Government mechanism in waste electrical and electronic equipment recycling among prospect theory and evolutionary game theory

Abstract

Improper disposal of waste electrical and electronic equipment (WEEE) poses challenges for resource loss and public environmental pollution. Government policies are required to regulate WEEE recycling. To better explore the governance mechanism in the WEEE recycling network, a tripartite evolutionary game model composed of the government, recyclers, and manufacturers under bounded rationality and uncertainty was established using evolutionary game theory and prospect theory by underscoring the WEEE processing fee (WPF) exemption to adjust the behaviour of the WEEE recycling network participants. The research found that nine equilibrium points and eight possible ESSs exist and that the evolutionary stability strategies of tripartite decisions mainly depend on the trade-off between costs and revenues. Through numerical simulations, it was observed that the psychological perception of risk for recyclers and manufacturers can affect the behaviour and strategies of WEEE recycling stakeholders in uncertain environments. As WEEE recycling stakeholders increase their pursuit of safety risks, both recyclers’ and manufacturers’ willingness to engage in green behaviour is increasing. In addition, under the WPF exemption, the government can achieve the goal of encourage recyclers and manufacturers to choose green behaviour at a relatively low cost. We suggest that the government should strive to establish a stable WEEE recycling environment, establish an appropriate WPF exemption mechanism, and fully consider the cross-impact of recyclers and manufacturers to promote the green behaviour and efficiency of WEEE recycling.

Keywords

Waste electrical and electronic equipment tripartite evolutionary game model government mechanism prospect theory policy implications

Introduction

The amount of waste electrical and electronic equipment (WEEE) increased sharply and has become the fastest-growing type of garbage.^1,2 In 2019, the global WEEE generation was 53.6 Mt, only 17.4% was formally recycled. Improper handling of WEEE not only causes resource loss and environmental pollution,^3,4 but also harmful to public health.^5,6 To increase the qualified recycling rate and recovery of rare materials, the WEEE Directive and Extended Producer Responsibility (EPR) are formulated to promote the reuse of WEEE.^7,8 However, many challenges remain in the WEEE recycling industry, particularly in terms of policy. In current WEEE recycling system, the government can adjust policies to incentivise and guide participants in choosing green behaviours. Specifically, owing to the relatively high WEEE processing fee, manufacturers are more focused on their profits and can change their strategy at any time to maximise their profits. For recyclers, government subsidies and manufacturers’ formal recycling can incentivise them to choose a qualified recycling strategy, while government pressure on subsidies will be delayed, and manufacturers’ strategy will influence the chosen strategy. To address this circumstance, a WEEE processing fee (WPF) exemption using an evolutionary game model and prospect theory was designed in this study to adjust the behaviour of participants in WEEE.

Compared with traditional game theory, evolutionary game theory emphasises participants’ bounded rationality and evolutionary results’ dynamic equilibrium, and is applicable to research the evolutionary path and influencing factors of participants’ decisions in WEEE recycling.^9,10 However, due to incomplete information, the government is uncertain about recyclers’ and manufacturers’ behaviours, such as recycling methods and production modes, and needs to constantly adjust policy to seek an optimal strategy. Thus, prospect theory is useful for research on risk decision-making attitudes under uncertain information conditions.¹¹

According to the above description, a tripartite evolutionary model is constructed based on rational participants and uncertain conditions, the psychological perceived value of a certain event is calculated based on prospect theory, the evolutionary path of the tripartite decision is constructed, and the factors that influence participants’ decisions are analysed. The main contributions of this paper are as follows. (1) Most previous literature emphasized the implications of subsidy on WEEE recycling but ignored the subsidy variability. To bridge this gap, we both consider the variable subsidy in WEEE tripartite evolutionary game model and analyze the interaction between the three stakeholders to better understand the governance mechanism. (2) Different from the literature on optimal decisions of manufacturers and collectors in operations management, this study designed the WPF waiver policy to alleviate financial pressure for government, providing a new perspective for analyzing WEEE recycling network. (3) Previous literature extensively argued that how government should regulate the formal recycling channel in WEEE recycling. However, fully subsidy will result in mass financial pressure and decreased recycling efficiency. This study enriches the theoretical research on formal recycling channel by exploring the influencing factors of recyclers’ strategies to obtain policy suggestions on promoting formal recycling.

The structure of this study is as follows: Section “Literature review” is the literature review. Section “Construction of the evolutionary game model” is assumptions and triangular evolutionary game model construction. In Section “Analysis of the evolutionary game model,” the ESS is obtained by Jacobean matrix and the corresponding to different stages of the WEEE recycling is discussed. Section “Numerical Simulation” is numerical simulation. Section “Discussion and management implications” is discussion and the managerial implications. Section “Conclusion” summarized the research and future directions.

Literature review

The existing research on WEEE recycling networks adopts a combination of various modeling methods, such as life cycle method,¹² input–output method,¹³ multi-objective optimization method,¹⁴ and simultaneously, it focuses on numerical simulation of the model.¹⁵ Research has found that, in terms of environmental and social benefits, WEEE recycling and reuse is considered as the best choice, but unreasonable WEEE recycling can have negative impacts.¹⁶ Therefore, an optimal WEEE recycling treatment method needs to be explored for the construction of the current WEEE recycling network. Some scholars have used Stackelberg games to analyse the optimal decisions of WEEE recycling network entities and the impact of various variables on the decisions,¹⁷ but this method only remains at the static analysis level. Through evolutionary games, the decision-making results of agents can be used to analyse the evolution of agent selection over time, and the impact of various influencing factors on the evolution process can be analysed based on the evolution results.¹⁸ Moreover, the WEEE recycling network must consider the mutual relationships between the interests and cooperation methods of various entities to establish a stable cooperative relationship and maximise the overall benefits. However, each subject also has the characteristic of maximising their own interests. When one subject hopes to improve their own interests through technological investment and other measures, other subjects will also gain higher benefits. Therefore, the government must formulate reasonable policies and cooperation methods to achieve overall revenue growth for WEEE network subjects.

