Evolutionary Game Analysis Between Large Power Consumers and Power Sellers in the Context of Big-Data-Driven Value-Added Services

Abstract

As power system reforms deepen, direct trading with large power consumers has emerged as a crucial aspect of opening up the power sales market. In light of this trend, power sales enterprises should accelerate their digital transformation in response to the growing demand for personalized services from large consumers and continuous advancements in energy digitalization and smart technologies. In particular, big data technology is critical for power enterprises to satisfy users and increase profitability as it can help enterprises gain deeper insights into user needs and behavioral characteristics. The application of big data to provide customized value-added services for large power consumers has become a key development focus. In this paper, we develop a two-party evolutionary game model involving power sellers using big data technology to profile large consumers and offer them customized value-added power packages. We conduct a detailed analysis of the local stability of equilibrium points and employ MATLAB.R2021a to examine the impact of changes in the benefits of value-added services for large consumers and the cost coefficients associated with big data on the system’s evolutionary outcomes. The study results indicate that big data technology can enhance the competitiveness of power sellers in the market. Value-added services based on user-profiling using big data have become a crucial factor in influencing the decision-making behavior of large consumers. Additionally, the investment cost in big data infrastructure by power sellers impacts system evolution, with the cost coefficient being inversely proportional to their willingness to offer customized services.

1. Introduction

The reform of the international power system represents a crucial global effort aimed at enhancing energy supply security, promoting economic growth and development, and adapting to energy transitions. It also focuses on mitigating climate change, improving energy efficiency and resource optimization, and fostering international cooperation and interconnectivity. These reforms provide a solid foundation for sustainable development and global energy governance. Different countries and regions have adopted various approaches to reforming the international power system. For instance, to establish an integrated European power market, the European Union (EU) requires its member states to liberalize their power markets to encourage cross-border power trading, while separating power transmission from power generation. Additionally, the EU encourages the development of renewable energy and the enhancement of energy efficiency. In the United States, power market reforms mainly occur at the state level. The country is also actively modernizing its power system, including the implementation of smart grids and distributed energy. Meanwhile, the United Kingdom has undertaken extensive power system reforms by introducing a competitive power market and implementing policies for renewable energy and low-carbon power generation. It also encourages consumers to choose their energy suppliers and has established independent regulatory agencies for market supervision [1,2,3,4].

In China, a new round of power system reform has officially launched following the issuance of the Opinions of the CPC Central Committee and the State Council on Further Deepening the Reform of the Power System. This issuance explicitly mentions the “liberalization of consumers’ right of choice”, allowing large users who meet access criteria to directly negotiate contracts with power generation enterprises. This enables multi-party independent transactions, aiming to reduce power costs for enterprises. Concurrently, the establishment of a stable bilateral negotiation and trading model is actively encouraged [3,4,5]. This means that with the liberalization of the power sales market, large consumers can independently choose their power sellers, thereby enjoying better services at a lower price through market-based transactions. Currently, China’s power sales market is liberalizing at an accelerated pace, having already reached a considerable market size with numerous participants.

The continuous advancement of power system reform is creating a more open power sales market, resulting in a more competitive environment for power enterprises [5,6,7]. Simultaneously, with the rapid growth of the “experience economy”, traditional static customer services can no longer meet the increasing expectations and needs of power consumers for personalized services. Consequently, power enterprises must adjust and upgrade their existing service models, focusing on implementing differentiated services. They also should classify their customers and provide more precise and personalized services to meet user demands.

With the increasing number of power sales enterprises participating in market-based transactions, users enjoy a more diverse range of choices. Consequently, the market share of these enterprises is directly influenced by user preferences, which play a crucial role in shaping the decision-making processes of these enterprises. When selecting a power seller, large consumers typically consider various factors, including the seller’s credibility, service quality, power price, power stability, and energy management technology. However, the most critical considerations remain power price and value-added services [8,9,10].

In this context, to attract more large customers, power sales enterprises need to utilize user data accumulated on information technology platforms and conduct in-depth analysis using big data technology to understand users’ needs and behavioral characteristics. This analysis can help power sellers better understand the needs and behaviors of their users, enabling them to offer power packages that are better suited to different types of users. With the support of big data technology, power sales enterprises can offer users a variety of personalized value-added services, such as power consumption forecasts, optimization suggestions for power usage, precise bill customization, intelligent energy management, and smart home solutions. These value-added services will assist power sales enterprises in enhancing their service quality and customer satisfaction, thereby increasing their profitability [11,12,13].

In addition, as the power industry is a crucial pillar of the national economy and a key employer, the digital transformation of power enterprises has also become a focal point for government attention. On 28 March 2023, the National Energy Administration of China released the Opinions on Accelerating the Development of Energy Digitization and Intelligence, highlighting that the integration of the energy industry with digital technology is a vital driver for advancing the foundation of China’s energy industry and modernizing the industrial chain in the new era [14,15]. User-profiling based on big data, as a significant application in the digital economy, builds user profiles by collecting, aggregating, and analyzing user data. This technology provides personalized services and support across various sectors of the digital economy.

