# Effective Boundary of Innovation Subsidy: Searching by Stochastic Evolutionary Game Model

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## Abstract

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## 1. Introduction

- (1)
- Taking a game scenario, the replication dynamic equation is solved.
- (2)
- A stochastic disturbance system is constructed to simulate the random disturbance in the game process.
- (3)
- The existence and stability theorem for trivial solutions are used to solve the boundary conditions of cooperative strategy.
- (4)
- A set of data is used to solve the boundary conditions of game players’ cooperation.

## 2. Methodology

#### 2.1. The Brief Introduction of the Game Scenario

#### 2.2. Replicator Dynamic Equations

#### 2.3. Stochastic Interference System

#### 2.4. Existence and Stability of Trivial Solution

- (1)
- If there is a positive constant $\mathsf{\gamma}$ that lets $LV\left(t,x\right)\le -\gamma V\left(t,x\right)$, $t\ge 0$, then the p-th moment exponential of the zero solution of equation (13) is stable, and $E{\left|x\left(t,{x}_{0}\right)\right|}^{p}<\left(\frac{{c}_{2}}{{c}_{1}}\right){\left|{x}_{0}\right|}^{p}{e}^{-\gamma t}$, $t\ge 0$.
- (2)
- If there is a positive constant $\mathsf{\gamma}$ that lets $LV\left(t,x\right)\ge -\gamma V\left(t,x\right)$, $t\ge 0$, then the p-th moment exponential of the zero solution of equation (10) is not stable, and $E{\left|x\left(t,{x}_{0}\right)\right|}^{p}\ge \left(\frac{{c}_{2}}{{c}_{1}}\right){\left|{x}_{0}\right|}^{p}{e}^{-\gamma t}$, $t\ge 0$.

## 3. An Application of Boundary Analysis of Variables

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

Game Parameters | Symbol |
---|---|

Public incentives in the form of subsidies | S |

Share of S received by the private sector | δ |

Cost of recharging infrastructures | C_{i} |

Share of C_{i} imputed to the public sector | ρ |

Coordination benefit of selling/buying a vehicle | b |

Increments to EV technology | i |

Punishment imposed on the private sector by citizen activism and product boycotting | P_{b} |

Punishment imposed on the public sector through citizen activism | P |

Public-civil synergistic effects | ∆1 |

Private-civil synergistic effects | ∆2 |

Green taxes over the benefits of private defectors | γ |

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**Figure 2.**The evolution process of the proportion of noncooperative strategy chosen by the public sector.

**Figure 3.**The evolution process of the proportion of noncooperative strategy chosen by the private sector.

**Figure 4.**The evolution process of the proportion of noncooperative strategy chosen by the civil sector.

$\mathit{\delta}$ | ${\mathit{C}}_{\mathit{i}}$ | $\mathit{\rho}$ | $\mathit{b}$ | $\mathit{i}$ | ${\mathit{P}}_{\mathit{b}}$ | $\mathit{P}$ | $\mathbf{\Delta}1$ | $\mathbf{\Delta}2$ | ${\mathit{Z}}_{\mathit{s}\mathit{i}\mathit{z}\mathit{e}}$ | $\mathit{\gamma}$ | $\mathit{x}$ | $\mathit{y}$ | $\mathit{z}$ |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

0.8 | 0.1 | 0.5 | 0.8 | 0.6 | 0.9 | 3 | 0.51 | 1 | 5 | 0.9 | 0.5 | 0.5 | 0.5 |

$\mathit{\delta}$ | ${\mathit{C}}_{\mathit{i}}$ | $\mathit{\rho}$ | $\mathit{b}$ | $\mathit{i}$ | ${\mathit{P}}_{\mathit{b}}$ | $\mathit{P}$ | $\mathbf{\Delta}1$ | $\mathbf{\Delta}2$ | ${\mathit{Z}}_{\mathit{s}\mathit{i}\mathit{z}\mathit{e}}$ | $\mathit{\gamma}$ | $\mathit{x}$ | $\mathit{y}$ | $\mathit{z}$ | Effective Boundary and Scope of Subsidy | Range of Change | Ranking of Regulatory Effectiveness | |
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