Policies and regulations introduced by the government have a guiding role in WEEE recycling decision-making.^19,20 Existing research has constructed game models involving multiple stakeholders, such as the government, recyclers, and manufacturers, to analyse the economic and environmental benefits of WEEE recycling under government regulations.^21–23 The results show that the WEEE recycling network is influenced by government macroeconomic regulations and that the constraint effect on the WEEE recycling network will be changed by government regulatory policies.^16,24 Recycling regulations affect manufacturers’ recycling and remanufacturing rates. Moreover, policy formulations should consider the environmental impact of WEEE recycling networks. In addition, research on government recycling regulations often refers to WEEE directives, and recycling network structures and market competition levels often vary among different regions. In addition, the government can use financial incentives to promote WEEE recycling. Relevant scholars have studied the impact of government subsidies on the balanced decision-making of recyclers and remanufacturers, and found that implementing subsidy policies for WEEE recycling stimulates market demand, increases the number of WEEE recycling projects, and benefits all members of the WEEE recycling network. When the government only provides subsidies to remanufacturers, an increase in manufacturer profits leads to a decrease in remanufacturer profits; both parties are in a perfectly competitive relationship. Therefore, the government should not only subsidise remanufacturing but also provide subsidies to recyclers and manufacturers simultaneously to achieve common profit growth and recycling rate improvement for both parties. However, government incentives may not necessarily enable stakeholders to participate in WEEE recycling. According to the above discussion, governments face financial pressure in some countries such as China. Consequently, the government may not provide timely subsidies to recyclers, and the profits of manufacturers are reduced by the EPR policy, which is not conducive to WEEE production innovation.

There are two types of recyclers: formal and informal. Formal recyclers have government-licenced recycling qualifications, while informal recyclers are mainly individuals and workshops that use simple equipment and outdated technology for recycling.²⁵ Moreover, owing to the huge potential profits and low investment in technology, there are many informal recyclers, especially in developing countries such as China.²⁶ Informal recyclers are advantageous in terms of price and convenience in meeting consumer demand, which poses a challenge to formal recyclers.^27,28 Although government subsidies can support formal recyclers, their marginal effects are not always promising.²⁹ Thus, government policies can incentivise formal recycling to generate environmental benefits, whereas current subsidies in many countries may not effectively reduce informal recycling behaviours. Manufacturers play an important role in WEEE recycling, and their product behaviours are largely influenced by EPR in many countries, such as China.³⁰ According to the EPR, manufacturers pay for WEEE processing, which aggregates their cost burden.³¹ Moreover, manufacturers can reuse components and materials to alleviate resource shortages and generate economic and environmental value.^32,33 However, in current WEEE recycling, a larger amount of reusable components and materials cannot be recycled qualitatively, and the manufacturer's production methods may influence other subjects’ behaviours, such as recycling methods and the degree of supervision.

Construction of the evolutionary game model

In a WEEE recycling network, decisions are influenced by factors such as individual differences and risk preferences because of the limited rationality of subjects and information imbalance between them, resulting in different information outcomes obtained by the subjects. For recyclers and processors of WEEE, the choice of recycling method affects the quality of WEEE recycling. Government subsidies for formal recycling can increase formal recycling intention, whereas government regulations constrain informal recycling behaviour. However, as the main body of the WEEE recycling network, manufacturers’ production methods will affect the recycling methods and profits of recyclers. Government policies towards manufacturers will affect their interests under different production methods. In addition, as the regulator of WEEE recycling and policymakers, its policies will affect the interests of recyclers and manufacturers, and the government will change its regulatory policies based on the behaviour of recyclers and manufacturers. However, owing to the uncertainty of information, governments, manufacturers, and recyclers may not be able to make an optimal strategy from the beginning but instead change their own strategies through continuous dynamic adjustments to achieve equilibrium. Evolutionary game theory is studied under the assumption of bounded rationality and incomplete information, which is more in line with the reality of the behavioural strategy selection of WEEE recycling network subjects. At the same time, evolutionary game theory emphasizes the strategic choices and dynamic changes of the subject over time, emphasizing that each participant continuously evolves dynamically by adjusting their own strategies, ultimately achieving a stable state of evolution. Therefore, evolutionary game theory was used in this study to analyse the factors influencing subsidy policies on the choice of triangular behaviour strategies and to provide relevant suggestions for policy improvement.

Based on the evolutionary game theory, the following assumptions are made in our model: The parameters and descriptions of the proposed model are defined in Table 1.

Table 1.

Parameters and descriptions.

Parameters		Description
recyclers	$C_{1}$	The initial cost of recycling
	$Δ C$	Additional investment for formal recycling
	$R_{a}$	The profit of FR
	$R_{u}$	The profit of IR
	$v (R_{m})$	Psychologically perceived when using FR
	$v (F, - m)$	Psychologically perceived when using IR
	$θ$	Proportion rate when using IR
	$F$	The penalty when using IR
	$m$	The penalty rate when using IR
manufacturers	$P_{i}$	Manufacturer's initial profit
	$P_{g}$	Manufacturer's corporate social image benefit when GP
	$c_{m i}$	Saving material cost when using IR
	$c_{m f}$	Saving material cost when using FR
	$c_{g}$	Extra expenses for GP
	$C$	Processing FR for suitable production
	$v (- C)$	Perceived value of manufacturer for processing FR
	$v (T, b)$	Manufacturers psychological perceived on WPF exemption
government	$E_{m}$	Additional expenditure when NP
	$b$	WEEE processing fee waive rate for manufacturers
	$v (F, m)$	Penalty value of IF
	$v (T, - b)$	Psychological perceived on WPF exemption
	$G$	Positive supervision benefit
	$S$	Subsidy to recyclers
	$H_{1}$	Formal recycling benefit
	$H_{2}$	Social benefit to government when GP
	$T$	WEEE processing fee for manufacturers
	$E_{r}$	Extra expenditure when IF

Assumption 1. There are two ways of recycling: formal recycling (FR) and informal recycling (IR). FR expends additional investment on WEEE recycling technology and equipment which can reduce the damage on environment and improve components and materials recovery rate. With advances in technology and equipment, FR can obtain WEEE processing qualifications licenced by the government and subsidised.

Assumption 2. There are many informal recyclers; they perform rough WEEE recycling and processing with simple equipment, and the government needs to pay for the extra recovery of environmental expenditure. The government has two choices: positive supervision (PS) and normal supervision (NP). As the government conducts PS, it discovers IR behaviour and penalises IR for unqualified recycling behaviour.