Therefore, under the premise of gradually opening up the power sales market, providing customized value-added services using big data from a fresh perspective, it is meaningful to explore the evolution path of the game between power sellers and large consumers. On the one hand, this can help power sellers improve their stickiness with large consumers and their competitiveness, formulate more targeted market strategies, optimize product and service design, and enhance their market position. Additionally, it benefits large consumers by allowing them to comprehensively evaluate and choose the optimal power supplier, thereby maximizing their interests. On the other hand, it can elucidate how various factors influence and interact with the decision-making behaviors of the two parties involved in bilateral power transactions amidst the wave of digital transformation. This has significant practical implications for the formulation and revision of market rules, policies, and regulatory measures. It also provides valuable references for the power industry and fosters the industry’s advancement towards digitization. Thus, it fills a gap in game theory by addressing the new competitive landscape of the digitally transformed power market.

2. Literature Review

Numerous studies have utilized game theory to investigate issues related to power markets. Shilov et al. integrates Nash equilibrium with peer-to-peer power markets to examine the privacy aspects of information-sharing within the power market [16]. Oliveira et al. investigates the strategies of power sellers for power purchases in the spot market and the bilateral contract market, as well as their strategies for selling power to residential customers [17]. It establishes a two-layer game model of the profit of sellers’ power purchases and sales, and provides a Nash equilibrium solution. Xie et al. develops a stochastic evolutionary game market-clearing model for market regulators and risk units, to perform an evolutionary dynamic analysis of the strategic evolution space and market risk variations among spot market participants [18]. Additionally, Wang et al. introduces an incomplete evolutionary information game approach to analyze bidding strategies in power markets with price-elastic demand [19]. Kumar presents a game-theoretic approach-based bidding strategy decision-making through a case study to help the GENCOs make a judicious selection among the possible available strategies. These analyses and studies have collectively affirmed the applicability and feasibility of game theory in the power market [20].

Research on competitive strategies in the power sales market has yielded significant results. K. Huang examines the power pricing strategy of various power sales entities in the context of energy reforms using the Cournot model [21]. W. Zhang et al. applies the Bertrand model to the pricing behavior of power sales entities [22]. Anderson et al. explores a bidding decision model that minimizes power purchasing costs in both pre-equilibrium and short-term equilibrium markets [23]. Chai B et al. focuses on assessing user selection of power sales enterprises based on price sensitivity, utilizing a power consumption function [24,25,26].

In recent years, there has been a growing body of research and applications focused on value-added services provided by power sellers [27,28,29]. Q. Cao et al. proposes various value-added services and examines those suitable for advanced users based on their characteristics [30]. Additionally, Dong et al. describes potential business models from the perspectives of the user center of the Energy Internet, value-added information, and technological innovation, and offers relevant suggestions for the development of the business models and market mechanisms of the Energy Internet [31,32,33,34]. This clearly demonstrates that to adapt to development and enhance customer satisfaction and enterprise competitiveness, power grid enterprises should leverage their advantages to implement innovative value-added services within the context of the Energy Internet, thereby maximizing value gains.

The review of the above literature indicates that most scholars currently focus on power price and consumption when exploring the competitive strategies of power sales enterprises. However, in the context of a liberalized power sales market and the trend of value-added services emerging as competitive advantages, there is a notable gap in research that considers both power price and value-added services—two crucial factors influencing users’ decision-making. The interaction between these two factors is also often deemed minor. Only Wei et al. employs the Stackelberg game model and proposes a strategy of “price competition and value-added service differentiation” as the primary approach for power sales entities to respond to the current market situation [8].

Meanwhile, it is assumed in most existing research that power enterprises operate with complete rationality. However, the actual transactions between buyers and sellers in the power market are dynamic and long-term processes. Most power companies face difficulties in fully understanding the decision-making approaches of competitors and users, resulting in individual power companies possessing limited rationality. Evolutionary game theory, which effectively characterizes long-term dynamic processes with bounded rationality, is more suitable than other methods for analyzing the decision-making processes in the trading game between power enterprises and sellers from a micro perspective.

Moreover, current studies primarily focus on users’ price sensitivity and the pricing mechanisms of power sellers, leaving a gap in research from the perspective of big-data-driven differentiation in value-added services. This paper takes a novel approach by recognizing the positive role that big data plays in enabling power sellers to compete in the market. It also employs simulation modeling to verify whether this positive impact is constrained by the cost–benefit ratio, thereby filling a gap in the game theory regarding the new competitive landscape of the power market in the context of digitalization.

Therefore, in the context of power system reform, this paper, from a fresh perspective, explores the game between power sellers and large consumers within a bilateral framework, focusing on the use of big data to provide customized value-added services. The study integrates a big data analysis coefficient into its modeling approach and analyzes the mechanism of value-added services and pricing factors influencing bilateral transactions in the context of big data technology, taking into account the cost of value-added services. Utilizing the evolutionary game model, it examines the evolutionary decision-making paths of both parties to identify the optimal equilibrium point. This approach enriches the game model involving large consumers and power sellers, filling existing gaps in the literature. Additionally, it aims to provide new perspectives and theoretical support for the application of big data technology in the power market and the development of the power sales market.

