A new energy vehicle supply chain member game based on triangular fuzzy numbers

Abstract

The competition in the new energy vehicle industry has intensified with the rapid development of the industry. In order to create innovative products, many businesses are now seeking cooperation with their supply chain members. Previous research on the new energy vehicle supply chain has mainly focused on government policies, supply chain retailers and with consumer gaming issues. This manuscript examines the problem of cooperation decisions between members of the new energy vehicle supply chain, namely a battery manufacturer and vehicle producer. The benefits of the two members are analyzed by constructing two models, one with non-incentives and the other with government incentives. The model uses the triangular fuzzy number (TFN) instead of parameters in numerical calculations, taking complete account of the influence of uncertain environmental factors and using the triangular structured element method. The numerical examples result that government incentives positively promote cooperation between the two players, but the incentives should be as equal as possible. Finally, we aim to encourage supply chain members to cooperate and promote the development of the new energy vehicle industry. This study has positive implications for future supply chain member cooperation issues.

Keywords

Energy vehicle supply chain triangular fuzzy number (TFN)nash equilibrium triangular structured element method

1 Introduction

China is currently focusing on developing a new energy vehicle industry to address environmental and energy scarcity to achieve ‘Double Carbon’ targets [1]. As competition in the industry intensifies, more and more supply chain members choose to work collaboratively to achieve mutual success. The new energy vehicle sector is also witnessing significant research on supply chain cooperation. Extensive research has been conducted on government subsidies, but less attention has been given to cooperative innovation within supply chain members. To aid in the advancement of the new energy industry, this paper delves into the topic of government incentives and cooperation among new energy members. In the subsequent sections, we offer a comprehensive overview of the current research.

Research on the strategic choices made by supply chain members for new energy vehicles has mostly focused on government policies. Liu et al. [2] constructed an evolutionary game model of new energy vehicle manufacturers and the government to study the impact of government subsidy policies on the dynamic development of the new energy vehicle industry. The effect of various factors on industry development was analyzed, and it was determined that dynamic taxation and static subsidy policies are more effective. Shi et al. [3] developed a three-way game model that involved the government, consumers, and manufacturers of new energy vehicles. They used simulation analysis to demonstrate that each party impacts the others in their strategic decisions, albeit to varying degrees. Zhang et al. [4] constructed an evolutionary game model consisting of the government, manufacturers and consumers to discuss the problem of recycling used batteries from new energy vehicles. They found that the government’s strategy of choosing between support, tiered usage and consumer participation can achieve equilibrium.

The effects of government subsidies on supply chain members are currently a popular research topic. Specifically, researchers are exploring how different subsidy policies impact the strategies of supply chain members. Liu et al. [5] analyzed the recycling of used batteries. They developed a three-party supply chain model that included the government, battery suppliers, and new energy vehicle manufacturers. The goal was to determine the best strategies for battery suppliers and vehicle manufacturers based on various government subsidy policies. Zhao et al. [6] investigated the market for new energy vehicles through government subsidies. They developed a model to distribute profits among members of a closed supply chain, comparing scenarios with and without government subsidy. They assessed the effects of various subsidy targets on the supply chain’s competitiveness. According to a study by Zhao et al. [7], a closed-loop supply chain model was developed for individual manufacturers, retailers, and third-party recyclers under various government subsidies. Jin et al. [8] investigated ways to boost the growth of new energy vehicles with minimal regulatory expenses following the withdrawal of financial incentives. They created an evolutionary game model between the government and automobile manufacturers, which led them to conclude that the government and businesses should prioritize innovative technologies. To examine the distinct inclinations of regional authorities and car makers regarding the collection techniques for carbon taxes, Liao et al. [9] developed and examined an evolutionary game model which suggested that governments should implement varying carbon tax policies and subsidies to encourage low-carbon consumption across different stages of industrial development effectively. Sun et al. [10] examined the most effective subsidy policy for various R&D stages of new energy vehicles by utilizing the Stackelberg game model and Nash equilibrium to develop a subsidy strategy. Cheng et al. [11] assessed the effects of governmental policies on the decrease in electric vehicles and formulated a game model for the supply chain involving electric vehicle (EV) producers and sellers. They also took into account other factors that could influence the decisions of both parties and determined that the policy of reducing subsidies would not significantly affect the electric vehicle markets. Dai et al. [12] examined the impact of production and consumer subsidies offered by the government on the game behavior of manufacturers and remanufacturers in the supply chain of new energy vehicles. Jafari et al. [13] examined how government subsidies can serve as incentives for manufacturers of new energy vehicles. They found that when manufacturers invest in improving the energy efficiency of their products, it stimulates more investment and demand for their vehicles, leading to increased profits for all members of the supply chain. Zhou et al. [14] categorized government subsidies as static and dynamic incentives and examined their effects on the growth of the new energy vehicle industry. Through an evolutionary game model, they found that the game is more sustainable when the government implements dynamic incentives.

Furthermore, there is a current focus on studying the strategic decisions undertaken by participants within the supply chain of new energy vehicles. Zhao et al. [15] developed a game model involving battery manufacturers, vehicle manufacturers, and third-party recyclers. They suggested a coordination contract and a power battery slope usage system to tackle the issue of used batteries. For example, Rasti-Barzoki et al. [16] studied South Korea’s sustainable development goals. They created a Stackelberg game model to calculate the most effective pricing strategy for new energy vehicle manufacturers and fuel producers. They were able to solve the problem and find the best solution. Pu et al. [17] developed a supply chain model comprising a new energy vehicle manufacturer and a retailer, which consisted of two stages. They analyzed the business strategy problem of both firms using dynamic Stackelberg and static Nash structure models. Liu et al. [18] delved into the pricing strategies of supply chain members for new energy vehicles, considering the impact of consumers’ low-carbon preferences. They proposed a game model for manufacturers and retailers in the context of dual policy implementation and concluded that vehicle pricing is positively related to the unit carbon price and subsidy amount. Sun et al. [19] utilized the Stackelberg game model to examine how carbon trading policies, electric battery range, and advertising impact the selection of recycling channels for electric batteries. They determined the most effective recycling model while considering various external environmental factors. Ma et al. [20] conducted a study on the level and yield of fuel economy improvements for both conventional internal combustion engines and new energy vehicles in the context of China’s implementation of a dual credit policy. They also explored the options and performance of a coordinated contract.

The literature reviews examine the decision-making processes of individuals within the new energy vehicle supply chain in a specific environment. However, the complexity of the environment may influence the decisions of the supply chain members, particularly in the analysis of the game model parameters. In this situation, the uncertain parameters can be approximated by the judgement and experience of the decision maker and can be expressed as fuzzy variables [21]. Zadeh proposed the concept of fuzzy sets [22], and different types of fuzzy sets [23] have been extended to different contexts. Fuzzy numbers have also been considerably extended. Dubois and Prade proposed fuzzy numbers in 1978 [24], based on which studies such as intuitionistic fuzzy numbers have been carried out, triangular fuzzy numbers and interval fuzzy numbers [25] have also been rapidly developed, triangular fuzzy numbers A = (a₁, a₂, a₃) [26] have a great advantage in representing exact numbers and are more focused and realistic.

Numerous studies have been conducted on games incorporating fuzzy numbers in new energy vehicles’ supply chain. Xu et al. [27] explored non-cooperative games that involved fuzzy payoff functions. They merged game theory with fuzzy set theory to devise a fuzzy Nash equilibrium for these types of games. Meng et al. [28] presented an extended version of fuzzy countermeasures. They based it on the restricted difference of fuzzy numbers and analyzed its axiomatic system. As a result, they could provide an explicit expression for the Shapley value. Zhang et al. [29] introduced a comprehensive relationship for fuzzy numbers, utilizing the expectation value of fuzzy numbers. They also demonstrated the existence of at least one fuzzy payment vector in the undifferentiated fuzzy kernel. Fuzzy games can be a valuable tool in addressing various supply chain issues. Liu et al. [30] analyzed a pricing problem in a supply chain involving a manufacturer and two rival retailers. They incorporated fuzzy variables representing manufacturing expenses and product demand and formulated an expectation value model. Sang [31] analyzed the price competition between two manufacturers in the fuzzy environment, substituted fuzzy numbers for the demand function and manufacturing cost, and found the optimal pricing strategy. Lou et al. [32] identified the limitations of traditional supply chain management. They put forth a novel method of supply chain management that leverages fuzzy big data and large-scale group decision-making. A sustainable multi-channel supply chain model was created by Mahmoudi [33] using an intuitive fuzzy game approach to address uncertain business environments. The model also focused on designing a sustainable supply chain with government and third-party logistics involvement. In an actual e-commerce model, Zhou et al. [34] employed a game-theoretic approach to examine the pricing strategy and coordination of interests among participants in a closed-loop supply chain.