Assumption 3. There are two choices of manufacturers: green production (GP) and non-green production (NP). The manufacturers’ GP decisions provide social benefits to the government. When manufacturers choose GP, recyclers with qualified recycling methods can reduce costs by selling components and materials to compete with informal recyclers. As manufacturers choose NP, they also save materials. FR may not achieve the standard of manufactured production; manufacturers must pay additional costs for processing FR.

Assumption 4. According to Extended Producers Responsibility, manufacturers must pay WEEE processing fees for their products. To reduce manufacturers’ cost pressure and incentivise manufacturers to choose green produce, the government waives the WEEE processing fee obtained from the manufacturers’ psychological perception.

Using the above assumptions, we obtain a tripartite profit matrix, as shown in Tables 2 and 3.

Table 2.

Tripartite profit under government positive supervision.

		Manufacturers
		GP ( $y$ )	NP ( $1 - y$ )
Recyclers	FR ( $x$ )	$R_{a} + S - C_{1} - Δ C + v (R_{m})$	$R_{a} + S - C_{1} - Δ C$
		$P_{i} + P_{g} + c_{m i} - T + v (T, b) - c_{g}$	$P_{i} + c_{m i} - T$
		$G - c_{p} - S + v (T, - b) + H_{1} + H_{2} + T$	$G - c_{p} - S + H_{1} - E_{m} + T$
	IR ( $1 - x$ )	$θ (R_{u} - C_{1}) + v (F, - m)$	$R_{u} - C_{1} + v (F, - m)$
		$P_{i} + P_{g} - T + v (T, b) - c_{g}$	$P_{i} - T + v (- C) + c_{m f}$
		$G - c_{p} + v (T, - b) + H_{2} + T + v (F, m) - E_{r}$	$G - c_{p} + T + v (F, m) - E_{r} - E_{m}$

Table 3.

Tripartite profit under government normal supervision.

		Manufacturers
		GP( $y$ )	NP ( $1 - y$ )
Recyclers	FR ( $x$ )	$R_{a} - C_{1} - Δ C + v (R_{m})$	$R_{a} - C_{1} - Δ C$
		$P_{i} + P_{g} + c_{m i} - T - c_{g}$	$P_{i} + c_{m i} - T$
		$H_{1} + H_{2} + T$	$H_{1} - E_{m} + T$
	IR ( $1 - x$ )	$θ (R_{u} - C_{1})$	$R_{u} - C_{1}$
		$P_{i} + P_{g} - T - c_{g}$	$P_{i} - T + c_{m f} + v (- C)$
		$H_{2} + T - E_{r}$	$T - E_{r} - E_{m}$

Analysis of the evolutionary game model

Replicator dynamic equations analysis of the three stakeholders

The expected profit of recycler is:

π_{11} = z (y (R_{a} + S - C_{1} - Δ C + v (R_{m})) + (1 - y) (R_{a} + S - C_{1} - Δ C)) + (1 - z) (y (R_{a} - C_{1} - Δ C + v (R_{m})) + (1 - y) (R_{a} - C_{1} - Δ C))

(1)

π_{12} = z (y (θ R_{u} - θ C_{1} + v (F, - m)) + (1 - y) (R_{u} - C_{1} + v (F, - m))) + (1 - z) (y (θ R_{u} - θ C_{1}) + (1 - y) (R_{u} - C_{1}))

(2)

Where

π_{11}

and

π_{12}

represent the revenues of recyclers choosing qualified or unqualified recycling, respectively. According to equations (1) and (2), we calculated and derived the average recycler revenue, and the replicated dynamic equations are denoted as

\bar{π_{1}}

and

F (x)

, respectively, as follows:

\bar{π_{1}} = x π_{11} + (1 - x) π_{12}

(3)

F (x) = x (1 - x) (v (R_{m}) y + S z - Δ C + y (- 1 + θ) C_{1} + R_{a} + (- 1 + y - y θ) R_{u} - z v (F, - m))

(4)

Similarly, the manufacturer's revenue to decide whether to conduct green production is calculated as follows:

π_{21} = z (x (P_{i} + P_{g} + c_{m i} - T + v (T, b) - c_{g}) + (1 - x) (P_{i} + P_{g} - T + v (T, b) - c_{g})) + (1 - z) (x (P_{i} + P_{g} + c_{m i} - T - c_{g}) + (1 - x) (P_{i} + P_{g} - T - c_{g}))

(5)

π_{22} = z (x (P_{i} + c_{m i} - T) + (1 - x) (P_{i} - T + v (- C) + c_{m f})) + (1 - z) (x (P_{i} + c_{m i} - T) + (1 - x) (P_{i} + c_{m f} - T + v (- C)))

(6)

\bar{π_{2}} = y π_{21} + (1 - y) π_{22}

(7)

F (y) = (- 1 + y) y (c_{g} - (- 1 + x) c_{m f} - P_{g} - (- 1 + x) v (- C) - z v (T, b))

(8)

Where

π_{21}

and

π_{22}

represent the revenues of the manufacturer choosing green or non-green production,

\bar{π_{2}}

represents the average revenue of the manufacturer, and the replicated dynamic equation is denoted as

F (y)

The benefit for government choosing positive guide is:

\begin{aligned} π_{31} = x (y (G - c_{p} - S + v (T, - b) + H_{1} + H_{2} + T) + (1 - y) (G - c_{p} - S + H_{1} - E_{m} + T)) \\ + (1 - x) (y (G - c_{p} + v (F, m) + v (T, - b) + H_{2} - E_{r} + T) + (1 - y) (G - c_{p} + v (F, m) - E_{m} - E_{r} + T)) \end{aligned}

(9)

The benefit for government choosing negative supervision is:

π_{32} = x (y (H_{1} + H_{2} + T) + (1 - y) (H_{1} - E_{m} + T)) + (1 - x) (y (H_{2} - E_{r} + T) + (1 - y) (- E_{r} - E_{m} + T))

(10)

The average benefit of government is:

\bar{π_{3}} = z π_{31} + (1 - z) π_{32}

(11)

The replicate dynamic equation of government is:

F (z) = z (1 - z) (G - S x - c_{p} - (- 1 + x) v (F, m) + y v (T, - b))

(12)