3. Modeling

3.1. Model Assumptions

Assumption 1: The strategy of power sellers includes using big data technology to offer customers two types of power packages: one with customized value-added services and another that is ordinary (hereinafter referred to as VSP and OP). The value-added service package provides personalized services, with a focus on service quality and customer care. In contrast, the ordinary package offers standardized services at unified prices, instead of personalized services. Large power consumers can choose to trade with this power seller or with other sellers. Both power sellers and large consumers exhibit bounded rationality and make decisions that favor their own development [35,36].

Assumption 2: Once a power seller has invested in deploying big data technology, it is assumed to have the capability to provide value-added services. Therefore, the ongoing costs of providing these services to large consumers are not considered.

Assumption 3: When power sellers provide ordinary packages, they use a uniform scheme and pricing by default. Therefore, the cost of offering these packages is not considered.

Assumption 4: Large consumers must purchase power to maintain their operations. Therefore, their decision-making involves either purchasing from the power seller or alternative sellers, with no option to refrain from purchasing.

Assumption 5: The primary factors influencing the choice of large consumers are price and value-added services, with other factors temporarily disregarded.

Assumption 6: Other power sellers only have a competitive advantage of package pricing, with no personalized services offered. Therefore, large consumers can purchase power from other sellers at a lower price, provided that the power seller only offers the ordinary package. However, when big data are used to provide value-added services, the price difference between purchasing power from the seller and other sellers becomes uncertain. The reason is that, for large consumers, the power price is no longer their primary focus for customized value-added services provided by the power seller, making them less sensitive to price. In contrast, the power seller might be willing to set a lower price, considering the additional revenue from the value-added services. It is also assumed that the total sales revenue from power packages with value-added services exceeds the revenue from ordinary packages.

Assumption 7: Large consumers highly prefer value-added services provided by power sellers and will purchase these services if provided by the seller.

Based on these assumptions, the game flow chart is as follows (Figure 1):

3.2. Parameter Design

When a power seller uses big data to provide customized services, the parameter design is as follows: P—revenue from value-added services; θ—the level of big data analysis, $0<\theta <1$; k—the cost coefficient; $C_n={\displaystyle \frac{1}{2}}k{\theta }^2$—the investment cost in big data deployment; Q₁—the cost for the large consumer to purchase the power package/the revenue for the power seller from selling the power package; M—the cost for the large consumer to purchase the value-added services; and W—the positive benefit brought by the value-added services. When the power seller provides the ordinary package, Q₂ represents the sales revenue for the seller/the cost for the large consumer to purchase the power package. The power cost for the power seller, regardless of the decision made, is C_e. If the large consumer chooses to purchase from other sellers, the cost is represented as Q₃, which is less than Q₂. The benefits generated from the large user’s production and operations after purchasing power is E, regardless of the decision made.

It is assumed that the proportions of power sellers providing value-added service packages, sellers providing ordinary packages, large consumers purchasing power from the power seller, and consumers purchasing power from other sellers are α, β, $(1-\alpha )$, and $(1-\beta )$, respectively. Based on the parameters described above, a game payoff matrix can be constructed (see

Table 1. Payoff matrix of game between power sellers and large consumers.

Game Agent		Large Consumer Y
		Purchasing Power from the Power Seller β	$\mathbf{Purchasing} \mathbf{Power} \mathbf{from} \mathbf{Other} \mathbf{Power} \mathbf{Sellers} (1-\mathit{\beta })$
Power seller X	Value-added service package α	$Q_1+P-C_e-C_n$ $W+E-M-Q_1$	$-C_n$ $E-Q_3$
	$\mathrm{Ordinary} \mathrm{package} (1-\alpha )$	$Q_2-C_e$ $E-Q_2$	0 $E-Q_3$

).

4. Evolutionary Game Analysis Between Power Sellers and Large Consumers

Based on the above, the expected payoff for power sellers providing value-added service packages is as follows:

$${E\left(X\right)}_{\alpha }=\beta \left(Q_1+P-C_e-C_n\right)+\left(1-\beta \right)\left(-C_n\right)=\beta Q_1+\beta P+\beta C_e-C_n;$$

(1)

The expected payoff for power sellers providing ordinary packages is as follows:

$${E\left(X\right)}_{1-\alpha }=\beta \left(Q_2-C_e\right)=\beta Q_2-\beta C_e;$$

(2)

The average payoff for power sellers providing value-added service packages and ordinary packages is as follows:

$$E\left(X\right)=\alpha {E\left(X\right)}_{\alpha }+\left(1-\alpha \right){E\left(X\right)}_{1-\alpha }=\alpha \left(\beta Q_1+\beta P+\beta C_e-C_n\right)+\left(1-\alpha \right)\left(\beta Q_2-\beta C_e\right)$$

(3)

The replicator dynamics equation for power sellers is as follows:

$${\displaystyle \frac{d\alpha }{dt}}=\alpha \left[{E\left(X\right)}_{\alpha }-E\left(X\right)\right]=\alpha \left(1-\alpha \right)\left(\beta Q_1+\beta P-C_n-\beta Q_2\right);$$

(4)

The expected payoff for large consumers purchasing power from the power seller is as follows:

$${E\left(Y\right)}_{\beta }=\alpha \left(W+E-M-Q_1\right)+\left(1-\alpha \right)\left(E-Q_2\right)=\alpha W-\alpha M-\alpha Q_1-\left(1-\alpha \right)Q_2+E;$$