The fuzzy analysis involves manipulating fuzzy numbers and function values by examining all elements within a state. In practical operations, it can be highly inconvenient. To address this issue, Guo [35] introduced the concept of fuzzy structure elements. This has effectively resolved the problem of analytical representation of fuzzy numbers and fuzzy value function operations. As a result, it provides a valuable tool for simplifying fuzzy analysis calculations. Zhang et al. [36] investigated coalition cooperation policies that utilized fuzzy participation rates in a fuzzy structural meta-representation model. Zhou et al. [37] employed the structural element method to resolve the Bayesian Nash equilibrium of fuzzy returns. This approach simplified the solution process of the initial problem.

Based on the preceding analysis, this paper analyzes the collaboration strategies of supply chain members in a fuzzy environment. The innovation points can be summarized as follows:

(1) Construct a game model of cooperation among supply chain members. Current research on the new energy vehicle supply chain focuses primarily on the government’s role in the game between supply chain members. While the government has implemented various policies to promote industry growth, the literature primarily investigates the policies that have been implemented. Some studies examine how government actions impact variables such as pricing strategy and profit analysis among supply chain members. This paper, however, explores the strategic decisions made by battery and car manufacturers as supply chain members, including the decision to collaborate on research and development.

(2) Utilize a static Nash equilibrium method. In previous supply chain studies, dynamic game models were utilized to analyze participant behavior. These models considered sequential actions, with followers basing their decisions on the leader’s choices. However, the order of decisions could impact the final results. This paper takes a different approach, using a static Nash equilibrium method. Participants make decisions simultaneously, considering the optimal strategies of others involved in the chain.

(3) Consider the impact of the fuzzy environment on the analysis process. To achieve accurate results in games, parametric analysis is essential. However, existing studies have often relied on expert experience and judgment, neglecting the effects of environmental uncertainties. While some studies have treated game analysis in a fuzzy environment as a pricing problem, none have explored the issue of cooperative strategies among supply chain members. This paper introduces triangular fuzzy numbers to capture the uncertain environment. Rather than determining parameters through numerical analysis, we fully consider the potential impact of environmental uncertainties on the outcomes.

(4) Use the triangular structured element method for the solution. Numerous research studies have been conducted on the precise resolution of fuzzy numbers, as indicated by references 42 and 43. However, current techniques face computational challenges and lack comprehensive discriminative ability. Although some methods are simple to compute, they result in excessive loss of information. To address this issue, this paper introduces the triangular fuzzy element method to handle the triangular fuzzy numbers in the model. This approach is combined with a rational human decision criterion to simplify the fuzzy operation and accurately express the original information, thus ensuring the model’s rationality.

The rest of the paper is organized as follows. In Section 2, we review the mathematical concepts and basic principles. In section 3, we build a game model in a fuzzy environment. In section 4, we perform operations given specific values. In section 5, we conclude and identify further improvements that need to be made.

2 Preliminaries

2.1 The definition of TFN

Definition 1. [24] A real fuzzy number A is more precisely described as any fuzzy subset of the R (real line), whose membership function is a continuous mapping from R to the closed interval [0,1].

Definition 2. [26] Assume the triangular fuzzy number (TFN) A is defined as a fuzzy set on R (real line), whose membership function is: $μ_{A} = {\begin{matrix} (x - a_{1}) / - (a_{2} - a_{1}), a_{1} < x ⩽ a_{2} \\ (a_{3} - x) / - (a_{3} - a_{2}), a_{2} < x ⩽ a_{3} \\ 0, otherwise \end{matrix}$

With a₁ ⩽ a₂ ⩽ a₃, the TFN is defined as A = (a₁, a₂, a₃).

Definition 3. [26] Let A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃) be two triangular fuzzy numbers (TFNs), then the operations on the TFNs are defined as:

Addition: A ⊕ B = (a₁ + b₁a₂ + b₂a₃ + b₃).

Multiplication: A ⊗ B = (a₁b₁a₂b₂a₃b₃).

Division: A/ B = (a₁/ b₃, a₂/ b₂, a₃/ b₁).

Number multiplication: λ · A = λa₁, λa₂, λa₃.

Reciprocal: (A) ^-1 = (1/a₃, 1/a₂, 1/a₁).

Definition 4. [38] Let A = (a₁, a₂, a₃) and B = (b₁, b₂, b₃) be two TFNs, possible degrees of a A > B is: $\begin{matrix} V (A > B) = \\ {\begin{matrix} 0, a_{3} ⩽ b_{1} \\ (a_{3} - b_{1}) / - [(a_{3} - b_{1}) + (b_{2} - a_{2})], a_{3} > b_{1} and a_{2} < b_{2} \\ 1, a_{2} ⩾ b_{2} \end{matrix} \end{matrix}$

Definition 5. [39] For a TFN, the optimistic and pessimistic values of A = (a₁, a₂, a₃) are expressed as: ${\begin{matrix} ɛ_{α}^{L} = a_{2} α + a_{1} (1 - α) \\ ɛ_{α}^{U} = a_{2} α + a_{3} (1 - α) \end{matrix}$

The finite expected value of A is expressed as: $E (ɛ) = \frac{1}{2} \int_{0}^{1} ɛ_{α}^{L} + ɛ_{α}^{U} d α = \frac{a_{1} + 2 a_{2} + a_{3}}{4}$

2.2 TFN-based equilibrium strategy solution for the game

The TFN-based game model is denoted as [40]: $G = {N_{1}, N_{2}; S_{1}, S_{2}; P_{ij}}$

Where N₁, N₂ are the two players involved in the game, S₁ = { α₁, α₂, . . . , α_m } , S₂ = { β₁, β₂, . . . , β_n } are the set of strategies of N₁, N₂, and P_ij is the triangular fuzzy benefit function of the strategy j taken by the game player i. The Nash equilibrium existence theorem states that every finite game has at least one Nash equilibrium. If an optimal solution does not exist for pure strategy, a mixed strategy optimal solution becomes necessary. For a two-by-two game, the mixed strategy of the player N₁ is σ₁ = (x, 1 - x), the player N₁ chooses strategy α₁ with probability (x) and strategy α₂ with probability (1 - x). The mixed strategy of the player N₂ is σ₂ = (y, 1 - y), the player N₂ chooses strategy β₁ with probability (y) and strategy β₂ with probability (1 - y). The game payment matrix based on TFN is shown in Table 1.

Table 1
Gaming payment matrix

N ₂

Strategy β₁(y) β₂(1 - y)

N ₁ α₁(x) (A₁₁, B₁₁) (A₁₂, B₁₂)

α₂(1 - x) (A₂₁, B₂₁) (A₂₂, B₂₂)

		N ₂
	Strategy	β₁(y)	β₂(1 - y)
N ₁	α₁(x)	(A₁₁, B₁₁)	(A₁₂, B₁₂)
	α₂(1 - x)	(A₂₁, B₂₁)	(A₂₂, B₂₂)

There is a unique optimal mixed strategy solution to the game of players N₁, N₂, as follows from Nash theorem. ${\begin{matrix} x^{*} = \frac{B_{22} - B_{21}}{B_{11} - B_{12} - B_{21} + B_{22}} \\ y^{*} = \frac{A_{22} - A_{12}}{A_{11} - A_{12} - A_{21} + A_{22}} \end{matrix}$

2.3 Principle of the triangular structure element method

Definition 6. Let E be a fuzzy set over a real number field R and the affiliation function be denoted E (x), x ∈ R. If E (x) satisfies the following properties:

E (0) =1.