Combining equations (4), (8), and (12), we obtain the three-dimensional replicate dynamic equation of the tripartite evolutionary game as follows:

{\begin{matrix} F (x) = x (1 - x) (v (R_{m}) y + S z - Δ c + y (- 1 + θ) c_{1} + R_{a} + (- 1 + y - y θ) R_{u} - z v (F, - m)) \\ F (y) = y (1 - y) (- c_{g} - (1 - x) c_{m f} + P_{g} - (1 - x) v (- C) + z v (T, b)) \\ F (z) = z (1 - z) (G - S x - c_{p} - (- 1 + x) v (F, m) + y v (T, - b)) \end{matrix}

(13)

Evolutionary game model equilibrium points

According to evolutionary game theory, the decision maker is bounded rationally and their decision would change over time, while as per the prospect theory,³⁴ in uncertain and risky situations, people make decisions that differ from certain situations, people's attitude towards risk influences their decisions, and people's behaviour is predictable. Therefore, we considered the variability of the tripartite decision and calculated the evolutionary stability strategies (ESSs) that the tripartite decisions did not vary with time. To obtain the ESSs, we first calculate the stable points of the tripartite replicate dynamic equation and let equation (13) equal 0, as follows:

{\begin{matrix} F (x) = x (1 - x) (v (R_{m}) y + S z - Δ c + y (- 1 + θ) c_{1} + R_{a} + (- 1 + y - y θ) R_{u} - z v (F, - m)) = 0 \\ F (y) = (- 1 + y) y (c_{g} - (- 1 + x) c_{m f} - P_{g} - (- 1 + x) v (- C) - z v (T, b)) = 0 \\ F (z) = z (1 - z) (G - S x - c_{p} - (- 1 + x) v (F, m) + y v (T, - b)) = 0 \end{matrix}

(14)

We obtained the stable points

E_{1} (0, 0, 0)

E_{2} (0, 0, 1)

E_{3} (0, 1, 0)

E_{3} (0, 1, 1)

E_{4} (1, 0, 0)

E_{1} (1, 0, 1)

E_{1} (1, 1, 0)

E_{1} (1, 1, 1)

and

E_{9} (x *, y *, z *)

, in

E_{9} (x *, y *, z *)

, the value of

x *

y *

and

z *

satisfy the conditions as follows:

{\begin{matrix} (v (R_{m}) y + S z - Δ C + y (- 1 + θ) C_{1} + R_{a} + (- 1 + y - y θ) R_{u} - z v (F, - m)) = 0 \\ (c_{g} - (- 1 + x) c_{m f} - P_{g} - (- 1 + x) v (- C) - z v (T, b)) = 0 \\ (G - S x - c_{p} - (- 1 + x) v (F, m) + y v (T, - b)) = 0 \end{matrix}

(15)

We simplified equation (15) and solved it to obtain the values of

x *

y *

, and

z *

as follows:

\begin{aligned} x * = \frac{- (v (R_{m}) + (- 1 + θ) C_{1} - (- 1 + θ) R_{u}) (G - c_{p} + v (F, m)) v (T, b)}{(c_{m f} + v (- C)) (S - v (F, - m)) v (T, - b) - (v (R_{m}) + (- 1 + θ) C_{1} - (- 1 + θ) R_{u}) (S + v (F, m)) v (T, b)} \\ + \frac{+ v (T, - b) ((c_{g} + c_{m f} - P_{g} + v (- C)) (S - v (F, - m)) - (Δ C - R_{a} + R_{u}) v (T, b))}{(c_{m f} + v (- C)) (S - v (F, - m)) v (T, - b) - (v (R_{m}) + (- 1 + θ) C_{1} - (- 1 + θ) R_{u}) (S + v (F, m)) v (T, b)} \end{aligned}

(16)

\begin{aligned} y * = \frac{(S - v (F, - m)) (c_{m f} (- G + S + c_{p}) - S P_{g} - G v (- C) + S v (- C) + c_{p} v (- C) - P_{g} v (F, m) + c_{g} (S + v (F, m)))}{(c_{m f} + v (- C)) (S - v (F, - m)) v (T, - b) - (v (R_{m}) + (- 1 + θ) C_{1} - (- 1 + θ) R_{u}) (S + v (F, m)) v (T, b)} \\ - \frac{- (Δ C - R_{a} + R_{u}) (S + v (F, m)) v (T, b)}{(c_{m f} + v (- C)) (S - v (F, - m)) v (T, - b) - (v (R_{m}) + (- 1 + θ) C_{1} - (- 1 + θ) R_{u}) (S + v (F, m)) v (T, b)} \end{aligned}

(17)

\begin{aligned} z * = \frac{(v (R_{m}) + (- 1 + θ) C_{1} - (- 1 + θ) R_{u}) (c_{m f} (- G + S + c_{p}) - S P_{g} - G v (- C) + S v (- C) + c_{p} v (- C) - P_{g} v (F, m) + c_{g} (S + v (F, m)))}{(- c_{m f} - v (- C)) (S - v (F, - m)) v (T, - b) + (v (R_{m}) + (- 1 + θ) C_{1} - (- 1 + θ) R_{u}) (S + v (F, m)) v (T, b)} \\ - \frac{- (Δ C - R_{a} + R_{u}) (c_{m f} + v (- C)) v (T, - b)}{(- c_{m f} - v (- C)) (S - v (F, - m)) v (T, - b) + (v (R_{m}) + (- 1 + θ) C_{1} - (- 1 + θ) R_{u}) (S + v (F, m)) v (T, b)} \end{aligned}

(18)

Owing to the value range of the tripartite decision probability,

(x, y, z)

[0, 1] \times [0, 1] \times [0, 1]

. As the parameter values change, the values of

x *

y *

, and

z *

may not be limited to the solution domain

[0, 1] \times [0, 1] \times [0, 1]

, and the tripartite ESSs is asymptotically stable, as it is a pure strategy. Otherwise, the strategy would be unstable; therefore, this study did not consider

E_{9} (x *, y *, z *)

