of $C_{3}$ variations on the evolution of the model, as shown in Figure 7. It is observed that, with the increase of $C_{3}$ , the rates at which $x$ , $y$ , and $z$ approach 1 all decelerate. When management costs reach a certain threshold, the states of $x$ and $z$ transition into periodic fluctuations, while $y$ is more sensitive to changes in $C_{3}$ , eventually stabilizing at 0.

Figure 7.

The evolution of tripartite behavior under different management costs incurred by property companies ( $C_{3}$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

This data posits that, during the embryonic phase of PCPSM, the augmented management costs linked to the sharing model could dissuade ambivalent property companies. Furthermore, the inclination of pile owners and EV users to participate in this model profoundly correlates with the endorsement from property companies. As the value of $y$ approaches 0, the growth dynamics and rhythmicity of both $x$ and $z$ undergo significant alterations. Such nuances suggest that exorbitant property management expenses might induce property companies to refrain from participation, instigating a tactical dynamic between pile owners and EV users. This dynamic prompts cyclical patterns, impeding the momentum of PCPSM and intensifying systemic complexities with periodic strategy adjustments. Therefore, reducing property management costs or providing subsidies to property companies could be an effective strategy to promote PCPSM.

Parking Fees (C₄)

The parameter $C_{4}$ was set to 0, 1.0, 2.0, 4.0, and 6.0 for simulation purposes. To facilitate a clearer observation of their impact on the system’s evolution, the observation period was set to 100 months, as depicted in Figure 8. As $C_{4}$ increases, the convergence speed rates of $x$ , $y$ , and $z$ toward stable state 1 decrease, subsequently leading to fluctuations in their stabilization patterns. Specifically, when $C_{4} = 4.0$ , $x$ initially rises but soon shows oscillations before gravitating toward the stable state 1. In contrast, $y$ and $z$ demonstrate ongoing fluctuations within a high probability range after their initial ascents. At $C_{4} = 6.0$ , $x$ exhibits continuous oscillatory behavior, $y$ exhibits minor fluctuations around the stable state 0, and $z$ experiences amplified fluctuations.

Figure 8.

The evolution of tripartite behavior under different parking fees ( $C_{4}$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

Considering practical implications, parking fees directly affect property companies and EV users, thereby magnifying the magnitude of their responses to changes in $C_{4}$ . For property companies, an increase in $C_{4}$ enhances the attractiveness of abstaining from supporting PCPSM, slowing its adoption rate and even swaying them toward non-support. For EV users, a surge in $C_{4}$ values lessens the appeal of choosing the sharing private piles, inducing oscillations or declines after an initial uptick. The oscillatory intensity is directly proportional to the parking fee. With property companies becoming less supportive, and erratic preferences among EV users, the strategic decisions of pile owners are also influenced. In summary, $C_{4}$ influences the decision-making of all three parties involved in the game, albeit in different ways and to varying extents. Therefore, higher parking fees constitute a critical barrier to the development of PCPSM.

Electricity and Service Fees for EV Users Using Public Charging Piles (R₀)

This subsection varied $R_{0}$ across 7.0, 7.5, 8.0, 8.5, and 9.0 to examine its effect on evolutionary outcomes, as depicted in Figure 9. The analysis indicates that, when $R_{0}$ is at a lower level (not exceeding 7.5), the rates at which $x$ , $y$ , and $z$ approach the stable state of 0 are accelerated, suggesting that lower public charging pile pricing diminishes the appeal of PCPSM, leading to a reduced propensity for participation. Conversely, as $R_{0}$ incrementally increases, the entities within the system begin to shift toward actively engaging in the sharing model, with evolutionary outcomes gravitating toward a stable state of 1. The higher the $R_{0}$ , the quicker the convergence to state 1.

Figure 9.

The evolution of tripartite behavior under different electricity and service fees for electric vehicle users using public charging piles ( $R_{0}$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

The study delineates the critical role of public charging pile pricing in shaping the competitive landscape of PCPSM. Competitive pricing at public charging piles, when set below a certain low threshold, undermines the market position of private sharing alternatives. This necessitates that the sharing model implements enhancements in service quality and convenience to maintain user interest. Conversely, higher public pricing enhances the relative appeal of the sharing model, resulting in increased user engagement. Therefore, when formulating policies, it is essential to fully consider the impact of public charging pile pricing on the market competitiveness of PCPSM. A reasonable pricing strategy should strike a balance between ensuring the rational use of public resources and fostering the healthy development of PCPSM.

Electricity and Service Fees for EV Users Using Shared Private Charging Piles (R₁)

In this section, $R_{1}$ is adjusted to 3.0, 4.0, 5.0, 6.0, and 7.0 to analyze its impact on system evolution, as shown in Figure 10. As $R_{1}$ increases, $x$ and $y$ initially accelerate toward 1. Beyond a certain $R_{1}$ threshold, $x$ and $y$ experience fluctuations, with $x$ eventually oscillating at a lower probability and $y$ moving toward a stable state of 0. For $z$ , its rate of approaching stability slows with increasing $R_{1}$ , ultimately fluctuating continuously. Fluctuations first appear in $z$ , followed by $x$ , and lastly in $y$ .

Figure 10.

The evolution of tripartite behavior under different electricity and service fees for electric vehicle users using shared private charging piles ( $R_{1}$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

It becomes apparent that $R_{1}$ is crucial for the evolution of the game’s stakeholders. When $R_{1} \leq 5.0$ , higher pricing boosts profits for pile owners and property companies, increasing their engagement in PCPSM. For EV users, higher $R_{1}$ slows their adoption rate, which is expected. Excessive pricing leads to game outcomes transitioning from an ESS to fluctuating states, primarily affecting EV users. This causes pile owners and property companies to reduce their involvement because of the market’s instability. Remarkably, at $R_{1} = 6.0$ , initial fluctuations lead to a return to ESS (1,1,1), suggesting increased participation by pile owners encourages property company support, enhancing EV users’ confidence and participation. The engagement of all parties is influenced by $R_{1}$ , highlighting the importance of reasonable pricing for PCPSM’s stability.

Cost of Risk When Property Companies Do Not Support Sharing (L₀)

In this section, we designate $L_{0}$ at values of 0.4, 0.7, 1.0, 1.3, and 1.6. The outcomes of this sensitivity analysis are depicted in Figure 11. With a decline in $L_{0}$ , the rates at which $x$ , $y$ , and $z$ converge to the stable state 1 diminish. Notably, for $L_{0}$ at 0.4, $y$ achieves a stable state of 0, whereas $x$ and $z$ demonstrate volatile dynamics. Property companies, directly affected by $L_{0}$ , may be disinclined to support PCPSM given the potential uncertainties of its benefits at lower $L_{0}$ values. Without their endorsement, the sharing model, encompassing pile owners and EV users, is susceptible to inherent instability and continuous oscillation. This finding aligns with the sensitivity analysis for $C_{2}$ , underscoring the criticality of property companies in stabilizing PCPSM. A surge in $L_{0}$ might induce property companies to reassess their stance, potentially fostering more robust backing for the sharing model, which, in turn, could mitigate uncertainties for both pile owners and EV users, hastening the convergence rates of $x$ and $z$ to 1.