(5)

The expected payoff for large consumers purchasing power from other power sellers is as follows:

$${E\left(Y\right)}_{1-\beta }=\alpha \left({E-Q}_3\right)+\left(1-\alpha \right)\left({E-Q}_3\right)={E-Q}_3;$$

(6)

The average payoff for large consumers purchasing power from the power seller and other sellers is as follows:

$$E\left(Y\right)=\beta {E\left(Y\right)}_{\beta }+\left(1-\beta \right){E\left(Y\right)}_{1-\beta }=\beta \left[\alpha W-\alpha M-\alpha Q_1-\left(1-\alpha \right)Q_2+E\right]+\left(1-\beta \right)\left({E-Q}_3\right);$$

(7)

The replicator dynamics equation for large consumers is as follows:

$${\displaystyle \frac{d\beta }{dt}}=\beta \left[{E\left(Y\right)}_{\beta }-E\left(Y\right)\right]=\beta \left(1-\beta \right)\left[\alpha W-\alpha M-\alpha Q_1-\left(1-\alpha \right)Q_2+Q_3\right];$$

(8)

4.1. Evolutionary Paths and Stabilization Strategies for Power Sellers and Large Consumers

4.1.1. Trends in the Evolutionary Game of Power Sellers

Let $F\left(\alpha \right)={\displaystyle \frac{d\alpha }{dt}}=\alpha \left[{E\left(X\right)}_{\alpha }-E\left(X\right)\right]=\alpha \left(1-\alpha \right)\left(\beta Q_1+\beta P-C_n-\beta Q_2\right)$.

Derive to obtain $F{\left(\alpha \right)}^{\mathrm{\prime }}=\left(1-2\alpha \right)\left({\beta Q}_1+\beta P-C_n-{\beta Q}_2\right)$.

Let $F\left(\alpha \right)={\displaystyle \frac{d\alpha }{dt}}=0$, to obtain ${\alpha }_1=0$, ${\alpha }_2=1$, ${\beta }^{\mathrm{*}}={\displaystyle \frac{C_n}{P+Q_1-Q_2}}$.

Proposition 1. 

For power sellers: When ${\beta }^{*}={\displaystyle \frac{C_n}{P+Q_1-Q_2}}$, they will not change their initial strategies; when ${\beta }^{*}>{\displaystyle \frac{C_n}{P+Q_1-Q_2}}$, they will actively utilize big data to provide value-added services; and when ${\beta }^{*}<{\displaystyle \frac{C_n}{P+Q_1-Q_2}}$, they will participate in the market in a traditional way.

Proof. 

Based on the nature of evolutionarily stable strategy [36], when $F\left(\alpha \right)=0$ and $F{\left(\alpha \right)}^{\prime }<0$, α is an evolutionarily stable strategy. When ${\beta }^{\mathrm{*}}={\displaystyle \frac{C_n}{P+Q_1-Q_2}}$, $F\left(\alpha \right)=0$, $F{\left(\alpha \right)}^{\prime }=0$, all values of α are in a steady state. This means that the probability of the power seller providing value-added service packages or ordinary packages remains stable when the probability that large consumers purchasing power from the seller reaches ${\beta }^{\mathrm{*}}={\displaystyle \frac{C_n}{P+Q_1-Q_2}}$. □

When ${\beta }^{*}\ne {\displaystyle \frac{C_n}{P+Q_1-Q_2}}$, due to $C_n>0$ and $P+Q_1-Q_2>0$, the discussions are divided into two cases:

(1) If ${\beta }^{\mathrm{*}}>{\displaystyle \frac{C_n}{P+Q_1-Q_2}}$, regarding ${\alpha }_1=0$ and ${\alpha }_2=1$, there are $F\left(0\right)=F\left(1\right)=0$, $F^{\prime }\left(0\right)>0$, and $F^{\prime }\left(1\right)<0$. Thus, it is evident that ${\alpha }_2=1$ is an evolutionarily stable strategy. Specifically, when the probability that large consumers purchase power from the power seller is greater than ${\displaystyle \frac{C_n}{P+Q_1-Q_2}}$, it is optimal for the seller to provide customized value-added services.

(2) If ${\beta }^{\mathrm{*}}<{\displaystyle \frac{C_n}{P+Q_1-Q_2}}$, regarding ${\alpha }_1=0$ and ${\alpha }_2=1$, there are $F\left(0\right)=F\left(1\right)=0$, $F^{\prime }\left(0\right)<0$, and $F^{\prime }\left(1\right)>0$. Thus, it is evident that ${\alpha }_2=0$ is an evolutionarily stable strategy. Specifically, when the probability that large consumers purchase power from the power seller is smaller than ${\displaystyle \frac{C_n}{P+Q_1-Q_2}}$, it is optimal for the seller to provide ordinary packages.

Based on the above analysis, a replicator dynamics phase diagram (Figure 2) for power sellers can be derived.

4.1.2. Trends in the Evolutionary Game of Large Consumers

Proposition 2. 