Stability strategies of equilibrium points in evolutionary game mode

In order to verify the stability of $E_{1} (0, 0, 0)$ , $E_{2} (0, 0, 1)$ , $E_{3} (0, 1, 0)$ , $E_{4} (0, 1, 1)$ , $E_{5} (1, 0, 0)$ , $E_{6} (1, 0, 1)$ , $E_{7} (1, 1, 0)$ , $E_{8} (1, 1, 1)$ , we adopted Lyapunov's method and calculated the Jacobian matrix for each stable point. According to this method, the necessary and sufficient condition for the system to achieve ESSs is that all eigenvalues of the Jacobian matrix have negative real parts. Therefore, we first calculated the Jacobian matrix of ESSs using (13) as follows:

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

The elements on the main diagonal among matrix are:

\frac{\partial F (x)}{x} = (1 - 2 x) (v (R_{m}) y + S z - Δ C + y (- 1 + θ) C_{1} + R_{a} + (- 1 + y - y θ) R_{u} - z v (F, - m))

\frac{\partial F (y)}{y} = (- 1 + 2 y) (c_{g} - (- 1 + x) c_{m f} - P_{g} - (- 1 + x) v (- C) - z v (T, b))

\frac{\partial F (z)}{\partial z} = (1 - 2 z) (G - S x - c_{p} - (- 1 + x) v (F, m) + y v (T, - b))

The remaining elements and the specific calculation process for each equilibrium-point Jacobian matrix can also be conducted. The stable conditions of the tripartite evolutionary game system are listed in Table 4.

Table 4.

The stable condition of the tripartite evolutionary game.

Equilibrium point	Evolutionary stability	Stable condition
$E_{1} (0, 0, 0)$	Condition (1)	$\begin{aligned} - Δ C + R_{a} - R_{u} < 0 \\ - c_{g} - c_{m f} + P_{g} - v (- C) < 0 \\ G - c_{p} + v (F, m) < 0 \end{aligned}$

$E_{2} (0, 0, 1)$	Condition (2)	$\begin{aligned} S - Δ C + R_{a} - R_{u} - v (F, - m) < 0 \\ - c_{g} - c_{m f} + P_{g} - v (- C) + v (T, b) < 0 \\ - G + c_{p} - v (F, m) < 0 \end{aligned}$

$E_{3} (0, 1, 0)$	Condition (3)	$\begin{aligned} v (R_{m}) + (- 1 + θ) C_{1} - Δ C + R_{a} - θ R_{u} < 0 \\ c_{g} + c_{m f} - P_{g} + v (- C) < 0 \\ G - c_{p} + v (F, m) + v (T, - b) < 0 \end{aligned}$

$E_{4} (0, 1, 1)$	Condition (4)	$\begin{aligned} S + v (R_{m}) + R_{a} - Δ C - θ R_{u} + (- 1 + θ) C_{1} - v (F, - m) < 0 \\ c_{g} + c_{m f} - P_{g} + v (- C) - v (T, b) < 0 \\ - G + c_{p} - v (F, m) - v (T, - b) < 0 \end{aligned}$

$E_{5} (1, 0, 0)$	Condition (5)	$\begin{aligned} Δ C - R_{a} + R_{u} < 0 \\ - c_{g} + P_{g} < 0 \\ G - S - c_{p} < 0 \end{aligned}$

$E_{6} (1, 0, 1)$	Condition (6)	$\begin{aligned} - S + Δ C - R_{a} + R_{u} + v (F, - m) < 0 \\ - c_{g} + P_{g} + v (T, b) < 0 \\ - G + S + c_{p} < 0 \end{aligned}$

$E_{7} (1, 1, 0)$	Condition (7)	$\begin{aligned} - v (R_{m}) + Δ C - (- 1 + θ) C_{1} - R_{a} + θ R_{u} < 0 \\ c_{g} - P_{g} < 0 \\ G - S - c_{p} + v (T, - b) < 0 \end{aligned}$

$E_{8} (1, 1, 1)$	Condition (8)	$\begin{aligned} - S - v (R_{m}) + Δ C - (- 1 + θ) C_{1} - R_{a} + θ R_{u} + v (F, - m) < 0 \\ c_{g} - P_{g} - v (T, b) < 0 \\ - G + S + c_{p} - v (T, - b) < 0 \end{aligned}$

Tripartite evolutionary path of different life cycle stage

Based on the industry life cycle theory,³⁵ we divided the life cycle of WEEE recycling into three stages: initial, middle, and mature.

In the initial stage, owing to lower recycling costs and higher revenue from unqualified recycling, the penalty for unqualified recycling is less, recyclers’ psychological perceived value of unqualified recycling is lower, and the revenue from IR is higher than that from FR. For manufacturers, the revenue of NP is higher than that of GP, and the manufacturers’ psychological perceived value of the WEEE process cost is less than that of NP. Thus, the government can obtain more benefits for PS, and the benefits are higher than NP. Moreover, recyclers and manufacturers form a stable transaction relationship in which recyclers sell recycled materials to manufacturers, which can save material costs for FR. In summary, the tripartite ESS is (IR, NP, NS). We contracted the evolutionary path of tripartite decision's probability (as shown in Figure 1) with the value of parameters as follows: $R_{a} = 15$ , $R_{u} = 20$ , $S = 2$ , $c_{1} = 5$ , $Δ c = 8$ , $θ = 0.2$ , $F = 15$ , $m = 0.2$ , $P_{g} = 5$ , $c_{m f} = 6$ , $c_{m i} = 4$ , $b = 0.3$ , $c_{g} = 5$ , $C = 2$ , $G = 15$ , $c_{p} = 10$ , $T = 10$ , $R_{m} = 2$ , $α_{1} = 0.88$ , $α_{2} = 0.88$ , $α_{3} = 0.88$ , $λ_{1} = 1.2$ , $λ_{2} = 1.2$ , $λ_{3} = 1.2$ , which satisfy condition (2).

Figure 1.

The evolutionary path of tripartite evolutionary stable strategy $(0, 0, 1)$ .