When x ∈ [-1, 0), E (x) is a continuous monotonically increasing function, when x ∈ [0, 1), E (x) is a continuously monotonically decreasing function.

When x ∈ (- ∞ , -1) or x ∈ (1, + ∞), E (x) =0, thus it is called a fuzzy structure element on R. When E (x) satisfies the following equation, the fuzzy set is a fuzzy triangular structured element: $E (x) = {\begin{matrix} 1 + x, x \in [- 1, 0) \\ 1 - x, x \in [0, 1) \\ 0, otherwise \end{matrix}$

Definition 7. [41] For a given fuzzy triangular structured element and any fuzzy number A, there always exists a function f (x) such that A = f (E), where f (x) is a monotonically bounded function on the interval [–1,1], and the affiliation function of the fuzzy number A is E [f^-1 (x)], f^-1 (x) is a rotationally symmetric function of f (x) with respect to the variable x, y. For a triangular fuzzy number A = (a₁, a₂, a₃), the corresponding monotone bounded function is: $f (x) = {\begin{matrix} (a_{2} - a_{1}) x + a_{2} \\ (a_{3} - a_{2}) x + a_{2} \end{matrix}$

Definition 8. [35] For a fuzzy monotone matrix $A_{m \times n} = (a_{ij})_{m \times n}, if A_{m \times n}^{-} = (a_{ij}^{-})_{m \times n}, \bar{a_{ij}} = \frac{1}{2} \int_{- 1}^{1} f_{ij} (x) (1 - x) dx$ , then $A_{m \times n}^{-}$ is said to be the projection matrix of A_m×n.

3 Optimal strategy analysis of a fuzzy game considering government incentives

3.1 Model construction and basic assumptions

3.1.1 Problem description

With the new energy vehicle market expanding quickly, product development cycles have become shorter, and competition has intensified. This has prompted supply chain members to seek cooperative research and development opportunities to reduce costs, shorten cycle times, and enhance competitiveness. Research on the new energy vehicle supply chain mainly focuses on pricing, profitability, and recovery strategies. However, there is a lack of research on collaboration strategies among supply chain members. The new energy industry also faces funding challenges, and government subsidies primarily incentivize sales and buyers, potentially affecting the R&D efforts of supply chain members. To summarize, this paper presents a game model for cooperative R&D within the new energy vehicle supply chain. The model involves upstream battery manufacturers and downstream vehicle manufacturers as participants, and considers their decision-making process in a fuzzy environment. Furthermore, a separate analysis is conducted on how government incentives affect the decision-making process of each party, with the corresponding model illustrated in Fig. 1.

Fig. 1

Supply chain members game model.

3.1.2 Basic assumptions

In this manuscript, the cooperation among supply chain participants is impacted by various elements such as government policies, consumer demands, and market conditions. For the purpose of our study, we have identified two key players: the producer of new energy vehicle batteries and the manufacturer of new energy vehicles. Without changing the essence of the problem, we simplify some of the more complex conditions and propose the following assumptions:

Hypothesis 1: The new energy vehicle industry studies supply chain members as a group. In this paper, we identify a battery manufacturer (N₁) and a vehicle producer (N₂) as the supply chain members for the study and investigate whether the two players cooperate. The analysis of Nash equilibrium involves examining the benefits, inputs, and penalties of cooperation between two players to determine the optimal outcome for each player. The goal for both players is to maximize their interests.

Hypothesis 2: In this study, we are operating under the assumption that there are only mixed Nash equilibria between the two players. This ensures that the resulting outcomes meet the following conditions. ${\begin{matrix} A_{11} - A_{12} - A_{21} + A_{22} \neq 0 \\ B_{11} - B_{12} - B_{21} + B_{22} \neq 0 \\ (B_{22} - B_{21}) \cdot (B_{11} - B_{12} - B_{21} + B_{22}) > 0 \\ (A_{22} - A_{12}) \cdot (A_{11} - A_{12} - A_{21} + A_{22}) > 0 \end{matrix}$

Due to the lack of complete information, it is difficult for both the battery manufacturer and the vehicle producer to accurately assess their respective benefits. N₁ can choose the strategy of cooperation α₁ with probability (x) and non-cooperation α₂ with probability (1 - x). Similarly, N₂ can choose the strategy of cooperation β₁ with probability (y) and non-cooperation β₂ with probability (1 - y).

Hypothesis 3: In a game scenario, if neither player selects cooperation, they make a standard profit. However, if they both agree to cooperate, they must have resources. The battery manufacturer is accountable for developing batteries, and the vehicle producer handles the assembly of vehicles. Additionally, the partnership has resulted in some revenue growth for both players. Whether or not to cooperate is influenced by the cost of collaboration, the severity of the penalty, and the rate of benefits increase.

Hypothesis 4: The ‘free-rider’ phenomenon occurs when one player chooses to cooperate while the other chooses not. Furthermore, the non-cooperative player will be penalized to compensate for the loss incurred by the other player.

Hypothesis 5: This paper also investigates if the government is part of the partnership. If both parties decide to collaborate, the government offers incentives such as tax breaks and loans. However, if they do not cooperate, the government does not intervene.

The strategy combinations for a game between battery manufacturer (N₁) and vehicle producer (N₂) are shown in Table 2.

Table 2
Gaming strategy portfolio

N ₂

Strategy Cooperation (y) Non-cooperation (1 - y)

N ₁ Cooperation (x) (Cooperation, Cooperation) (Cooperation, Non-cooperation)

Non-cooperation (1 - x) (Non-cooperation, Cooperation) (Non-cooperation, Non-cooperation)

		N ₂
	Strategy	Cooperation (y)	Non-cooperation (1 - y)
N ₁	Cooperation (x)	(Cooperation, Cooperation)	(Cooperation, Non-cooperation)
	Non-cooperation (1 - x)	(Non-cooperation, Cooperation)	(Non-cooperation, Non-cooperation)

Based on the above assumptions and descriptions, the symbols used in this paper are summarized in Table 3.

Table 3

Parameter descriptions

Parameters	Description	Limit
π_i (i = 1, 2)	The normal profit of N₁, N₂	π_i > 0
C_i (i = 1, 2)	Input costs for two players’ cooperation	C_i > 0
λ_i (i = 1, 2)	Rate of revenue growth from cooperation	λ_i > 0
I_i (i = 1, 2)	Probability of both sides choosing to ‘free-ride’	I_i > 0
P	Punishment for the ‘free-rider’
r_i (i = 1, 2)	Government incentives for cooperation between the two players	r_i > 0

3.2 The Non-incentive game model

When taking into account non-incentives, the advantages for both players consist of regular profit, extra incremental rates, and input costs, as per the aforementioned assumptions. When both players choose to cooperate, they can both benefit from increased revenue and reduced input costs. In the event that both players do not cooperate, their profits will remain regular. However, under certain circumstances, free-riding and penalties may be imposed. The game payment matrix for both players is shown in Table 4.

