# Enhancement of Cooperation and Reentrant Phase of Prisoner’s Dilemma Game on Signed Networks

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## Abstract

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## 1. Introduction

## 2. Models

## 3. Results and Discussion

## 4. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A

## References

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**Figure 1.**Prisoner’s dilemma game on a signed square lattice with eight neighbors. Two players with a positive link play with the original payoff matrix. However, two players with a negative link play with a different payoff matrix.

**Figure 2.**Mean-field results of the order parameter ${\rho}_{c}$in the mixed spatial game on a signed square lattice. Two players with a positive link play with the original payoff matrix of the PD game. However, two players with a negative link play with a different payoff matrix of the reverse PD game.

**Figure 3.**Ratio of cooperators as a function of payoff parameter $b$ for a fixed ratio of negative links. When the ratio of the negative links is less than 0.3, the dynamics are similar to those of the original PD game. Around ${r}_{-}=0.6$, we observe a reentrant phase of cooperators at the high value of $b>1.9$. At ${r}_{-}>0.7$, the system shows a cooperating phase, regardless of temptation parameter $b$.

**Figure 4.**Order parameter and snapshot of the configuration on the game: (

**a**) the order parameter versus control parameter b for ${r}_{-}=0.61$ (blue), $0.64$ (orange), and $0.67$ (green); (

**b**) a snapshot of the configuration in the steady state at ${r}_{-}$ = 0.61 (green: cooperators, red: defectors). It shows a configuration for the mixture of cooperators and defectors at ${r}_{-}$ = 0.61.

**Figure 5.**The fraction of cooperators versus the ratio of negative links for a fixed payoff parameter.

**Table 1.**Payoff matrix of the prisoner’s dilemma game. A defector obtains a bigger payoff than a cooperator (b > 1).

Strategy | Cooperation (C) | Defection (D) |
---|---|---|

Cooperation (C) | 1 | 0 |

Defection (D) | $b$ | 0 |

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**MDPI and ACS Style**

Choi, J.H.; Lee, S.; Lee, J.W.
Enhancement of Cooperation and Reentrant Phase of Prisoner’s Dilemma Game on Signed Networks. *Entropy* **2022**, *24*, 144.
https://doi.org/10.3390/e24020144

**AMA Style**

Choi JH, Lee S, Lee JW.
Enhancement of Cooperation and Reentrant Phase of Prisoner’s Dilemma Game on Signed Networks. *Entropy*. 2022; 24(2):144.
https://doi.org/10.3390/e24020144

**Chicago/Turabian Style**

Choi, Jae Han, Sungmin Lee, and Jae Woo Lee.
2022. "Enhancement of Cooperation and Reentrant Phase of Prisoner’s Dilemma Game on Signed Networks" *Entropy* 24, no. 2: 144.
https://doi.org/10.3390/e24020144