For large consumers: When ${\alpha }^{*}={\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}$ or ${\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}>1$, they will not change their initial strategies; when ${\alpha }^{*}<{\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}<1$, they purchase power from other power sellers; and when ${\alpha }^{*}>{\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}$, they purchase power from the power seller.

Proof. 

Let $F\left(\beta \right)={\displaystyle \frac{d\beta }{dt}}=\beta \left[{E\left(Y\right)}_{\beta }-E\left(Y\right)\right]=\beta \left(1-\beta \right)[\alpha W-\alpha M-\alpha Q_1-\left(1-\alpha \right)Q_2+Q_3]$. □

Derive to obtain $F\prime \left(\beta \right)=\left(1-2\beta \right)\left[\alpha W-\alpha M-\alpha Q_1-\left(1-\alpha \right)Q_2+Q_3\right]$.

Let $F\left(\beta \right)={\displaystyle \frac{d\beta }{dt}}=0$, to obtain ${\beta }_1=0$, ${\beta }_2=1$, ${\alpha }^{*}={\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}$.

Based on the nature of evolutionarily stable strategy [37], when $F\left(\beta \right)=0$ and $F{\left(\beta \right)}^{\prime }<0$, β is the evolutionarily stable strategy.

When ${\alpha }^{*}={\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}$, $F\left(\beta \right)=0$, and $F{\left(\beta \right)}^{\prime }=0$, all the values of β are in a steady state. Specifically, when the probability of power sellers providing ordinary packages reaches ${\alpha }^{*}={\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}$, the probability of large consumers purchasing power from the power seller or other sellers remains stable.

When ${\alpha }^{*}\ne {\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}$:

(1) If ${\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}>1$, $W+E-M-Q_1<E-Q_3$. Specifically, when power sellers provide value-added service packages, the payoff large consumers obtain from purchasing power from the power seller is less than what they obtain from purchasing power from other sellers. It is always the case that ${\beta }^{*}<{\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}$. For ${\beta }_1=0$ and ${\beta }_2=1$, there are $F\left(0\right)=F\left(1\right)=0$, $F^{\prime }\left(0\right)<0$, and $F^{\prime }\left(1\right)>0$. Thus, ${\beta }_1=0$ is an evolutionarily stable strategy. This means that regardless of the strategies chosen by power sellers, the strategy of large consumers opting to purchase power from other power sellers will remain unchanged.

(2) If $0<{\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}<1$, there are $W+E-M-Q_1>E-Q_3$ and $E-Q_2<E-Q_3$. Specifically, when a power seller provides value-added service packages, the payoff that large consumers obtain from purchasing power from the seller exceeds what they obtain from purchasing power from other sellers. In contrast, when a power seller provides traditional packages, the payoff that large consumers obtain from purchasing power from other sellers exceeds what they obtain from purchasing power from the power seller. Therefore, the discussions are divided into two cases. ① When ${\alpha }^{*}<{\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}$, regarding ${\beta }_1=0$ and ${\beta }_2=1$, there are $F\left(0\right)=F\left(1\right)=0$, $F^{\prime }\left(0\right)<0$, and $F^{\prime }\left(1\right)>0$. Thus, ${\beta }_1=0$ is an evolutionarily stable strategy. It is optimal for large consumers to purchase power from other power sellers rather than from the power seller. ② When ${\alpha }^{*}>{\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}$, for ${\beta }_1=0$ and ${\beta }_2=1$, there are $F\left(0\right)=F\left(1\right)=0$, $F^{\prime }\left(0\right)>0$, and $F^{\prime }\left(1\right)<0$. Therefore, ${\beta }_1=1$ is an evolutionarily stable strategy. It is optimal for large consumers to purchase power from the power seller [38].

Based on the above analysis, a replicator dynamics phase diagram (Figure 3) of large consumers can be derived.

4.2. Evolutionary Stability Analysis

Proposition 3. 

The equilibrium points of this replicator dynamics system consist of five local equilibrium solutions: (0, 0), (1, 0), (0, 1), (1, 1), and (x*, y*).

Proof. 

By linking Equations (4) and (8) into a system of replicator dynamics equations, solving this system reveals that there exist five replicator dynamics equilibrium points between power sellers and large consumers in the game process: (0, 0), (0, 1), (1, 0), (1, 1), (x*, y*). Among these, $x^{*}={\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}$ and $y^{*}={\displaystyle \frac{C_n}{P+Q_1-Q_2}}$. □

Proposition 4. 

The replicator dynamics system exhibits four definite evolutionarily stable strategies: (0, 0), (1, 0), (0, 1), and (1, 1).

(1) When ${\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}>1$, the evolutionary game of the system tends to a local stable point (0, 0).

(2) When $0<{\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}<1$ and $Q_1+P-Q_2<C_n$, the local stabilization equilibrium point is (0, 0).

(3) When $0<{\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}<1$ and $Q_1+P-Q_2>C_n$, the evolution strategy of the system is stable at either (0, 0) or (1, 1), and the final state of evolution depends on the speed of learning and adjustment of power sellers and large consumers.

Proof. 

According to Friedman’s stability discrimination method, stability can be evaluated by performing a local stability analysis of the Jacobian matrix corresponding to the five equilibrium points (

Table 2. Det(J) and Tr(J) results of the Jacobian matrix.

