# Dynamic Research on Three-Player Evolutionary Game in Waste Product Recycling Supply Chain System

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## Abstract

**:**

## 1. Introduction

#### 1.1. Background of The Work

#### 1.2. Motivation of The Work

#### 1.3. Contributions of the Work

## 2. Literature Review

## 3. Model and Assumption

#### 3.1. Problem Definition

#### 3.2. Decision Framework

#### 3.3. Assumptions

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Assumption**

**4.**

**Assumption**

**5.**

#### 3.4. Using the Evolutionary Stability Strategy of Replication Dynamic Equations to Solve

## 4. Asymptotic Stability of the Equilibrium Points and Evolutionary Stability Strategies

**Theorem**

**1.**

**Theorem**

**2.**

**Theorem**

**3.**

## 5. Numerical Simulation

#### 5.1. The Stability Simulation of Equilibrium Point

#### 5.2. Impacts of Changes in Government Subsidies and Advertising on Evolutionary Paths

#### 5.3. Impacts of Changes in Tax Coefficient on Evolutionary Paths

#### 5.4. Impacts of Changes in Liquidated Damages on Evolutionary Paths

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 3.**Phase diagram of Theorem 1. (Note: The direction of the arrow in the figure represents the evolution path of the strategies, and the intersection of all lines indicates the evolution stability strategy (ESS)).

**Figure 4.**Phase diagram of Theorem 2. (Note: The direction of the arrow in the figure represents the evolution path of the strategies, and the intersection of all lines indicates the evolution stability strategy (ESS)).

**Figure 5.**Phase diagram of Theorem 3. (Note: The direction of the arrow in the figure represents the evolution path of the strategies, and the intersection of all lines indicates the evolution stability strategy (ESS)).

Literatures | Closed-Loop Supply Chain | Cooperation for Online and Offline | Waste Products Recovery | Evolutionary Game | Government Participation |
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[40] | √ | √ | |||

[41] | √ | √ | |||

[37] | √ | √ | √ | ||

[39] | √ | √ | √ | ||

[43] | √ | ||||

[44] | √ | √ | √ | ||

[49] | √ | √ | √ | ||

[42] | √ | √ | |||

[45] | √ | ||||

[46] | √ | ||||

[36] | √ | √ | √ | ||

[48] | √ | ||||

[47] | √ | ||||

Our study | √ | √ | √ | √ | √ |

P | |||
---|---|---|---|

Cooperate | No-Cooperate | ||

M | Cooperate | ${R}_{G}-A-S$ | ${R}_{G}-A-S+\delta K$ |

${R}_{M}+\alpha \left(R+A\right)-t\left(C-S\right)$ | ${R}_{M}-t\left(C-S\right)+\alpha A$ + V | ||

${R}_{P}+\left(1-\alpha \right)\left(R+A\right)-$ $\left(1-t\right)\left(C-S\right)$ | ${R}_{P}-\delta K-V$ | ||

No-cooperate | ${R}_{G}-A-S+\beta K$ | ${R}_{G}-A-S+\delta K$+$\beta K$ | |

${R}_{M}-\beta K-W$ | ${R}_{M}-\beta K$ | ||

${R}_{P}+\left(1-\alpha \right)A-$ $\left(1-t\right)\left(C-S\right)+W$ | ${R}_{P}-\delta K$ |

P | |||
---|---|---|---|

Cooperate | No-Cooperate | ||

M | Cooperate | $b{R}_{G}$ | $b{R}_{G}$ |

${R}_{M}+\alpha R-tC$ | ${R}_{M}-tC+V$ | ||

${R}_{P}+\left(1-\alpha \right)R-\left(1-t\right)C$ | ${R}_{P}-V$ | ||

No-cooperate | $b{R}_{G}$ | $b{R}_{G}$ | |

${R}_{M}-W$ | ${R}_{M}$ | ||

${R}_{P}-\left(1-t\right)C+W$ | ${R}_{P}$ |

Equilibrium | $\mathbf{Eigenvalue}{\mathit{\lambda}}_{1}$ | $\mathbf{Eigenvalue}{\mathit{\lambda}}_{2}$ | $\mathbf{Eigenvalue}{\mathit{\lambda}}_{3}$ |
---|---|---|---|

${E}_{1}\left(0,0,0\right)$ | $\left(1-b\right){R}_{G}-S-A+\beta K+K\delta $ | $V-tC$ | $C\left(-1+t\right)+W$ |

${E}_{2}\left(0,1,0\right)$ | $\left(1-b\right){R}_{G}-S-A+K\delta $ | $-\left(V-tC\right)$ | $V+\left(1-\alpha \right)R-\left(1-t\right)C$ |

${E}_{3}\left(0,0,1\right)$ | $\left(1-b\right){R}_{G}-S-A+\beta K$ | $W+\alpha R-tC$ | $\left(1-t\right)C-W$ |

