# Research on Repeated Quantum Games with Public Goods under Strong Reciprocity

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## Abstract

**:**

## 1. Introduction

## 2. Quantum Game Analysis of Repeated Supply of Public Goods under Strong Reciprocity

#### 2.1. Repeated Game Based on Strong Reciprocity Public Goods

- (i)
- Game scenario strategy assumption: In the game process, both the private enterprises and the state-owned enterprises have only cooperation and betrayal strategies. After the first game, a tit for tat update mechanism is adopted in the repeated game. In the sub game, the game population only has two strategy choices, that is, hypothesis C strategy and D strategy. We will not consider the escape strategy of the game population for the time being.
- (ii)
- Game process parameter assumption: Consider the strong reciprocal punishment that affects the game strategy in the repeated game process as $\delta (0<\delta <1)$. So, when the player chooses the defector strategy, he will pay $\delta $ as a cost of defector. The payoffs on defector decreases with the increase in strong reciprocity $\delta $, which is reflected in the repeated quantum game payoffs ${B}_{E}$ and ${C}_{G}$ in 2.2 below.
- (iii)
- Game result assumption: Consider the time value of returns in repeated game returns as $\rho (0<\rho <1)$. It is the discount factor and the probability of repeated games in the next stage. So, $1-\rho $ is the probability of game ending.

- (1)
- In the second stage, we assume that the state-owned enterprises and private enterprises always choose to cooperate, except when opponents choose to defect, then the payoffs of private enterprises are ${A}_{1}{\sum}_{i=0}^{\infty}{\rho}^{i}=\frac{{A}_{1}}{1-\rho},({A}_{1}=(1-\beta )b-\frac{{\gamma}_{1}}{2});$ and the payoffs of the state-owned enterprises are ${A}_{2}{\sum}_{i=0}^{\infty}{\rho}^{i}=\frac{{A}_{2}}{1-\rho},({A}_{2}=\beta b-\frac{{\gamma}_{2}}{2}).$
- (2)
- In the second stage, we assume that the state-owned enterprises with cooperation will always choose to defect in the future if the private enterprises respond by choosing to defect; then, the payoffs of the private enterprises are 0, and the payoffs of the state-owned enterprises are $\frac{{C}_{2}(1-\rho )-\delta \rho}{1-\rho},({C}_{2}=\frac{{\gamma}_{2}}{2}).$
- (3)
- This is similar to $\left(2\right)$—the private enterprises with cooperation will always choose to defect in the future if the state-owned enterprises respond by choosing to defect; then, the payoffs of the private enterprises are $\frac{{B}_{1}(1-\rho )-\delta \rho}{1-\rho},({B}_{1}=\frac{{\gamma}_{1}}{2}).$ and the payoffs of the state-owned enterprises are 0.
- (4)
- In this repeated game, a tit for tat update strategy was adopted. Once a player chooses a defector strategy in the current stage, the other player will choose a defector (never cooperate) strategy in the following stages. That is to say, both sides of the game have chosen a defector strategy, and the payoffs on the game is 0.

#### 2.2. Public Goods Repeated Quantum Game

## 3. Entanglement of Quantum States

#### 3.1. The Entanglement without States

**Theorem 1.**

- (i)
- The private enterprises’ payoffs $E{E}_{U}$ raise the increase in effort degree ${e}_{1}$ if and only if ${A}_{E}{cos}^{2}\frac{{\theta}_{2}}{2}+{B}_{E}{sin}^{2}\frac{{\theta}_{2}}{2}>0;$
- (ii)
- the state-owned enterprises’ payoffs $E{G}_{U}$ lower the increase in effort degree ${e}_{2}$ if and only if ${A}_{G}{cos}^{2}\frac{{\theta}_{1}}{2}+{C}_{G}{sin}^{2}\frac{{\theta}_{1}}{2}>0.$

**Proof.**

#### 3.2. The Entanglement of States

**Theorem 2.**

- (i)
- when the private enterprises adopt maximal quantum strategys $({\phi}_{1}=\frac{\pi}{2})$ and ${sin}^{2}{\phi}_{2}{cos}^{2}\frac{{\theta}_{2}}{2}>0,$ the private enterprises’ payoffs $E{E}_{U}$ raise the increase in effort degree ${\theta}_{1},\phantom{\rule{4pt}{0ex}}\delta ;$
- (ii)
- when the private enterprises adopt maximal non quantum strategy $({\phi}_{2}=\frac{\pi}{2})$ and ${sin}^{2}{\phi}_{1}{cos}^{2}\frac{{\theta}_{1}}{2}>0,$ the state-owned enterprises’ payoffs $E{G}_{U}$ raise the increase in effort degree ${\theta}_{2},\phantom{\rule{4pt}{0ex}}\delta .$

