The better choice of green product development strategy in supply chain under optimistic preference

Abstract:Considering the supply chain composed of upstream manufacturers and downstream retailers, and the cognitive bias of supply chain members affected by optimistic preference in green R&D, this paper constructs a game model for manufacturers or retailers who overestimate the impact of green product R&D on demand, and studies the impact of optimistic preference on the game equilibrium results of different R&D subjects in the supply chain. The results show that: (1) Affected by optimistic preference, the greenness of products will be higher than the concentration level of rational decision-making within a certain threshold, and manufacturers are more sensitive to the influence of optimistic preference. (2) Optimistic preference may hurt individual profits, but it can promote higher returns for the supply chain under a certain threshold. (3) When the manufacturer has an optimistic preference behavior, the income of the supply chain depends on the optimistic preference coefficient when the two parties conduct green product research and development respectively, and the manufacturer's green product research and development within a certain threshold can increase the income of the supply chain, on the contrary, the retailer is better for green products; When manufacturers have rational behavior, regardless of whether the retailer has a rational or optimistic preference behavior, it is better for manufacturers to conduct green product development. (4) Although the retailer's optimistic preference behavior can make the supply chain's revenue close to the manufacturer's green product R&D effect within a certain threshold, with the increase of retailer's optimistic preference behavior, retailer R&D makes the supply chain's revenue lower. (5) When the optimistic preference of the R&D subject is not known to the partners, both the manufacturers and the retailers with the optimistic preference can obtain higher benefits, and from the perspective of the supply chain, the overall revenue of the supply chain increases when the manufacturer implements green R&D, while the retailer's implementation of green R&D is reflected in the transfer of revenue within the supply chain. The results of this paper provide some insights into the optimistic preference of supply chain green R&D choices, so that supply chain members can make more rational decisions.

Key words: Optimism; Green product development; Game theory; Decision analysis.

1. Introduction

The rapid development of the global economy has also brought about a series of environmental problems, such as a gradually warming climate and environmental pollution. A variety of measures have been taken at the government and business levels to effectively address environmental issues. At the government level, countries such as China, the United States, the European Union, and Japan have introduced policies such as carbon taxes and carbon trading that are considered effective in controlling CO2/pollutant emissions (Xu et al., 2022; Evro S et al., 2024; Zhang et al., 2025)。 In addition, due to the increasing environmental awareness of consumers, they are more inclined to buy green products when choosing products (Li and Wang, 2023), and the proportion of willingness to buy is even more than 80 percentage points (Hong and Guo, 2019) In order to gain a competitive advantage in the market, the enterprise level actively conducts green product research and development, and adopts a green supply chain management model to achieve transformation and upgrading (Yao et al.). , 2024; Yu et al.,2025)。 Influenced by consumers' green preferences and government policies, manufacturers adopt green R&D measures including the use of recyclable materials, investment in renewable energy, green emission reduction technologies, and green product design (Chen et al., 2021; Zhang et al., 2023; Ren et al., 2024; Li et al., 2025)。 For example, Volvo, an automobile manufacturer, plans to replace plastic materials with sustainable materials, and achieve a target of 3.0% substitution by 2025.1 In an effort to eliminate plastic pollution, sporting goods manufacturer Adidas recycles its branded ^products2.

In the context of sustainable development, there are also many typical examples of retailers using green R&D measures to achieve environmental protection goals. For example, H&M, a fashion retail brand, uses green emission reduction technologies in the manufacturing process to achieve energy conservation and emission ^reduction3; Department stores such as M&S and Mango use environmentally friendly organic cotton to seek sustainable operations (Guo et al., 2020); JD Logistics uses environmentally friendly and biodegradable materials in the packaging process to reduce pollution in the transportation ^process4; IKEA invests €3 billion in green technology research and development projects (Yao et al., 2024); As early as 2005, Walmart's CEO set an ambitious goal to eventually go green across its global supply chain, and more than 200 suppliers volunteered to offer products with the lowest possible environmental impact.

The primary goal of implementing green R&D is to obtain economic value, and the main body of R&D tasks may be manufacturers upstream of the supply chain or retailers downstream. The inevitable investment in green product research and development may cause investors to lose money. Therefore, the responsibility of green R&D by different actors will affect the balance between the costs and benefits of participating members in the green supply chain. However, in order to remain competitive, companies often choose to go green and innovate (Adnan et al.). , 2023; Bataineh et al., 2024 ), and most business managers maintain an optimistic attitude towards green strategies, but this expectation may be too optimistic and fall into the trap of decision-making (Li &Tang, 2010), Because optimism bias is a common phenomenon in the field of psychology. Research on managerial optimism shows a mixed view, with Accenture and the United Nations Global Compact's study of CEOs' attitudes towards business development, which shows that nearly 99% of senior executives are optimistic about the success of companies that implement sustainable ^strategies5. Managers with an optimistic attitude are more likely to be hired in business operations management because they are more likely to facilitate the realization of innovative projects (Galasso & Simcoe, 2011; Hirshleifer et al., 2012), such as Apple CEO Paul Jobs's optimism in business activities made the company a great success. However, in the survey of green product development and consumer willingness to pay, it was found that although 65% of consumers are willing to buy green products, only 2 6% of them are implemented in the actual purchase process (White & Habib ,2019)。 Managers may have a certain bias against consumers' purchasing behavior and overestimate consumers' desire for green purchase in decision-making, which has a significant impact on enterprises' green operation decisions, supply chain management, and corporate responsibility.

Based on the above examples and theoretical research, this paper uses game theory to analyze and verify the influence of optimism bias on the selection and influence of green product development strategies. We assume that the bearer of green product R&D can be the manufacturer or retailer, and that the green product R&D manager has a rational reality or green optimistic preference for consumers' green product purchase propensity. When the subject of green R&D is rational reality, it can objectively assess the impact of green product R&D on market demand. When the main body of R&D is green optimistic, it will overestimate the impact of green product R&D on market demand. In this context, this article mainly answers the following questions:

(1) Compared with rational manufacturers or retailers in green product research and development, can optimistic preference behavior increase the benefits of both parties, and is the product greener?

