# Evolutionary Game Analysis between Local Government and Enterprises on Bridge Employment from the Perspective of Dynamic Incentive and Punishment

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. Theoretical Basis

**Definition**

**1**

**Definition**

**2**

**.**Let $F(i|s)$ of a strategy be an estimate of the growth rate ${r}_{i}$, then the RD can be written as

#### 3.2. Problem Description

#### 3.3. Model Establishment

## 4. Results

#### 4.1. Static Equilibrium Analysis

#### 4.1.1. Static Equilibrium Analysis of Local Government

**Proposition**

**1.**

**Proof**

**of**

**Proposition**

**1.**

**Proposition**

**2.**

**Proposition**

**3.**

**Proof**

**of**

**Propositions**

**2**

**and**

**3.**

- (1)
- If $y>{e}_{f}/({g}_{c1}+{e}_{f})$, we can have $\frac{\partial F\left(x\right)}{\partial x}{|}_{x=0}<0$, and $\frac{\partial F\left(x\right)}{\partial x}{|}_{x=1}>0$, then $x=0$ is the stable point.
- (2)
- If $y<{e}_{f}/({g}_{c1}+{e}_{f})$, we can have $\frac{\partial F\left(x\right)}{\partial x}{|}_{x=0}>0$, and $\frac{\partial F\left(x\right)}{\partial x}{|}_{x=1}<0$, then $x=1$ is the stable point.

#### 4.1.2. Static Equilibrium Analysis of Enterprises

**Proposition**

**4.**

**Proof**

**of**

**Proposition**

**4.**

**Proposition**

**5.**

**Proposition**

**6.**

**Proof**

**of**

**Propositions**

**5**

**and**

**6.**

- (1)
- If ${e}_{c}-{e}_{b}>{g}_{c1}+{e}_{f}$, we can have $0\le x\le 1<\left({e}_{c}-{e}_{b}\right)/({g}_{c1}+{e}_{f})$, then $y=0$ is the stable point.
- (2)
- If ${e}_{c}-{e}_{b}<0$, we can have $x>\left({e}_{c}-{e}_{b}\right)/({g}_{c1}+{e}_{f})$, then $y=1$ is the stable point.
- (3)
- If $0<{e}_{c}-{e}_{b}<{g}_{c1}+{e}_{f}$, there are two scenarios to discuss:
- (i)
- if $x<\left({e}_{c}-{e}_{b}\right)/({g}_{c1}+{e}_{f})$, then $\frac{\partial F\left(y\right)}{\partial y}{|}_{y=0}<0$, $\frac{\partial F\left(y\right)}{\partial y}{|}_{y=1}>0$, $y=0$ is the stable point.
- (ii)
- if $x>\left({e}_{c}-{e}_{b}\right)/({g}_{c1}+{e}_{f})$, then $\frac{\partial F\left(y\right)}{\partial y}{|}_{y=0}>0$, $\frac{\partial F\left(y\right)}{\partial y}{|}_{y=1}<0$, $y=1$ is the stable point.

#### 4.1.3. Static Equilibrium Analysis of the System

**Proposition**

**7.**

- (1)
- The system always has four fixed equilibrium points, namely $\left(0,0\right)$, $\left(0,1\right)$, $\left(1,0\right)$, and $\left(1,1\right)$.
- (2)
- If $0\le \frac{{e}_{c}-{e}_{b}}{{g}_{c1}+{e}_{f}}\le 1$ and $0\le \frac{{e}_{f}}{{g}_{c1}+{e}_{f}}\le 1$, the system has another equilibrium point $\left({x}^{\ast},{y}^{\ast}\right)$, where ${x}^{\ast}=\frac{{e}_{c}-{e}_{b}}{{g}_{c1}+{e}_{f}}$, ${y}^{\ast}=\frac{{e}_{f}}{{g}_{c1}+{e}_{f}}$.

