# The Superiority of Quantum Strategy in 3-Player Prisoner’s Dilemma

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## Abstract

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## 1. Introduction

## 2. The General Case

## 3. The 2-Player Prisoner’s Dilemma

## 4. The 3-Player Prisoner’s Dilemma

#### 4.1. The Separated Case

#### 4.2. The Entanglement Parameter

#### 4.2.1. The Initial State ${\widehat{J}}_{x}|000\rangle $

#### 4.2.2. The Initial State ${\widehat{J}}_{x}|111\rangle $

#### 4.3. The Case of the Other Two Players Choosing $\mathrm{i}{\widehat{\sigma}}_{y}$

#### 4.4. The Entanglement Gate

## 5. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

- Myerson, R.B. Game Theory; Harvard University Press: Cambridge, MA, USA, 2013. [Google Scholar]
- Tadelis, S. Game Theory: An Introduction; Princeton University Press: Princeton, NJ, USA, 2013. [Google Scholar]
- Klarreich, E. Playing by quantum rules. Nature
**2001**, 414, 244–245. [Google Scholar] [CrossRef] - Benjamin, S.C.; Hayden, P.M. Multiplayer quantum games. Phys. Rev. A
**2001**, 64, 030301. [Google Scholar] [CrossRef] [Green Version] - Andronikos, T.; Sirokofskich, A.; Kastampolidou, K.; Varvouzou, M.; Giannakis, K.; Singh, A. Finite automata capturing winning sequences for all possible variants of the PQ penny flip game. Mathematics
**2018**, 6, 20. [Google Scholar] [CrossRef] [Green Version] - Giannakis, K.; Theocharopoulou, G.; Papalitsas, C.; Fanarioti, S.; Andronikos, T. Quantum conditional strategies and automata for Prisoners’ Dilemmata under the EWL scheme. Appl. Sci.
**2019**, 9, 2635. [Google Scholar] [CrossRef] [Green Version] - Accardi, L.; Boukas, A. Von Neumann’s minimax theorem for continuous quantum games. J. Stoch. Anal.
**2020**, 1, 5. [Google Scholar] [CrossRef] - Andronikos, T.; Sirokofskich, A. The Connection between the PQ Penny Flip Game and the Dihedral Groups. Mathematics
**2021**, 9, 1115. [Google Scholar] [CrossRef] - Dowling, J.P.; Milburn, G.J. Quantum technology: The second quantum revolution. Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci.
**2003**, 361, 1655–1674. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Nielsen, M.A.; Chuang, I.L. Quantum Computation and Quantum Information; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Eisert, J.; Wilkens, M.; Lewenstein, M. Quantum games and quantum strategies. Phys. Rev. Lett.
**1999**, 83, 3077–3080. [Google Scholar] [CrossRef] [Green Version] - Benjamin, S.C.; Hayden, P.M. Comment on “Quantum Games and Quantum Strategies”. Phys. Rev. Lett.
**2001**, 87, 069801. [Google Scholar] [CrossRef] [PubMed] [Green Version] - de Sousa, P.B.M.; Ramos, R.V. Multiplayer quantum games and its application as access controller in architecture of quantum computers. Quantum Inf. Process.
**2008**, 7, 125–135. [Google Scholar] [CrossRef] [Green Version] - Khan, F.S.; Solmeyer, N.; Balu, R.; Humble, T.S. Quantum games: A review of the history, current state, and interpretation. Quantum Inf. Process.
**2018**, 17, 1–42. [Google Scholar] [CrossRef] [Green Version] - Cheng, H.M.; Luo, M.X. Tripartite Dynamic Zero-Sum Quantum Games. Entropy
**2021**, 23, 154. [Google Scholar] [CrossRef] [PubMed] - Szopa, M. Efficiency of Classical and Quantum Games Equilibria. Entropy
**2021**, 23, 506. [Google Scholar] [CrossRef] [PubMed] - Li, Y.; Zhao, Y.; Fu, J.; Xu, L. Reducing food loss and waste in a two-echelon food supply chain: A quantum game approach. J. Clean. Prod.
**2021**, 285, 125261. [Google Scholar] [CrossRef] - Du, J.; Li, H.; Xu, X.; Zhou, X.; Han, R. Entanglement enhanced multiplayer quantum games. Phys. Lett. A
**2002**, 302, 229–233. [Google Scholar] [CrossRef] [Green Version] - Du, J.; Li, H.; Xu, X.; Shi, M.; Wu, J.; Zhou, X.; Han, R. Experimental realization of quantum games on a quantum computer. Phys. Rev. Lett.
**2002**, 88, 137902. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Dong, Z.; Zhang, G.; Amini, N.H. Single-photon quantum filtering with multiple measurements. Int. J. Adapt. Control Signal Process.
**2018**, 32, 528–546. [Google Scholar] [CrossRef] [Green Version]

**Figure 1.**The payoff matrix of the 2-player Prisoner’s Dilemma. The first number in the parenthesis denotes the payoff of Alice, the second number denotes the payoff of Bob.

**Figure 2.**The red line denotes the payoff of Alice with the strategic set of $(\mathrm{i}{\widehat{\sigma}}_{x},\mathrm{i}{\widehat{\sigma}}_{x},\mathrm{i}{\widehat{\sigma}}_{x})$; while the blue curve is that with the strategic set of $(\mathrm{i}{\widehat{\sigma}}_{y},\mathrm{i}{\widehat{\sigma}}_{y},\mathrm{i}{\widehat{\sigma}}_{y})$.

**Figure 3.**The payoff of Alice with respect to the strategy parameters $\theta $ and $\varphi $, $0\le \theta \le \pi $, $0\le \varphi \le \frac{\pi}{2}$.

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Dong, Z.; Wu, A.-G.
The Superiority of Quantum Strategy in 3-Player Prisoner’s Dilemma. *Mathematics* **2021**, *9*, 1443.
https://doi.org/10.3390/math9121443

**AMA Style**

Dong Z, Wu A-G.
The Superiority of Quantum Strategy in 3-Player Prisoner’s Dilemma. *Mathematics*. 2021; 9(12):1443.
https://doi.org/10.3390/math9121443

**Chicago/Turabian Style**

Dong, Zhiyuan, and Ai-Guo Wu.
2021. "The Superiority of Quantum Strategy in 3-Player Prisoner’s Dilemma" *Mathematics* 9, no. 12: 1443.
https://doi.org/10.3390/math9121443