Table 4
Non-incentive payment matrix

N ₂

Strategy Cooperation (y) Non-cooperation (1 - y)

N ₁ Cooperation (x) π₁ (1 + λ₁) - C₁, π₂ (1 + λ₂) - C₂ π₁ - C₁ + P, π₂ (1 + I₂) - P

Non-cooperation (1 - x) π₁ (1 + I₁) - P, π₂ - C₂ + P π₁, π₂

		N ₂
	Strategy	Cooperation (y)	Non-cooperation (1 - y)
N ₁	Cooperation (x)	π₁ (1 + λ₁) - C₁, π₂ (1 + λ₂) - C₂	π₁ - C₁ + P, π₂ (1 + I₂) - P
	Non-cooperation (1 - x)	π₁ (1 + I₁) - P, π₂ - C₂ + P	π₁, π₂

According to Table 4, the expected utility functions for the battery manufacturer (N₁) and the vehicle producer (N₂) can be obtained as follows. $\begin{matrix} E_{1} (x, y) = xy [π_{1} (1 + λ_{1}) - C_{1}] + x \cdot (1 - y) \\ \cdot (π_{1} - C_{1} + P) + y \cdot (1 - x) \cdot [π_{1} (1 + I_{1}) - P] \\ + (1 - x) \cdot (1 - y) \cdot π_{1} \end{matrix}$ $\begin{matrix} E_{2} (x, y) = xy [π_{2} (1 + λ_{2}) - C_{2}] + y \cdot (1 - x) \cdot \\ (π_{2} - C_{2} + P) + x \cdot (1 - y) \cdot [π_{2} (1 + I_{2}) - P] \\ + (1 - x) \cdot (1 - y) \cdot π_{2} \end{matrix}$

The probability of the optimal mixed strategy is obtained by applying the calculus to the above expected utility functions and solving for the extrema, respectively: $\frac{\partial E_{1} (x, y)}{\partial x} = y \cdot π_{1} \cdot (λ_{1} - I_{1}) + P - C_{1}$ $\frac{\partial E_{2} (x, y)}{\partial y} = x \cdot π_{2} \cdot (λ_{2} - I_{2}) + P - C_{2}$ ${\begin{matrix} x^{*} = \frac{C_{2} - P}{(λ_{2} - I_{2}) π_{2}} \\ y^{*} = \frac{C_{1} - P}{(λ_{1} - I_{1}) π_{1}} \end{matrix}$

According to the conditions for mixed Nash equilibrium, the above result should satisfy the following conditions. The probability of the battery manufacturer (N₁) and the vehicle producer (N₂) choose to cooperate is influenced by the other player at the same time, λ₁ > I₁, λ₂ > I₂ hold at the same time if C₁, C₂ > P, it means that when the cost C₁, C₂ of N₁, N₂ choosing to cooperate is greater than the penalty P for free-riding, then the additional return to N₁, N₂ is greater than the corresponding probability of free-riding. Similarly, if C₁, C₂ < P, the above applies accordingly. ${\begin{matrix} λ_{1} \neq I_{1} \\ λ_{2} \neq I_{2} \\ (λ_{1} - I_{1}) (C_{1} - P) > 0 \\ (λ_{2} - I_{2}) (C_{2} - P) > 0 \end{matrix}$

The factors influencing the choice of cooperation between the two players are analyzed separately below.

(1) Analysis of factors affecting x^*.

x^* is an increasing function concerning C₂ and I₂. If N₂ chooses to cooperate with an increase in input costs, N₁ is more likely to choose to cooperate. In other words, if N₁ knows N₂’s input cost (C₂), the size of C₂ determines N₁’s strategy: a larger C₂ indicates that it is more likely to engage in resource sharing, and therefore N₁ is also more likely to choose to cooperate. If the probability of N₂ selecting to free-ride (I₂) is higher, N₁ is more inclined to choose cooperation to achieve win-win cooperation.

x^* is a decreasing function concerning λ₂ and P. N₂’s revenue growth rate λ₂ influences N₁’s decision, and if N₁ knows that λ₂ is larger, N₁’s probability of choosing cooperation becomes smaller. Looking at this factor alone, when both players choose the cooperation strategy, the probability of choosing cooperation becomes smaller when N₁ learns that N₂’s larger revenue growth will result in a smaller payoff for itself and a disadvantage in the two-party game. P is another influencing factor that constrains N₁’s choice of cooperation, and the probability of choosing cooperation becomes smaller when P is larger. If N₁ learns that the penalty P of free-riding is greater, the probability of choosing not to cooperate is greater.

(2) Analysis of factors affecting y^*.

In the same way as x^*, y^* also influenced by C₁, I₁, λ₁ and P. y^* is an increasing function concerning C₁ and I₁. If N₁ chooses to cooperate with an increase in input costs, N₂ is more likely to choose to cooperate. Similarly, when the probability I₁ of N₁ wanting to free-ride is higher, N₂ is more inclined to choose cooperation.

y^* is a decreasing function concerning λ₁ and P. The revenue growth (λ₁) of N₁ will also affect the decision of N₂ and will have the same outcome as N₁. The constraint effect of P on N₁, N₂ is the same, the larger P is, the more likely N₂ will make a conservative decision.

In summary, the value of x^* and y^* is influenced by a combination of facilitating and constraining factors. Both are influenced by each other’s decision parameters, so the values of optimal decision probabilities are mutually constrained.

3.3 The government incentive game model

When government incentives are considered, the government will consider additional incentives r₁, r₂ for N₁, N₂ respectively when both players choose the cooperation strategy, which will eventually affect the probability of both parties choosing to cooperate, and the game payment matrix is shown in Table 5.

Table 5
Incentive payment matrix

N ₂

Strategy Cooperation (y) Non-cooperation (1 - y)

N ₁ Cooperation (x) π₁ (1 + λ₁) - C₁ + r₁, π₂ (1 + λ₂) - C₂ + r₂ π₁ - C₁ + P, π₂ (1 + I₂) - P

Non-cooperation (1 - x) π₁ (1 + I₁) - P, π₂ - C₂ + P π₁, π₂

		N ₂
	Strategy	Cooperation (y)	Non-cooperation (1 - y)
N ₁	Cooperation (x)	π₁ (1 + λ₁) - C₁ + r₁, π₂ (1 + λ₂) - C₂ + r₂	π₁ - C₁ + P, π₂ (1 + I₂) - P
	Non-cooperation (1 - x)	π₁ (1 + I₁) - P, π₂ - C₂ + P	π₁, π₂

According to Table 4, the expected utility functions for the battery manufacturer (N₁) and the vehicle producer (N₂) can be obtained as follows: $\begin{matrix} E_{1}^{,} (x, y) = xy [π_{1} (1 + λ_{1}) - C_{1} + r_{1}] + x \cdot \\ (1 - y) \cdot (π_{1} - C_{1} + P) + y \cdot (1 - x) [π_{1} (1 + I_{1}) \\ - P] + (1 - x) \cdot (1 - y) \cdot π_{1} \end{matrix}$ $\begin{matrix} E_{2}^{,} (x, y) = xy [π_{2} (1 + λ_{2}) - C_{2} + r_{2}] + y \cdot \\ (1 - x) \cdot (π_{2} - C_{2} + P) + x \cdot (1 - y) \cdot [π_{2} (1 + I_{2}) \\ - P] + (1 - x) \cdot (1 - y) \cdot π_{2} \end{matrix}$

The probability of the mixed optimal strategy is obtained by applying the calculus to the above expected utility functions and solving for the extrema, respectively: $\frac{\partial E_{1}^{,} (x, y)}{\partial x} = y [(λ_{1} - I_{1}) π_{1} + r_{1}] + P - C_{1}$ $\frac{\partial E_{2}^{,} (x, y)}{\partial y} = x [(λ_{2} - I_{2}) π_{2} + r_{2}] + P - C_{2}$ ${\begin{matrix} x^{, *} = \frac{C_{2} - P}{(λ_{2} - I_{2}) π_{2} + r_{2}} \\ y^{, *} = \frac{C_{1} - P}{(λ_{1} - I_{1}) π_{1} + r_{1}} \end{matrix}$

Compared to the anarchic incentive game model, the results of the optimal solution of the mixed Nash equilibrium are also affected by government incentives. The above result should satisfy the following conditions. ${\begin{matrix} π_{1} (λ_{1} - I_{1}) + r_{1} \neq 0 \\ π_{2} (λ_{2} - I_{2}) + r_{2} \neq 0 \\ (C_{1} - P) \cdot [π_{1} (λ_{1} - I_{1}) + r_{1}] > 0 \\ (C_{2} - P) \cdot [π_{2} (λ_{2} - I_{2}) + r_{2}] > 0 \end{matrix}$

The probability of N₁ and N₂ choosing to cooperate when government incentives are present is influenced by another player and government incentives. Then we analyze x^,* and y^,*.