Equilibrium Point	Det(J)	Tr(J)
α = 0, β = 0	$\left(-C_n\right)\times \left(-Q_2+Q_3\right)$	$\left(-C_n\right)+\left(-Q_2+Q_3\right)$
α = 1, β = 0	$C_n\times \left(W-M-Q_1+Q_3\right)$	$C_n+\left(W-M-Q_1+Q_3\right)$
α = 0, β = 1	$\left(Q_1+P-C_n-Q_2\right)\times \left(Q_2-Q_3\right)$	$\left(Q_1+P-C_n-Q_2\right)+\left(Q_2-Q_3\right)$
α = 1, β = 1	$\left(Q_1+P-C_n-Q_2\right)\times \left(W-M-Q_1+Q_3\right)$	$-\left(Q_1+P-C_n-Q_2\right)-\left(W-M-Q_1+Q_3\right)$
α = x*, β = y*	0	0

). That is, [38] and are judged based on the signs (+/−) of the determinant (Det) and trace (Tr) of the Jacobian matrix. □

The Jacobian matrix for the replicator dynamics equations is as follows:

$$J=\left(\begin{array}{cc}(1-2\alpha )(\beta Q_1+\beta P-C_n-\beta Q_2)& \alpha (1-\alpha )(P+Q_1-Q_2)\\ \beta (1-\beta )(W-M-Q_1+Q_2)& \left(1-2\beta \right)[\alpha W-\alpha M-\alpha Q_1-\left(1-\alpha \right)Q_2+Q_3]\end{array}\right)$$

The corresponding determinant and trace of the Jacobian matrix are as follows:

$$Det\left(J\right)=\left\{\left(1-2\alpha \right)\left({\beta Q}_1+\beta P-C_n-{\beta Q}_2\right)\times \left(1-2\beta \right)\left[\alpha W-\alpha M-\alpha Q_1-\left(1-\alpha \right)Q_2+Q_3\right]\right\}$$

$$Tr\left(J\right)=\left\{\left(1-2\alpha \right)\left({\beta Q}_1+\beta P-C_n-{\beta Q}_2\right)+\left(1-2\beta \right)\left[\alpha W-\alpha M-\alpha Q_1-\left(1-\alpha \right)Q_2+Q_3\right]\right\}$$

If Tr(J) < 0 and Det(J) > 0, this equilibrium point constitutes an evolutionarily stable strategy (ESS).

According to the Tr(J) and Det(J) results of the Jacobian matrix, at (α*, β*), Tr(J) = 0 and Det(J) = 0. Therefore, it is sufficient to discuss the stability of the four points (0, 0), (1, 0), (0, 1), and (1, 1). Stability analysis results of each equilibrium point are shown in

Table 3. Stability analysis results of local equilibrium points.

Case	Equilibrium Point	Det(J) Notation	Tr(J) Notation	Result
$W-M-Q_1+Q_3<0$	(0, 0)	+	−	ESS
	(1, 0)	−	Uncertain	Saddle point
	(0, 1)	$\mathrm{When} C_n<P+Q_1-Q_2$, + $\mathrm{When} C_n>P+Q_1-Q_2$, −	+ Uncertain	Unstable Saddle point
	(1, 1)	$\mathrm{When} C_n<P+Q_1-Q_2$, − $\mathrm{When} C_n>P+Q_1-Q_2$, +	Uncertain +	Saddle point Unstable
$W-M-Q_1+Q_3>0$	(0, 0)	+	−	ESS
	(1, 0)	+	+	Unstable
	(0, 1)	$\mathrm{When} C_n<P+Q_1-Q_2$, + $\mathrm{When} C_n>P+Q_1-Q_2$, −	+ Uncertain	Unstable Saddle point
	(1, 1)	$\mathrm{When} C_n<P+Q_1-Q_2$, + $\mathrm{When} C_n>P+Q_1-Q_2$, −	− Uncertain	ESS Saddle point

below.

Case 1: When ${\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}>1$, $W+E-M-Q_1<E-Q_3$. When power sellers employ big data technology, large consumers gain higher payoffs by purchasing power from other power sellers than from the power seller, and it is always the case that $E-Q_2<E-Q_3$. Therefore, regardless of the strategy of the power seller, large consumers tend to purchase power from other power sellers. For power sellers, providing ordinary packages yields greater profits when large consumers purchase power from other sellers. Through continuous learning and evolution in the game, large consumers finally opt to purchase power from other power sellers, while sellers choose the traditional approach, leading the evolutionary game towards a local stable point (0, 0).

Case 2: When $0<{\displaystyle \frac{Q_3-Q_2}{M+Q_1-Q_2-W}}<1$, $W+E-M-Q_1>E-Q_3$. When the power seller employs big data technology, large consumers gain higher payoffs by purchasing power from the seller compared to other sellers. Conversely, when the power seller adopts traditional approaches, large consumers gain higher payoffs by purchasing power from other sellers compared to the seller.