Figure 11.

The evolution of tripartite behavior under different cost of risk when property companies do not support sharing ( $L_{0}$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

It can be deduced that, as the perceived risk cost for property companies increases, they become more amenable to PCPSM, even when considering the associated managerial costs $C_{3}$ . Under prospect theory, even though $L_{0}$ and $C_{3}$ are set at the same parameter levels, $L_{0}$ has a more significant impact on system stability. For the widespread adoption of PCPSM, unwavering support from property companies is essential. Consequently, should property companies perceive a reduced risk cost, more compelling incentives may be required to galvanize their active participation in the sharing model.

The Value of Public Charging Piles to EV Users (U₀)

The parameter $U_{0}$ was set at values of 8.0, 9.0, 10.0, 11.0, and 12.0 to observe its impact on the evolution of the system. Figure 12 indicates that, as $U_{0}$ increases, the rates of convergence of $x$ , $y$ , and $z$ toward 1 decelerate. Notably, when $U_{0}$ escalates from 10.0 to 11.0, the system’s ESS shifts from (1,1,1) to (0,0,0). This infers that, as public charging piles gain increased perceived value among EV users, a pronounced inclination toward these piles ensues, potentially curtailing the expansion of PCPSM. The drivers behind this trend could be traced to pile owners and property companies witnessing diminished user demand, leading them to reconsider participation in the sharing paradigm.

Figure 12.

The evolution of tripartite behavior under different values of public charging piles to electric vehicle users ( $U_{0}$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

In conclusion, $U_{0}$ emerges as a decisive determinant shaping the progression of PCPSM. An elevated $U_{0}$ accentuates the allure of public charging piles for EV users, posing challenges for the sharing model. In light of the proliferating public charging networks, strategizing to amplify the competitive edge of the sharing private charging pile—through technological advancements, business strategy overhauls, and regulatory endorsements—is of paramount importance to foster its market penetration.

The Value of Sharing Private Charging Piles to EV Users (U₁)

Based on empirical observations, EV users generally perceive the value of sharing private charging piles to be inferior to public charging piles. Assuming $U_{0} = 10$ , this study sets $U_{1}$ at discrete values of 5.0, 6.0, 7.0, 8.0, and 9.0. Subsequently, the effect of these different perceived values on the system’s evolutionary stability is demonstrated in Figure 13.

Figure 13.

The evolution of tripartite behavior under different values of sharing private charging piles to electric vehicle users ( $U_{1}$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

With an incremental increase in $U_{1}$ , the rates of convergence for variables $x$ , $y$ , and $z$ to 1 exhibit a noticeable acceleration. Intriguingly, $z$ ’s growth trajectory, compared with $x$ and $y$ , is characterized by a complex pattern, undergoing fluctuations before reaching a stable phase. The intensity of these fluctuations is inversely related to the $U_{1}$ value. The findings underscore the pronounced influence of $U_{1}$ on EV users, while its ramifications for pile owners and property companies appear relatively muted. Furthermore, as the perceived value of sharing private piles decreases for EV users, they need to meticulously evaluate the utilities of both charging models, leading to oscillations in their propensity to choose the sharing private pile strategy. Such variations potentially impede the decision-making velocity of the remaining stakeholders.

Therefore, by amplifying the perceived utility of private piles for EV users, via technological enhancements and strategic promotions, the proliferation of PCPSM can be fostered, enriching the array of options and augmenting convenience in the EV landscape.

Coefficient of the Management Fee Paid by the Charging Pile Owners to the Property Companies (s₁)

This section builds on the earlier sensitivity analysis related to the management fee, designating $C_{3}$ as 1.5. Furthermore, considering $s_{1}$ falls within the range $[0, 1]$ , $s_{1}$ is assigned values of 0.00, 0.25, 0.50, 0.75, and 1.00 to study its effect on the system’s evolution, as depicted in Figure 14. The analysis demonstrates that participants in the tripartite game are highly sensitive to changes in $s_{1}$ . An increase of $s_{1}$ from 0.00 to 0.25 significantly boosts system stability, with $x$ and $z$ moving toward stability and $y$ achieving a stable state of 1. Further, as $s_{1}$ rises to 1.00, the likelihood of all parties adopting PCPSM grows, supporting the first incentive hypothesis. Pile owners, who benefit most from PCPSM, offer a moderate management fee as an incentive, enhancing property companies’ support for the sharing model and leading to a more stable sharing system.

Figure 14.

The evolution of tripartite behavior under different coefficients of the management fee paid by the charging pile owners to the property companies ( $s_{1}$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

In summary, economic incentives have played a positive role in encouraging pile owners to participate in the sharing model, with the effective management fee coefficient for pile owners suggested to be between 0.5 and 1. By collecting appropriate management fees, property companies can ensure the normal operation of charging piles, enhance the attractiveness of PCPSM to both EV users and pile owners, and achieve a win-win situation as far as economic and social benefits are concerned.

Coefficient of Government Subsidy to Property Companies (s₂)

Similarly, to investigate the influence of governmental subsidies to property companies within the incentive mechanism on system evolution, $C_{3}$ is set to 1.5. Furthermore, $s_{2}$ is designated values of 0, 0.25, 0.50, 0.75, and 1 to discern its effect on the system’s progression, as illustrated in Figure 15.

Figure 15.

The evolution of tripartite behavior under different coefficients of government subsidy to property companies ( $s_{2}$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

The analysis shows that the impact of $s_{2}$ on system evolution is similar to $s_{1}$ , with both influencing the system toward stability as they increase from 0 to 1. However, the transition of the system from an unstable to a stable state occurs as $s_{2}$ increases from 0.25 to 0.5, and, for $s_{1}$ , this shift happens as it grows from 0 to 0.25. This indicates that the management fee incentive ( $s_{1}$ ) implemented by pile owners has a more pronounced effect than the direct subsidies ( $s_{2}$ ) provided by the government. This difference likely arises because $s_{1}$ affects both pile owners and property companies, creating diverse responses, whereas $s_{2}$ represents direct government intervention aimed solely at property companies. The interaction between the game’s stakeholders becomes more complex, reflecting the nuanced dynamics of external incentives versus internal management strategies.

In conclusion, both $s_{2}$ and $s_{1}$ act as accelerants in system evolution, yet their operational avenues and impact intensities may differ. An effective government subsidy coefficient should range between 0.5 and 1. To drive the advancement of PCPSM, it is essential to judiciously weigh these elements in practical scenarios, seeking the most potent combination of incentives.