(2) Does optimism have the same effect on the decision-making variables (i.e., the level of green R&D of products and the price decision) of manufacturers or retailers when developing green products?

(3) When there is an optimistic preference among manufacturers or retailers, who is better to undertake the research and development of green products in the supply chain?

To solve these problems, we built a Stackelberg game model in which manufacturers and retailers conduct green product development, respectively. Firstly, we consider that the manufacturers and retailers who implement green product R&D are the basic models under rational conditions, secondly, we introduce optimistic preference into different green product R&D models, analyze the impact of optimistic preference on manufacturers and retailers in green product R&D and the choice of green product R&D implementers, and finally, compare the game equilibrium strategies of manufacturers and retailers under different green product R&D scenarios. Here are a few things we learned:

Firstly, the level of green R&D of products will increase when the optimistic preference behavior of manufacturers and retailers is higher than that of rational reality when the optimistic preference exceeds a certain threshold, and even higher than that of the centralized situation when both parties are rational, and the level of green R&D of products during the manufacturer's R&D is more sensitive to the influence of optimistic preference.

Secondly, if manufacturers have optimistic preference behaviors when developing green products, their profits will be harmed, while if retailers have optimistic behaviors when developing green products, their profits will increase first and then decrease. From the perspective of the supply chain, whether manufacturers or retailers are engaged in green R&D, optimistic preference behavior can increase the overall revenue of the supply chain within a certain threshold.

Thirdly, in the case of green product research and development by manufacturers or retailers, the green R&D level and sales price of the product change trend with the influence of optimistic preference, that is, they both increase with the increase of optimistic preference, and the green level of the manufacturer's R&D is higher when the degree of optimistic preference is the same. The wholesale price of products varies with the change of optimistic preference, that is, when the manufacturer undertakes the research and development of green products, the wholesale price of the product will increase with the increase of optimistic preference, while when the retailer undertakes the research and development of green products, the wholesale price of the product will increase first and then decrease with the increase of optimistic preference.

Furthermore, when manufacturers and retailers have optimistic preferences and rational realistic behaviors, respectively, the total supply chain revenue generated by the R&D of green products by manufacturers with optimistic preferences is not necessarily higher than that of rational retailers. When manufacturers and retailers have rational reality and optimistic preference behaviors, respectively, the total supply chain revenue generated by green product research and development by retailers with optimistic preference is lower than that generated by rational manufacturers when they conduct green product research and development. When both manufacturers and retailers have optimistic preference behaviors, if the degree of optimistic preference is lower than a certain threshold, the manufacturer's green product research and development will bring more benefits, and on the contrary, the retailer's green product research and development will bring more benefits.

Finally, when the optimistic preference of the R&D subject is not known to the partners, both the manufacturers and retailers with the optimistic preference can obtain higher benefits, and from the perspective of the supply chain, the overall revenue of the supply chain increases when the manufacturer implements green R&D, while the implementation of green R&D by retailers is reflected in the transfer of revenue within the supply chain.

The remainder of this article is organized as follows. The second part reviews the literature related to this paper. The third part describes the model and gives the basic model. In the fourth part, we build a game model in which manufacturers and retailers implement green product development under optimistic preferences. In the fifth part, we make a comparative analysis of the equilibrium results. In the sixth part, we expanded the model to mean that the optimistic preference behavior of manufacturers (retailers) considering the implementation of green product development is not known to collaborators. The seventh part is a summary of the full text.

2. Literature review

This paper mainly studies the selection of green product R&D strategies in the upstream and downstream of the supply chain under optimistic preference. Therefore, the research related to this paper mainly includes two aspects, namely green product development and behavioral operation management, especially the literature on managers' optimistic attitude in operation management.

2.1. Green product development

Green product research and development is aimed at reducing environmental pollution, investing relevant costs in the production process to design and reengineer products, and finally achieve the production of products with higher environmental performance than similar products or less pollution than the original products (Pujari et al., 2003; Huang & Wu ,2010)。 Green product R&D can enable companies to gain a competitive advantage, and has become one of the hot research topics in sustainable supply chain management research (Kleindorfer et al., 2005; Choi, 2019; Ahmad et al., 2024)。 In the research literature on green product development or the improvement of the green performance of products, many scholars have conducted a large number of studies with manufacturers taking this task as the starting point (Du et al,2015; Gao et al.,2018; Zhang et al., 2019; Shen et al.,2020; Li et al., 2021; Yuan et al., 2022; Wang et al., 2024; Xia et al., 2025)。 In fact, retailers' green R&D can also have an important impact on social and environmental well-being (Nyilasy et al., 2016). For example, Ghosh and Shah (2012) conducted research on the decision-making of manufacturers and retailers in green product R&D under different power structures. Dong et al. (2019) A comparative analysis was conducted on whether manufacturers and retailers make green R&D decisions in the second phase and which R&D model is more beneficial. Li et al. (2021) introduced the risk preference of decision-makers into the model when the task of green product R&D is undertaken by manufacturers or retailers, and explored the influence of risk factors on the selection of green product R&D. Wang et al. (2023) compared the advantages and disadvantages of low-carbon R&D strategies and social welfare issues of upstream manufacturers, downstream retailers or upstream and downstream joint efforts in the context of carbon tax, and explored the impact of carbon tax on game equilibrium. Yao et al., (2024) examines the equilibrium of green design implementation by different supply chain members under different power structures. In addition, there are also studies on the procurement of green products, the problem of green inventory, and the problem of subsidies for green products (Liu et al., 2019; Gao et al. 2024 ,Gao et al. 2024)。

We have studied similar literatures such as Ghosh and Shah (2012), Dong et al (2019), Wang et al. (2023) and Yao et al. (2024), all of which consider the development of green products in the upstream and downstream of the supply chain. Unlike them, this paper takes into account the differences in decision-makers' expectations of the consumer market, that is, the influence of optimistic preference attitudes on different decision-making outcomes.

