# Stability Analysis of Low-Carbon Technology Innovation Cooperation under a Reward and Punishment Mechanism

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## Abstract

**:**

## 1. Introduction

## 2. Review of the Literature

#### 2.1. A Study of the Evolution of LCTI

#### 2.2. Research Methodology of LCTI

## 3. Model Construction

#### 3.1. Basic Assumptions

**Assumption**

**1.**

**Assumption**

**2.**

**Assumption**

**3.**

**Assumption**

**4.**

#### 3.2. Model Establishment

#### 3.3. Replicator Dynamic Analysis

#### 3.3.1. Stability Analysis of Enterprise A

#### 3.3.2. Stability Analysis of Enterprise B

#### 3.3.3. Stability Analysis of Government

#### 3.4. Stability Analysis of Equilibrium Points

- ${n}_{11}=\left(2x-1\right)\left\{\left(1-y\right)\Delta I-z\left. [{L}_{1}+\alpha \beta W\left({P}_{1}-{P}_{2}\right)\right]-\alpha W\left({P}_{1}-{P}_{2}\right)-\Delta W\right\}$
- ${n}_{12}=x\left(1-x\right)\Delta I$
- ${n}_{13}=x\left(1-x\right)\left. [{L}_{1}+\alpha \beta W\left({P}_{1}-{P}_{2}\right)\right]$
- ${n}_{21}=y\left(1-y\right)\Delta I$
- ${n}_{22}=\left(2y-1\right)\left\{\left(1-x\right)\Delta I-z\left. [{L}_{2}+\beta W\left({P}_{1}-{P}_{2}\right)\left(1-\alpha \right)\right]+\Delta W+W\left({P}_{1}-{P}_{2}\right)\left(\alpha -1\right)\right\}$
- ${n}_{23}=y\left(1-y\right)\left. [{L}_{2}+\beta W\left({P}_{1}-{P}_{2}\right)\left(1-\alpha \right)\right]$
- ${n}_{31}=-z\left(1-z\right)\left({L}_{1}+{S}_{1}y\right)$
- ${n}_{32}=z\left(1-z\right)\left. [{S}_{1}-{S}_{1}x-{L}_{2}\right]$
- ${n}_{33}=\left(2z-1\right)\left\{y\left. [{L}_{2}+{S}_{1}\left(x-1\right)\right]+\left(x-1\right){L}_{1}-{L}_{2}-E\right\}$

## 4. Numerical Simulation and Results

#### 4.1. The Dynamic Evolution of ESS

#### 4.2. Punishment Analysis

#### 4.3. Parametric Analysis of Coefficient Parameters

#### 4.4. Reward Analysis

#### 4.5. Cost Analysis

## 5. Discussion and Implications

#### 5.1. Reward and Punishment Mechanism of LCTI

#### 5.2. Policy Implications

## 6. Conclusions and Limitation

#### 6.1. Conclusions

#### 6.2. Limitation

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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Parameters | Meanings |
---|---|

$W$ | Market value of LCTI. |

$\Delta W$ | Extra gains earned by firms from total market returns when they choose opportunism (hence also the losses incurred by reciprocal firms). |

$I$ | Total cost of investment required for corporate LCTI. |

$\Delta I$ | Additional cost to companies when they choose reciprocity for LCTI. |

$\alpha $ | Proportion of the share of enterprise A in the market. |

$\beta $ | Government’s carbon tax subsidy for LCTI. |

${P}_{1}$ | Probability of LCTI success under reciprocity. |

${P}_{2}$ | Probability of LCTI success under opportunism. |

$E$ | Environmental and economic gains for the government when participating in LCTI. |

${E}_{11}$ | Additional benefits to the government when both enterprises A and B practice reciprocity. |

${E}_{10}$ | Additional benefits to the government when only one enterprise chooses reciprocity. |

${L}_{1}$ | Government penalties for enterprise A when it chooses opportunism. |

${L}_{2}$ | Government penalties for enterprise B when it chooses opportunism. |

${S}_{1}$ | Environmental and economic losses incurred by the government when one business chooses opportunism and the other reciprocity. |

${S}_{2}$ | Environmental and economic gains for the government when both enterprises choose opportunism. |

Enterprise A | Enterprise B | Government | |
---|---|---|---|

Participation | Non-Participation | ||

Reciprocity | Reciprocity | $\left({A}_{x},{B}_{y},{G}_{z}\right)$ | $\left({A}_{x},{B}_{y},{G}_{1-z}\right)$ |