Pricing Proportion of Revenue for Charging Pile Owners (k)

In the context of PCPSM, not all fees paid by EV users for using shared private piles translate directly into profits for pile owners. These fees also cover electricity costs paid to power companies and service fees to the sharing platform. Therefore, this models the actual revenue of pile owners as $k R_{1}$ , where $k$ is the pricing profit ratio, ranging from 0 to 1. To explore the impact of varying $k$ on system evolution, values of 0.3, 0.4, 0.5, 0.6, and 0.7 were simulated, with results shown in Figure 16.

Figure 16.

The evolution of tripartite behavior under different pricing proportions of revenue for charging pile owners ( $k$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

Data analysis indicates that, as $k$ increases, the convergence speed of both $x$ and $y$ toward the equilibrium value of 1 accelerates, although the rate at which the convergence accelerates slows down over time. This trend implies that, when pile owners can secure a greater share of the revenue from EV user fees, it can potentially incentivize more pile owners and property companies to integrate into PCPSM. However, there exists a saturation point for this influence. Beyond a certain $k$ threshold, incremental gains in $k$ offer diminishing promotional benefits. Concurrently, $z$ demonstrates an initial surge, followed by oscillations. The magnitude of these oscillations intensifies with higher $k$ values but eventually stabilizes rapidly to the equilibrium state of 1. This pattern suggests potential system instability when the profit margin ratio of pile owners approaches a certain threshold, attributed to EV users’ perceived unfairness of pile owners’ elevated profits and initial market realignments. Nonetheless, over an extended period, as the other two entities exhibit heightened participation interest, attracting more EV users, $z$ exhibits a trend toward stabilization, steering the system toward a balance.

Overall, an optimal pricing profit ratio lies between 0.3 and 0.5. To maintain the stability of the private pile sharing market, policies guiding stakeholders, such as pile owners and sharing platforms to set reasonable pricing profit ratios, are necessary.

Proportion of Costs and Benefits (m)

When both pile owners and property companies participate in sharing, the cost and revenue are distributed in a $m : 1 - m$ ratio. Typically, as the providers of charging piles, pile owners incur higher costs and profits than property companies. Accordingly, this section sets $m$ at 0.3, 0.5, 0.7, 0.9, and 1.0 to explore its impact on system evolution, as shown in Figure 17. With an increase in $m$ , the rate at which $x$ approaches 1 first accelerates then decreases, with the fastest speed at $m = 0.7$ ; $y$ and $z$ exhibit a similar pattern, peaking at $m = 0.5$ , though oscillations occur at $m = 1.0$ .

Figure 17.

The evolution of tripartite behavior under different proportions of costs and benefits ( $m$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

These results highlight that the profit and cost distribution coefficient significantly influences the system’s evolutionary speed and stability. An increase in $m$ indicates greater net gains for pile owners but reduced net gains for property companies. Higher profits for pile owners incentivize their participation in the sharing model, yet overly high profits can diminish property companies’ enthusiasm, affecting EV users’ choices and the model’s stability. At $m = 1.0$ , when property companies do not share costs or profits with pile owners, their willingness to join fluctuates, destabilizing the system. Therefore, an optimal cost and revenue distribution ratio lies between 0.5 and 0.7. Establishing a balanced economic model between pile owners and property companies is crucial for the rapid and stable growth of PCPSM.

The summary of the sensitivity analysis results for the aforementioned parameters is presented in Table 9.

Table 9.

Analysis of Influencing Parameters of the Game Model

Parameter	Whether it affects the ESS (1,1,1)	Parameter control strategy
Prospect theory parameters
$β$	Yes	Reduce to lessen perceived profit uncertainty in PCPSM.
$λ$	No	Increase to heighten users’ sensitivity to personal losses.
Cost
$M$	No	Maintain at an appropriate level to balance profits with sharing equipment’s performance and lifespan.
$C_{2}$	No	Maintain within property companies’ acceptable range to encourage participation.
$C_{3}$	Yes	Minimize to reduce property company costs, which have a significant impact on system equilibrium.
$C_{4}$	Yes	Reduce to attract EV users, influencing property company support.
Pricing
$R_{0}$	Yes	Avoid too low; more significant impact on EV users attraction than $R_{1}$ .
$R_{1}$	Yes	Lower to attract EV users while ensuring profits for pile owners and property companies.
Perceived cost
$L_{0}$	Yes	Increase to motivate property companies’ support for sharing.
$U_{0}$	Yes	Moderate level; high value hinders the development of PCPSM.
$U_{1}$	No	Maximize to accelerate the development of PCPSM.
Ratio
$s_{1}$	Yes	Enhance to promote development; effective incentives between 0.5 and 1.0.
$s_{2}$	Yes	Like $s_{1}$ (effective incentives between 0.5 and 1.0), but less impactful at the same levels.
$k$	No	Optimal range: 0.3–0.5, to keep the system stable.
$m$	Yes	Optimal range: 0.5–0.7, for a balanced distribution model.

Note: ESS = evolutionarily stable strategies; EV = electric vehicle; PCPSM = private charging pile sharing model.

Impact of Prospect Theory Applications on Parameters

In this research, prospect theory was employed to assess how perceived costs influence the robustness of the system. Comparative analyses of scenarios, both inclusive and exclusive of prospect theory, facilitated an exploration of evolutionary dynamics across diverse psychological frameworks, as depicted in Figure 18.

Figure 18.

The evolution of tripartite behavior under different parameter levels with and without the incorporation of prospect theory: (a) prospect theory in the value of public charging piles to electric vehicle (EV) users ( $U_{0}$ ), (b) non-prospect theory in $U_{0}$ , (c) prospect theory in the value of sharing private charging piles to EV users ( $U_{1}$ ), (d) non-prospect theory in $U_{1}$ , (e) prospect theory in pile owners gaining insight into the costs of time and effort spent on private charging pile sharing model (PCPSM) ( $C_{1}$ ), (f) non-prospect theory in $C_{1}$ , (g) prospect theory in gains in resident goodwill and improved company image that property companies bring when they support PCPSM ( $R_{2}$ ), (h) non-prospect theory in $R_{2}$ , (i) prospect theory in cost of risk to be borne when property companies do not support PCPSM ( $L_{0}$ ), (j) non-prospect theory in $L_{0}$ , (k) prospect theory in risk costs to EV users choosing to use sharing private charging piles when property companies do not support PCPSM ( $L_{1}$ ), (l) non-prospect theory in $L_{1}$ .

