# Non-Cooperative Power Control Game in D2D Underlying Networks with Variant System Conditions

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## Abstract

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## 1. Introduction

- We formulate a non-cooperative game for multiple (D2D) users, where each pair of users can communicate with each other without exchanging information from a base station, and determine the Nash equilibrium of the game based on extreme value theorem.
- We describe the existence of Nash solution of the proposed game and prove the uniqueness of this Nash equilibrium. We show that the analysis in [10] is also valid in the D2D system. In addition, we derive the dependence of the Nash equilibrium on the distance between the transmitter device and receiver.
- We prove that the Nash equilibrium of the game may not exist in some variant system conditions. Unlike previous studies, we assume that the cellular user does not change its transmit power, and we proved through extensive simulations that the existence of Nash equilibrium of the power control game is also depend on other system parameters, such as the target SINR, distance, sigmoid, and pricing coefficient parameters.

## 2. Related Work

## 3. System Model

## 4. Non-Cooperative Power Control Game

#### 4.1. Nash Equilibrium

**Propostion**

**1.**

**Proof.**

- The action set ${P}_{i}$ is a nonempty, convex and compact subset of some Euclidean space ${\mathcal{R}}^{N}$
- The cost function ${J}_{i}={p}_{i}\to \mathcal{R}$ is continuous in ${p}_{i}=[0,{P}_{i}^{max}]$.

#### 4.2. Convergence

- Positivity $f\left(p\right)\ge 0$
- Monotonicity $p\ge {p}^{\prime}\Rightarrow f\left(p\right)\ge f\left({p}^{\prime}\right)$
- Scalability ∀$\alpha \ge 1;\alpha f\left(p\right)\ge f\left(\alpha p\right)$

#### 4.3. Dependence of Equilibrium on Distance

## 5. Simulation Results

## 6. Conclusions

## Author Contributions

## Conflicts of Interest

## Abbreviations

D2D | Device-to-Device |

SINR | Signal-to-Interference-and-Noise-Ratio |

QoS | Quality of service |

CDMA | Code Division Multiple Access |

MTC | Machine type communications |

NOMA | Nonorthogonal multiple access |

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## Share and Cite

**MDPI and ACS Style**

Al-Gumaei, Y.A.; Aslam, N.; Al-Samman, A.M.; Al-Hadhrami, T.; Noordin, K.; Fazea, Y. Non-Cooperative Power Control Game in D2D Underlying Networks with Variant System Conditions. *Electronics* **2019**, *8*, 1113.
https://doi.org/10.3390/electronics8101113

**AMA Style**

Al-Gumaei YA, Aslam N, Al-Samman AM, Al-Hadhrami T, Noordin K, Fazea Y. Non-Cooperative Power Control Game in D2D Underlying Networks with Variant System Conditions. *Electronics*. 2019; 8(10):1113.
https://doi.org/10.3390/electronics8101113

**Chicago/Turabian Style**

Al-Gumaei, Yousef Ali, Nauman Aslam, Ahamed Mohammed Al-Samman, Tawfik Al-Hadhrami, Kamarul Noordin, and Yousef Fazea. 2019. "Non-Cooperative Power Control Game in D2D Underlying Networks with Variant System Conditions" *Electronics* 8, no. 10: 1113.
https://doi.org/10.3390/electronics8101113