${E}_{4}\left(0,1,1\right)$ | $\left(1-b\right){R}_{G}-A-S$ | $-\left(W+\alpha R-tC\right)$ | $-\left(V+\left(1-\alpha \right)R-\left(1-t\right)C\right)$ |

${E}_{5}\left(1,0,0\right)$ | $\left(-1+b\right){R}_{G}+S+A-\beta K-\delta K$ | $-t\left(C-S\right)+V+\alpha A+\beta K$ | $\left(1-\alpha \right)A-\left(1-t\right)\left(C-S\right)+W+\delta K$ |

${E}_{6}\left(1,1,0\right)$ | $\left(-1+b\right){R}_{G}+S+A-\delta K$ | $\left(C-S\right)t-V-\alpha A-\beta K$ | $\left(1-\alpha \right)\left(R+A\right)-\left(1-t\right)\left(C-S\right)+V+K\delta $ |

${E}_{7}\left(1,0,1\right)$ | $-\left[\left(1-b\right){R}_{G}-S-A+\beta K\right]$ | $-t\left(C-S\right)+W+\left(R+A\right)\alpha +\beta K$ | $-\left[\left(1-\alpha \right)A-\left(1-t\right)\left(C-S\right)+W+\delta K\right]$ |

${E}_{8}\left(1,1,1\right)$ | $-\left[\left(1-b\right){R}_{G}-S-A\right]$ | $-\left(\alpha \left(R+A\right)-t\left(C-S\right)+W+\beta K\right)$ | $-\left[\left(1-\alpha \right)\left(R+A\right)-\left(1-t\right)\left(C-S\right)-V+\delta K\right]$ |

Equilibrium | Case 1 | Case 2 | Case 3 | |||
---|---|---|---|---|---|---|

${\mathit{\lambda}}_{1},{\mathit{\lambda}}_{2},{\mathit{\lambda}}_{3}$ | Stability | ${\mathit{\lambda}}_{1},{\mathit{\lambda}}_{2},{\mathit{\lambda}}_{3}$ | Stability | ${\mathit{\lambda}}_{1},{\mathit{\lambda}}_{2},{\mathit{\lambda}}_{3}$ | Stability | |

${E}_{1}\left(0,0,0\right)$ | $\left(+,+,+\right)$ | Saddle point | $\left(+,-,-\right)$ | Unstable point | $\left(+,-,-\right)$ | Unstable point |

${E}_{2}\left(0,1,0\right)$ | $\left(+,-,+\right)$ | Unstable point | $\left(+,+,+\right)$ | Saddle point | $\left(+,+,+\right)$ | Saddle point |

${E}_{3}\left(0,0,1\right)$ | $\left(+,+,-\right)$ | Unstable point | $\left(+,+,+\right)$ | Saddle point | $\left(+,+,+\right)$ | Saddle point |

${E}_{4}\left(0,1,1\right)$ | $\left(+,-,-\right)$ | Unstable point | $\left(+,-,-\right)$ | Unstable point | $\left(+,-,-\right)$ | Unstable point |

${E}_{5}\left(1,0,0\right)$ | $\left(-,+,+\right)$ | Unstable point | $\left(-,-,-\right)$ | ESS | $\left(-,+,+\right)$ | Unstable point |

${E}_{6}\left(1,1,0\right)$ | $\left(-,-,+\right)$ | Unstable point | $\left(-,+,+\right)$ | Unstable point | $\left(-,-,+\right)$ | Unstable point |

${E}_{7}\left(1,0,1\right)$ | $\left(-,+,-\right)$ | Unstable point | $\left(-,+,+\right)$ | Unstable point | $\left(-,+,-\right)$ | Unstable point |

${E}_{8}\left(1,1,1\right)$ | $\left(-,-,-\right)$ | ESS | $\left(-,-,-\right)$ | ESS | $\left(-,-,-\right)$ | ESS |

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**MDPI and ACS Style**

Xie, B.; An, K.; Cheng, Y.
Dynamic Research on Three-Player Evolutionary Game in Waste Product Recycling Supply Chain System. *Systems* **2022**, *10*, 185.
https://doi.org/10.3390/systems10050185

**AMA Style**

Xie B, An K, Cheng Y.
Dynamic Research on Three-Player Evolutionary Game in Waste Product Recycling Supply Chain System. *Systems*. 2022; 10(5):185.
https://doi.org/10.3390/systems10050185

**Chicago/Turabian Style**

Xie, Bo, Keyu An, and Yingying Cheng.
2022. "Dynamic Research on Three-Player Evolutionary Game in Waste Product Recycling Supply Chain System" *Systems* 10, no. 5: 185.
https://doi.org/10.3390/systems10050185