Simulations assessing the perceived value ( $U_{0}$ ) of public charging infrastructure among EV users demonstrated that prospect theory’s integration markedly modifies the system’s ESS. As $U_{0}$ increases, the system incorporating prospect theory trends toward ESS (1,1,1), whereas, without it, ESS shifts to (0,0,0) with greater $U_{0}$ accelerating the change. This is because of users’ heightened sensitivity to losses over gains under prospect theory, leading them to undervalue public charging and prefer private charging options. However, for perceived value ( $U_{1}$ ) of sharing private piles, prospect theory’s impact on system evolution is minimal, altering evolutionary paths without changing the final state, especially slowing transition to ESS (1,1,1) at lower $U_{1}$ levels, aligning with the theory’s lower sensitivity to gains.

For pile owners, understanding the time and effort cost ( $C_{1}$ ) under prospect theory does not change the final state, but affects the perception of costs, therefore the evolutionary path. High $C_{1}$ magnifies loss aversion, slowing the rate of joining the sharing model because of perceived losses outweighing gains, especially when support from property companies and EV users’ choice of shared piles ( $y$ ) is low. As support ( $y$ and $z$ ) increases, perceived potential benefits of sharing by pile owners rise, accelerating participation. Yet, at lower $C_{1}$ , the impact is limited, with decisions more influenced by direct benefits of sharing. Similarly, prospect theory doesn not alter the stable state across parameter levels for the benefits of resident goodwill and company image improvement ( $R_{2}$ ), nor perceived risk costs ( $L_{0}$ and $L_{1}$ ) for property companies and EV users, but it does affect their evolutionary paths.

Prospect theory asserts that individuals exhibit an increased sensitivity to losses over gains, especially when evaluating substantial costs or benefits. The study further demonstrates the considerable influence of this behavioral trait on the decision-making of EV users. Simulation results indicate a higher $U_{0}$ critically affects EV users’ choices by enhancing cost sensitivity under high perceived value, amplifying loss aversion in prospect theory, and thus exerting a more significant effect on system evolution. In contrast, $C_{1}$ , a minor cost component for pile owners, does not trigger strong loss aversion, affecting decisions more by direct economic losses than psychological loss perception. Risk cost perceptions, such as $L_{0}$ and $L_{1}$ , further confirm this, with $L_{0}$ ’s impact on system evolution and its sensitivity to prospect theory exceeding $L_{1}$ ’s because of its larger proportion in property company costs.

The extent of prospect theory’s impact varies with the actual value level of parameters and their significance in decision-makers’ cost strategies. A greater sensitivity to losses can foster the development of PCPSM. Therefore, policymaking should comprehensively account for the psychological reactions elicited by various parameters across different value levels, as well as how these reactions influence the entire system’s evolutionary stability. A thorough comprehension of these psychological and behavioral traits can facilitate the formulation of more efficacious strategies to guide the system toward the desired evolutionary stable state.

The Optimization Plan of the Incentive Mechanism

Sensitivity analysis of parameters underscores that the incentive mechanisms proposed within our model framework have a favorable impact on the stable evolution of PCPSM. Nevertheless, the effectiveness of these incentives is contingent on the management costs levied by property companies; high expenses can undermine system stability. This research, therefore, recommends an optimized approach to the extant incentive structures, advocating for a management fee from pile owners to property companies that is inversely correlated with the degree of support for the sharing initiative. When support, denoted by $y$ , is minimal, a heightened management fee could incentivize property companies. Conversely, as $y$ augments, a gradual reduction in management fee subsidies by pile owners is advised. In parallel, government subsidies should reciprocally adjust to the support provided by property companies. Enhanced subsidies at lower $y$ levels could catalyze broader participation by property companies. As PCPSM reaches maturity and $y$ amplifies, a systematic withdrawal of government subsidies would be prudent, mirroring the prevailing dynamics of government support in the new-energy sector ( 34 , 55 ).

Consequently, on the existing framework of subsidy factors, we propose phase-out rates, $r_{1}$ and $r_{2}$ , affecting $s_{1}$ and $s_{2}$ , respectively, as delineated in Equations 13 and 14.

s_{1} = r_{1} (1 - y)

(13)

s_{2} = r_{2} (1 - y)

(14)

The optimized SD model, incorporating revised incentive mechanisms, is depicted in Figure 19. This illustration expands on the SD model from Figure 1 by introducing two external variables: $r_{1}$ and $r_{2}$ . Furthermore, it establishes the relationships among $y$ , $r_{1}$ , and $r_{2}$ with $s_{1}$ and $s_{2}$ , transitioning $s_{1}$ and $s_{2}$ from external to intermediary variables.

Figure 19.

The system dynamics model under the optimized incentive mechanism.

Stability of the Evolutionary System After Optimizing the Incentive Mechanism

Sensitivity analysis of the parameters suggests that the stability of the system is critically dependent on the variability of subsidy incentives, particularly as property management costs, denoted as $C_{3}$ , increase. To evaluate the effectiveness of the optimized incentive mechanisms proposed here, $C_{3}$ is varied across a spectrum of values: 1.0, 2.0, 3.0, 4.0, 5.0, and 6.0. The analysis identifies a phase-out rate proximate to 1 as conducive to the system’s evolution. Consequently, both $r_{1}$ and $r_{2}$ are calibrated to this optimal rate to determine the potential for enhanced system stability. The evolutionary outcomes are presented in Figure 20.

Figure 20.

The evolution of tripartite behavior under optimal incentive mechanism and different management costs incurred by property companies when they support the private charging pile sharing model ( $C_{3}$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

Contrasting the findings with Figure 7, the pre-optimization model, with static subsidy factors, demonstrates heightened sensitivity to cost-induced perturbations at higher $C_{3}$ thresholds, specifically $C_{3} \geq 2.0$ . This sensitivity is evidenced by the trend toward 0 in the proactive strategy probability $y$ for property companies and persistent fluctuations in the strategic choice probabilities $x$ and $z$ for pile owners and EV users. The refined model, by contrast, introduces dynamic adjustments to subsidy factors, achieving pronounced evolutionary stability even when subjected to increased $C_{3}$ levels, and resulting in $x$ and $z$ swiftly aligning to a stable state of 1, while the strategic orientation of property companies maintains a consistently high probability.

The data affirm that the refined incentive mechanisms markedly fortify the system’s evolutionary stability, showing resilience even amidst the strain of elevated property management fees. This resilience is indicative of the mechanisms’ capacity to reflect the dynamic economic realities of PCPSM. It facilitates more engaged participation across all stakeholders, particularly incentivizing property companies to support the sharing model, thereby fostering the sustainability and growth of the system as a whole.

Impact of Different Incentives on the System

The involvement of diverse stakeholders in the dual incentive mechanisms necessitates a nuanced understanding of their impacts on the system. Our analysis indicates that, when the management fee subsidy from pile owners to property companies, denoted as $s_{1} C_{3}$ , becomes significant, it may lead to pile owners withdrawing from PCPSM. Consequently, it is crucial to discern the differential impacts of these two distinct incentive approaches after optimization on the system.