When $Q_1+P-Q_2<C_n$, meaning that the benefit obtained by the power seller from providing value-added service packages is less than the additional costs incurred, and the cost investment in big data is excessively high, this leads the power seller to opt for ordinary packages. Therefore, the local stable equilibrium point at this time is (0, 0). In a competitive market where other sellers hold a pricing advantage, if the return from implementing big data is less than the cost invested, and the advantages of big data are not fully leveraged compared to other competitors, then it is prudent for the seller to opt for ordinary packages.

When $Q_1+P-Q_2>C_n$ and $W+E-M-Q_1>E-Q_3>E-Q_2$, if large consumers choose to purchase power from the power seller, the seller gains more from offering value-added service packages than from offering ordinary packages. Conversely, if large consumers choose to purchase power from other power sellers, the benefit is smaller compared to the traditional approach. The final state of the evolution depends on the speed of learning and adjustment of both power sellers and large consumers.

5. Numerical Simulation

To visualize the specific relationship between the way the power sellers provide packages and the transaction choices made by large consumers, as well as the evolutionary path, this paper employs MATLAB software to simulate the trends in the evolutionary game under various conditions. The initial values of α and β are set to 0 and the step size to 0.1, with the evolution ending when they reach 1. During this process, addressing the issue of numerical selection in simulations, Sterman emphasizes the necessity for simulation models to be accurate and practical, to reveal the regular characteristics of changes. Regarding evolutionary game models, while the initial values to some extent determine the speed and magnitude of system evolution, they do not affect the overall trend and direction [39]. Therefore, based on the literature review and equation balancing methods, relevant parameters in the payment matrix are assigned values that meet the model’s requirements.

Case 1: $W-M-Q_1+Q_3<0$.

Assume that P = 100, θ = 0.6, k = 400, $C_n={\displaystyle \frac{1}{2}}k{\theta }^2$, Q₁ = 500, Q₂ = 500, Q₃ = 470, M = 100, W = 120, C_{e =} 330, and E = 1000. The simulation results are shown in Figure 4 below.

In the figure, the lines of different colors represent the dynamic evolution paths of the main body’s strategy choices after multiple simulations. Through observing the trend of the line changes, the evolution process and direction of the entire system can be analyzed and obtained. As shown in Figure 4, the proportion of power sellers providing value-added service packages initially trends towards 1. However, with continuous evolutionary iterations, this proportion eventually trends towards 0. Concurrently, the proportion of large consumers purchasing power from the power seller decreases, with the strategy β gradually approaching 0. Therefore, in this case, large consumers gain more by purchasing power from other sellers than from the power seller. No matter what decisions power sellers make, even if they offer highly personalized value-added services to cater to users’ specific needs, they might still fail to capture the attention of large consumers. These users are inclined to make decisions based primarily on maximizing their own benefits. Consequently, as the power seller’s investment in big data fails to yield returns, it will gradually cease investment and abandon this strategy.

Case 2: $W-M-Q_1+Q_3>0$.

Assume that P = 100, θ = 0.6, k = 400, $C_n={\displaystyle \frac{1}{2}}k{\theta }^2$, Q₁ = 500, Q₂ = 500, Q₃ = 470, M = 100, W = 150, C_e = 330, and E = 1000. The simulation results are shown in Figure 5 below.

As shown in Figure 5, the evolutionary system can evolve to a local evolutionary stable state under various initial proportion conditions. The high cost of big data investment by power sellers and the low benefits obtained by large consumers from value-added services will drive the initial proportion of both parties choosing the traditional approach and large consumers opting to purchase power from other sellers to evolve to (0, 0). Conversely, the low cost of big data investment by power sellers and the high benefits obtained by large consumers from value-added services will drive the initial proportion of both parties choosing value-added service packages and large consumers opting to purchase power from the power seller to gradually evolve to (1, 1).

To further analyze the impact of the benefits of value-added services on the evolution of large consumers, the value of W was adjusted to 150, 350, and 650 for simulation, with the results shown in Figure 6 below.

As shown in Figure 6, with the increasing benefits from value-added services, the evolutionary system gradually converges to the stable point (1, 1). The high benefits of value-added services generated by big data technology significantly attract large consumers to make deals. Thus, these consumers are more inclined to cooperate with specific power sellers, thereby maximizing their benefits. Meanwhile, this cooperation provides power sellers the opportunity to increase their value-added service revenues, further encouraging them to provide customized value-added services, thereby evolving to the stable point (1, 1). Therefore, if a power seller can provide substantial value-added service benefits to large consumers, the likelihood that these consumers choose to transact with the seller will increase, fostering a mutually beneficial, win–win relationship.

Based on this, the effects of changes in W/Cn, or the input–output ratio of big data on decision-making, are further investigated. Specifically, the value of W/Cn is set to 0.5, 1, 2, and 4 for simulation, with the results shown in Figure 7.

As shown in the figure, the evolutionary system tends towards the stable point (0, 0) at the input–output ratios of 0.5 and 1, which are unacceptable for power sellers and make them reluctant to provide value-added services. When the ratio reaches 2, power sellers barely accept the ratio but gradually lose confidence in deciding to offer value-added services due to high costs and low expected returns, ultimately making the system continue to evolve towards the stable point (0, 0). However, when the ratio reaches 4, it is acceptable for power sellers and encourages them to actively employ big data to increase transaction rates, attracting large consumers into transactions. Finally, the system continues to evolve towards the stable point (1, 1).