(1) Analysis of factors affecting x^,*.

As with the analysis of x^* in 3.2, x^,* is an increasing function of C₂ and I₂. The probability that N₁ chooses to cooperate is greater if the input cost (C₂) of N₂ and the probability that N₂ chooses to free-ride I₂ are greater. The probability that N₁ ultimately decides to cooperate is determined by the parameters of N₂.

In contrast to 3.2, x^,* is a decreasing function concerning λ₂, r₂ and P. If the government incentive for N₂ is large, the probability that N₁ will choose to cooperate is smaller. It means that N₁ may perceive itself as disadvantaged when it learns that the government incentive for both players to cooperate is large, therefore the smaller the strategy of choosing to cooperate.

(2) Analysis of factors affecting y^,*.

Similar to the analysis of x^,*, y^,* is an increasing function of C₁ and I₁, and a decreasing function of λ₁, r₁ and P. If the probability that N₁ chooses the cost of cooperation and free-riding is greater, the probability that N₂ chooses to cooperate is greater. The probability that N₂ chooses to cooperate becomes smaller when N₁ chooses a higher return to cooperation, government incentives and penalties for free-riding.

In summary, the probabilities that affect N₁, N₂’s decision to cooperate are all influenced by the parameters of another player’s decision. In addition to the influence of their factors, the degree of punishment and incentive must be captured if the players ultimately cooperate.

3.4 Mixed strategy equilibrium analysis

Our desired result is that both players achieve cooperation. Based on the analysis of x^* and y^* in 3.2 and 3.3, we find that the outcome of the decisions made by both players will be influenced by another player. In the non-incentive model, the desire for both players to adopt a cooperative strategy requires a reduction in the penalty for free-riding. If the difference between the revenue growth rate (λ_i) and the probability of free-riding (I_i) is not large, the fact that the benefits of both players choosing to cooperate are greater than non-cooperating will encourage players to tend to cooperate. In the government incentive model, in addition to the above influences, there are government incentives, and the probability of one player cooperating is also influenced by another player’s reward, so the government should reward both players equally. The results of the payment matrix showed that when the two players choose to cooperate, the benefits after the government incentive are greater than the benefits without the government incentive.

In the solution process, we use triangular fuzzy numbers (TFNs) instead of exact numbers to account for the comprehensiveness of the information involved. The payment matrix represented by the TFNs is obtained based on the operation. In the next step, we use the triangular structured element method to solve and transform the original information into crisp numbers. Finally, we can obtain the strategy for both players.

Then we perform a numerical simulation and analysis to obtain the probability of both players choosing the cooperation strategy under different parameters.

4 Numerical examples

In this section, we perform a numerical analysis of the new energy vehicle members making cooperative decisions. In 4.1 and 4.2, we perform numerical calculations for the non-incentive and government incentive models. The difference between the two models is the incentive, so 4.3 we base our estimates on the original data from 4.2 without changing it. Through the numerical analysis, the introduction of incentives by the government can facilitate the decision of both players to cooperate. However, if one player knows that another’s incentive is greater than its own, the probability of choosing to cooperate becomes smaller.

4.1 The non-incentive calculation

Based on the above analysis, the fuzzy values for each parameter are assumed to be as shown in Table 6.

Table 6
Non-incentive parameter fuzzy values

Parameters Fuzzy values Parameters Fuzzy values

C ₁ (6, 7, 9) C ₂ (6, 8, 9)

π₁ (8, 9, 19/2) π₂ (7, 8, 9)

λ₁ (9/20, 1/2, 7/10) λ₂ (1/2, 11/20, 7/10)

I ₁ (3/10, 7/20, 2/5) I ₂ (1/5, 3/10, 7/20)

P (4, 5, 11/2)

Parameters	Fuzzy values	Parameters	Fuzzy values
C ₁	(6, 7, 9)	C ₂	(6, 8, 9)
π₁	(8, 9, 19/2)	π₂	(7, 8, 9)
λ₁	(9/20, 1/2, 7/10)	λ₂	(1/2, 11/20, 7/10)
I ₁	(3/10, 7/20, 2/5)	I ₂	(1/5, 3/10, 7/20)
P	(4, 5, 11/2)

Step 1: Calculate the non-incentive payment matrix. According to the rules of triangular fuzzy number operation, the payment matrix of the game between the battery manufacturer (N₁) and the vehicle producer (N₂) in the absence of government incentives is obtained, as shown in Table 7.

Table 7

Non-incentive payment value

		N ₂
	Strategy	Cooperation (y)	Non-cooperation (1 - y)
N ₁	Cooperation (x)	(13/5, 13/2, 203/20), (3/2, 22/5, 93/10)	(3, 7, 9) (29/10, 27/5, 163/20)
	Non-cooperation (1 - x)	(49/10, 143/20, 93/10) (2, 5, 17/2)	(8, 9, 19/2) (7, 8, 9)

We can translate the above matrix into the following equation. $A_{2 \times 2} = [\begin{matrix} (\frac{13}{5}, \frac{13}{2}, \frac{203}{20}) & (3, 7, 9) \\ (\frac{49}{10}, \frac{143}{20}, \frac{93}{10}) & (8, 9, \frac{19}{2}) \end{matrix}]$ $B_{2 \times 2} = [\begin{matrix} (\frac{3}{2}, \frac{22}{5}, \frac{93}{10}) & (\frac{29}{10}, \frac{27}{5}, \frac{163}{20}) \\ (2, 5, \frac{17}{2}) & (7, 8, 9) \end{matrix}]$

Step 2: Generate linear triangular structured elements. According to the definition of triangular fuzzy elements, the elements in the fuzzy matrix can be linearly composed of triangular structured elements.

A_2×2 = (f_ij (E)) _2×2, B_2×2 = (g_ij (E)) _2×2, and f_ij, g_ij is a homogeneous monotonic function, we obtain f_ij (x) , g_ij (y) as: $f_{11} (x) = {\begin{matrix} \frac{39}{10} x + \frac{13}{2}, x \in [- 1, 0) \\ \frac{73}{20} x + \frac{13}{2}, x \in [0, 1] \end{matrix} f_{12} (x) = {\begin{matrix} 4 x + 7, x \in [- 1, 0) \\ 2 x + 7, x \in [0, 1] \end{matrix}$ $f_{21} (x) = {\begin{matrix} \frac{9}{4} x + \frac{143}{20}, x \in [- 1, 0) \\ \frac{43}{20} x + \frac{143}{20}, x \in [0, 1] \end{matrix} f_{22} (x) = {\begin{matrix} x + 9, x \in [- 1, 0) \\ \frac{1}{2} x + 9, x \in [0, 1] \end{matrix}$ $g_{11} (y) = {\begin{matrix} \frac{29}{10} y + \frac{22}{5}, y \in [- 1, 0) \\ \frac{49}{10} y + \frac{22}{5}, y \in [0, 1] \end{matrix} g_{12} (y) = {\begin{matrix} \frac{5}{2} y + \frac{27}{5}, y \in [- 1, 0) \\ \frac{11}{4} y + \frac{27}{5}, y \in [0, 1] \end{matrix}$ $g_{21} (y) = {\begin{matrix} 3 y + 5, y \in [- 1, 0) \\ \frac{7}{2} y + 5, y \in [0, 1] \end{matrix} g_{22} (y) = {\begin{matrix} y + 8, y \in [- 1, 0) \\ y + 8, y \in [0, 1] \end{matrix}$

Step 3: Generate the projection matrix. According to Definition 8, we can obtain the projection matrix ${\bar{A}}_{2 \times 2}, {\bar{B}}_{2 \times 2}$ of A_2×2, B_2×2. ${\bar{A}}_{2 \times 2} = [\begin{matrix} \frac{1243}{240} & \frac{65}{12} \\ \frac{767}{120} & \frac{69}{8} \end{matrix}] {\bar{B}}_{2 \times 2} [\begin{matrix} \frac{18}{5} & \frac{367}{80} \\ \frac{97}{24} & \frac{23}{3} \end{matrix}]$

