# Stochastic Game Analysis of Cooperation and Selfishness in a Random Access Mechanism

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## Abstract

**:**

## 1. Introduction

- We propose a novel stochastic game model incorporating cooperative and non-cooperative players in the same game.
- We develop a bi-dimensional Markov chain to determine the system’s state in the stationary regime.
- We show that the game admits an equilibrium solution that integrates the Nash and social optimality concepts.
- We explore different performance metrics, such as throughput, delay, number of backlogged packets, and the equilibrium retransmission policy.
- We undertake a comparative study of two game scenarios with different levels of cooperation and selfishness.

## 2. Related Work

#### 2.1. Cooperative Game Models

#### 2.2. Non-Cooperative Game Models

#### 2.3. Mixed Game Models

## 3. SAZD Overview

- 0: when the medium is idle;
- 1: when one packet is successfully transmitted without interference;
- ZigZag: when the AP senses a simultaneous transmission of two packets;
- C: when three or more stations transmit at the same time slot.

## 4. Problem Formulation

#### 4.1. Model Description

#### 4.2. Analytical Model

**Theorem**

**1.**

**Proof.**

#### 4.3. Performance Evaluation

**Proposition**

**1.**

**Proof.**

**Remark**

**1.**

**Proposition**

**2.**

**Proof.**

**Corollary**

**1.**

**Proof.**

**Proposition**

**3.**

**Proof.**

**Proposition**

**4.**

**Proof.**

**Corollary**

**2.**

**Proof.**

**Proposition**

**5.**

**Proof.**

## 5. Stochastic Game Formulation

- Players: The sets of cooperative and non-cooperative players are defined, respectively, as ${S}_{M}$ and ${S}_{N}$. In what follows, we refer to the players as users.
- Strategy space: The set of strategies is the set of users’ actions. For each user i, we define the set of pure strategies as ${A}_{i}=\{T,W\}$, where T represents the action “Transmit”, and W is the action “Wait”. Thus, at a given time slot, a user holding a packet can choose one action in ${A}_{i}$. Furthermore, we define the mixed strategies as the set of all the distributions over ${A}_{i}$, which is ${\varphi}_{i}=\{{q}_{c}^{i},1-{q}_{c}^{i}\}$ for a cooperative user i, and ${\psi}_{j}=\{{q}_{nc}^{j},1-{q}_{nc}^{j}\}$ for a non-cooperative user j.
- Utility: The utility function corresponds to the user’s level of satisfaction, which can be, in the case of our study, the average throughput, the access delay, or any other performance metric of interest. Let ${u}_{i}:{\psi}_{i}\times {\psi}_{-i}\to \mathbb{R}$ denote the utility function of user i in ${S}_{N}$. ${u}_{i}$ depends on ${q}_{r}^{i}$ the transmission probability of user i and the vector ${q}_{r}^{-i}=[{q}_{r}^{1},{q}_{r}^{2},\dots ,{q}_{r}^{i-1},{q}_{r}^{i+1},\dots ]$ of others’ transmission probabilities. Thus, each non-cooperative user possesses his own utility function. On the other hand, let ${U}_{g}$ be the common utility function of all cooperative users among the set ${S}_{M}$, which corresponds to the overall system performance.
- Game information: We assume that all players share a common knowledge, which is: the total number of players in the game, their own strategy space, and the strategy space of others, their utility, and the utility of others. On the other hand, we assume that cooperative players do not have the knowledge of the existence of selfish players among them. As a result, they behave cooperatively assuming that others will behave similarly. However, selfish users assume that everyone in the game is selfish, and therefore they behave selfishly.

#### 5.1. Basic Assumptions

**Assumption**

**1.**

**Assumption**

**2.**

#### 5.2. Characterization of the Game Equilibrium

**Theorem**

**2.**

**Proof.**

## 6. Numerical Results

## 7. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Transition Probabilities

## References

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**Figure 1.**A scenario of a wireless network where M cooperative users share the same medium with N selfish users.

**Figure 2.**Transition diagram of the Markov chain. The straight line corresponds to either a successful transmission or an increase in backlogged packets, whereas the dashed line represents a successful transmission with ZigZag.

**Figure 4.**Normalized throughput for different retransmission policies. (

**a**) $M=3$, $N=2$, and (

**b**) $M=2$, $N=3$.

**Figure 5.**Number of backlogged users for different retransmission policies. (

**a**) $M=3$, $N=2$, and (

**b**) $M=2$, $N=3$.

**Figure 6.**Delay of transmitted packets for different retransmission policies. (

**a**) $M=3$, $N=2$, and (

**b**) $M=2$, $N=3$.

**Figure 7.**Delay of backlogged packets for different retransmission policies. (

**a**) $M=3$, $N=2$, and (

**b**) $M=2$, $N=3$.

**Table 1.**Performance evaluation in the case of $M=10$ and $N=2$. ${P}_{col}$ is the system collision probability, $T{H}_{c}^{i}$ and $T{H}_{nc}^{j}$ are the individual throughputs of a cooperative user i and a selfish user j, respectively.