Additionally, to analyze the impact of the big data cost coefficient k on the evolution, k was adjusted to 200, 400, and 800 for simulation, with the results shown in Figure 8 below.

Figure 8 shows that as the cost coefficient increases, the probability of power sellers providing ordinary packages and large consumers purchasing power from other sellers gradually increases, causing the evolution system to converge towards the stable point (0, 0). As the cost coefficient of big data investment increases, the profit margin obtained by power sellers decreases, making it unable to offset the costs incurred. Considering profitability, sellers will gradually reduce or even abandon their investment in big data technology, reverting to the traditional approach. Therefore, relevant departments can subsidize sellers’ big data costs, promoting digital transformation within the power industry. Currently, some studies have explored the relationship between government subsidies and the digital transformation of enterprises, focusing on increasing senior management attention, removing constraints on technological innovation, and alleviating digital resource constraints [40,41,42,43]. It can be confirmed that while the outcomes of government subsidies are diverse, they are both inevitable and essential for driving scientific and technological innovation forward. However, it is worth noting that there is a wide variety of government subsidy types, each potentially producing different effects on enterprises [42]. Although subsidies for digital transformation are increasingly recognized as a crucial form of financial support, research in this area has only recently begun to emerge. Additionally, some studies have indicated that insufficient government subsidies fail to motivate enterprises to initiate digital transformation.

6. Conclusions and Recommendations

This paper explores the interaction mechanism of strategy selection between large power consumers and power sellers based on big data user-profiling technology through evolutionary game theory. By considering the bounded rationality of power sellers and large consumers, the study emphasizes the learning and dynamic nature of the game participants and draws the following conclusions:

First, big data technology can serve as a competitive advantage for power sellers in the market. By analyzing consumers’ behavior data, transaction data, and social media data, big data technology can provide more precise customer profiles, enabling power sellers to deliver personalized services and targeted marketing. This, in turn, enhances customer satisfaction and boosts the competitiveness of enterprises in the market.

Second, value-added services based on big data technology significantly affect the decision-making behavior of large consumers. Over a long-term evolution, if the benefits provided by these value-added services to large consumers are less than the costs incurred and other sellers offer a price advantage, large consumers will not choose to transact with the power seller. However, as the benefits from value-added services increase, the probability of large consumers transacting with the power seller also rises.

Third, power sellers must consider the costs associated with investing in big data technology, with the input–output ratio serving as a crucial factor influencing their decision-making. If the costs of big data technology are too high and the expected benefits are limited, the willingness of power sellers to adopt such technology will decrease. Therefore, power sellers must conduct a comprehensive assessment to ensure a balance between costs and expected benefits, thereby providing strong support for their competitiveness in the market. Based on the above conclusions, this paper offers the following recommendations from the perspectives of economics and management:

(1) For large consumers

Large consumers should consider whether the value-added services provided by power sellers can bring sufficient benefits when making their choice. Value-added services are designed to increase the added value of power sellers, thereby attracting more large consumers and increasing transaction volume and profit. Large consumers should evaluate the value-added services based on their own needs and interests, and select sellers that best meet their needs. In addition, large consumers can negotiate with sellers for more tailored value-added services to achieve greater benefits.

(2) For power sellers

Power sellers should conduct a comprehensive benefit assessment, weighing the costs and profits of big data technology, as well as its impact on their competitive advantage. Big data technology can offer multiple benefits. However, its application requires significant investment in technology, manpower, and data collection and processing, which can create cost pressures. Power sellers should thoroughly evaluate these factors to balance costs and profits. In addition, power sellers should use big data user-profiling technology to understand the needs and behaviors of large consumers, thereby offering more tailored services and improving satisfaction.

(3) For relevant government departments

Compared with governments of many other market economies, China, as a socialist market economy, features more significant economic interventions and regulations by the government, which plays a crucial role in economic activities with greater decision-making power and influence.

For example, relevant Chinese government departments can support power sellers by formulating policies that encourage investment in big data technology, thereby enhancing their market competitiveness. The government can adopt policy instruments such as financial subsidies and tax incentives to encourage power sales enterprises to invest in and apply big data technology, thus enhancing their market competitiveness. Additionally, the government should strengthen regulations to ensure power sellers comply with relevant regulations and standards when using big data technology, safeguarding the privacy and data security of large consumers. The government can also promote cooperation and information sharing among power sellers by establishing a data-sharing mechanism to improve market efficiency and competitiveness. Furthermore, the government should foster a supportive environment for the digital economy, providing comprehensive support and guarantee for the digital transformation of power enterprises and the application of user-profiling technology. This includes encouraging cooperation between power enterprises and other sectors and deepening the integration of the real and digital economies, thereby advancing the digital economy’s development and continuously enhancing its overall benefits and social value.

This paper provides theoretical recommendations for large consumers engaging in market transactions and for power sellers to participate in market competition. However, the model in this study only considers two participants: power sellers and large consumers. As mentioned above, the government has also become a significant participant in the bilateral power market. Future research could explore how the strategies of all three participants evolve when the third party is introduced into the evolutionary game model, building on the findings of this paper.