**Proof.**

**Theorem 3.**

- (i)
- when the private enterprises adopts a non quantum strategy $({\phi}_{1}=0)$ and ${A}_{E}{cos}^{2}{\phi}_{2}{cos}^{2}\frac{{\theta}_{2}}{2}-{B}_{E}{sin}^{2}\frac{{\theta}_{2}}{2}\ge 0,$ the private enterprises’ payoffs $E{E}_{U}$ raise the increase in effort degree ${\theta}_{1},\phantom{\rule{4pt}{0ex}}\delta ;$
- (ii)
- when the state-owned enterprises adopt the non quantum strategy $({\phi}_{2}=0)$ and ${A}_{G}{sin}^{2}{\phi}_{1}{cos}^{2}\frac{{\theta}_{1}}{2}+{C}_{G}{sin}^{2}\frac{{\theta}_{1}}{2}>0,$ the the state-owned enterprises’ payoffs $E{E}_{U}$ raise the increase in effort degree ${\theta}_{2},\phantom{\rule{4pt}{0ex}}\delta .$

**Proof.**

**Theorem 4.**

- (i)
- when the private enterprises adopts a quantum strategy $({\phi}_{1}\in (0,\frac{\pi}{2}))$ and ${A}_{E}{cos}^{2}({\phi}_{1}+{\phi}_{2}){cos}^{2}\frac{{\theta}_{2}}{2}+{B}_{E}{cos}^{2}{\phi}_{1}{sin}^{2}\frac{{\theta}_{2}}{2}\ge 0,$ the private enterprises’ payoffs $E{E}_{U}$ raise the increase in effort degree ${\theta}_{1},\phantom{\rule{4pt}{0ex}}\delta ;$
- (ii)
- when the state-owned enterprises adopt a quantum strategy $({\phi}_{2}\in (0,\frac{\pi}{2}))$ and ${A}_{G}{cos}^{2}({\phi}_{1}+{\phi}_{2}){cos}^{2}\frac{{\theta}_{1}}{2}+{C}_{G}{cos}^{2}{\phi}_{2}{sin}^{2}\frac{{\theta}_{1}}{2}>0,$ then the state-owned enterprises’ payoffs $E{G}_{U}$ raise the increase in effort degree ${\theta}_{2},\phantom{\rule{4pt}{0ex}}\delta .$

**Proof.**

## 4. Example Numerical Analysis

#### 4.1. Regardless of Entanglement of States

- (1)
- When $30\le {e}_{2}\le {e}_{2}^{*}$, $F\left({e}_{2}\right)$ is negative, and $E{E}_{U}$ is a monotone decreasing function of ${e}_{1}$; the optimal choice of the private enterprises is ${e}_{1}^{*}=40$, which is the minimum investment;
- (2)
- When ${e}_{2}^{*}\le {e}_{2}\le 60$, $F\left({e}_{2}\right)$ is positive, and $E{E}_{U}$ is a monotone increasing function of ${e}_{1}$; the optimal choice of the private enterprises is ${e}_{1}^{*}=80$, which is the maximum investment.

#### 4.2. Considering Entanglement of States

## 5. Conclusions

- Because of quantifiable explicit discount indicators, the signing of strong reciprocity entanglement contracts can better improve the correlation of returns between the game parties in public goods industry cooperation projects and can better promote the cooperation efforts of the game parties. From the analysis in Section 3.2, after considering the entanglement of states, adopting a complete quantum strategy makes it easier to achieve an increase in returns with the increase in one’s own efforts. The cost of defection by the other party is no longer borne by the striving party, which solves the prisoner’s dilemma problem in classical games to some extent.
- In the analysis in Section 3, due to the entanglement of states, the cooperative game parties in the public goods industry need to entrust a third party to determine the strong reciprocity entanglement contract index before the project implementation, which ensures that there is no motivation for the game parties to adopt non quantum strategies. Then, only the fully quantum strategy with the maximum degree of cooperation is optimal.
- It is reasonable to analyze the repeated game of public goods from the perspective of quantum games. Due to the complexity of implementing cooperation projects in the public goods industry, the strategies of the game players are not a pure set of cooperative or non cooperative strategies. Because of many influencing factors, the degree of cooperation between the game players should be a continuous set of strategies, similar to the superposition of states in quantum games. The superposition of states and entanglement of states in quantum games can better reflect the cooperative degree of repeated games in public goods.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Three-dimensional image of the enterprises’ payoffs $E{E}_{U}$ (