In the middle stage, the government's benefit from PS is still higher than that of NS, that is, $G - S - c_{p} + v (T, - b) > 0$ . Under PS, WPF is reduced, and manufacturers can obtain a higher corporate social image and benefit from GP, $P_{g} - c_{g} + v (T, b) > 0$ . For recyclers, because manufacturers choose GP and FR, they have more psychological perceived value for qualified recycling. Moreover, government provides PS that subsidizes recyclers for their FR, recycler's revenue of FR is higher than IR, $R_{a} + S - Δ C + v (R_{m}) - C_{1} > θ R_{u} - θ C_{1} + v (F, - m)$ , then recyclers will choose IR. We contract tripartite evolutionary path with parameters as follows: $R_{a} = 15$ , $R_{u} = 20$ , $S = 2$ , $C_{1} = 5$ , $Δ C = 6$ , $θ = 0.2$ , $F = 15$ , $m = 0.2$ , $P_{g} = 11$ , $c_{mf} = 6$ , $c_{mi} = 4$ , $b = 0.2$ , $c_{g} = 5$ , $C = 2$ , $G = 18$ , $c_{p} = 10$ , $T = 15$ , $R_{m} = 2$ , $a_{1} = 0.88$ , $a_{2} = 0.88$ , $a_{3} = 0.88$ , $λ_{1} = 1.2$ , $λ_{2} = 1.2$ , $λ_{3} = 1.2$ , which satisfies condition (8) and the evolutionary path of $E_{8} (1, 1, 1)$ , as shown in Figure 2. As shown in Figure 2, manufacturers choose GP at extremely high speeds and the decision is GP. While the probability of recyclers first decreases and then increases, the decision of recyclers to be stable is FR at the end. The probability of a PS increases rapidly in the beginning and the government's decision is PS for a relatively long time.

Figure 2.

The evolutionary path of tripartite evolutionary stable strategy $(1, 1, 1)$ .

In the mature stage, manufacturers can obtain more corporate social image by GP, that is, $P_{g} - c_{g} > 0$ , manufacturers and recyclers form an environmental-friendly and resource-friendly partnership. Manufacturers will choose FR as recyclers generate more psychological perceived value on FR, the additional investment on FR is less compared to higher revenue. Therefore, recyclers’ profit of FR is higher than IR, that is, $R_{a} - C_{1} - Δ C + v (R_{m}) > θ R_{u} - θ C_{1}$ . For government, as WEEE recycling developed into mature stage, manufacturers will choose GP and form a green partnership with recyclers. Thus, the benefit of government's PS is less than NS, that is, $G - S - c_{p} + v (T, - b) < 0$ . Therefore, the tripartite ESS is (FR, GP, NS). We contract the evolutionary path of tripartite decision with the parameter values as follows: $R_{a} = 15.5$ , $R_{u} = 20$ , $S = 2$ , $C_{1} = 5$ , $Δ C = 6$ , $θ = 0.2$ , $F = 15$ , $m = 0.2$ , $P_{g} = 11$ , $c_{m f} = 6$ , $c_{m i} = 4$ , $b = 0.2$ , $c_{g} = 5$ , $C = 2$ , $G = 13$ , $c_{p} = 10$ , $T = 15$ , $R_{m} = 3$ , $α_{1} = 0.88$ , $α_{2} = 0.88$ , $α_{3} = 0.88$ , $λ_{1} = 1.2$ , $λ_{2} = 1.2$ , $λ_{1} = 1.2$ ,which satisfies condition (7), and the initial probability of tripartite decision is set as 0.5, 0.5 and 0.5, the evolutionary path is shown in Figure 3. As shown in Figure 3, the decision of manufacturers achieves a stable situation at an extremely fast speed, and manufacturers choose GP. Recyclers use a relatively fast speed to achieve stability, and recyclers’ decisions are FR, while the government needs a relatively long time to reach stability, the probability of government for NS increases and then decreases. Finally, the government selects the NS strategy.

Figure 3.

The evolutionary path of tripartite evolutionary stable strategy $(1, 1, 0)$ .

Numerical simulation

To explore the validity of the evolutionary game model and the impact of the main model parameters on tripartite behavioural strategies through sensitivity analysis, the evolutionary path under different parameters, including the WPF exemption rate b, subsidy S, positive guide cost $c_{p}$ , and the marginal diminishing degree of the value function $α$ , are constructed in this section. The parameter values were primarily based on the sales price of Hisense Televisions in JD and the Chinese government's policies in the WEEE area, such as Anhui, Guangdong, and Chongqing. The main sources of the data are the white paper released by China Household Electrical Appliances Research Institute, the Ministry of Environmental Protection of China, and the Ministry of Industry and Information Technology of China, which are all the most authoritative and effective document in the field of household appliances. The refrigerator market has the characteristics of high demand and moderate product replacement time. Scrapped household refrigerators belong to the typical WEEE, and their core component, the refrigeration engine, has reuse value. Considering the timeliness of the data, 2021 data from the refrigerator industry is selected. The MATLAB 2021a software was used to conduct the numerical simulations.

The impact of WPF exemption rate on tripartite decision

First, we set the value of the WPF exemption rate as follows: $b = 0.2$ , $b = 0.4$ , $b = 0.6$ , and $b = 0.8$ , The other parameters are set as shown in Table 5.

Table 5.

The values of parameters.

Parameter	$R_{a}$	$R_{u}$	$S$	$C_{1}$	$Δ C$	$θ$	$F$	$m$	$P_{g}$	$c_{m f}$	$c_{m i}$
Value	15	20	2	5	8	0.2	15	0.2	5	6	4
Parameters	$c_{g}$	$C$	$G$	$c_{p}$	$T$	$R_{m}$	$α_{1}$	$α_{2}$	$α_{3}$	$λ_{1}$	$λ_{2}$
Value	5	2	15	10	10	2	0.88	0.88	0.88	1.2	1.2

As shown in Figure 4, when WPF exemption rate is 0.2, the probability of recyclers choosing FR and manufacturers choosing GP decreases and the probability of government choosing PS increases. When $b = 0.4$ , FR and GP still decreases and PS increases, but the time for tripartite decision to achieve stable is longer than $b = 0.2$ . When WPF exemption rate $b = 0.6$ , FR and GP increases and PS decreases, which means manufacturers are willing to GP strategy because manufacturers can obtain expected profit generated from the relatively high WPF exemption rate. As manufacturers prefer FR, recyclers can obtain more revenue from FR. As recyclers and manufacturers choose green strategy, WEEE recycling achieves optimal situation, the profit of government PS is less than NS and government can choose NS. As WPF exemption rate b reaches 0.8, government's subsidy capacity decreases and recyclers’ intention of FR decreases due to extremely high WPF exemption, but the revenue generated from FR makes recyclers still obtain more revenue from FR than IR. Therefore, recyclers use relatively longer time for FR strategy. For manufacturers, due to extremely higher WPF exemption, the additional profit of GP is less than WPF exemption, hence, manufacturers’ probability of GP decreases.