Opportunism | $\left({A}_{x},{B}_{1-y},{G}_{z}\right)$ | $\left({A}_{x},{B}_{1-y},{G}_{1-z}\right)$ | |

Opportunism | Reciprocity | $\left({A}_{1-x},{B}_{y},{G}_{z}\right)$ | $\left({A}_{1-x},{B}_{y},{G}_{1-z}\right)$ |

Opportunism | $\left({A}_{x1-},{B}_{1-y},{G}_{z}\right)$ | $\left({A}_{1-x},{B}_{1-y},{G}_{1-z}\right)$ |

Strategy Combination | Enterprise A | Enterprise B | Government |
---|---|---|---|

$\left({A}_{x},{B}_{y},{G}_{z}\right)$ | $\left(1+\beta \right)\alpha {P}_{1}W-\alpha I$ | $\left(1+\beta \right)(1-\alpha ){P}_{1}W-\left(1-\alpha \right)I$ | $E+{E}_{11}$ |

$\left({A}_{x},{B}_{y},{G}_{1-z}\right)$ | $\alpha {P}_{1}W-\alpha I$ | $(1-\alpha ){P}_{1}W-\left(1-\alpha \right)I$ | ${E}_{11}$ |

$\left({A}_{x},{B}_{1-y},{G}_{z}\right)$ | $\left(1+\beta \right)\alpha {P}_{1}W-\alpha I-\Delta W-\Delta I$ | $\left(1+\beta \right)(1-\alpha ){P}_{2}W-\left(1-\alpha \right)I+\Delta W-{L}_{2}$ | $E+{E}_{10}+{L}_{2}-{S}_{1}$ |

$\left({A}_{x},{B}_{1-y},{G}_{1-z}\right)$ | $\alpha {P}_{1}W-\alpha I-\Delta W-\Delta I$ | $(1-\alpha ){P}_{2}W-\left(1-\alpha \right)I+\Delta W$ | ${E}_{10}-{S}_{1}$ |

$\left({A}_{1-x},{B}_{y},{G}_{z}\right)$ | $\left(1+\beta \right)\alpha {P}_{2}W-\alpha I+\Delta W-{L}_{1}$ | $\left(1+\beta \right)(1-\alpha ){P}_{1}W-\left(1-\alpha \right)I-\Delta W-\Delta I$ | $E+{E}_{10}+{L}_{1}-{S}_{1}$ |

$\left({A}_{1-x},{B}_{y},{G}_{1-z}\right)$ | $\alpha {P}_{2}W-\alpha I+\Delta W$ | $(1-\alpha ){P}_{1}W-\left(1-\alpha \right)I-\Delta W-\Delta I$ | ${E}_{10}-{S}_{1}$ |

$\left({A}_{x1-},{B}_{1-y},{G}_{z}\right)$ | $\left(1+\beta \right)\alpha {P}_{2}W-\alpha I-{L}_{1}$ | $\left(1+\beta \right)(1-\alpha ){P}_{2}W-\left(1-\alpha \right)I-{L}_{2}$ | $E+{L}_{1}+{L}_{2}-{S}_{2}$ |

$\left({A}_{1-x},{B}_{1-y},{G}_{1-z}\right)$ | $\alpha {P}_{2}W-\alpha I$ | $(1-\alpha ){P}_{2}W-\left(1-\alpha \right)I$ | $-{S}_{2}$ |

Equilibrium Points | Eigenvalues | Eigenvalue Symbol | State | Stability Condition |
---|---|---|---|---|

${E}_{1}\left(0,0,0\right)$ | $\alpha W\left({P}_{1}-{P}_{2}\right)-\Delta I-\Delta W$ | uncertain | Unstable point | — |

$W\left({P}_{1}-{P}_{2}\right)\left(1-\alpha \right)-\Delta I-\Delta W$ | uncertain | |||

${L}_{1}+{L}_{2}+E$ | $+$ | |||

${E}_{2}\left(0,0,1\right)$ | ${L}_{1}+\alpha W\left({P}_{1}-{P}_{2}\right)\left(\beta +1\right)-\Delta I-\Delta W$ | uncertain | ESS | ① |

${L}_{2}-\Delta I-W\left({P}_{1}-{P}_{2}\right)\left(1-\alpha \right)\left(1-\beta \right)-\Delta W$ | uncertain | |||

$-{L}_{1}-E$ | $-$ | |||

${E}_{3}\left(0,1,0\right)$ | $\alpha W\left({P}_{1}-{P}_{2}\right)-\Delta W$ | uncertain | Unstable point | — |