Step 4: Obtain the expected utility functions of N₁ and N₂. $\begin{matrix} E_{1} (x, y) = (x, 1 - x) [\begin{matrix} \frac{1243}{240} & \frac{65}{12} \\ \frac{767}{120} & \frac{69}{8} \end{matrix}] (\begin{matrix} y \\ 1 - y \end{matrix}) \\ = \frac{479}{240} xy - \frac{37}{24} x - \frac{67}{30} y + \frac{69}{8} \end{matrix}$ $\begin{matrix} E_{2} (x, y) = (y, 1 - y) [\begin{matrix} \frac{18}{5} & \frac{367}{80} \\ \frac{97}{24} & \frac{23}{3} \end{matrix}] (\begin{matrix} x \\ 1 - x \end{matrix}) \\ = \frac{211}{80} xy - \frac{87}{24} x - \frac{539}{240} y + \frac{23}{3} \end{matrix}$

Step 5: Calculate the optimal mixing strategy. A first order derivative of E₁ (x, y) and E₂ (x, y) respectively gives: $\frac{\partial E_{1} (xy)}{\partial x} = \frac{\partial (\frac{479}{240} xy - \frac{37}{24} x - \frac{67}{30} y + \frac{69}{8})}{\partial x} = \frac{479}{240} y - \frac{37}{24} = 0$ $\frac{\partial E_{2} (xy)}{\partial y} = \frac{\partial (\frac{211}{80} xy - \frac{87}{24} x - \frac{539}{240} y + \frac{23}{3})}{\partial y} = \frac{211}{80} x - \frac{539}{240} = 0$

Further solving for this results in the following: $x^{*} = \frac{539}{633} \approx 0.85, y^{*} = \frac{370}{479} \approx 0, 77$

Therefore, the optimal strategy for N₁ and N₂ is: ${\begin{matrix} (x, 1 - x) = (0 . 85, 0 . 15) \\ (y, 1 - y) = (0 . 77, 0 . 23) \end{matrix}$

Step 6: Calculate the fuzzy payoffs. Based on the optimal strategy solution, the fuzzy payoffs for N₁ and N₂ are found to be: $\begin{matrix} V_{1} = (0.85, 0.15) \cdot [\begin{matrix} (\frac{13}{5}, \frac{13}{2}, \frac{203}{20}) & (3, 7, 9) \\ (\frac{49}{10}, \frac{143}{20}, \frac{93}{10}) & (8, 9, \frac{19}{2}) \end{matrix}] \cdot (\begin{matrix} 0.77 \\ 0.23 \end{matrix}) \\ = (3, 6.7, 9.7) \end{matrix}$ $\begin{matrix} V_{2} = (0.77, 0.23) \cdot [\begin{matrix} (\frac{3}{2}, \frac{22}{5}, \frac{93}{10}) & (\frac{29}{10}, \frac{27}{5}, \frac{163}{20}) \\ (2, 5, \frac{17}{2}) & (7, 8, 9) \end{matrix}] \cdot (\begin{matrix} 0.85 \\ 0.15 \end{matrix}) \\ = (1.9, 4.8, 9) \end{matrix}$

Through the above series of solutions, we find that the probability of N₁ and N₂ choosing to cooperate in the non-incentive case is x^* = 0.85, y^* = 0, 77 and the final payoff for each side is V₁ = (3, 6.7, 9.7) and V₂ = (1.9, 4.8, 9). The value of the payoff for N₁ is greater than that for N₂. Next, we discuss the model in the incentive case, assuming in 4.2 that the incentive for N₂ is greater than that for N₁.

4.2 The government incentive calculation

When both players choose the cooperative strategy, the government will consider an additional incentive r₁, r₂ for N₁, N₂ separately, assuming r₂ > r₁, the fuzzy values for each parameter as shown in Table 8:

Table 8
Incentive parameter fuzzy values

Parameters Fuzzy values Parameters Fuzzy values

C ₁ (6, 7, 9) C ₂ (6, 8, 9)

π₁ (8, 9, 19/2) π₂ (7, 8, 9)

λ₁ (9/20, 1/2, 7/10) λ₂ (1/2, 11/20, 7/10)

I ₁ (3/10, 7/20, 2/5) I ₂ (1/5, 3/10, 7/20)

r ₁ (5/2, 3, 4) r ₂ (2, 7/2, 9/2)

P (4, 5, 11/2)

Parameters	Fuzzy values	Parameters	Fuzzy values
C ₁	(6, 7, 9)	C ₂	(6, 8, 9)
π₁	(8, 9, 19/2)	π₂	(7, 8, 9)
λ₁	(9/20, 1/2, 7/10)	λ₂	(1/2, 11/20, 7/10)
I ₁	(3/10, 7/20, 2/5)	I ₂	(1/5, 3/10, 7/20)
r ₁	(5/2, 3, 4)	r ₂	(2, 7/2, 9/2)
P	(4, 5, 11/2)

Step 1: Calculate the incentive payment matrix. According to the triangular fuzzy number operation rules, the game payment matrix of N₁ and N₂ at the time of government incentive is obtained as shown in Table 9.

Table 9

Incentive payment value

		N ₂
	Strategy	Cooperation (y)	Non-cooperation (1 - y)
N ₁	Cooperation (x)	(51/10, 19/2, 283/20), (7/2, 79/10, 138/10)	(3, 7, 9) (29/10, 27/5, 163/20)
	Non-cooperation (1 - x)	(49/10, 143/20, 93/10) (2, 5, 17/2)	(8, 9, 19/2) (7, 8, 9)

Step 2: Generate linear triangular structured elements. $A_{2 \times 2}^{,} = (f_{ij}^{,} (E))_{2 \times 2}, B_{2 \times 2}^{,} = (g_{ij}^{,} (E))_{2 \times 2}$ , and $f_{ij}^{,}, g_{ij}^{,}$ is a homogeneous monotonic function, we obtain $f_{ij}^{,} (x), g_{ij}^{,} (y)$ as: $f_{11}^{,} (x) = {\begin{matrix} \frac{22}{5} x + \frac{19}{2}, x \in [- 1, 0) \\ \frac{93}{20} x + \frac{19}{2}, x \in [0, 1] \end{matrix} f_{12}^{,} (x) = {\begin{matrix} 4 x + 7, x \in [- 1, 0) \\ 2 x + 7, x \in [0, 1] \end{matrix}$ $f_{21}^{,} (x) = {\begin{matrix} \frac{9}{4} x + \frac{143}{20}, x \in [- 1, 0) \\ \frac{43}{20} x + \frac{143}{20}, x \in [0, 1] \end{matrix} f_{22}^{,} (x) = {\begin{matrix} x + 9, x \in [- 1, 0) \\ \frac{1}{2} x + 9, x \in [0, 1] \end{matrix}$ $g_{11}^{,} (y) = {\begin{matrix} \frac{22}{5} y + \frac{79}{10}, y \in [- 1, 0) \\ \frac{59}{10} y + \frac{79}{10}, y \in [0, 1] \end{matrix} g_{12}^{,} (y) = {\begin{matrix} \frac{5}{2} y + \frac{27}{5}, y \in [- 1, 0) \\ \frac{11}{4} y + \frac{27}{5}, y \in [0, 1] \end{matrix}$ $g_{21}^{,} (y) = {\begin{matrix} 3 y + 5, y \in [- 1, 0) \\ \frac{7}{2} y + 5, y \in [0, 1] \end{matrix} g_{22}^{,} (y) = {\begin{matrix} y + 8, y \in [- 1, 0) \\ y + 8, y \in [0, 1] \end{matrix}$

Step 3: Generate the projection matrix. $A_{2 \times 2}^{'}, B_{2 \times 2}^{'}$ generates the respective projection matrix $A_{2 \times 2}^{-^{'}}, B_{2 \times 2}^{-^{'}}$ as: $A_{2 \times 2}^{-^{'}} = [\begin{matrix} \frac{1933}{240} & \frac{65}{12} \\ \frac{767}{120} & \frac{69}{8} \end{matrix}] B_{2 \times 2}^{-^{'}} = [\begin{matrix} \frac{787}{120} & \frac{367}{80} \\ \frac{97}{24} & \frac{23}{3} \end{matrix}]$