2.2. Behavioral operations management

The influence of human behavior on strategy development in the field of operations management is a non-negligible factor, but many studies often assume that human behavior is a rational actor in the modeling process (Gino & Pisano, 2008). Scholars in the study of behavioral operation management have analyzed the decision-making problems under bounded rationality, such as strategic waiting, regret psychology and reference effect in consumer behavior factors, based on the behavioral factors of consumers or managers (Men et al., 2022; Gu et al., 2024; Berendtet al., 2024 )； Risk aversion, fairness preference, and altruistic preference in managerial behavioral factors (Li et al. 2021; Kumar et al. 2023; Zhao., 2024) and so on.

The literature related to this study is mainly reflected in the bias of market expectations, which mainly includes optimistic preference or overconfident behavior. In terms of inventory setting and product purchasing decisions, the optimistic preference behavior of decision-makers may lead to the deviation of the plans made by managers (Jain et al., 2018; Ren et al. 2013)。 Executives in the professional field are more influenced by this behavior when making decisions, and they have higher expectations for the return of investment and financing (Mahajan, 1992; Pikulina et al.2017)。 For example, Simon & Shrader, 2012, found that optimistic managers invest more in innovation to increase the number of patents and patent citations, and firms are more likely to gain a first-mover advantage. Tang et al.According to the 2015 study, the performance of the company will be affected by the attitude of the manager, and the overconfidence behavior of the enterprise manager can make the enterprise innovation take a new direction. Jiang and Liu (2019a) used the game theory model to find that when competitors make decisions, unilateral optimism can benefit both parties, while bilateral optimism is the opposite. Cheng et al. 2022 The study found that remanufacturers achieve a win-win situation for OEMs and independent remanufacturers under certain conditions by overestimating consumers' willingness to pay for remanufactured goods. In fact, under certain conditions, optimistic preference can compensate for information weakness (Ma et al., 2024) and give a competitive advantage in duopoly (Li & Tayi., 2024) and improving the profitability of green manufacturing (Sim &Kim., 2024).

Although the above literature examines the innovation investment of policymakers in the context of their optimistic preference for the market, they mainly focus on the impact of optimistic behavior on the innovation of upstream manufacturers, which is different from our study. We not only consider the impact of optimistic behavior on the green product R&D of upstream manufacturers, but also consider the impact of this behavior on the green product R&D of downstream retailers, and at the same time, we make a horizontal and vertical comparison, that is, the difference between optimistic behavior and rational green product R&D, and the choice of green product R&D investment under the influence of optimistic behavior.

2.3. Research gap

According to the above-mentioned literature on green product development and operation management under optimistic preference, we found that there are the following vacancies in this field. First of all, there are many research literatures on green R&D as the main body of green R&D, although there are also scholars who focus on the research and development of green products for retailers, such as considering the power structure, stage selection, carbon trading and risk appetite of green product R&D strategy. However, there are few relevant literatures on the R&D choices of green products under different expectations of managers. In order to better make up for the shortcomings in this aspect, this paper constructs a model of green product research and development for manufacturers and retailers under the rational behavior of managers, and compares and analyzes the game equilibrium. In addition, the optimistic preference attitude of decision-makers is rarely introduced into the decision-making model of downstream retailers in the product green R&D literature. In this paper, optimistic preference factors are introduced into the construction of the two product green R&D models, the influence of optimistic preference factors on the two R&D models is analyzed, and the difference between rational behavior and optimistic preference behavior is compared and analyzedand the impact of this behavior on the choice of R&D strategy. Finally, because optimistic preference behavior is a manager's personal prediction of market expectations. In the study of optimistic preference behavior, we compare and analyze the impact of green product development strategies when the behavior is not known to collaborators. This paper introduces optimistic preference behavior into the selection of green product R&D strategies, which enriches the research in the field of green product R&D, and can also put forward some theoretical decisions for enterprises in the process of green practice.

3. Basic model without optimistic preference

This section considers a rational two-tier green supply chain structure that includes an upstream manufacturer and a downstream retailer as a benchmark model. In the process of production and sales of products, manufacturers and retailers determine the wholesale and retail prices of products respectively. In order to meet the requirements of consumers' green preferences, and supply chain members are affected by both power structure and decision-making behavior in the game decision-making process, we construct two Steinberg games in which upstream manufacturers and downstream retailers undertake green R&D, which are denoted as model M and model R, respectively. The two decision models can be solved in three stages, and the specific event process is shown in Figure 1. First, the manufacturer and retailer make decisions on the level of green R&D, then the manufacturer decides the wholesale price of the product, and finally the retailer decides the retail price of the product. The costs of manufacturers and retailers in the production and sales of green products are respectively and in order to ensure that the benefits of supply chain members are positive. Similar to the study with Du et al., 2021 and Gao et al., 2024, we assume that the green R&D cost of the product is the last time, indicating that it is increasingly difficult to increase the demand for the product due to the enhancement of product greenness.

Fig. 1. The Event sequences.

On the one hand, consumer demand for products is negatively affected by price, and on the other hand, it is positively affected by the degree of green R&D,2019; Li et al. 2021; Wang et al. (2023)), which defines the demand function as, where is the degree of consumer preference for green products. In addition, due to the uncertainty of the degree to which the product will be accepted by technology and market (Du et al. 2021), we define the consumer green preference coefficient as a random variable within the range, and the mean and variance of the random variable are and and has a cumulative distribution function and a probability distribution function of and are respectively.