**Proof**

**of**

**Proposition**

**7.**

**Proposition**

**8.**

**Proof**

**of**

**Proposition**

**8.**

#### 4.1.4. Simulation Analysis

#### 4.2. Dynamic Equilibrium Analysis

#### 4.2.1. Dynamic Incentive Mechanism

**Proposition**

**9.**

- (1)
- The dynamic incentive system always has three fixed equilibrium points, namely $\left(0,0\right)$, $\left(1,0\right)$, and $\left(1,1\right)$.
- (2)
- For $\forall {x}_{i}\in \left[0,1\right]$, $\left({x}_{i},1\right)$ are equilibrium points of the dynamic incentive system.
- (2)
- If $0\le \frac{{e}_{c}-{e}_{b}}{i}\le 1$ and $0\le \frac{{e}_{f}}{i}\le 1$, the dynamic incentive system has another equilibrium point $\left({x}_{i}^{\ast},{y}_{i}^{\ast}\right)$, where ${x}_{i}^{\ast}=\frac{{e}_{c}-{e}_{b}}{i}$, ${y}_{i}^{\ast}=\frac{{e}_{f}}{i}$.

**Proof**

**of**

**Proposition**

**9.**

**Proposition**

**10.**

**Proof**

**of**

**Proposition**

**10.**

**Proposition**

**11.**

- (i)
- $\frac{\partial {x}_{i}^{\ast}}{\partial {e}_{c}}>0$, $\frac{\partial {x}_{i}^{\ast}}{\partial {e}_{b}}<0$, and $\frac{\partial {x}_{i}^{\ast}}{\partial i}<0$.
- (ii)
- $\frac{\partial {y}_{i}^{\ast}}{\partial {e}_{f}}>0$and $\frac{\partial {y}_{i}^{\ast}}{\partial i}<0$.

#### 4.2.2. Dynamic Punishment Mechanism

**Proposition**

**12.**

- (1)
- The system always has four fixed equilibrium points, namely $\left(0,0\right)$, $\left(0,1\right)$, $\left(1,0\right)$, and $\left(1,1\right)$.
- (2)
- If $0\le \frac{2\left({e}_{c}-{e}_{b}\right)}{{g}_{c1}+\sqrt{{g}_{c1}{}^{2}+4p{g}_{c1}}}\le 1$ and $-1\le \frac{{g}_{c1}}{2p}-\frac{\sqrt{+4p{g}_{c1}}}{2p}\le 0$, the system has another equilibrium point $\left({x}_{p}^{\ast},{y}_{p}^{\ast}\right)$, where ${x}_{p}^{\ast}=\frac{2\left({e}_{c}-{e}_{b}\right)}{{g}_{c1}+\sqrt{{g}_{c1}{}^{2}+4p{g}_{c1}}}$, ${y}_{p}^{\ast}=1+\frac{{g}_{c1}}{2p}-\frac{\sqrt{{g}_{c1}{}^{2}+4p{g}_{c1}}}{2p}$.

**Proof**

**of**

**Proposition**

**12.**

**Proposition**

**13.**

**Proof**

**of**

**Proposition**

**13.**

**Proposition**

**14.**

- (i)
- $\frac{\partial {x}_{p}^{\ast}}{\partial {e}_{c}}>0$, $\frac{\partial {x}_{p}^{\ast}}{\partial {e}_{b}}<0$, $\frac{\partial {x}_{p}^{\ast}}{\partial {g}_{c1}}<0$, and $\frac{\partial {x}_{p}^{\ast}}{\partial p}<0$.
- (ii)
- $\frac{\partial {y}_{p}^{\ast}}{\partial {g}_{c1}}<0$and $\frac{\partial {y}_{p}^{\ast}}{\partial p}>0$.