Step 4: Obtain the expected utility functions of N₁ and N₂. $\begin{matrix} E_{1}^{'} (x, y) = (x, 1 - x) [\begin{matrix} \frac{1933}{240} & \frac{65}{12} \\ \frac{767}{120} & \frac{69}{8} \end{matrix}] (\begin{matrix} y \\ 1 - y \end{matrix}) \\ = \frac{903}{80} xy - \frac{77}{8} x - \frac{268}{120} y + \frac{69}{8} \end{matrix}$ $\begin{matrix} E_{2}^{'} (x, y) = (y, 1 - y) [\begin{matrix} \frac{787}{120} & \frac{367}{80} \\ \frac{97}{24} & \frac{23}{3} \end{matrix}] (\begin{matrix} x \\ 1 - x \end{matrix}) \\ = \frac{1343}{240} xy - \frac{87}{24} x - \frac{739}{240} y + \frac{23}{3} \end{matrix}$

Step 5: Calculate the optimal mixing strategy. A first order derivative of $E_{1}^{,} (x, y)$ and $E_{2}^{,} (x, y)$ respectively gives: $\begin{matrix} \frac{\partial E_{1}^{'} (x, y)}{\partial x} = \frac{\partial (\frac{903}{80} xy - \frac{77}{8} x - \frac{268}{120} y + \frac{69}{8}}{\partial x}) \\ = \frac{903}{80} y - \frac{77}{8} = 0 \end{matrix}$ $\begin{matrix} \frac{\partial E_{2}^{'} (x, y)}{\partial y} = \frac{\partial (\frac{1343}{240} xy - \frac{87}{24} x - \frac{739}{240} y + \frac{23}{3})}{\partial y} \\ = \frac{1343}{240} x - \frac{739}{240} = 0 \end{matrix}$

Further solving for this results in the following: $x^{, *} = \frac{739}{1343} \approx 0.55, y^{, *} = \frac{770}{903} \approx 0.85$

Therefore, the optimal strategy for N₁ and N₂ is: ${\begin{matrix} (x^{'}, 1 - x^{'}) = (0 . 55, 0 . 45) \\ (y^{,}, 1 - y^{,}) = (0 . 85, 0 . 15) \end{matrix}$

Step 6: Calculate the fuzzy payoffs. Based on the optimal strategy solution, the fuzzy payoffs for N₁ and N₂ are found to be: $\begin{matrix} V_{1}^{,} = (0 . 55, 0 . 45) \cdot [\begin{matrix} (\frac{51}{10}, \frac{19}{2}, \frac{283}{20}) & (3, 7, 9) \\ (\frac{49}{10}, \frac{143}{20}, \frac{93}{10}) & (8, 9, \frac{19}{2}) \end{matrix}] \cdot (\begin{matrix} 0.85 \\ 0.15 \end{matrix}) \\ = (5.1, 8.3, 11.6) \end{matrix}$ $\begin{matrix} V_{2}^{,} = (0 . 85, 0 . 15) \cdot [\begin{matrix} (\frac{7}{2}, \frac{79}{10}, \frac{138}{10}) & (\frac{29}{10}, \frac{27}{5}, \frac{163}{20}) \\ (2, 5, \frac{17}{2}) & (7, 8, 9) \end{matrix}] \cdot (\begin{matrix} 0.55 \\ 0.45 \end{matrix}) \\ = (3 . 4, 6 . 7, 12 . 6) \end{matrix}$

Through the above series of solutions, we obtain the probability of N₁ and N₂ choosing to cooperate in the incentive case as x^,* = 0.55, y^,* = 0, 85 and the fuzzy utility value as V₁ = (5.1, 8.3, 11.6) and V₂ = (3.4, 6.7, 12.6). We find that the probability of N₁ choosing to cooperate decreases when the reward for N₂ is greater than that for N₁. It is possible that N₁ believes that the choice of cooperation is different from the investment and benefits. As for N₂, the probability of choosing to cooperate is greater than the result in the non-incentive model, which can prove that government incentives positively impact encouraging both players to choose cooperation. The results of the two models are analyzed in more detail below.

4.3 Comparative analysis

4.3.1 Analysis of gaming cooperation

According to the above analysis, the probability that N₁ and N₂ choose to cooperate in the non-incentives is x^* = 0.85, y^* = 0, 77, and the probability that they choose to cooperate in the incentives is x^,* = 0.55, y^,* = 0.85.

For battery manufacturer (N₁), the probability of choosing to cooperate decreases after the government incentive, and for the vehicle producer (N₂), the probability of choosing to cooperate increases after the government incentive. Therefore, the outcome of the cooperation between the two parties cannot be looked at in isolation from the numerical results, which are analyzed as follows.

No government incentives: $S = \frac{1}{2} (1 - x^{*}) + \frac{1}{2} (1 - y^{*}) = \frac{1}{2} (1 - \frac{C_{2} - P}{λ_{2} - I_{2} π_{2}}) + \frac{1}{2} (1 - \frac{C_{1} + - P}{λ_{1} - I_{1} π_{1}})$

Government incentives: $\begin{matrix} S^{,} = \frac{1}{2} (1 - x^{, *}) + \frac{1}{2} (1 - y^{, *}) = \frac{1}{2} (1 - \frac{C_{2} - P}{λ_{2} - I_{2} π_{2} + r_{2}}) + \\ \frac{1}{2} (1 - \frac{C_{1} - P}{λ_{1} - I_{1} π_{1} + r_{1}}) \end{matrix}$

Taking the partial derivative of r₁, r₂ yields: $\frac{\partial S^{,}}{\partial r_{1}} = \frac{C_{2} - P}{{[(λ_{2} - I_{2}) π_{2} + r_{2}]}^{2}}$ $\frac{\partial S^{,}}{\partial r_{2}} = \frac{C_{1} - P}{{[(λ_{1} - I_{1}) π_{1} + r_{1}]}^{2}}$

If the input cost (C_i) of choosing to cooperate is greater than the punishment (P) for free-riding, S is an increasing function over r₁, r₂. Thus, government incentives can encourage the choice to cooperate, but the incentives must be essentially the same for both players. If C_i < P, then S is a decreasing function over r₁, r₂. In this case, both players will still choose not to cooperate, no matter how large r_i is.

4.3.2 Revenue analysis

Benefits to both parties in the absence of government incentives: ${\begin{matrix} V_{1} = (3, 6.7, 9.7) \\ V_{2} = (1.9, 4.8, 9) \end{matrix}$

Finding the expected value gives: $E_{1} = 6.901, E_{2} = 4.583$

Benefits for both parties when government incentives are available: ${\begin{matrix} V_{1}^{,} = (5.1, 8.3, 11.6) \\ V_{2}^{,} = (3.4, 6.7, 12.6) \end{matrix}$

Finding the expected value gives: $E_{1}^{,} = 6.71, E_{2}^{,} = 5.672$

Based on the above results, we can find that $E_{1}^{,} > E_{1}, E_{2}^{,} > E_{2}$ . Both players’ returns increase when the government takes incentives. The increase in N₂ is greater than N₁, related to the degree of government incentives and confirms our analysis of the incentive model in Section 3.

5 Conclusion and prospect

As the competition in the new energy vehicle industry becomes more intense, many related enterprises are seeking a path of cooperation for mutually beneficial development. While most research on supply chain members focuses on the impact of government subsidies and the supply-demand relationship, there still needs to be more research on cooperation and sharing among internal members. This paper explores game models examining whether a battery manufacturer and a vehicle producer, both involved in the new energy vehicle supply chain, will collaborate on developing a new product. To account for the complex environment, we utilize numerical computations using triangular fuzzy numbers (TFNs) rather than exact numbers. We use the triangular structured element method to calculate the final probability of both players deciding to cooperate. Additionally, we examine two game models, one with non-incentives and another with government incentives. We analyze the various factors influencing the participants’ decisions in both scenarios. Overall, this paper provides a comprehensive analysis of the research findings.