To enhance the observation of evolutionary trends, the observation period was extended to 100 months, with the property management cost variable $C_{3}$ set to 3.0, to evaluate the impact of diverse incentive mechanisms on system evolution. Specifically, this study methodically investigated four distinct scenarios: pre-optimization, optimization of $s_{1}$ only, optimization of $s_{2}$ only, and joint optimization of both $s_{1}$ and $s_{2}$ . As indicated in Figure 21, under scenarios of high property management costs, an incentive strategy optimizing $s_{1}$ alone fails to stabilize the system, while an approach focusing solely on optimizing $s_{2}$ facilitates rapid stabilization among all three game participants. Combined optimization of both incentive mechanisms accelerates the stabilization of pile owners and EV users to state 1 and quickly moves property companies to a higher level of equilibrium. Therefore, the most effective incentive mechanism identified involves fixing $s_{1}$ while dynamically modulating $s_{2}$ in accordance with variations in $y$ . This approach maintains the proportion of management fees that pile owners pay to property companies and adjusts government subsidies according to the support ratio of property companies. Such a strategy significantly enhances the willingness of property companies to engage in PCPSM, rapidly directing the system toward ESS (1,1,1).

Figure 21.

Simulation results for various optimization scenarios: (a) pre-optimization, (b) optimization of the coefficient of the management fee paid by the charging pile owners to the property companies ( $s_{1}$ ) only, (c) optimization of the coefficient of government subsidy to property companies ( $s_{2}$ ) only, and (d) joint optimization of both $s_{1}$ and $s_{2}$ .

The optimized incentive mechanism demonstrates a direct correlation with the probability of property companies adopting proactive strategies. In light of actual operational dynamics, the research assessed how the system evolved when the three participating entities—pile owners, property companies, and EV users—were subjected to various initial strategic probabilities, categorized as low, medium, high, and mixed. This assessment entailed setting initial probabilities at (0.1, 0.1, 0.1), (0.4, 0.4, 0.4), (0.7, 0.7, 0.7), and (0.7, 0.3, 0.4), to discern the effect of diverse initial conditions on the system’s evolution, as depicted in Figure 22. The results conclusively show that, independent of the participants’ initial strategic choices, with the implementation of the optimal incentive mechanism, the system invariably gravitates toward the ESS of (1,1,1). Comparing these findings with the system evolution outcomes under fixed incentives with varying initial probabilities in Figure 2, it further confirms the effectiveness of the optimized incentive mechanism.

Figure 22.

Simulation results for the most effective optimization scenarios of different initial probability: (a) low, (b) medium, (c) high, and (d) mixed.

To delve into the specific impact of the gradual phase-out of government subsidies on the evolutionary stability of PCPSM, this study designed a range of experimental parameters based on real-world conditions, as shown in Table 3. Specifically, we adjusted the phase-out rates ( $r_{2}$ ) to values of 0.6, 0.7, 0.8, 0.9, 1.0, and 1.1, while maintaining a coefficient of the management fee ( $s_{1}$ ) of 0.5 and a property management cost ( $C_{3}$ ) of 3.0. The simulation duration was extended to 200 months to accurately track the system’s evolutionary trajectory, with results shown in Figure 23.

Figure 23.

The evolution of tripartite behavior under different phase-out rates ( $r_{2}$ ): (a) $x$ , (b) $y$ , and (c) $z$ .

Findings indicate that at lower phase-out rates ( $r_{2} \leq 0.7$ ), the equilibrium solution of PCPSM declines, potentially driving property companies to exit the sharing model, resulting in market instability. This is attributed to the low participation level of property companies during the early development stages of the sharing model. A low incentive phase-out rate, reflecting insufficient initial subsidy levels, fails to cover the costs of property companies participating in the sharing model, triggering a negative feedback loop that limits model expansion and development. Conversely, higher phase-out rates ( $r_{2} \geq 0.8$ ) encourage stakeholders to adopt more positive strategies. Thus, under conditions with $C_{3} = 3.0$ , pairing $s_{1} = 0.5$ with $r_{2} = 0.8$ constitutes an effective strategy for managing fees and subsidy phase-out rates. From a policymaking perspective, it is essential to find the minimum phase-out rate that effectively promotes system stability, based on actual operational costs and management fee contributions. This approach aims to reduce government expenditures while efficiently fostering the sustainable development of PCPSM.

Results and Discussion

This research adopts an innovative approach centered on the collaborative dynamics of property companies, employing evolutionary game theory and incorporating psychological aspects. This involves a comprehensive analysis of path evolution and sensitivity, with the objective of identifying effective incentives to tackle the diffusion challenges of PCPSM.

1) Initial probabilities indicate the preferences of the three key stakeholders. When these initial probabilities are distributed randomly, in scenarios of localized stability, stakeholders demonstrate a propensity toward two distinct strategy combinations: {participate in sharing, support sharing, opt for sharing} and {not participate in sharing, not support sharing, not opt for sharing}. Observations indicate that higher initial probabilities among the three entities tend to lead the system toward a positive stable state, whereas lower initial probabilities predispose the system to evolve into a negative stable state. Therefore, it is necessary to reinforce the collaborative intent between pile owners and property companies, as well as foster an innovation-oriented mindset among EV users, to facilitate rapid and sustained development of PCPSM, thus more effectively tackling the challenges in the EV charging market.

2) The integration of prospect theory provides a new lens, facilitating the observation of the behavior of the three parties in the game and their nuanced responses to alterations in risk preferences. The simulations reveal that, with increasing risk preferences coefficient ( $β$ ) and diminishing loss aversion coefficient ( $λ$ ), there is a deceleration in the growth rate of PCPSM. In reality, pile owners, property companies, and EV users, as entities driven by economic considerations, display a propensity for conservative decision-making and an increased sensitivity to potential losses pertaining to the emerging PCPSM in the marketplace. This observation indicates that considerable uncertainty about stakeholder benefits can pose challenges to the adoption of PCPSM. Consequently, the implementation of industry standards and the provision of risk education are recommended to mitigate the associated risks of the sharing model, thereby fostering rapid and stable market development.