The model in this section assumes that the members of the supply chain are rational and only consider the maximization of their own monetary returns in the decision-making process. Drawing on (Ghosh and Shah.),2012; Yao et al. 2021) and combined with the above assumptions, we use the superscript MN to indicate that the manufacturer conducts green R&D under the condition that the supply chain members are completely rationalm and r represent the manufacturer and retailer, respectively, and the revenue functions of the manufacturer and retailer in the M model can be expressed as:

(1)

(2)

Without prejudice to the conclusions of this study, we assume that the manufacturing cost and the cost of sales are both zero for manufacturers and retailers, at which point (1) and (2) and rewrite as:

(3)

(4)

3.1 Centralized case

We start by establishing a baseline model in which manufacturers and retailers make decisions from the supply chain as a whole (Model C). The profit function of the supply chain in this model is as follows:

(5)

It is easy to find the equilibrium retail price and product development level under the centralized decision-making model, i.e.:

(6)

Then substituting and substituting into equation (5), the profit value of the supply chain in this decision model is:

(7)

The superscript in the formula represents the centralized decision model and represents the optimal solution under the model. Since, and both are positive, we can easily get.

3.2 Decentralized case

In this scenario, both manufacturers and retailers aim to maximize their own profits. Since the green R&D of products can be implemented by different entities, we consider the two situations of manufacturers and retailers conducting green R&D respectively. When a manufacturer conducts green R&D (i.e., the M model), the profit function of the manufacturer and retailer is shown in equations (3) and (4). When a retailer conducts green R&D (i.e., the R model), the profit function expression for the manufacturer and retailer at this time is as follows:

(8)

(9)

Among them, the superscript RN indicates that the retailer conducts green R&D under the condition that the supply chain members are completely rational. Using the inverse induction method to solve the model M and model R respectively, we list the optimal solution and profit value under the equilibrium decision, as shown in Table 1.

Corollary 1. When the members of the supply chain are rational, we can get the following results according to Table 1:

（1）； ； ；

（2）； ； ；

（3）； 。

The results of Corollary 1 are similar to those of Li et al. 2021, and when combined with Table 1, it can be concluded that the level of green R&D and supply chain benefits in the supply chain integration decision-making (model C) are higher than those in the equilibrium results of the manufacturers and retailers in the independent R&D, and the manufacturers conduct green R&D (model M).The greenness of the product is higher, and the sales price of the product is the smallest under centralized decision-making, followed by the retailer when it conducts green R&D, and the largest when the manufacturer conducts green R&D. At the same time, the wholesale price of the product is higher when the manufacturer conducts green R&D, and the income obtained by the manufacturer itself is greater than that obtained by the retailer when it conducts green R&D, and the income obtained by the retailer when the manufacturer conducts green R&D is smaller than the income obtained by the manufacturer when the retailer conducts R&D.

Table 1 The equilibrium solutions without optimistic preference

4. Green product development with optimistic preference

Draw on (Du et al ,2021; Jin et al. 2021), managers with optimistic preference have higher expectations of market demand than the actual level, and we define the demand function under optimistic preference as , where is a continuous random variable with a value range and the mean isThe variance is the manager's optimistic preference coefficient for the green product market, which indicates the manager's overestimation of the market demand. The value is also greater when the manager's optimistic preference is stronger. In order to analyze the impact of decision-makers' optimistic preference on green product R&D investment, we also considered two models of green product R&D by manufacturers and retailers, in which we recorded model M O when manufacturers with optimistic preference conducted green R&D, and model R O when retailers conducted green product R&D.

4.1 Mode MO

When manufacturers with optimistic preferences conduct green R&D, they pursue utility maximization as the decision-making goal, while retailers' profit functions are similar to those when they are rational. Thus, we can get the decision equations for both sides of Eq. (10) and Eq. (11): respectively

(10)

(11)

Lemma 1.In the MO model, we still use the inverse induction method to solve the game problem, and it is easy to find out that the wholesale price and the green R&D level of the manufacturer in the model are:

(12)

The equilibrium results of the retail price of the product and the profit of the manufacturer and the retailer are as follows:

(13)

(14)

In order to ensure that the model conforms to reality, it is necessary to meet the requirements of , , , and , then the known value range is: where: .

Corollary 2. When manufacturers with optimistic preferences go green, we can draw the following conclusions

(1)； ； ；

(2)； When,; When,; among others.

Corollary 2(1) shows that when manufacturers have optimistic preference behavior, they overestimate the market demand brought by green R&D with the enhancement of this behavior, so the level of green R&D increases with the increase of optimistic preference. As the level of green R&D increases, more costs need to be invested, and the unit production cost of green products also increases, so manufacturers will increase the wholesale price of products to obtain stable profits, which will increase the total cost of the products ordered by retailers, so the selling price of retailers will also increase with the increase of optimistic preference level. Corollary 2(2) shows that manufacturers with optimistic preferences invest too much in green technology R&D, but do not achieve an increase in product profits, so their profits decrease with the increase of excessive confidence level. For retailers, although manufacturers can attract more consumers by conducting green R&D, increasing the selling price of their products will weaken the increase in market demand, so under the interaction of the two factors, retailers' profits will increase first and then decrease with the increase of optimistic preference level.

4.1 Mode RO

Retailers with optimistic preferences also pursue the goal of maximizing utility when conducting green R&D, while the profit function of manufacturers is similar to that of their rational approach. In this case, we can get the decision equations for both sides of equations (15) and equations (16): respectively

(15)

(16)

Lemma 2.In the RO model, similar to the MO model solving method, the wholesale price and the green R&D level of the retailer in the model can be obtained:

(17)

The equilibrium results of the retail price of the product and the profit of the manufacturer and the retailer are as follows:

(18)

(19)

According to the actual situation, we have , , , and the easily obtained values range are: where: .

Corollary 3. When retailers with optimistic preferences go green, we can draw the following conclusions

(1)； ； when,when,when,where;

(2) when,when,when,; When,,When,; One of them is the correct solution.