Figure 4.

The impact of WPF exemption rate on tripartite decision. (a) Evolutionary path of recyclers. (b) Evolutionary path of manufacturers. (c) Evolutionary path of government. (d) Evolutionary path of tripartite decision system.

In summary, the WPF exemption rate should be appropriate. An extremely low WPF exemption cannot incentivise manufacturers and recyclers to choose GP and FR. However, an extremely high WPF exemption rate will reduce the government's financial capacity to subsidise recyclers, and manufacturers’ decisions will be influenced by the government's supervision attitude. When the WPF exemption rate is appropriate, the manufacturers choose GP which can lead recyclers to choose FR by increasing the demand for FR, and the tripartite decision evolves into an optimal strategy.

The impact of subsidy on tripartite decision

To verify the impact of the subsidy on the tripartite decision, the values of the parameters are set as listed in Table 6, which meet the conditions of the initial periods that correspond to $(0, 0, 1)$ , and the values of the subsidy are set as $S = 1$ , $S = 2$ , $S = 3$ , $S = 4$ , and $S = 5$ .

Table 6.

The values of parameters.

Parameter	$R_{a}$	$R_{u}$	$C_{1}$	$Δ C$	$θ$	$F$	$m$	$P_{g}$	$c_{m f}$	$c_{m i}$	$b$
Value	15	20	5	8	0.2	15	0.2	5	6	4	0.3
Parameters	$c_{g}$	$C$	$c_{p}$	$T$	$R_{m}$	$α_{1}$	$α_{2}$	$α_{3}$	$λ_{1}$	$λ_{2}$	$λ_{3}$
Value	5	2	10	10	2	0.88	0.88	0.88	1.2	1.2	1.2

As shown in Figure 5(a)–(c), when $S = 1$ and $S = 2$ , compared to unqualified recycling, the profit of qualified recycling with less subsidy is lower, the subsidy can’t motive recyclers to carry out qualified recycling. The recyclers’ perceived psychological value of the penalty for IR is lower than that for IR, and recyclers prefer to choose IR for more profit. Due to more profit generated from NP and manufacturers’ psychological perceived value of the WPF exemption, manufacturers choose NP. Moreover, the government chooses PS because the benefits of PS are greater than those of NS.

Figure 5.

The impact of subsidy on tripartite decision. (a) Evolutionary path of recyclers. (b) Evolutionary path of manufacturers. (c) Evolutionary path of government. (d) Evolutionary path of tripartite decision system.

When $S = 3$ , recyclers will choose FR and manufacturers will choose GP because of more subsidy and higher psychological perceived value on the penalty of IR, the government's psychological perceived value on subsidising recyclers, and waiving manufacturers’ WPF is low, and the government will still choose PS. When $S = 4$ , recyclers choose FR, and manufacturers choose GP. As recyclers and manufacturers choose green behaviour, the government will choose NM to maintain the current situation. When $S = 5$ , recyclers will choose FR. However, the extremely high subsidy reduces the government's financial capacity to waive manufacturers’ WPF. Manufacturers cannot obtain the expected profit, and the probability of manufacturers choosing GP decreases. However, because recyclers choose FR, manufacturers can obtain more profit with FR than with the NP strategy. Therefore, the probability of a manufacturer choosing a GP is stable from zero to one. Because recyclers and manufacturers choose green behaviour, the government chooses the NS strategy.

The impact of PS cost on tripartite decision

To verify the impact of the positive guide cost on the tripartite decision, the value of the parameter is set as follows: $c_{p} = 6, 11, 16, 21$ , and the values of the other parameters are set as listed in Table 7.

Table 7.

The values of parameters.

Parameter	$R_{a}$	$R_{u}$	$S$	$Δ C$	$θ$	$F$	$m$	$P_{g}$	$c_{m f}$	$c_{m i}$	$b$
Value	15	20	2	8	0.2	15	0.2	5	6	4	0.3
Parameters	$c_{g}$	$C$	$G$	$T$	$R_{m}$	$α_{1}$	$α_{2}$	$α_{3}$	$λ_{1}$	$λ_{2}$	$λ_{3}$
Value	5	2	15	10	2	0.88	0.88	0.88	1.2	1.2	1.2

As shown in Figure 6, when S, used as a positive guide, increased from 6 to 21, the time required for FR and GP to evolve to a stable state decreased. This is because when the positive guiding cost is higher, the government's financial capacity to subsidise recyclers and exemption manufacturers’ WPF is weakened, and recyclers and manufacturers have little time to choose IR and NP with relatively little subsidy. For the government, as the positive guide cost increases, the benefit of PS becomes less than that of NS, and the government chooses NS.

Figure 6.

The impact of PS cost on tripartite decision.

The impact of decision maker's security risk pursuit degree

To explore the impact of the decision-maker's degree of security risk pursuit on the tripartite decision, the evolutionary path of the tripartite is constructed using $α_{1} = α_{2} = α_{3} = 0.1, 0.3, 0.5, 0.7, 0.9$ , and the other parameters are set as listed in Table 8.

Table 8.

The values of parameters.

Parameter	$R_{a}$	$R_{u}$	$S$	$C_{1}$	$Δ C$	$θ$	$F$	$m$	$P_{g}$	$c_{m f}$	$c_{m i}$	$b$
Value	15	20	2	5	8	0.2	15	0.2	5	6	4	0.3
Parameters	$c_{g}$	$C$	$G$	$c_{p}$	$T$	$R_{m}$	$α_{1}$	$α_{2}$	$α_{3}$	$λ_{1}$	$λ_{2}$	$λ_{3}$
Value	5	2	15	10	10	2	V	V	V	1.2	1.2	1.2

As shown in Figure 7, when the decision-maker's security risk pursuit degree varies from 0.1 to 0.9, FR and GP use more time to reach a stable state, while the trend of PS has not changed. This means that as the decision-maker's security risk pursuit degree increases, the psychological perceived value of recyclers and manufacturers on risk increases and they prefer to take risks. However, the government needs to subsidise recyclers as FR and penalise recyclers as IR, and the government reduces manufacturers’ WPF as GP; thus, the benefit of the government is not influenced. Therefore, the evolutionary path of government decisions has not changed.

