Evolutionary game analysis of WEEE recycling tripartite stakeholders under variable subsidies and processing fees

Abstract

The standardization of formal recycling and rational subsidy plays an important role in waste electrical and electronic equipment recycling. In order to explore the tripartite decision and evolution path of waste electrical and electronic equipment recycling in different time periods, a tripartite evolutionary game model consisting of recyclers, manufacturers, and government are presented. Moreover, the evolution stability strategies and conditions in each period are calculated by replicating the dynamic equation and Jacobian matrix. Numerical simulations on tripartite evolution stability strategies corresponding to different stages of industry development are used to verify the rationality of the model. The results indicate that there is existed an indirect effect between tripartite decisions, and the indirect effect can expand the slack of tripartite decisions’ thresholds of waste electrical and electronic equipment recycling. The variable subsidy in waste electrical and electronic equipment recycling proposed in this paper is useful to incentive recyclers to choose a formal recycling strategy, and manufacturers also choose production with recycled materials as subsidy varies. Besides, the appropriate waste electrical and electronic equipment processing fee is a conducive indirect effect for the tripartite decision to the optimal evolutionary stability strategy in waste electrical and electronic equipment recycling and can promote manufacturers to produce with the recycled materials. The research can assist in benefit coordination and behavior adjustment of waste electrical and electronic equipment recycling members and provide a theoretical basis for the government to formulate appropriate recycling subsidies to promote the formal recycling of electronic waste recycling.

Appendices

Appendix 1

For the manufacturer, it is assumed that E₁ represents the expected benefit of the manufacturer, E₁₁ and E₁₂ were selected to represent the expected benefit of the manufacturer products goods with the recycled materials and output with the new materials. The expected benefit of the manufacturer is E₁ = xE₁₁ + (1 − x)E₁₂.

And the expected benefit of producing with recycled materials is

$${E}_{11}=I2+I1y-I2y+f\left(-1+z+r2\left(-1+y\right)z-r1 yz\right).$$

The expected benefit of producing with new materials is

$${E}_{12}=-f+I3+y\left(I1-I3-f\left(-1+r1\right)z\right)\theta$$

Then E₁ = I3 − I3x + x(I2 + I1y − I2y) + f(−1 + x(1 + r2(−1 + y) − r1y)z) + (−1 + x)y(−I1 + I3 + f(−1 + r1)z)θ

Similarly for the recycler, the parameter E₂ represents the expected benefit of the recycler, and it is assumed that E₂₁ and E₂₂were selected to represent the expected benefit of the recycler choosing formal recycling and informal recycling.

The expected benefit of recycler is E₂ = xE₂₁ + (1 − x)E₂₂.The expected benefit of formal recycling is

$${E}_{21}=R1\left(\zeta +\theta -x\left(-1+\zeta +\theta \right)\right)-k{\varphi}^2-\left(-\zeta -\theta +x\left(-1+\zeta +\theta \right)\right)\left(1+ z\varphi \right){f}_c$$

And the expected benefit of informal recycling is E₂₂ =  − Fmz + R2(x + ζ − xζ)

Then,E₂ = (−1 + y)(−R2x + Fmz + R2(−1 + x)ζ) + y(R1(ζ + θ − x(−1 + ζ + θ)) − kφ² − (−ζ − θ + x(−1 + ζ + θ))(1 + zφ)f_c)

Finally, same as the manufacturer and the recycler, it is assumed the expected benefit of the government is E₃.

$${E}_3={zE}_{31}+\left(1-z\right){E}_{32}$$

Where the parameters E₃₁ and E₃₂ is the expected benefit of government active management and the expected benefit of government negative management.

$${\displaystyle \begin{array}{l}{E}_{31}=\left(\begin{array}{l}-C1- Er+f+G+ Fm+\left( Er+L1+f\left(-1+r2\right)\right)x+ Ee\left(-1+y\right)\\ {}+y\left(H1- Fm+f\left(r1-r2\right)x-f\left(-1+r1\right)\left(-1+x\right)\theta \right)+y\left(-\zeta -\theta +x\left(-1+\zeta +\theta \right)\right)\left(1+\varphi \right){f}_c\end{array}\right)\\ {}{E}_{32}=- Er+f+\left( Er+L1\right)x+ Ee\left(-1+y\right)+H1y+y\left(-\zeta -\theta +x\left(-1+\zeta +\theta \right)\right){f}_c\end{array}}$$

Then,

$${E}_3=\left(\begin{array}{l}f+ Er\left(-1+x\right)+L1x+ Ee\left(-1+y\right)+\left(-C1+G+ Fm+f\left(-1+r2\right)x\right)z\\ {}+y\left(H1+z\left(- Fm+f\left(r1-r2\right)x-f\left(-1+r1\right)\left(-1+x\right)\theta \right)\right)+y\left(-\zeta -\theta +x\left(-1+\zeta +\theta \right)\right)\left(1+ z\varphi \right){f}_c\end{array}\right)$$

Appendix 2

The calculation process of other elements in the matrix is shown as follows:

$${\displaystyle \begin{array}{l}\frac{\partial F(x)}{\partial y}=\left(-1+x\right)x\left(I2+I1\left(-1+\theta \right)-I3\theta + fz\left(r1-r2+\theta -r1\theta \right)\right)\\ {}\frac{\partial F(x)}{\partial z}=\left(-1+x\right)x\left(f\left(-1+r2\right)+ fy\left(r1-r2+\theta -r1\theta \right)\right)\\ {}\begin{array}{l}\frac{\partial F(y)}{\partial x}=\left(-1+y\right)y\left(\left(R1-R2\right)\left(-1+\zeta \right)+R1\theta +\left(-1+\zeta +\theta \right)\left(1+ z\varphi \right){f}_c\right)\\ {}\frac{\partial F(y)}{\partial z}=\left(-1+y\right)y\left(- Fm+\left(-\zeta -\theta +x\left(-1+\zeta +\theta \right)\right)\varphi {f}_c\right)\\ {}\begin{array}{l}\frac{\partial F(z)}{\partial x}=\left(-1+z\right)z\left(f\left(1+r2\left(-1+y\right)-r1y\right)+f\left(-1+r1\right) y\theta +y\left(1-\zeta -\theta \right)\varphi {f}_c\right)\\ {}\frac{\partial F(z)}{\partial y}=\left(-1+z\right)z\left( Fm+f\left(-r1+r2\right)x+f\left(-1+r1\right)\left(-1+x\right)\theta +\left(\zeta +\theta -x\left(-1+\zeta +\theta \right)\right)\varphi {f}_c\right)\end{array}\end{array}\end{array}}$$