$\Delta I+\Delta W-W\left({P}_{1}-{P}_{2}\right)\left(1-\alpha \right)$ | uncertain | |||

${S}_{1}+E$ | $+$ | |||

${E}_{4}\left(0,1,1\right)$ | ${L}_{1}+\alpha W\left({P}_{1}-{P}_{2}\right)\left(\beta +1\right)-\Delta W$ | uncertain | ESS | ② |

$\Delta I-{L}_{2}+W\left({P}_{1}-{P}_{2}\right)\left(1-\alpha \right)\left(1-\beta \right)+\Delta W$ | uncertain | |||

$-{L}_{1}-{S}_{1}-E$ | $-$ | |||

${E}_{5}\left(1,0,0\right)$ | $\Delta I-\alpha W\left({P}_{1}-{P}_{2}\right)+\Delta W$ | uncertain | Unstable point | — |

$W\left({P}_{1}-{P}_{2}\right)\left(1-\alpha \right)-\Delta W$ | uncertain | |||

${L}_{2}+E$ | $+$ | |||

${E}_{6}\left(1,0,1\right)$ | $\Delta I-{L}_{1}-\alpha W\left({P}_{1}-{P}_{2}\right)\left(\beta +1\right)+\Delta W$ | uncertain | ESS | ③ |

${L}_{2}-W\left({P}_{1}-{P}_{2}\right)\left(1-\alpha \right)\left(1-\beta \right)-\Delta W$ | uncertain | |||

$-{L}_{2}-E$ | $-$ | |||

${E}_{7}\left(1,1,0\right)$ | $\Delta W-\alpha W\left({P}_{1}-{P}_{2}\right)$ | uncertain | Unstable point | — |

$\Delta W-W\left({P}_{1}-{P}_{2}\right)\left(1-\alpha \right)$ | uncertain | |||

$E$ | $+$ | |||

${E}_{8}\left(1,1,1\right)$ | $\Delta W-{L}_{1}-\alpha W\left({P}_{1}-{P}_{2}\right)\left(\beta +1\right)$ | uncertain | ESS | ④ |

$\Delta W-{L}_{2}+W\left({P}_{1}-{P}_{2}\right)\left(1-\alpha \right)\left(1-\beta \right)$ | uncertain | |||

$-E$ | $-$ | |||

${E}_{9}$ | eigenvalues with different valence | Saddle point |

$\mathit{W}$ | $\Delta \mathit{W}$ | $\mathit{I}$ | $\Delta \mathit{I}$ | $\mathit{\alpha}$ | $\mathit{\beta}$ | ${\mathit{P}}_{1}$ | ${\mathit{P}}_{2}$ | ${\mathit{L}}_{1}$ | ${\mathit{L}}_{2}$ | $\mathit{E}$ | ${\mathit{S}}_{1}$ | ${\mathit{S}}_{2}$ | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Case 1 | 4 | 0.5 | 2 | 1 | 0.6 | 0.4 | 0.55 | 0.45 | 0.7 | 0.5 | 1 | 1 | 2 |

Case 2 | 3.5 | 0.7 | 2 | 0.5 | 0.6 | 0.6 | 0.6 | 0.4 | 1.2 | 1 | 1 | 1 | 2 |

Case 3 | 4 | 1.5 | 2 | 0.5 | 0.6 | 0.5 | 0.6 | 0.2 | 2.5 | 0.5 | 1 | 1 | 2 |

Case 4 | 4 | 0.4 | 2 | 1.8 | 0.6 | 0.5 | 0.6 | 0.3 | 1.5 | 1 | 1 | 1 | 2 |

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## Share and Cite

**MDPI and ACS Style**

Wang, M.; He, R.; Ren, K.; He, Y.; Zhou, J.
Stability Analysis of Low-Carbon Technology Innovation Cooperation under a Reward and Punishment Mechanism. *Systems* **2022**, *10*, 118.
https://doi.org/10.3390/systems10040118

**AMA Style**

Wang M, He R, Ren K, He Y, Zhou J.
Stability Analysis of Low-Carbon Technology Innovation Cooperation under a Reward and Punishment Mechanism. *Systems*. 2022; 10(4):118.
https://doi.org/10.3390/systems10040118

**Chicago/Turabian Style**

Wang, Min, Runxin He, Kai Ren, Youshi He, and Jianya Zhou.
2022. "Stability Analysis of Low-Carbon Technology Innovation Cooperation under a Reward and Punishment Mechanism" *Systems* 10, no. 4: 118.
https://doi.org/10.3390/systems10040118