Corollary 3 (1) shows that when retailers have optimistic preference behavior, they also overestimate the increase in market demand brought about by green R&D, so the level of R&D increases with the increase of optimistic preferenceAt the same time, retailers will increase the selling price in order to make up for the cost of green R&D, and manufacturers will increase wholesale prices to obtain more benefits because of the stimulation of market demand for green R&D when the retailer's optimistic preference level is below a certain threshold, but when the retailer's optimistic preference is greater than a certain threshold, the increase in products brought about by its green R&D is not obvious, and the manufacturer wants to maintain it The sales volume of the product will reduce the wholesale price and keep the profit stable. Corollary 2(2) shows that the profits of supply chain members are all concave functions of the level of retailer optimism, indicating that retailers' optimism will not damage the performance of supply chain members at a certain level. When the optimistic preference level is greater than a certain threshold, the performance of supply chain members will be compromised by the retailer's optimistic preference level.

5. Comparative analysis

In this section, we make a comparative analysis from two aspects: one is the impact of optimistic preference behavior on the green R&D investment of manufacturers and retailers, and we compare and analyze the game equilibrium solution and profit value under the optimistic preference and rational behavior of manufacturers and retailers; The second is the impact of optimistic preference behavior on the choice of green R&D strategy in the supply chain, and we compare and analyze the game equilibrium solution and profit value when manufacturers and retailers undertake green R&D tasks respectively.

Corollary 4.Compared with the centralized model, when manufacturers and retailers have optimistic preference behaviors, we can get the following relationship between green R&D levels

(1) when, wherein;

(2) when, wherein;

(3) When,.

Fig. 2. The change of with

In order to visualize the impact of optimistic preference behavior on green R&D, we combined with numerical simulation for analysis, referring to the study of Du et al., 2021, we set the parameters, and we have similar settings below, and the relationship between optimistic preference and R&D level is shown in Figure 2. Based on Figure 2 and Corollary (4), we conclude that the gap between the green level of products gradually decreases when manufacturers and retailers have optimistic preference behaviors and when they are rational in both R&D alone With the gradual increase of optimistic preference behavior, the independent R&D level of manufacturers is higher than that of products under centralized decision-making, and the level of independent R&D of retailers is also higher than that of products under centralized R&D decision-making. The results show that the optimistic preference behavior of green R&D implementers can improve the level of green R&D of products, and even higher than the level of product green in the rational supply chain under centralized decision-making. The main reason is that the R&D implementer's R&D investment is higher than the market increase expectation brought about by centralized decision-making when both parties are rational, and may obtain more profit value as a result. In a nutshell, whether manufacturers or retailers are developing green products, optimistic preference behaviors can improve the overall green level of products in the supply chain.

Corollary 5.When green R&D is undertaken by the manufacturer, we can draw the following conclusions by comparing and analyzing the equilibrium solution and profit of the game when the manufacturer has rational and optimistic preference behavior

(1) ； ； ；

(2) ； At that time, at that time, at that time, which was the correct solution;

(3) When,,When,,,where is the correct solution,.

Fig. 3. The change of with Fig. 4. The change of with

From Corollary 5 (1), we can conclude that the effect of optimistic preference behavior on all decision variables is positive relative to rational manufacturers. Due to the optimistic preference behavior of manufacturers, their market increase brought by green R&D is higher than the estimation under rational behavior, and the green R&D investment is also higher than that of rational decision-making. As a result of increasing investment in green R&D, manufacturers will also increase the wholesale price of their products to balance the excessive cost of green R&D. For retailers, the cost of ordering products is greatly influenced by the optimistic preference behavior of manufacturers, which also increases the selling price of products to obtain profit equilibrium. Another reason for the increase in selling prices when manufacturers have optimistic preference behaviors may be that manufacturers have a higher level of green R&D, which makes consumers more motivated to buy. From inferences 5(2) and (3), we can see that the optimistic preference of the manufacturer does not increase its profit, which may be due to the optimistic preference and rational behavior of the manufacturer

Compared with the actual sales volume of green R&D, the manufacturer's profit is reduced even if the wholesale price of the product is not enough to compensate for the cost of increasing the green R&D. For retailers, in order to more intuitively see the impact of the manufacturer's optimistic preference level on it, we carried out numerical simulations, and the results are shown in Figure 3, that is, when the manufacturer's optimistic preference level is at a certain threshold, the increase of its green R&D level will increase the demand for the whole product, and the retailer will increase the product price appropriately, so that the retailer can benefit from the manufacturer's optimistic preference behavior, but as the manufacturer's optimistic preference level increases, the retailer will spend more money when ordering. Moreover, the market increment brought by green R&D is not significant, and retailers cannot benefit from the optimistic preference behavior of manufacturers. The numerical simulation of the impact of manufacturers' optimistic preference behavior on the revenue of the whole supply chain is shown in Figure 4, combined with inference 3(2), the optimistic preference behavior can increase the performance of the entire supply chain within a certain threshold, and the overall profit of the supply chain beyond a certain threshold is less than the profit of the supply chain under its rational behavior.

Corollary 6.When the green R&D is undertaken by the retailer, we can draw the following conclusions by comparing and analyzing the game equilibrium solution and profit when the retailer has rational and optimistic preference behaviors

(1) ； ；；

(2) ； when,when,,where is the correct solution;

(3) When, when, , where is the correct solution.

Fig. 5. The change of with Fig. 6. The change of with

Fig. 7 The change of with

From Corollary 6 (1), we can see that the optimistic preference behavior of a rational retailer has a positive effect on all decision variables, which is similar to Inference 5 (1). At this time, the retailer has optimistic behavior and bears the investment in green R&D, and the optimistic preference behavior also makes the retailer higher than the level of green R&D under the rational situation, and in order to make up for the cost of increasing the investment in green R&D, the retailer will increase the selling price of the product. While the retailer's optimistic preference level is above a certain threshold, the manufacturer will reduce the wholesale price, but it is still higher than the wholesale price if the retailer is rational. According to inferences 6 (2) and (3) and in combination with inferences 3 (2With the increase of retailers' optimistic preference behavior, the increase in wholesale price and sales volume of the retailer compared with the retailer's rational behavior does not affect the final benefit of the manufacturer, as shown in Figure 5. For retailers, in order to more intuitively see the impact of optimistic preference level on them, we carried out numerical simulations, and the results are shown in Figure 6, when the optimistic preference behavior exceeds a certain threshold, the green R&D cost is greater than the resulting sales growth, although it increases the selling price of the product, the retailer's profit will also increase first and then decrease compared with its rational behavior. For the whole supply chain, the numerical simulation results are shown in Figure 7, when the retailer's optimistic preference behavior is within a certain threshold, the profit of the whole supply chain will also increase, and when the retailer's optimistic preference behavior exceeds a certain threshold, the positive growth of the retailer's optimistic preference behavior on the manufacturer's profit stimulation is less than the decrease of the retailer's profit.