Figure 7.

Impact of decision maker's security risk pursuit degree on tripartite decision.

Discussion and management implications

Discussion

The proposed approach with the triangular evolutionary game model can facilitate the issue of the WEEE recycling network under variable subsidy policies and enable an understanding of the stakeholders responsible for the behavioural strategy of the WEEE recycling network. This can eventually be implemented in the subsidy mechanism and government guide towards corporate integration. For example, referring to Section 4, when government formulates policy to subsidize recyclers or manufacturers, the decision of others will also be impacted.^36,37 However, such systems may not perform equally well in areas where WEEE recycling technology and customer satisfaction are also important.³⁸

In addition, we found that when the government exemption manufacturers’ WPF, and as the exemption rate increases, the direct effect is that manufacturers choose the NP and GP strategies in the end. The indirect effect is that the recyclers’ decisions changed, the trends for recyclers choosing IR slowed, and as the wavier rate of WPF reached an appropriate level, the recyclers chose FR. As the exemption rate of WPF is extremely high, the probability of manufacturers choosing GP decreases, and the probability of recyclers choosing FR requires a longer time to reach a stable state. Similarly, as the value of the subsidy increases, recyclers would choose FR, and manufacturers would choose GP. As the subsidy given to recyclers increases (FR, GP, NS), recyclers and manufacturers would choose green behaviours, and the government will choose NM to maintain the current situation. However, the stakeholders responsible for WEEE recycling networks generally differ and are less sensitive to geographical dimensions.³⁹

We also found that when the government subsidises recyclers or manufacturers, others’ decisions are influenced.^40–42 Moreover, when the government subsidises recyclers or manufacturers appropriately, both the recyclers and manufacturers choose green behaviour. This is because there is an interactive relationship of interest between recyclers and manufacturers, and one decision influences the other's benefits and decisions. According to the decision-maker's degree of security risk pursuit in tripartite decisions, when recycling is regulated, the government has a strong tolerance for risks, and the decision-maker's psychology perceives that the value of risks has little influence on their decision-making. As profit-oriented units with weak risk-taking ability, recyclers and manufacturers prefer a safe and stable profit strategy and avoid risk which can be found in the analysis of the impact of the decision-maker's degree of security risk pursuit. However, the subsidy mechanism may not be effective for open-loop WEEE recycling.⁴³

Additionally, as recyclers’ psychological perceived value of penalty increases, recyclers and manufacturers prefer to choose green behaviours, and their decisions eventually evolve and stabilise at FR and GP.^3,44,45 The proposed subsidy mechanism can motivate motivation for sustainable recycling of WEEE. In all, the variable subsidy mechanisms and cooperation strategies between recyclers and manufacturers are dynamic and can vary across the quantity and quality of WEEE under different government policies. The triangular evolution game theoretic approach could be applied in the future to identify the optimal strategy for governments in WEEE based on different recycling stages and to assist in developing recycling networks for WEEE. However, putting this numerical approach into practice requires a tracking system to enable the calculation by answering the questions, such as what recycling channel and mechanism is opted for by recyclers and government and the flow of WEEE in different recycling stage.

Management implications

Based on our research and the discussion above, the following managerial implications are proposed.

Adjust subsidy policies to alleviate financial pressure. Specifically, the government can consider the mutual influence between the decisions of recyclers and manufacturers by adjusting the subsidy policies and subsidy intensity of recycling entities to incentivise a certain network entity to choose green behaviour and improve the green behaviour choices of other network entities.

The government should establish reasonable subsidies for recyclers, impose appropriate carbon taxes and WEEE treatment fees on manufacturers, and adopt effective regulations for WEEE recycling networks. Government should attempt to reduce active regulation costs as much as possible while relying solely on strengthening regulatory efforts, which is not conducive to the development of WEEE recycling networks. Excessive regulatory costs are not conducive to incentivising recyclers and manufacturers to adopt green behaviours and improve green standards.

Maintaining the stability of the WEEE recycling network and improving its risk-bearing capacity The government should strive to build a safe and stable WEEE recycling network environment and provide reasonable and timely subsidies, WEEE processing costs, and carbon taxes to recyclers and manufacturers to improve the risk-bearing capacity of network subjects and encourage recyclers and manufacturers to adopt green behaviour.

Conclusions

All in all, this study found that there is an interaction between the decision-making processes of the WEEE tripartite recycling network. Constructing a cost-sharing contract and formulating appropriate cost-sharing coefficients have a positive effect on refining the green level of both parties, and the interests of both parties can also be improved. It was also found that changing subsidy policies can effectively motivate WEEE recycling network participants to choose green behaviour strategies and improve their green levels. In addition, it was determined that when the government subsidises one of the recyclers or manufacturers with reasonable subsidy values, the behavioural strategy of the other party evolves towards a green behavioural strategy.

However, this study had some limitations. It is assumed that the government, recyclers, and manufacturers have the same degree of security risk pursuit and risk aversion coefficient. In future research, the degree of decision-making security risk pursuit and the risk aversion coefficient of the government, recyclers, and manufacturers can be assumed to have different values to explore the impact of tripartite decisions. In addition, further research could focus on the consumer green awareness affecting behavioural strategy on WEEE recycling systems.

Footnotes

Correction (March 2024):

The funding number was missing in the online version. The article has now been updated.

Declaration of conflicting interests

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

Funding

The authors disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: This work is supported by Key R&D Program (Soft Science Project) of Shandong Province (2023RKY05017).

ORCID iD

Qiang Sun

Yujiao Sui received her BS degree in Accounting from Shandong University of Technology in 2020. She is currently working toward an MS degree in Industrial Engineering with Shandong University of Technology from 2022. Her research interests include waste management and supply chain management

Qiang Sun received his BS degree in Transportation from Ludong University in 2003 and a PhD degree in Management Science and Engineering from Liaoning Technical University in 2017. His research interests include waste management and supply chain management.

Shuhao Li received his BS degree in Industrial Engineering from Shandong University of Technology in 2019 and an MS degree in Management Science and Engineering from Shandong University of Technology in 2023. His research interests include waste management and supply chain management.

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