3) Effective management of key parameters is vital for attaining an optimal level in this study. Firstly, concerning cost control, keeping equipment upgrade and maintenance costs ( $M$ ) at a moderate level is essential. This ensures pile owners’ profitability and optimizes the performance and lifespan of shared equipment, thereby increasing the attractiveness of sharing private charging piles to EV users. It is important to keep negotiation costs with sharing platforms ( $C_{2}$ ) and management fees for property companies ( $C_{3}$ ) within reasonable limits to decrease the chances of their withdrawal from the sharing model. Minimizing parking fees ( $C_{4}$ ) aims to encourage EV users to choose the sharing private charging piles regardless of property company support, while also reducing income for property companies not supporting the sharing model. Secondly, with regard to pricing, lower pricing markedly enhances the participation enthusiasm of EV users, while, concurrently, balanced pricing attracts a greater number of EV users and safeguards the interests of both pile owners and property companies. Simulation analysis reveals that the pricing of public charging piles ( $R_{0}$ ) exerts a more substantial influence on user attraction than that of sharing private charging piles ( $R_{1}$ ). Consequently, enhancing the market competitiveness of PCPSM requires not only strategic pricing but also leveraging other competitive advantages. Considering perceived costs, an increase in perceived risks ( $L_{0}$ ) associated with adverse strategies by property companies significantly heightens their inclination to endorse sharing. Higher user-perceived value ( $U_{0}$ ) of public charging piles directly impedes the growth of PCPSM, while, conversely, an enhanced perceived value ( $U_{1}$ ) of sharing private piles expedites its development. Finally, as far as system parameters are concerned, in scenarios with low management fees for property companies backing PCPSM, escalating the economic incentives ( $s_{1}$ , $s_{2}$ ) offered by pile owners and the government to property companies can effectively facilitate rapid and stable development of the system. However, the impact varies in nature and degree: with high management fees, $s_{1}$ should be maintained at a level tolerable to pile owners. The pricing factor ( $k$ ) for pile owners needs to be optimized to not initiate the threshold effect, balancing their profitability against EV users’ perceptions of fairness. The cost and profit sharing ratio ( $m$ ) between pile owners and property companies should strike a balance, fostering an equitable economic distribution between the two.

4) In the context of PCPSM, the influence of perceived cost parameters varies, influenced by behavioral characteristics delineated in prospect theory. Empirical simulations reveal that high-value parameters, such as the perceived value of public charging piles ( $U_{0}$ ), which are directly associated with substantial economic benefits, can significantly influence the psychological responses of decision-makers. Under the influence of prospect theory, subtle changes in these parameters can result in notable alterations in the system’s evolutionary path and stability, especially in scenarios involving higher costs or benefits for game participants. Conversely, parameters with lower values, such as pile owners’ perceived costs of time and effort for sharing ( $C_{1}$ ), exert a reduced impact, as they are deemed less significant by decision-makers and are not substantial enough to activate loss aversion mechanisms. This insight provides a novel perspective for property companies in decision-making processes supporting PCPSM, alongside theoretical underpinnings for policy formulation aimed at optimizing the sharing economy ecosystem.

5) This research underscores the inadequacy of existing incentive policies in bolstering the system’s robustness. With the escalation of property management fees, there is a marked trend of property companies disengaging from PCPSM, resulting in perturbations in market supply and demand dynamics. To mitigate this issue, the study delved into the ramifications of dynamic subsidies on the behavioral evolution of the three key stakeholders involved in the game-theoretic framework. It was discerned that establishing a constant subsidy factor ( $s_{1}$ ) from pile owners to property companies, coupled with the strategic adjustment of the government’s subsidy factor ( $s_{2}$ ) in response to the property companies’ support ratio ( $y$ ), significantly fosters their engagement in the sharing model. This methodology concurrently attenuates the variability in strategic decisions made by pile owners and EV users, thereby anchoring the system at an ESS of (1,1,1). Additionally, the initial heightened subsidy factor plays a pivotal role in stimulating proactive strategies among the stakeholders. Nonetheless, in the execution of subsidy policies, it is imperative for the government to judiciously weigh the trade-off between fiscal efficiency and systemic stability, aiming to identify an optimally minimal yet effective phase-out rate that guarantees both policy endurance and system robustness. It is noteworthy that the isolated dynamic adjustment of the subsidy factor $s_{1}$ is insufficient in fostering systemic stability and could potentially exacerbate fluctuations in the system’s evolution because of increased costs of pile owners in the nascent stages of PCPSM. Consequently, this highlights the necessity for a holistic approach in promoting the sharing model—one that judiciously balances the obligations of pile owners with the interests of property companies, transcending reliance on mere subsidy-driven incentives.

This research applies an evolutionary game theory framework to extensively analyze the behavioral evolution of key stakeholders in PCPSM: pile owners, property companies, and EV users. The study employs simulation analysis, grounded in psychological behavior, to augment our comprehension of stakeholder responses under varying incentive mechanisms, thereby offering a foundation for governmental and market entities to craft efficacious policies. It highlights the critical role of property companies’ support in ensuring the stable expansion of the pile-sharing market. To this end, the study proposes an innovative economic incentive strategy to foster engagement from property companies, contributing novel insights to the new-energy-charging market’s theoretical discourse and establishing the groundwork for future empirical investigations.

Nevertheless, the research presents limitations. Firstly, it predominantly addresses the interactions among three primary groups, overlooking the broader spectrum of stakeholders involved in PCPSM. Subsequent research should delve into the broader stakeholder impact to holistically evaluate the model’s viability and efficacy. Additionally, the study also examines parameters linked to individual psychological perceptions, such as the effort and time costs ( $C_{1}$ ) for pile owners, reputational benefits ( $R_{2}$ ) for property companies, and perceived value ( $U_{0}$ ) of public charging piles for EV users. Future research endeavors, incorporating field studies and interviews, are planned to secure more robust data, thereby enhancing the model’s practical relevance. Lastly, while this research primarily scrutinizes economic incentives, future explorations may also consider integrating punitive mechanisms and tax policies. A multifaceted approach utilizing diverse policy instruments could optimize the market dynamics for PCPSM, facilitating its healthy and stable growth.

Conclusions and Policy Recommendations

Conclusions

This research, centered on the pivotal role of property companies’ support, explores the critical challenges in the dissemination of PCPSM, to advance the proliferation of EVs and contribute to the realization of carbon peaking and carbon neutrality goals. By applying prospect theory and incorporating an evolutionary game model along with SD analysis, this study offers novel insights into the stability of equilibrium points and the robustness of the system under existing incentive structures. Such an analytical approach enables the formulation of targeted and efficacious incentive policies to overcome obstacles in the promotion of PCPSM. Key findings include:

1) During the expansion of PCPSM, conflicting interests among pile owners, EV users, and property companies create barriers to market stability. Our findings underscore the crucial role of property companies’ active engagement in resolving this deadlock. An increase in the support ratio from property companies correlates with a heightened preference among EV users for PCPSM, which in turn amplifies the involvement of pile owners. Therefore, the focus of future policy incentives should be on influencing the conduct of property companies.