(1) This paper presents two game models that aim to identify the optimal combination of strategies for two players. Initially, the payoffs for both players are identified, and a payoff matrix is constructed to determine the rewards for each strategy. Subsequently, we conduct a mixed Nash equilibrium analysis to ascertain the likelihood of cooperation between the players. Lastly, we discuss the factors influencing both players’ probability of cooperating. In the no-incentive model, we discovered that influencing factors can be categorized into two parts. The first part consists of parameters related to another player’s participation, such as their input, normal benefits, and the benefits of cooperating or free-riding. The cost of the other player’s input and free-riding benefits play a role in both players’ decision to cooperate. However, choosing to cooperate results in a negative gain for both players. Another part is a penalty for free-riding, and the likelihood that two parties will choose to cooperate is reduced if the penalty is too high. In the incentive model, government rewards are added to the two players when they agree to cooperate. We find that the government’s reward to another party also influences both players’ decisions and that too large a difference in the reward to both players is detrimental to their decision to cooperate.

(2) we used TFNs to solve the model and represent its parameters in the numerical examples. Based on our calculations, we discovered that government incentives for both players could motivate them to cooperate as long as the cost of inputs surpasses the penalty for free-riding. In cases where the government offers unequal incentives to both players, the likelihood of the player receiving the smaller incentive cooperating decreases. The findings presented in section 4 validate the analysis presented in section 3 regarding the government incentive model. The reason is that when there are varying government incentives, the player receiving the lesser incentive may view the situation as unjust, resulting in them being disadvantaged in the overall game between the two players. Therefore, to achieve this outcome of cooperation, the government should provide equal incentives to both players in the supply chain.

Based on the results of this study and the current state of development in the new energy vehicle supply chain industry, we suggest the following recommendations to both the members of the supply chain and the government:

Our ultimate goal is to ensure that all players involved in the game cooperate. The supply chain members can enhance product development and innovation by adopting cooperative measures. Numerous studies have already proven the effectiveness of such strategies in games involving supply chain members. By sharing information and technology, supply chain costs can be reduced. Additionally, combining members can help rationalize resource allocation, leading to a mutually beneficial outcome for all parties involved. For governments: To support the automotive supply chain, governments should provide greater incentives to battery manufacturers and vehicle producers. Currently, incentives primarily focus on retailers and consumers, neglecting upstream supply chain members. Encouraging upstream members to create superior products through incentives can positively impact the entire new energy vehicle industry. By enhancing technology and producing quality products, consumers will be more inclined to make purchases, which can lead to a broader market for the industry.

This research investigates the extent of cooperation among members of the new energy vehicle supply chain in a real-world setting. The study analyzes the game model parameters to identify the factors influencing the participants’ decisions. Although relevant conclusions are drawn through numerical calculations, there are still some limitations in the research process. This paper needs to include research on whether using triangular fuzzy numbers instead of exact numbers is the best choice in a fuzzy environment. It also needs to be determined if the size of government incentive policies affects the probability of cooperation between both parties. Further research is necessary to address these questions.

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A new energy vehicle supply chain member game based on triangular fuzzy numbers

Abstract

Keywords

1 Introduction

2 Preliminaries

2.1 The definition of TFN

2.2 TFN-based equilibrium strategy solution for the game

Table 1 Gaming payment matrix N 2 Strategy β1(y) β2(1 - y) N 1 α1(x) (A11, B11) (A12, B12) α2(1 - x) (A21, B21) (A22, B22)

3 Optimal strategy analysis of a fuzzy game considering government incentives

3.1 Model construction and basic assumptions

3.1.1 Problem description

Table 2 Gaming strategy portfolio N 2 Strategy Cooperation (y) Non-cooperation (1 - y) N 1 Cooperation (x) (Cooperation, Cooperation) (Cooperation, Non-cooperation) Non-cooperation (1 - x) (Non-cooperation, Cooperation) (Non-cooperation, Non-cooperation)

Table 4 Non-incentive payment matrix N 2 Strategy Cooperation (y) Non-cooperation (1 - y) N 1 Cooperation (x) π1 (1 + λ1) - C1, π2 (1 + λ2) - C2 π1 - C1 + P, π2 (1 + I2) - P Non-cooperation (1 - x) π1 (1 + I1) - P, π2 - C2 + P π1, π2

Table 5 Incentive payment matrix N 2 Strategy Cooperation (y) Non-cooperation (1 - y) N 1 Cooperation (x) π1 (1 + λ1) - C1 + r1, π2 (1 + λ2) - C2 + r2 π1 - C1 + P, π2 (1 + I2) - P Non-cooperation (1 - x) π1 (1 + I1) - P, π2 - C2 + P π1, π2

4 Numerical examples

4.1 The non-incentive calculation

Table 6 Non-incentive parameter fuzzy values Parameters Fuzzy values Parameters Fuzzy values C 1 (6, 7, 9) C 2 (6, 8, 9) π1 (8, 9, 19/2) π2 (7, 8, 9) λ1 (9/20, 1/2, 7/10) λ2 (1/2, 11/20, 7/10) I 1 (3/10, 7/20, 2/5) I 2 (1/5, 3/10, 7/20) P (4, 5, 11/2)

Table 8 Incentive parameter fuzzy values Parameters Fuzzy values Parameters Fuzzy values C 1 (6, 7, 9) C 2 (6, 8, 9) π1 (8, 9, 19/2) π2 (7, 8, 9) λ1 (9/20, 1/2, 7/10) λ2 (1/2, 11/20, 7/10) I 1 (3/10, 7/20, 2/5) I 2 (1/5, 3/10, 7/20) r 1 (5/2, 3, 4) r 2 (2, 7/2, 9/2) P (4, 5, 11/2)

4.3.1 Analysis of gaming cooperation

4.3.2 Revenue analysis

5 Conclusion and prospect

References

Table 1
Gaming payment matrix

N ₂

Strategy β₁(y) β₂(1 - y)

N ₁ α₁(x) (A₁₁, B₁₁) (A₁₂, B₁₂)

α₂(1 - x) (A₂₁, B₂₁) (A₂₂, B₂₂)

Table 2
Gaming strategy portfolio

N ₂

Strategy Cooperation (y) Non-cooperation (1 - y)

N ₁ Cooperation (x) (Cooperation, Cooperation) (Cooperation, Non-cooperation)

Non-cooperation (1 - x) (Non-cooperation, Cooperation) (Non-cooperation, Non-cooperation)

Table 4
Non-incentive payment matrix

N ₂

Strategy Cooperation (y) Non-cooperation (1 - y)

N ₁ Cooperation (x) π₁ (1 + λ₁) - C₁, π₂ (1 + λ₂) - C₂ π₁ - C₁ + P, π₂ (1 + I₂) - P

Non-cooperation (1 - x) π₁ (1 + I₁) - P, π₂ - C₂ + P π₁, π₂

Table 5
Incentive payment matrix

N ₂

Strategy Cooperation (y) Non-cooperation (1 - y)

N ₁ Cooperation (x) π₁ (1 + λ₁) - C₁ + r₁, π₂ (1 + λ₂) - C₂ + r₂ π₁ - C₁ + P, π₂ (1 + I₂) - P

Non-cooperation (1 - x) π₁ (1 + I₁) - P, π₂ - C₂ + P π₁, π₂

Table 6
Non-incentive parameter fuzzy values

Parameters Fuzzy values Parameters Fuzzy values

C ₁ (6, 7, 9) C ₂ (6, 8, 9)

π₁ (8, 9, 19/2) π₂ (7, 8, 9)

λ₁ (9/20, 1/2, 7/10) λ₂ (1/2, 11/20, 7/10)

I ₁ (3/10, 7/20, 2/5) I ₂ (1/5, 3/10, 7/20)

P (4, 5, 11/2)