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

^{1}

^{2}

^{3}

^{*}

## 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

- Abramson, N. The ALOHA system: Another alternative for computer communications. In Proceedings of the Fall Joint Computer Conference, 1970, AFIPS ’70 (Fall), Houston, TX, USA, 17–19 November 1970; pp. 281–285. [Google Scholar]
- Oku, T.; Kimura, T.; Cheng, J. Performance Evaluation of Hierarchical Slotted ALOHA for IoT Applications. In Proceedings of the 2020 IEEE International Conference on Consumer Electronics-Taiwan (ICCE-Taiwan), Taoyuan, Taiwan, 28–30 September 2020; pp. 1–2. [Google Scholar]
- Li, Y.; Zhan, W.; Dai, L. Rate-Constrained Delay Optimization for Slotted ALOHA. IEEE Trans. Commun.
**2021**, 69, 5283–5298. [Google Scholar] [CrossRef] - Boujnoui, A.; Zaaloul, A.; Haqiq, A. A stochastic game analysis of the slotted ALOHA mechanism combined with zigzag decoding and transmission cost. In International Conference on Innovations in Bio-Inspired Computing and Applications; Springer International Publishing: Cham, Swiztherland, 2018; Volume 735, pp. 102–112. [Google Scholar]
- Oinaga, M.; Ogata, S.; Ishibashi, K. ZigZag decodable coded slotted ALOHA. In Proceedings of the 2018 15th Workshop on Positioning, Navigation and Communications (WPNC), Bremen, Germany, 25–26 October 2018; pp. 1–6. [Google Scholar]
- Bianchi, G. Performance analysis of the IEEE 802.11 distributed coordination function. IEEE J. Sel. Areas Commun.
**2000**, 18, 535–547. [Google Scholar] [CrossRef] - Ahmetoglu, M.; Yavascan, O.T.; Uysal, E. MiSTA: Threshold-ALOHA with Mini Slots. In Proceedings of the 2021 IEEE International Black Sea Conference on Communications and Networking (BlackSeaCom), Bucharest, Romania, 24–28 May 2021; pp. 1–6. [Google Scholar]
- Wang, H.; Fapojuwo, A.O. Design and performance evaluation of successive interference cancellation-based pure ALOHA for Internet-of-Things networks. IEEE Internet Things J.
**2019**, 6, 6578–6592. [Google Scholar] [CrossRef] - Bankov, D.; Khorov, E.; Lyakhov, A. Mathematical model of LoRaWAN channel access with capture effect. In Proceedings of the 2017 IEEE 28th Annual International Symposium on Personal, Indoor, and Mobile Radio Communications (PIMRC), Montreal, QC, Canada, 8–13 October 2017; pp. 1–5. [Google Scholar]
- Boujnoui, A.; Zaaloul, A.; Haqiq, A. Enhanced Pricing Strategy for Slotted ALOHA with ZigZag Decoding: A Stochastic Game Approach. Int. J. Comput. Inf. Syst. Ind. Manag. Appl.
**2021**, 13, 160–171. [Google Scholar] - Dai, M.; Mao, B.; Gong, X.; Sung, C.W.; Zhuang, W.; Lin, X. Zigzag-Division Multiple Access for Wireless Networks With Long and Heterogeneous Delays. IEEE Trans. Aerosp. Electron. Syst.
**2019**, 55, 2822–2835. [Google Scholar] [CrossRef] - Boujnoui, A.; Zaaloul, A.; Haqiq, A. Mathematical model based on game theory and Markov chains for analysing the transmission cost in SA-ZD mechanism. Int. J. Comput. Inf. Syst. Ind. Manag. Appl.
**2018**, 10, 197–207. [Google Scholar] - Jin, Y.; Kesidis, G. Equilibria of a noncooperative game for heterogeneous users of an ALOHA network. IEEE Commun. Lett.
**2002**, 6, 282–284. [Google Scholar] - Gopal, S.; Kaul, S.K.; Chaturvedi, R.; Roy, S. A non-cooperative multiple access game for timely updates. In Proceedings of the IEEE INFOCOM 2020-IEEE Conference on Computer Communications Workshops (INFOCOM WKSHPS), Toronto, ON, Canada, 6–9 July 2020; pp. 924–929. [Google Scholar]
- Zhao, B.; Ren, G.; Zhang, H. Cooperative Contention Resolution Diversity Slotted ALOHA with Transmit Power Diversity for Multi-Satellite Networks. In Proceedings of the 2019 IEEE 90th Vehicular Technology Conference (VTC2019-Fall), Honolulu, HI, USA, 22–25 September 2019; pp. 1–5. [Google Scholar]
- Vaezi, K.; Ashtiani, F. Delay-optimal cooperation policy in a slotted ALOHA full-duplex wireless network: Static approach. IEEE Syst. J.
**2019**, 14, 2257–2268. [Google Scholar] [CrossRef] - Vaezi, K.; Akar, N.; Karasan, E. Age of Information in a Cooperative Slotted ALOHA Network: Marginal and Joint Distributions. TechRxiv. Preprint. 2021. Available online: https://www.techrxiv.org/articles/preprint/Age_of_Information_in_a_Cooperative_Slotted_Aloha_Network_Marginal_and_Joint_Distributions/16437330/1 (accessed on 1 February 2022).
- Liu, R.; Hong, T.; Ding, X.; Wang, Y.; Zhang, G. Multi-Satellite Cooperative Beamforming ALOHA for LEO Satellite IoT Networks. Front. Space Technol.