Corollary 7.When the manufacturer has an optimistic preference behavior and the retailer is rational, we can draw the following conclusions by comparing and analyzing the game equilibrium solution and profit when the two sides carry out green R&D respectively

(1) ，，；

(2) the correct solution of when, when, and wherein; ；

(3) When, when, when, where is the correct solution.

Fig. 8. The change of with Fig. 9. The change of with

Fig. 10. The change of with Fig. 11. The change of with

From Inference 7 (1) and in conjunction with Inference 1 (1) and Inference 5 (1) we know that there isCompared with the R&D of rational retailers, the wholesale price, green level and selling price of products carried out by manufacturers with optimistic preference behavior will further widen the gap between R&D and rational retailers on the basis of R&D by rational manufacturers. The reasons for the formation are easy to draw from inferences 1 (1) and 5 (1), and we will not go into too much detail. Based on inference 7 (2) and combined with inference 1 (3) and inference 5 (2We can see that when the two sides separately carry out green R&D, the benefits of manufacturers with optimistic preferences are not necessarily greater than those of retailers with rational behavior, as shown in Figure 8. The reason for this may be that although manufacturers are more familiar with green R&D and production of products, the market expectation brought by green R&D by manufacturers within a certain threshold is greater than the revenue of rational retailers' green R&D, although the cost of retailers' green inputs is smallerManufacturers with an optimistic preference still receive more benefits than rational retailers, and the opposite is true when manufacturers excessively exceed a certain threshold; In addition, the benefits obtained by retailers with optimistic preference for green R&D are smaller than those obtained by rational retailers when they manage green R&D, because manufacturers pay attention to the benefits brought by green R&D of products, and the wholesale price of products increases higher than the sales price of retailers in order to make up for the costs, as shown in Figure 9It shows that although the retailer increases the selling price of the product, it cannot offset the loss suffered by the increase in the order cost, and when the rational retailer conducts green research and development, the greenness of the product decreases at the same time, and the manufacturer's cost is also lower, and the profit obtained by the manufacturer is relatively higher, as shown in Figure 10. For the whole supply chain, the numerical simulation results are shown in Figure 11, the profit of the whole supply chain when the optimistic preference manufacturer conducts R&D is not necessarily higher than the profit of the supply chain when the rational retailer conducts R&D, and the optimistic preference coefficient is lower than that of the manufacturer when the optimistic preference is higher when the supply chain profit is higher, and the optimistic preference coefficient is greater than that When rational retailers conduct R&D, the supply chain is more profitable.

Corollary 8.When the retailer has optimistic preference behavior and the manufacturer is rational, we can draw the following conclusions by comparing and analyzing the game equilibrium solution and profit when the two sides carry out green R&D respectively

(1) ； When,When,;

When,,When,,Where

,；

(2) ； ； (3)。

Fig. 12. The change of with Fig. 13. The change of with

Fig. 14. The change of with

From inference 8 (1) and in conjunction with inference 1 (1) and inference 6 (1) we know that there isThe wholesale price of the product will further reduce the gap with the rational manufacturer on the basis of the rational retailer's R&D, while the green level and selling price of the product will gradually decrease and exceed the equilibrium results of the rational manufacturer's R&D on the basis of the rational retailer's R&D, as shown in Figure 12 shown. According to the explanation in Literature, Li et al., 2021, we can see that when both parties are rational when green R&D is carried out separately, the wholesale price and sales price of the manufacturer are higher than that of the retailer when the R&D is carried out, and the combined inference 1 (1) and inference 6 are combined (1) It is not difficult to understand that when a retailer has an optimistic preference behavior, its R&D efforts gradually increase, and when the optimistic preference level exceeds a certain threshold, it will exceed the R&D of rational manufacturers At this time, the R&D cost invested by the retailer gradually increases, and the retailer will gradually increase the sales price of the product and within a certain threshold higher than the equilibrium solution of the rational manufacturer when conducting green R&D, the purpose of which is also to make up for the cost of green product R&D. For the wholesale price of the product, the retailer's optimistic preference behavior can make the wholesale price of the product change within a certain range but still higher than the equilibrium result of the rational manufacturer when conducting R&D. From inference 8 (2) and in combination with inference 1 (3) and inference 6 (2) we knowWhen the two parties separately conduct green R&D, the benefits of the retailers with optimistic preference are smaller than those of the manufacturers with rational behavior, while the benefits of the retailers with optimistic preference are higher than those of the rational manufacturers. Although the optimistic preference behavior of retailers increases their own revenue within a certain threshold, the revenue is lower when the threshold is exceeded, and although the product can be greener, it will have to pay more R&D costsand unable to achieve the benefits obtained by rational manufacturers in research and development; At the same time, the optimistic preference of retailers can lead to more market demand, and although the manufacturer reduces the wholesale price to a certain extent, the revenue obtained at this time is still higher than that of the retailer when the rational manufacturer conducts R&D, as shown in Figure 13. The simulation results of the impact of optimistic preference on the revenue of the supply chain are shown in Figure 14, for the whole supply chain, the optimistic preference retailer conducts green product research and development is smaller than the supply chain benefit when the manufacturer conducts R&D.

Corollary 9.When both manufacturers and retailers have optimistic preference behaviors, we can draw the following conclusions by comparing and analyzing the game equilibrium solution and profit when the two sides carry out green R&D respectively

(1) ，，；

(2) the correct solution of when, when, , where is the correct solution; 。

(3) When, when, , where is the correct solution.

