# Hybrid Assessment Scheme Based on the Stern- Judging Rule for Maintaining Cooperation under Indirect Reciprocity

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## Abstract

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## 1. Introduction

## 2. Model of Hybrid-Assessment Scheme

#### 2.1. Game and Strategy

#### 2.2. Hybrid Assessment

## 3. Results

#### 3.1. Image Dynamics

#### 3.2. Replicator Dynamics

#### 3.3. Stability Analysis

## 4. Discussion

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**Replicator dynamics of hybrid assessments. Each triangle represents a simplex of the state space, where x, y, and z are non-negative real numbers denoting the frequencies of unconditional cooperators (AllC), unconditional defectors (AllD), and the stern judgers (SJ), respectively. The arrows in the triangles show the direction of replicator dynamics at each point. Trajectories following the dynamics are also drawn. If the top vertex $(x,y,z)=(0,0,1)$ is asymptotically stable, the average rate of cooperation (AveC) at that point is calculated and the basin of attraction is shown in yellow. The four triangles represent the results for different values of p. The parameter values are $b=3$, $c=1$, ${e}^{\prime}=1\%$, and $e=1\%$.

**Figure 2.**Phase transitions of p. The panels show parameter spaces in which the stern-judging rule is stable. The vertical axis shows p while the horizontal axis shows b (in the left panel, with ${e}^{\prime}=5\%$ and $e=5\%$), ${e}^{\prime}$ (in the center panel, with $b=3$ and $e=5\%$), and e (in the right panel, with $b=3$ and $e=5\%$). The borders of the yellow basin are $b=1.16$ in the left panel and $e=0.33$ in the right panel. The other parameter value is $c=1$.

Pattern | Contr. to Good Recipient | No Contr. to Good Recipient | Contr. to Bad Recipient | No Contr. to Bad Recipient |
---|---|---|---|---|

Assessment | Good | Bad | Bad | Good |

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**MDPI and ACS Style**

Okada, I.; Yamamoto, H.; Uchida, S.
Hybrid Assessment Scheme Based on the Stern- Judging Rule for Maintaining Cooperation under Indirect Reciprocity. *Games* **2020**, *11*, 13.
https://doi.org/10.3390/g11010013

**AMA Style**

Okada I, Yamamoto H, Uchida S.
Hybrid Assessment Scheme Based on the Stern- Judging Rule for Maintaining Cooperation under Indirect Reciprocity. *Games*. 2020; 11(1):13.
https://doi.org/10.3390/g11010013

**Chicago/Turabian Style**

Okada, Isamu, Hitoshi Yamamoto, and Satoshi Uchida.
2020. "Hybrid Assessment Scheme Based on the Stern- Judging Rule for Maintaining Cooperation under Indirect Reciprocity" *Games* 11, no. 1: 13.
https://doi.org/10.3390/g11010013