1 | 0.8 | 0.1 | 0.5 | 0.8 | 0.6 | 0.9 | 3 | 0.51 | 1 | 5 | 0.9 | 0.5 | 0.5 | 0.5 | [0.433, 1.005] | Initial group | |

2 | 0.88 | 0.11 | [0.431, 0.995] | −0.5%, −1% | 18 | ||||||||||||

3 | 0.88 | 0.1 | 0.55 | [0.410, 0.995] | −5.3%, −1% | 9 | |||||||||||

4 | 0.88 | 0.5 | 0.88 | [0.377, 1.041] | −12.9%, 3.6% | 5 | |||||||||||

5 | 0.88 | 0.8 | 0.66 | [0.415, 1.005] | −4.2%, 0% | 11 | |||||||||||

6 | 0.88 | 0.6 | 0.99 | [0.319, 1.005] | −26.3%, 0% | 1 | |||||||||||

7 | 0.88 | 0.9 | 3.3 | [0.415, 1.305] | −4.2%, 29.9% | 14 | |||||||||||

8 | 0.88 | 3 | 0.56 | [0.415, 0.980] | −4.2%, −2.5% | 10 | |||||||||||

9 | 0.88 | 0.51 | 1.1 | [0.362, 1.005] | −16.4%, 0% | 4 | |||||||||||

10 | 0.88 | 1 | 5.5 | [0.415, 1.005] | −4.2%, 0% | 11 | |||||||||||

11 | 0.88 | 5 | 0.99 | [0.377, 1.041] | −12.9%, 3.6% | 5 | |||||||||||

12 | 0.88 | 0.9 | 0.55 | [0.450, 1.005] | −3.9%, 0% | 15 | |||||||||||

13 | 0.88 | 0.5 | 0.55 | [0.415, 1.128] | −4.2%, 12.2% | 13 | |||||||||||

14 | 0.88 | 0.5 | 0.55 | [0.691, 0.767] | 59.6%, −23.7% | 19 | |||||||||||

15 | 0.88 | 0.11 | 0.55 | 0.5 | [0.425, 0.984] | −1.9%, −2.1% | 17 | ||||||||||

16 | 0.88 | 0.1 | 0.5 | 0.88 | 0.66 | [0.377, 1.041] | −12.9%, 3.6% | 5 | |||||||||

17 | 0.88 | 0.8 | 0.6 | 0.99 | 3.3 | [0.319, 1.305] | −26.3%, 29.9% | 2 | |||||||||

18 | 0.88 | 0.9 | 3 | 0.56 | 1.1 | [0.362, 0.980] | −16.4%, −2.5% | 3 | |||||||||

19 | 0.88 | 0.51 | 1 | 5.5 | 0.99 | [0.377, 1.041] | −12.9%, 3.6% | 5 | |||||||||

20 | 0.88 | 5 | 0.9 | 0.55 | 0.55 | [0.450, 1.128] | −3.9%, 12.2% | 16 |

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**MDPI and ACS Style**

Li, J.; Yi, J.; Zhao, Y.
Effective Boundary of Innovation Subsidy: Searching by Stochastic Evolutionary Game Model. *Symmetry* **2020**, *12*, 1531.
https://doi.org/10.3390/sym12091531

**AMA Style**

Li J, Yi J, Zhao Y.
Effective Boundary of Innovation Subsidy: Searching by Stochastic Evolutionary Game Model. *Symmetry*. 2020; 12(9):1531.
https://doi.org/10.3390/sym12091531

**Chicago/Turabian Style**

Li, Junqiang, Jingyi Yi, and Yingmei Zhao.
2020. "Effective Boundary of Innovation Subsidy: Searching by Stochastic Evolutionary Game Model" *Symmetry* 12, no. 9: 1531.
https://doi.org/10.3390/sym12091531