# Inspection—Corruption Game of Illegal Logging and Other Violations: Generalized Evolutionary Approach

^{1}

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## Abstract

**:**

## 1. Introduction

## 2. Preliminaries

## 3. Main Model

**Remark**

**1.**

Violator | |||

I | Refrain R | Violate V_{j} | |

H | $w,0$ | $w,(1-p){r}_{j}-pf\left({r}_{j}\right)$ | |

C | $w,0$ | $w+\alpha p\overline{r},{r}_{j}(1-p\alpha )$ |

Violator | |||

I | Refrain R | Violate V_{j} | |

H | $w,0$ | $w,-(\u03f5/v)f\left({r}_{j}\right)+(1-\u03f5/v)[(1-p){r}_{j}-pf\left({r}_{j}\right)]$ | |

C | $w,0$ | $p({w}_{0}-\mathbf{E}F\left(\alpha r\right))+(1-p)w,-(\u03f5/v)f\left({r}_{j}\right)+(1-\u03f5/v){r}_{j}(1-p\alpha )$ |

R | Violate ${V}_{j}$ | |

H | $w,0$ | $w,(1-\u03f5)[(1-p){r}_{j}-pf\left({r}_{j}\right)]-\u03f5f\left({r}_{j}\right)$ |

C | $w,0$ | $(1-v)(w+\alpha p\overline{r})+v[p({w}_{0}-\mathbf{E}F\left(\alpha r\right))+(1-p)w],(1-\u03f5){r}_{j}(1-p\alpha )-\u03f5f\left({r}_{j}\right)$ |

**Remark**

**2.**

## 4. Linear Fines

#### 4.1. Corruption Frontier

#### 4.2. Partial Kinetic Equations

**Remark**

**3.**

**Remark**

**4.**

**Lemma**

**1.**

**Proof.**

**Lemma**

**2.**

#### 4.3. Full Switching Dynamics for $J=1$

**Remark**

**5.**

**Theorem**

**1.**

**Proof.**

#### 4.4. A Numeric Example

#### 4.5. Full Switching Dynamics for Arbitrary J

**Remark**

**6.**

**Theorem**

**2.**

**Theorem**

**3.**

## 5. Power-Type Fines

**Remark**

**7.**

**Lemma**

**3.**

**Lemma**

**4.**

**Theorem**

**4.**

## 6. Further Perspectives

## Funding

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Kolokoltsov, V.N.
Inspection—Corruption Game of Illegal Logging and Other Violations: Generalized Evolutionary Approach. *Mathematics* **2021**, *9*, 1619.
https://doi.org/10.3390/math9141619

**AMA Style**

Kolokoltsov VN.
Inspection—Corruption Game of Illegal Logging and Other Violations: Generalized Evolutionary Approach. *Mathematics*. 2021; 9(14):1619.
https://doi.org/10.3390/math9141619

**Chicago/Turabian Style**

Kolokoltsov, Vassili N.
2021. "Inspection—Corruption Game of Illegal Logging and Other Violations: Generalized Evolutionary Approach" *Mathematics* 9, no. 14: 1619.
https://doi.org/10.3390/math9141619