2) This research underscores several pivotal factors that influence the behavioral evolution of the three key players in the game of PCPSM. Parameters such as equipment upgrade and maintenance costs ( $M$ ), the time and effort investment ( $C_{1}$ ) required by pile owners for sharing, the risk costs ( $L_{1}$ ) for EV users when property companies are non-supportive, the perceived value of sharing private pile ( $U_{1}$ ) for EV users, and the profit pricing ratio ( $k$ ) for pile owners, though influential, do not alter the stakeholders’ strategies for PCPSM. Nonetheless, judicious management of these parameters can expedite the system’s evolution toward a favorable and stable state of (1,1,1). In contrast, parameters such as the negotiation costs with sharing platforms ( $C_{2}$ ), management fees charged by property companies ( $C_{3}$ ), parking fees ( $C_{4}$ ), the costs of electricity and services for public charging piles ( $R_{0}$ ), the associated costs for private pile sharing services ( $R_{1}$ ), risk costs incurred when property companies withhold support ( $L_{0}$ ), the perceived value of public charging piles ( $U_{0}$ ) for EV users, and the subsidy factors ( $s_{1}$ , $s_{2}$ ), along with the profit-sharing ratio ( $m$ ), play a crucial role in shaping the strategic decisions of the game’s participants. Maintaining these parameters within optimal ranges is essential to mitigate potential negative impacts and foster a steady and positive evolution of the system.

3) This research meticulously examines the complexities involved in advancing PCPSM, focusing on the role of psychological factors under uncertainty in shaping the strategic choices of game participants. Employing prospect theory as a framework, we found that different degrees of risk preference and loss aversion considerably influence both the behavioral steadiness and the pace of evolution among participants. Notably, in high-risk preference scenarios, participants are likely to alter their decision-making approaches. A diminished extent of loss aversion leads to an increased diversity and dynamism in behavior patterns, which is particularly evident in how it tempers the enthusiasm of property companies for proactive strategy adoption, thereby delaying the collective shift toward an ESS. Consequently, attenuating risk preferences through educational initiatives and adopting standardized management practices to mitigate the intrinsic risks associated with PCPSM are pivotal in steering the system evolution toward an optimal state. Further, enhancing property companies’ awareness of potential losses is crucial for accelerating their engagement in proactive strategies. The study also underscores the role of PCPSM as an innovative model in the EV-charging market, characterized by significant uncertainty among stakeholders. Our comparative analysis reveals that prospect theory exerts a profound influence on decision-making in contexts involving high-cost or high-value parameters. In situations where perceived value is high, the effect of loss aversion critically shapes EV users’ receptiveness toward PCPSM. These findings lay the groundwork for the development of effective incentive mechanisms for PCPSM. Policymaking going forward should integrate psychological and behavioral considerations to promote the sustainable evolution of EV charging infrastructure.

4) Reliance solely on static subsidy incentives is inadequate for ensuring the stable progression of PCPSM, particularly under elevated management cost scenarios. The dynamic subsidy model introduced in this study, which sustains system equilibrium at (1,1,1) even amidst rising property management fees, demonstrates its efficacy in fostering property company engagement and market stability. It is imperative for governments to carefully weigh the subsidy amounts against system robustness, aiming to minimize the risks of market volatility. Moreover, as the sole dynamic adjustment of the subsidy factor $s_{1}$ may exacerbate market volatility, future research should investigate flexible subsidy mechanisms and integrate them with other policy instruments. This approach is intended to harmonize the interests of all involved stakeholders while enhancing overall system efficiency. Implementing such measures is anticipated to significantly aid in the evolution and consistent growth of PCPSM, as well as in the sustainable expansion of the EV charging infrastructure.

Policy Recommendations

This study, through analysis of behavioral dynamics and response to incentive mechanisms within PCPSM, proposes specific policy recommendations to foster the development and shape future policies of the sharing model:

1) Effective incentives cannot be achieved through stable subsidies when property companies face high management costs ( $C_{3}$ ). Establishing a dynamic incentive mechanism is crucial for the promotion of PCPSM. Firstly, it is essential to motivate pile owners to take on a portion of the management costs. The government can encourage cooperation between pile owners and property companies through economic measures, entrusting property companies with the unified management of external vehicle affairs. This approach not only mitigates the operational risks for property companies but also enhances the charging experience for EV users, thereby promoting the stability of PCPSM. Secondly, the government should provide appropriate subsidies to property companies for their management activities, ensuring they receive fair compensation for their services. Such a subsidy mechanism alleviates the pressure on property companies from increased management responsibilities and prevents inadequate support for PCPSM because of insufficient economic benefits. Thirdly, the intensity of government subsidies should be dynamically adjusted based on the actual support ratio of property companies. As their support increases, government subsidies can be gradually reduced until completely phased out, whereas subsidies for the management contributions from pile owners should be maintained to preserve the system’s interest balance and stability.

2) Governments should fully consider the implications of prospect theory, including risk preference and loss aversion, as well as nonlinear value perception on decision-making, and guide the public at a psychological level to promote the diffusion of PCPSM. Our research indicates that lower risk preferences coefficient ( $β$ ) and higher loss aversion coefficient ( $λ$ ) contribute to the diffusion of such sharing models. Therefore, the government should, on one hand, conduct targeted risk education to enhance the public’s awareness of decision-making risks and establish a risk compensation fund. This fund would provide economic compensation to pile owners or property companies who incur losses by participating in the sharing model, offering a necessary economic safety net. This measure aims to reduce participants’ perception of income uncertainty within PCPSM, thereby increasing its attractiveness. On the other hand, leveraging the principle of loss aversion, the government could emphasize the potential losses avoided by participating in PCPSM through public campaigns. This utilizes people’s strong preference for avoiding losses to encourage proactive engagement. Additionally, implementing policies in phases or initiating pilot projects, along with actively showcasing successful examples, can decrease public perception of loss associated with PCPSM.

3) It is crucial to control the factors affecting the revenue of property companies to prevent them from impeding the spread of PCPSM. This study identifies the costs of property companies associated with supporting the sharing model as a key factor influencing their strategic choices, particularly the liaison costs ( $C_{2}$ ), management costs ( $C_{3}$ ), and the psychological risk costs of not participating in the sharing model ( $L_{0}$ ). While implementing incentives, the government must also manage these costs effectively. On one hand, the government should issue regulatory policies to guide and coordinate the optimization of the cost structure. Through financial subsidies, tax reliefs, and similar measures, the operational and management pressures on property companies and pile owners can be alleviated. The enthusiasm of property companies to participate in sharing can be further enhanced by reducing costs $C_{2}$ and $C_{3}$ . On the other hand, by enacting relevant guiding policies that encourage residents to express their demands for private pile sharing to property management, the government can lead property companies to better understand the potential economic and social value they might miss by not participating, the social and environmental responsibility pressures faced, as well as the potential for a decrease in market competitiveness. By increasing the cost $L_{0}$ , property companies are prompted to reconsider their stance and actively engage in PCPSM.