Cooperative User | Non-Cooperative User | ||||||||
---|---|---|---|---|---|---|---|---|---|

${\mathit{p}}_{\mathit{a}}$ | ${\mathit{P}}_{\mathit{col}}$ | ${\mathit{q}}_{\mathit{c}}^{*}$ | ${\mathit{TH}}_{\mathit{c}}^{\mathit{i}}$ | ${\mathit{D}}_{\mathit{c}}$ | ${\mathit{S}}_{\mathit{c}}$ | ${\mathit{q}}_{\mathit{nc}}^{*}$ | ${\mathit{TH}}_{\mathit{nc}}^{\mathit{j}}$ | ${\mathit{D}}_{\mathit{nc}}$ | ${\mathit{S}}_{\mathit{nc}}$ |

0.0001 | 2.53 × 10^{−10} | 4.14 × 10^{−1} | 1.00 × 10^{−4} | 1.00 | 1.50 × 10^{−9} | 0.8787 | 1.00 × 10^{−4} | 1.00 | 1.55 × 10^{−10} |

0.1 | 2.14 × 10^{−1} | 1.61 × 10^{−1} | 5.29 × 10^{−2} | 7.38 | 3.38 | 0.9999 | 7.21 × 10^{−2} | 2.37 | 1.98 × 10^{−1} |

0.2 | 2.64 × 10^{−1} | 8.08 × 10^{−2} | 4.21 × 10^{−2} | 18.28 | 7.29 | 0.9999 | 1.30 × 10^{−1} | 2.21 | 3.17 × 10^{−1} |

0.3 | 3.05 × 10^{−1} | 7.07 × 10^{−2} | 3.26 × 10^{−2} | 27.24 | 8.57 | 0.9999 | 1.70 × 10^{−1} | 2.21 | 4.36 × 10^{−1} |

0.4 | 3.11 × 10^{−1} | 6.06 × 10^{−2} | 2.52 × 10^{−2} | 37.30 | 9.15 | 0.9999 | 2.20 × 10^{−1} | 2.10 | 4.95 × 10^{−1} |

0.5 | 3.47 × 10^{−1} | 6.06 × 10^{−2} | 1.88 × 10^{−2} | 51.19 | 9.48 | 0.9999 | 2.50 × 10^{−1} | 2.15 | 5.93 × 10^{−1} |

0.6 | 3.25 × 10^{−1} | 5.05 × 10^{−2} | 1.37 × 10^{−2} | 71.58 | 9.67 | 0.9999 | 3.00 × 10^{−1} | 1.97 | 5.88 × 10^{−1} |

0.7 | 2.92 × 10^{−1} | 4.04 × 10^{−2} | 9.02 × 10^{−3} | 109.65 | 9.81 | 0.9999 | 3.40 × 10^{−1} | 1.79 | 5.50 × 10^{−1} |

0.8 | 2.44 × 10^{−1} | 3.03 × 10^{−2} | 4.85 × 10^{−3} | 205.01 | 9.90 | 0.9999 | 3.90 × 10^{−1} | 1.60 | 4.76 × 10^{−1} |

0.9 | 9.99 × 10^{−4} | 1.00 × 10^{−4} | 1.14 × 10^{−5} | 8.70 × 10^{4} | 9.99 | 0.9999 | 4.97 × 10^{−1} | 1.00 | 1.99 × 10^{−3} |

0.9999 | 9.99 × 10^{−4} | 1.00 × 10^{−4} | 9.99 × 10^{−9} | 1.00 × 10^{8} | 10.00 | 0.9999 | 4.99 × 10^{−1} | 1.00 | 1.99 × 10^{−3} |

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**MDPI and ACS Style**

Boujnoui, A.; Zaaloul, A.; Orozco-Barbosa, L.; Haqiq, A.
Stochastic Game Analysis of Cooperation and Selfishness in a Random Access Mechanism. *Mathematics* **2022**, *10*, 694.
https://doi.org/10.3390/math10050694

**AMA Style**

Boujnoui A, Zaaloul A, Orozco-Barbosa L, Haqiq A.
Stochastic Game Analysis of Cooperation and Selfishness in a Random Access Mechanism. *Mathematics*. 2022; 10(5):694.
https://doi.org/10.3390/math10050694

**Chicago/Turabian Style**

Boujnoui, Ahmed, Abdellah Zaaloul, Luis Orozco-Barbosa, and Abdelkrim Haqiq.
2022. "Stochastic Game Analysis of Cooperation and Selfishness in a Random Access Mechanism" *Mathematics* 10, no. 5: 694.
https://doi.org/10.3390/math10050694