**2021**, 2, 9. [Google Scholar] [CrossRef] - Cheikh, I.; Sabir, E.; Aouami, R.; Sadik, M.; Roy, S. Throughput-Delay Tradeoffs for Slotted-ALOHA-based LoRaWAN Networks. In Proceedings of the 2021 International Wireless Communications and Mobile Computing (IWCMC), Harbin, China, 28 June–2 July 2021; pp. 2020–2025. [Google Scholar]
- Beltramelli, L.; Mahmood, A.; Österberg, P.; Gidlund, M. LoRa beyond ALOHA: An investigation of alternative random access protocols. IEEE Trans. Ind. Inform.
**2020**, 17, 3544–3554. [Google Scholar] [CrossRef] [Green Version] - Milarokostas, C.; Tsolkas, D.; Passas, N.; Merakos, L. A Comprehensive Study on LPWANs with a Focus on the Potential of LoRa/LoRaWAN Systems. TechRxiv. Preprint. 2021. Available online: https://www.techrxiv.org/articles/preprint/A_Comprehensive_Study_on_LPWANs_With_a_Focus_on_the_Potential_of_LoRa_LoRaWAN_Systems/16853893/1 (accessed on 1 February 2022).
- Tegos, S.A.; Diamantoulakis, P.D.; Lioumpas, A.S.; Sarigiannidis, P.G.; Karagiannidis, G.K. Slotted ALOHA with NOMA for the next generation IoT. IEEE Trans. Commun.
**2020**, 68, 6289–6301. [Google Scholar] [CrossRef] - Boujnoui, A.; Zaaloul, A.; Haqiq, A. Cooperative Slotted ALOHA with ZigZag Decoding and a Pricing Mechanism. In International Conference on Innovations in Bio-Inspired Computing and Applications; Springer: Berlin/Heidelberg, Germany, 2020; Volume 1372, pp. 111–119. [Google Scholar]
- Zhao, B.; Ren, G.; Zhang, H. Slotted ALOHA Game for Medium Access Control in Satellite Networks. In Proceedings of the 2019 IEEE/CIC International Conference on Communications in China (ICCC), Changchun, China, 11–13 August 2019; pp. 518–522. [Google Scholar]
- Cho, Y.; Tobagi, F.A. Cooperative and non-cooperative ALOHA games with channel capture. In Proceedings of the IEEE GLOBECOM 2008—2008 IEEE Global Telecommunications Conference, New Orleans, LA, USA, 30 November–4 December 2008; pp. 1–6. [Google Scholar]
- Hilbe, C.; Šimsa, Š.; Chatterjee, K.; Nowak, M.A. Evolution of cooperation in stochastic games. Nature
**2018**, 559, 246–249. [Google Scholar] [CrossRef] [PubMed] - Van Heesch, M.; Wissink, P.L.; Ranji, R.; Nobakht, M.; Den Hartog, F. Combining Cooperative With Non-Cooperative Game Theory to Model Wi-Fi Congestion in Apartment Blocks. IEEE Access
**2020**, 8, 64603–64616. [Google Scholar] [CrossRef] - Yang, Y.; Wu, Y.; Chen, N.; Wang, K.; Chen, S.; Yao, S. LOCASS: Local optimal caching algorithm with social selfishness for mixed cooperative and selfish devices. IEEE Access
**2018**, 6, 30060–30072. [Google Scholar] [CrossRef] - Gollakota, S.; Katabi, D. Zigzag decoding: Combating hidden terminals in wireless networks. In Proceedings of the ACM SIGCOMM 2008 conference on Data Communication, SIGCOMM ’08, Seattle, WA, USA, 22 August 2008; pp. 159–170. [Google Scholar]
- Allen, A.O. Probability, Statistics, and Queueing Theory; Academic Press: Cambridge, MA, USA, 2014. [Google Scholar]
- MacKenzie, A.B.; Wicker, S.B. Selfish users in ALOHA: A game-theoretic approach. In Proceedings of the IEEE 54th Vehicular Technology Conference, VTC Fall 2001, Proceedings (Cat. No. 01CH37211), Atlantic City, NJ, USA, 7–11 October 2001; Volume 3, pp. 1354–1357. [Google Scholar]
- Altman, E.; El Azouzi, R.; Jiménez, T. Slotted ALOHA as a game with partial information. Comput. Netw.
**2004**, 45, 701–713. [Google Scholar] [CrossRef] [Green Version] - El-Azouzi, R.; Sabir, E.; Jiménez, T.; Bouyakhf, E.H. Modeling slotted ALOHA as a stochastic game with random discrete power selection algorithms. J. Comput. Syst. Netw. Commun.
**2009**, 2009, 572650. [Google Scholar] [CrossRef] [Green Version] - El-Azouzi, R.; Jiménez, T.; Sabir, E.; Benarfa, S.; Bouyakhf, E.H. Cooperative and non-cooperative control for slotted ALOHA with random power level selections algorithms. In Proceedings of the 2nd International Conference on Performance Evaluation Methodologies and Tools & Workshops, ValueTools ’07, Nantes, France, 22–27 October 2007; pp. 1–10. [Google Scholar]
- Karouit, A. Efficient Incentive Scheme forWireless Random Channel Access with Selfish Users. In International Symposium on Ubiquitous Networking; Springer: Berlin/Heidelberg, Germany, 2015; Volume 366, pp. 27–38. [Google Scholar]
- Sabir, E.; El-Azouzi, R.; Hayel, Y. Hierarchy sustains partial cooperation and induces a Braess-like paradox in slotted ALOHA-based networks. Comput. Commun.
**2012**, 35, 273–286. [Google Scholar] [CrossRef]

**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} |

Publisher’s Note: MDPI stays neutral with regard to jurisdictional claims in published maps and institutional affiliations. |

© 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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