4) Pricing strategies and benefit distribution mechanisms are crucial components of PCPSM that require further optimization. Our research indicates that the pricing of public charging piles ( $R_{0}$ ) and the pricing of sharing private charging piles ( $R_{1}$ ) significantly influence EV users’ choices, thereby affecting the diffusion of PCPSM. In particular, lower pricing of public charging piles possesses strong appeal in the charging market. Therefore, on one hand, the government should regulate the pricing of public charging piles ( $R_{0}$ ) to ensure that it covers operational costs without hindering the development of PCPSM. By establishing reasonable pricing policies, a balanced competitive market environment between public charging piles and sharing private piles can be achieved. On the other hand, the pricing of sharing private charging piles ( $R_{1}$ ), typically set by the pile owners, can be guided by government-issued directives. These directives should encourage reasonable pricing and provide incentives such as electricity cost benefits and regulation of service fees on sharing platforms to reduce the cost of sharing private piles and attract EV users with lower prices. Moreover, as far as benefit distribution is concerned, the distribution coefficient ( $m$ ) should be maintained within a range of 0.5–0.7, ensuring that pile owners retain a significant share of the revenue from the sharing model, while also ensuring that property companies’ share of revenue is not too low. Additionally, this coefficient should be dynamically adjusted as the sharing model evolves to adapt to market changes. The government should establish a comprehensive monitoring and evaluation system that includes issuing guiding policies, clarifying the proportion and calculation methods of benefit distribution, and setting up a multi-party negotiation platform. This system aims to ensure the transparency of operational data and the fairness of benefit distribution, promoting the sustainable development of PCPSM.

5) To enhance the enthusiasm of both supply and demand sides for PCPSM, it is critical to control factors that affect the interests of pile owners and EV users. Factors related to the interests of pile owners have a minimal impact on the ultimate stability level of PCPSM, but measures can be taken to control these factors and expedite the adoption process by pile owners. For instance, providing operational training for pile owners can alleviate their burden on the sharing platform, while increasing the social awareness of the sharing model through media and public campaigns. For EV users, their perceived value of different charging options significantly influences system stability. The government can regulate this perception through the lens of prospect theory, for example, by increasing the cost of using public charging piles during peak times to decrease their perceived value ( $U_{0}$ ). Concurrently, leveraging its resource advantages, the government can attract more investors to private pile sharing projects through physical projects, enhancing the technological and service levels of PCPSM. This not only boosts the market competitiveness of PCPSM but also increases the perceived value ( $U_{1}$ ) of sharing private piles among EV users.

Supplemental Material

sj-docx-1-trr-10.1177_03611981241265846 – Supplemental material for Insights for Sustainable Urban Transport via Private Charging Pile Sharing in the Electric Vehicle Sector

Supplemental material, sj-docx-1-trr-10.1177_03611981241265846 for Insights for Sustainable Urban Transport via Private Charging Pile Sharing in the Electric Vehicle Sector by Jianming Cai, Zixin Zhou, Zhiqiang Zhao and Yaxin Wang in Transportation Research Record

Footnotes

Author Contributions

The authors confirm contribution to the paper as follows: study conception and design: J. Cai, Z. Zhou; data collection: Z. Zhou, Z. Zhao, Y. Wang; analysis and interpretation of results: J. Cai, Z. Zhou; draft manuscript preparation: J. Cai, Z. Zhou, Z. Zhao, Y. Wang. All authors reviewed the results and approved the final version of the manuscript.

Declaration of Conflicting Interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work was supported by the Natural Science Foundation of Hunan Province (2022JJ30763).

ORCID iDs

Jianming Cai

Zixin Zhou

Supplemental Material

Supplemental material for this article is available online.

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Supplementary Material

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Insights for Sustainable Urban Transport via Private Charging Pile Sharing in the Electric Vehicle Sector

Abstract

Keywords

Literature Review

Sharing Model of Private Charging Piles

Impact of Government Policies on the Development of EVCI

Application of Evolutionary Game Theory in the Field of EVCI

Evolutionary Game Model

Model Description

Model Assumptions

Construction of the Payoff Matrix Based on Prospect Theory

Stability Analysis of the Tripartite Game

Equilibrium Point Analysis in the Evolutionary Game Model

Incremental Stability Analysis of Unilateral Behavior Strategy

Strategy Stability Analysis of Charging Pile Owners

Strategy Stability Analysis of Property Companies

Strategy Stability Analysis of EV Users

Numerical Simulation of Tripartite Evolutionary Game

Evolutionary Game Model Based on System Dynamics (SD)

Simulation Design and Path Evolution

Parameter Sensitivity Analysis

Risk Preference Coefficient ( β )

Loss Aversion Coefficient ( λ )

Equipment Upgrade and Maintenance Costs ( M )

Negotiation Costs of Engaging with Sharing Platforms (C2)

Management Costs Incurred by Property Companies (C3)

Parking Fees (C4)

Electricity and Service Fees for EV Users Using Public Charging Piles (R0)

Electricity and Service Fees for EV Users Using Shared Private Charging Piles (R1)

Cost of Risk When Property Companies Do Not Support Sharing (L0)

The Value of Public Charging Piles to EV Users (U0)

The Value of Sharing Private Charging Piles to EV Users (U1)

Coefficient of the Management Fee Paid by the Charging Pile Owners to the Property Companies (s1)

Coefficient of Government Subsidy to Property Companies (s2)

Pricing Proportion of Revenue for Charging Pile Owners (k)

Proportion of Costs and Benefits (m)

Impact of Prospect Theory Applications on Parameters

The Optimization Plan of the Incentive Mechanism

Stability of the Evolutionary System After Optimizing the Incentive Mechanism

Impact of Different Incentives on the System

Results and Discussion

Conclusions and Policy Recommendations

Conclusions

Policy Recommendations

Supplemental Material

sj-docx-1-trr-10.1177_03611981241265846 – Supplemental material for Insights for Sustainable Urban Transport via Private Charging Pile Sharing in the Electric Vehicle Sector

Footnotes

Author Contributions

Declaration of Conflicting Interests

Funding

ORCID iDs

Supplemental Material

References

Supplementary Material

Risk Preference Coefficient ( $β$ )

Loss Aversion Coefficient ( $λ$ )

Equipment Upgrade and Maintenance Costs ( $M$ )

Negotiation Costs of Engaging with Sharing Platforms (C₂)

Management Costs Incurred by Property Companies (C₃)

Parking Fees (C₄)

Electricity and Service Fees for EV Users Using Public Charging Piles (R₀)

Electricity and Service Fees for EV Users Using Shared Private Charging Piles (R₁)

Cost of Risk When Property Companies Do Not Support Sharing (L₀)

The Value of Public Charging Piles to EV Users (U₀)

The Value of Sharing Private Charging Piles to EV Users (U₁)

Coefficient of the Management Fee Paid by the Charging Pile Owners to the Property Companies (s₁)

Coefficient of Government Subsidy to Property Companies (s₂)