Fig. 15. The change of with Fig. 16. The change of with

Fig. 17. The change of with

From Corollary 9 (1), we can draw the following conclusions: when both the manufacturer and the retailer have optimistic preference behavior, the wholesale price, green level and sales price of the product when the manufacturer conducts green R&D are higher, that is, the manufacturer is more sensitive to the optimistic preference behavior than the retailer, and the numerical simulation results are shown in Figure 2and Figure 1-5. According to Corollary 9(2), we can see that although manufacturers are more sensitive to optimistic preference behavior, the benefits obtained by manufacturers in R&D are not necessarily greater than those obtained by retailers in R&D, and the simulation results of the impact of optimistic preference on the profits of both parties in the case of R&D are shown in Figure 16It is shown that when the optimistic preference behavior exceeds a certain threshold, although the greenness of the product is lower when the retailer conducts R&D, the R&D cost is lower, and the wholesale price of the product is relatively low at this time, although the greenness of the product is higher when the manufacturer conducts green R&D, the cost is higher, and the increase in a certain wholesale price is still not enough to make up for the excessive investment in product R&D cost, so when the optimistic preference behavior exceeds a certain threshold, the manufacturer's own income is lower when the R&D is lower. At the same time, although the product is greener than the retailer's R&D, the retailer price of the product is higher but the wholesale price increase is higher at this time, and the manufacturer does not have to undertake the green R&D task when the retailer develops, so the retailer's income during the manufacturer's R&D is smaller than the manufacturer's income when the retailer develops. The impact of optimistic preference behavior on the profit of both parties when they conduct R&D is shown in Figure 17, and it is easy to see that when both parties are optimistic preference behaviors, the revenue of the supply chain during R&D depends on the level of optimistic preference. When the level of optimism is low, the supply chain returns are higher when the manufacturer conducts R&D, and conversely, the supply chain returns are higher when the retailer conducts R&D.

6. Extended Models

In this section, we extend the model to the case of the implementation of the optimistic preference behavior of a green R&D agent with personal private information. Drawing on the utility function settings of supply chain members in the research paper L i et al., 2017, this section still analyzes two cases of implementing green R&D, i.e., manufacturers undertaking green R&D of products and retailers undertaking green R&D. A superscript represents a situation where a manufacturer with an optimistic preference conducts green R&D and the level of optimism is not known to the retailer, and a superscript represents a manufacturer with an optimistic preferenceRetailers are doing green R&D and the level of optimism is not known to the manufacturer.

In this case, when the manufacturer's optimistic preference behavior is private information, and the retailer thinks that the selling price decision predicts the manufacturer's problem (3) by finding equation (4), we can obtain the retail price of the product as shown in Table 1. For manufacturers with optimistic preference behavior, the decision on the selling price of the product and the degree of green R&D investment can be derived according to equation (12) in Lemma 1. So we can get the manufacturer's equilibrium profit for

(20)

And the equilibrium profit of the retailer is

(21)

In the case where the retailer's optimistic preference behavior is private information when the manufacturer thinks that the wholesale price decision predicts the manufacturer's problem by finding equation (8) (9), we can obtain the wholesale price of the product as shown in Table 1. The equilibrium selling price of the retailer and the level of green R&D of the product are shown in equations (17) and (18) in Lemma 2. The equilibrium profit of manufacturers and retailers can be expressed as:

(22)

和

(23)

Due to the complexity of the profit function of the decision-maker under the two models, we choose the method of numerical simulation to study the impact of optimistic preference behavior on the implementation of green R&D decision-maker's private information.

Remark 1.When a manufacturer conducts green R&D and its optimistic preference behavior is personal private information, we can draw the following conclusions:

(1) ，。

(2) ； When,,When,.

(3), wherein.

As shown in Figures 18 and 19, we can see that when a manufacturer conducts green R&D and its optimistic preference behavior is personal private information, the benefits for the manufacturer are higher than those when the optimistic preference behavior is known to the retailerThe retailer's earnings are smaller than the gains made by the manufacturer's optimistic preference behavior. Combined with inference 2, we can see that when the optimistic preference behavior is known to the retailer, the manufacturer's profit is lost, but the manufacturer's optimistic preference behavior will prompt the manufacturer to invest more green R&D when personal information is induced, and the wholesale price is more than when the retailer knows the optimistic preference confident behavior. At the same time, because the retailer is not aware of the manufacturer's optimistic preference behavior, the lower retail price of the product and the higher greenness of the product ultimately leads to an increase in sales, so the manufacturer's profit is not only higher than the profit of its optimistic preference behavior, but also higher than the profit when both parties make rational decisions. For retailers who do not know the manufacturer's optimistic preference behavior, their benefits are lower than those of the manufacturer's optimistic preference behavior, but the trend is consistent, because the retailer's behavior of the manufacturer is unknown, the retailer's sales price will be lower, although it can attract more consumers at this time, but the ordering cost of the product is higher, compared with the situation where both parties are rational, although the initial income can be increased, and the manufacturer's optimistic preference level increases, the retailer's income is lower than that of both parties. From Figure 20, we can see that the manufacturer's optimistic preference behavior has a positive impact on the income of the entire supply chain, that is, when the manufacturer's optimistic preference is personal private information, the increase in revenue is greater than the loss to the retailer, and the manufacturer should set a reasonable revenue distribution strategy to ensure that both parties can obtain benefits, so as to ensure the stability of the supply chain.

Fig. 20 The change of with

Remark 2.When a retailer conducts green R&D and its optimistic preference behavior is personal private information, we can draw the following conclusions:

(1)，。

(2)； When,,When,.

(3)； When,,When,. Among them,.