#### 4.2.3. Dynamic Incentive and Punishment Mechanism

**Proposition**

**15.**

- (1)
- The system always has four fixed equilibrium points, namely $\left(0,0\right)$, $\left(0,1\right)$, $\left(1,0\right)$, and $\left(1,1\right)$.
- (2)
- If $0\le \frac{{e}_{c}-{e}_{b}}{i}\le 1$ and $-1\le \frac{p}{i+p}\le 0$, the system has another equilibrium point $\left({x}_{ip}^{\ast},{y}_{ip}^{\ast}\right)$, where ${x}_{ip}^{\ast}=\frac{{e}_{c}-{e}_{b}}{i}$, ${y}_{ip}^{\ast}=\frac{p}{i+p}$.

**Proposition**

**16.**

**Proof**

**of**

**Proposition**

**16.**

**Proposition**

**17.**

- (i)
- $\frac{\partial {x}_{ip}^{\ast}}{\partial {e}_{c}}>0$, $\frac{\partial {x}_{ip}^{\ast}}{\partial {e}_{b}}<0$, and $\frac{\partial {x}_{ip}^{\ast}}{\partial i}<0$.
- (ii)
- $\frac{\partial {y}_{ip}^{\ast}}{\partial i}<0$ and $\frac{\partial {y}_{ip}^{\ast}}{\partial p}>0$.

#### 4.2.4. Simulation Analysis

## 5. Discussion

#### 5.1. Management Implications

#### 5.2. Strengths, Limitations, and Future Research

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Parameters | Descriptions | Notes |
---|---|---|

$x$ | The probability that the government actively implements rewards and punishments. | $0\le x\le 1$ |

${g}_{b}$ | Additional benefits for the government. | ${g}_{b}\ge 0$ |

${g}_{c1}$ | Direct cost to the government. | ${g}_{c1}\ge 0$ |

${g}_{c2}$ | Indirect cost to the government. | ${g}_{c2}\ge 0$ |

${g}_{c3}$ | Additional cost to the government. | ${g}_{c3}\ge 0$ |

$y$ | The probability that enterprises actively respond to government policies. | $0\le y\le 1$ |

${e}_{p}$ | Profits earned by the enterprise. | ${e}_{p}\ge 0$ |

${e}_{b}$ | Indirect benefits to the enterprise. | ${e}_{b}\ge 0$ |

${e}_{c}$ | The cost consumed by the enterprise. | ${e}_{c}\ge 0$ |

${e}_{f}$ | Fines imposed on enterprises. | ${e}_{f}\ge 0$ |

Local Government | Enterprises | |
---|---|---|

Active (y) | Pasive (1 − y) | |

Active (x) | $(\begin{array}{c}{g}_{b}-{g}_{c1}-{g}_{c2},\\ {e}_{p}+{e}_{b}+{g}_{c1}-{e}_{c}\end{array})$ | $\left({e}_{f}-{g}_{c2}-{g}_{c3},{e}_{p}-{e}_{f}\right)$ |

Passive (1 − x) | $\left({g}_{b}-{g}_{c2},{e}_{p}+{e}_{b}-{e}_{c}\right)$ | $\left(-{g}_{c2}-{g}_{c3},{e}_{p}\right)$ |

Equilibrium Points | DetJ | Sign | TrJ | Sign | Result |
---|---|---|---|---|---|

$\left(0,0\right)$ | ${e}_{f}\left({e}_{b}-{e}_{c}\right)$ | $-$ | ${e}_{f}+{e}_{b}-{e}_{c}$ | Uncertain | Saddle point |

$\left(0,1\right)$ | ${g}_{c1}\left({e}_{b}-{e}_{c}\right)$ | $-$ | ${e}_{c}-{e}_{b}-{g}_{c1}$ | Uncertain | Saddle point |

$\left(1,0\right)$ | $-{e}_{f}\left({g}_{c1}+{e}_{f}+{e}_{b}-{e}_{c}\right)$ | $-$ | ${g}_{c1}+{e}_{b}-{e}_{c}$ | Uncertain | Saddle point |