**1a,2a,3a**). $E{E}_{U}$ and ${\theta}_{2}$ corresponds to projection on ${\theta}_{1}$ (

**1b,2b,3b**); $E{E}_{U}$ and ${\theta}_{1}$ corresponds to projection on ${\theta}_{2}$ (

**1c,2c,3c**).

**Figure 3.**Three-dimensional image of $E{E}_{U}$ When taking ${\phi}_{1}=\frac{\pi}{2}$ and ${\phi}_{2}=0$ (

**1a**) or ${\phi}_{1}=\frac{\pi}{2}$ and ${\phi}_{2}=\frac{\pi}{2}$ (

**2a**). $E{E}_{U}$ and ${\theta}_{2}$ corresponds to Projection on ${\theta}_{2}$ (

**1b**,

**2b**); $E{E}_{U}$ and ${\theta}_{1}$ corresponds to Projection on ${\theta}_{2}$ (

**1c**,

**2c**).

Payoffs | The State-Owned Enterprises | |
---|---|---|

The private enterprises | $C\left(\right|0\rangle )$ | $D\left(\right|1\rangle )$ |

$C\left(\right|0\rangle )$ | $(1-\beta )b-\frac{{\gamma}_{1}}{2},\beta b-\frac{{\gamma}_{2}}{2}$ | $-\frac{{\gamma}_{1}}{2},0$ |

$D\left(\right|1\rangle )$ | $0,-\frac{{\gamma}_{2}}{2}$ | $0,0$ |

Payoffs | The State-Owned Enterprises | |
---|---|---|

The private enterprises | $C\left(\right|0\rangle )$ | $D\left(\right|1\rangle )$ |

$C\left(\right|0\rangle )$ | $\frac{{A}_{1}}{1-\rho},\frac{{A}_{2}}{1-\rho}$ | $-{B}_{1}-\frac{\delta \rho}{1-\rho},0$ |

$D\left(\right|1\rangle )$ | $0,-{C}_{2}-\frac{\delta \rho}{1-\rho}$ | $0,0$ |

Payoff | The State Owned Enterprises | |||
---|---|---|---|---|

The private enterprises | ${\theta}_{2}=0,{\phi}_{2}=0$ | ${\theta}_{2}=0,{\phi}_{2}=\frac{\pi}{2}$ | ${\theta}_{2}=\pi ,{\phi}_{2}=0$ | ${\theta}_{2}=\pi ,{\phi}_{2}=\frac{\pi}{2}$ |

${\theta}_{1}=0,{\phi}_{1}=0$ | ${A}_{E},{A}_{G}$ | ${A}_{E},{A}_{G}$ | ${B}_{E},0$ | ${B}_{E},0$ |

${\theta}_{1}=0,{\phi}_{1}=\frac{\pi}{2}$ | ${A}_{E},{A}_{G}$ | ${A}_{E},{A}_{G}$ | ${B}_{E},0$ | ${B}_{E},0$ |

${\theta}_{1}=\pi ,{\phi}_{1}=0$ | $0,{C}_{G}$ | $0,{C}_{G}$ | $0,0$ | $0,0$ |

${\theta}_{1}=\pi ,{\phi}_{1}=\frac{\pi}{2}$ | $0,{C}_{G}$ | $0,{C}_{G}$ | $0,0$ | $0,0$ |

Payoffs | The State-Owned Enterprises | |||
---|---|---|---|---|

The private enterprises | ${\theta}_{2}=0,{\phi}_{2}=0$ | ${\theta}_{2}=0,{\phi}_{2}=\frac{\pi}{2}$ | ${\theta}_{2}=\pi ,{\phi}_{2}=0$ | ${\theta}_{2}=\pi ,{\phi}_{2}=\frac{\pi}{2}$ |

${\theta}_{1}=0,{\phi}_{1}=0$ | ${A}_{E},{A}_{G}$ | 0,0 | ${B}_{E},0$ | ${B}_{E},0$ |

${\theta}_{1}=0,{\phi}_{1}=\frac{\pi}{2}$ | 0,0 | ${A}_{E},{A}_{G}$ | $0,{C}_{G}$ | $0,{C}_{G}$ |

${\theta}_{1}=\pi ,{\phi}_{1}=0$ | $0,{C}_{G}$ | ${B}_{E},0$ | $0,0$ | $0,0$ |

${\theta}_{1}=\pi ,{\phi}_{1}=\frac{\pi}{2}$ | $0,{C}_{G}$ | ${B}_{E},0$ | $0,0$ | $0,0$ |

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**MDPI and ACS Style**

Sun, S.; Shu, Y.; Pi, J.; Zhou, D.
Research on Repeated Quantum Games with Public Goods under Strong Reciprocity. *Axioms* **2023**, *12*, 1044.
https://doi.org/10.3390/axioms12111044

**AMA Style**

Sun S, Shu Y, Pi J, Zhou D.
Research on Repeated Quantum Games with Public Goods under Strong Reciprocity. *Axioms*. 2023; 12(11):1044.
https://doi.org/10.3390/axioms12111044

**Chicago/Turabian Style**

Sun, Simo, Yadong Shu, Jinxiu Pi, and Die Zhou.
2023. "Research on Repeated Quantum Games with Public Goods under Strong Reciprocity" *Axioms* 12, no. 11: 1044.
https://doi.org/10.3390/axioms12111044