Similar to Remark 1, when a retailer conducts green R&D and its optimistic preference behavior is personal private information, the revenue for the retailer is higher than that when the manufacturer's optimistic preference behavior is known, and the manufacturer's benefit is smaller than the profit under the optimistic preference behavior of the informed retail, as shown in Figures 21 and 22. Combined with inference 3, it can be seen that when the optimistic preference behavior is known by the manufacturer, the profit of retail sales is similar to that of the retailer when it is not known, because the manufacturer does not know that the retailer's optimistic preference behavior will reduce the wholesale price, but the market demand does not change, and the retailer's ordering cost will decrease and the revenue will increase, but when the retailer's optimistic preference is greater than a certain threshold, its income is also lower than that under the rational situation of both parties. Similarly, for manufacturers, although the retailer will reduce wholesale prices without knowing the retailer's overconfidence behavior, the benefits will be slightly lower than when the retailer is aware of the retailer's optimistic preference behavior, but the benefits will increase due to increased market demand compared to the rational situation for both parties. At the same time, combined with Figure 23, it is easy to see that the manufacturer knows and does not know the retailer's behavior, but the total benefit of the supply chain does not change because the profit generated by the reduced wholesale price when the manufacturer is not aware of the retailer's optimism is transferred to the retailer.

Fig. 23. The change of with

Remark 3.When manufacturers and retailers carry out green R&D respectively, and the optimistic preference behavior of both is their own private information, we can draw the following conclusions: , .

According to Figure 1 8, Figure 1 9 and Remark 2 in Remark 1In Figure 21 and Figure 22, it is easy to conclude that Remark 3 shows that when the optimistic preference behavior is the private information of the manufacturer and the retailer, respectively, the manufacturer and the retailer undertake the green R&D task, and the manufacturer gains more benefits. The revenue of the retailer when the manufacturer conducts research and development is smaller than the revenue of the manufacturer when the retailer conducts research, similar to Corollary 1Both sides are the result of equilibrium under rational circumstances. This can be explained in two ways: on the one hand, when the optimistic preference is for the private information of green R&D implementers, the manufacturer's increased revenue comes from the increase in market demand, while the retailer's revenue comes from within the supply chain (transferred by the manufacturer), and the manufacturer's revenue increases more and is therefore easy to obtain; On the other hand, combined with Remark 1Figure 1-9 and Remark 2Figure 21 shows that when the manufacturer develops green and the optimistic preference behavior is personal private information, the damage to the retailer's revenue is greater than that when the retailer's R&D and optimistic preference behavior is personal private information, it is easy to know. Remark 3.It also shows that when manufacturers and retailers conduct R&D separately and optimistically prefer their own private information, although they will increase their own benefits, from the perspective of the overall benefits of the supply chain, manufacturers are more beneficial to the entire supply chain to carry out green R&D.

7. Conclusion

7.1. Key findings and managerial implications

Green product R&D is recognized as an effective approach in the practice of sustainable supply chains. Managers are optimistic about the consumer market in the decision-making process, and optimistic preference behavior is considered to be a behavioral bias, and the impact of this behavior is usually ignored in the rational decision-making model. In this paper, two game models of manufacturers and retailers are constructed to be the main body of green product R&D, and the optimistic preference behavior of decision-makers is introduced into the model to explore the influence and choice of optimism bias on green product development investment strategies. The main conclusions of this paper are as follows:

First, optimistic preference can motivate manufacturers and retailers to increase the R&D level of green products, within a certain threshold, or even higher than the level under centralized decision-making under the situation where both parties are rational, and optimistic preference has a greater impact on manufacturers' green R&D investment. While optimism has the potential to hurt individual profits, at a certain threshold, the overall benefits of the supply chain, whether manufacturers or retailers are developing green products, will increase. Secondly, when the manufacturer has optimistic preference behavior, regardless of whether the retailer is rational or optimistic preference behavior, the income of the supply chain depends on the optimistic preference coefficient when the two sides carry out green product research and development respectively, and the manufacturer's green product research and development within a certain threshold can increase the income of the supply chain, on the contrary, the retailer is better for green products. When manufacturers have rational behavior, regardless of whether the retailer has a rational or optimistic preference behavior, it is better for manufacturers to conduct green product development. Although the retailer's optimistic preference behavior can make the supply chain's revenue close to the manufacturer's green product R&D effect within a certain threshold, with the increase of retailer's optimistic preference behavior, retailer R&D makes the supply chain's revenue lower. From the perspective of the supply chain, when the manufacturer implements green R&D, the overall benefit of the supply chain increases, while the retailer's implementation of green R&D is reflected in the transfer of revenue within the supply chain.

From the research in this article, we can draw some management insights. First, in sustainable supply chain management, whether a manufacturer or a retailer undertakes the task of green product research and development, hiring a manager with an optimistic preference can improve the greenness of the product, and it will also be more conducive to the promotion of the sustainability process of the supply chain. At the same time, the optimistic preference of enterprise managers should be controlled within a certain range, which requires a closer connection between human resources and supply chain management. Second, in the process of enterprise green product development, the leadership of members of the supply chain will undergo certain changes, and the impact of optimistic preference on manufacturers and retailers will be different, so choosing the appropriate product development strategy is very important for supply chain sustainability. Finally, from the perspective of supply chain members and the supply chain as a whole, the effect of optimistic preference is not consistent, and the interests of the whole and individuals should be coordinated in the development process of green supply chain, so as to make the effect of green product research and development more effective.

7.2. Limitations and future research directions

This paper is the first attempt to examine the impact of optimistic preference on the choice of green product development strategy in the supply chain, but there are still some limitations in this study, which may be addressed in future research. First of all, this paper considers a single supply chain structure, and it will be very interesting to study how optimism affects the R&D decisions of green products in the supply chain in a competitive environment in future research. Secondly, we do not consider the cost coefficient when assuming the R&D costs of green products for manufacturers and retailers in our study, so it is necessary to explore the choice of R&D models under different green R&D costs. Thirdly, this paper only considers its own optimistic preference when studying the development of green products

The influence of collaborators' optimistic preference on decision-makers' R&D strategies is also worthy of further study.

Acknowledgement

We would like to express our sincere thanks to the anonymous reviewers and editors for their time and patience devoted to reviewing this paper. This research was funded by the.

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