$\left(1,1\right)$ | $-{g}_{c1}\left({g}_{c1}+{e}_{f}+{e}_{b}-{e}_{c}\right)$ | $-$ | ${e}_{c}-{e}_{b}-{e}_{f}$ | Uncertain | Saddle point |

$\left({x}^{\ast},{y}^{\ast}\right)$ | $\frac{{g}_{c1}{e}_{f}\left({e}_{c}-{e}_{b}\right)\left({g}_{c1}+{e}_{f}+{e}_{b}-{e}_{c}\right)}{{\left({g}_{c1}+{e}_{f}\right)}^{2}}$ | $+$ | 0 | 0 | Central point |

Local Government | Enterprises | |
---|---|---|

Active (y) | Passive (1 − y) | |

Active (x) | $\left(\begin{array}{c}{g}_{b}-{g}_{c1}\left(y\right)-{g}_{c2},\\ {e}_{p}+{e}_{b}+{g}_{c1}\left(y\right)-{e}_{c}\end{array}\right)$ | $\left({e}_{f}-{g}_{c2}-{g}_{c3},{e}_{p}-{e}_{f}\right)$ |

Passive (1 − x) | $\left({g}_{b}-{g}_{c2},{e}_{p}+{e}_{b}-{e}_{c}\right)$ | $\left(-{g}_{c2}-{g}_{c3},{e}_{p}\right)$ |

Local Government | Enterprises | |
---|---|---|

Active (y) | Passive (1 − y) | |

Active (x) | $(\begin{array}{c}{g}_{b}-{g}_{c1}-{g}_{c2},\\ {e}_{p}+{e}_{b}+{g}_{c1}-{e}_{c}\end{array})$ | $(\begin{array}{c}{e}_{f}\left(y\right)-{g}_{c2}-{g}_{c3},\\ {e}_{p}-{e}_{f}\left(y\right)\end{array})$ |

Passive (1 − x) | $\left({g}_{b}-{g}_{c2},{e}_{p}+{e}_{b}-{e}_{c}\right)$ | $\left(-{g}_{c2}-{g}_{c3},{e}_{p}\right)$ |

**Table 6.**Payment matrix of local government and enterprises based on dynamic incentive and punishment mechanism.

Local Government | Enterprises | |
---|---|---|

Active (y) | Passive (1 − y) | |

Active (x) | $(\begin{array}{c}{g}_{b}-{g}_{c1}\left(y\right)-{g}_{c2},\\ {e}_{p}+{e}_{b}+{g}_{c1}\left(y\right)-{e}_{c}\end{array})$ | $\left({e}_{f}\left(y\right)-{g}_{c2}-{g}_{c3},{e}_{p}-{e}_{f}\left(y\right)\right)$ |

Passive (1 − x) | $\left({g}_{b}-{g}_{c2},{e}_{p}+{e}_{b}-{e}_{c}\right)$ | $\left(-{g}_{c2}-{g}_{c3},{e}_{p}\right)$ |

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## Share and Cite

**MDPI and ACS Style**

Dong, J.; Yan, S.; Yang, X.
Evolutionary Game Analysis between Local Government and Enterprises on Bridge Employment from the Perspective of Dynamic Incentive and Punishment. *Systems* **2022**, *10*, 115.
https://doi.org/10.3390/systems10040115

**AMA Style**

Dong J, Yan S, Yang X.
Evolutionary Game Analysis between Local Government and Enterprises on Bridge Employment from the Perspective of Dynamic Incentive and Punishment. *Systems*. 2022; 10(4):115.
https://doi.org/10.3390/systems10040115

**Chicago/Turabian Style**

Dong, Junjie, Shumin Yan, and Xiaowei Yang.
2022. "Evolutionary Game Analysis between Local Government and Enterprises on Bridge Employment from the Perspective of Dynamic Incentive and Punishment" *Systems* 10, no. 4: 115.
https://doi.org/10.3390/systems10040115