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

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 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

- Nowak, M.A.; Sigmund, K. Evolution of indirect reciprocity by image scoring. Nature
**1998**, 393, 573–577. [Google Scholar] [CrossRef] [PubMed] - Leimar, O.; Hammerstein, P. Evolution of cooperation through indirect reciprocity. Proc. Natl. Acad. Sci. USA
**2001**, 268, 745–753. [Google Scholar] [CrossRef] [PubMed] - Nowak, M.A.; Sigmund, K. Evolution of indirect reciprocity. Nature
**2005**, 437, 1291–1298. [Google Scholar] [CrossRef] [PubMed] - Rockenbach, B.; Milinski, M. The efficient interaction of indirect reciprocity and costly punishment. Nature
**2006**, 444, 718–723. [Google Scholar] [CrossRef] [PubMed] - Sasaki, T.; Okada, I.; Nakai, Y. Indirect reciprocity can overcome free-rider problems on costly moral assessment. Biol. Lett.
**2016**, 12, 20160341. [Google Scholar] [CrossRef] [PubMed] - Milinski, M.; Semmann, D.; Krambeck, H.J. Reputation helps solve the ‘tragedy of the commons’. Nature
**2002**, 415, 424–426. [Google Scholar] [CrossRef] - Sommerfeld, R.D.; Krambeck, H.J.; Semmann, D.; Milinski, M. Gossip as an alternative for direct observation in games of indirect reciprocity. Proc. Natl. Acad. Sci. USA
**2007**, 44, 17435–17440. [Google Scholar] [CrossRef][Green Version] - McNamara, J.M.; Doodson, P. Reputation can enhance or suppress cooperation through positive feedback. Nat. Commun.
**2015**, 6, 6134. [Google Scholar] [CrossRef][Green Version] - Swakman, V.; Molleman, L.; Ule, A.; Egas, M. Reputation-based cooperation: Empirical evidence for behavioral strategies. Evol. Hum. Behav.
**2016**, 37, 230–235. [Google Scholar] [CrossRef] - Okada, I.; Yamamoto, H.; Sato, Y.; Uchida, S.; Sasaki, T. Experimental evidence of selective inattention in reputation-based cooperation. Sci. Rep.
**2018**, 8, 14813. [Google Scholar] [CrossRef] - Wedekind, C.; Milinski, M. Cooperation through image scoring in humans. Science
**2000**, 288, 850–852. [Google Scholar] [CrossRef] [PubMed] - Sigmund, K. The Calculus of Selfishness; Princeton University Press: Woodstock, UK, 2010. [Google Scholar]
- Pacheco, J.M.; Santos, F.C.; Chalub, F.A.C.C. Stern-judging: A simple, successful norm which promotes cooperation under indirect reciprocity. PLoS Comput. Biol
**2006**, 2, 1634–1638. [Google Scholar] [CrossRef] - Kandori, M. Social norms and community enforcement. Rev. Econ. Stud.
**1992**, 59, 63–80. [Google Scholar] [CrossRef][Green Version] - Ohtsuki, H.; Iwasa, Y. The leading eight: Social norms that can maintain cooperation by indirect reciprocity. J. Theor. Biol.
**2006**, 239, 435–444. [Google Scholar] [CrossRef] [PubMed] - Santos, F.P.; Pacheco, J.M.; Santos, F.C. Evolution of cooperation under indirect reciprocity and arbitrary exploration rates. Sci. Rep.
**2016**, 6, 37517. [Google Scholar] [CrossRef] [PubMed][Green Version] - Uchida, S.; Sasaki, T. Effect of assessment error and private information on stern-judging in indirect reciprocity. Chaos Solitons Fract.
**2013**, 56, 175–180. [Google Scholar] [CrossRef][Green Version] - Panchanathan, K.; Boyd, R. Indirect reciprocity can stabilize cooperation without the second-order free rider problem. Nature
**2004**, 432, 499–502. [Google Scholar] [CrossRef] [PubMed] - Brandt, H.; Sigmund, K. The logic of reprobation: Action and assessment rules in indirect reciprocity. J. Theor. Biol.
**2004**, 231, 475–486. [Google Scholar] [CrossRef][Green Version] - Brandt, H.; Sigmund, K. The good, the bad and the discriminator ? Errors in direct and indirect reciprocity. J. Theor. Biol.
**2006**, 239, 183–194. [Google Scholar] [CrossRef][Green Version] - Chalub, F.; Santos, F.C.; Pacheco, J.M. The evolution of norms. J. Theor. Biol.
**2006**, 241, 233–240. [Google Scholar] [CrossRef] - Masuda, N.; Ohtsuki, H. Tag-based indirect reciprocity by incomplete social information. Proc. R. Soc. B
**2007**, 274, 689–695. [Google Scholar] [CrossRef] [PubMed][Green Version] - Panchanathan, K. Two wrongs don’t make a right: The initial viability of different assessment rules in the evolution of indirect reciprocity. J. Theor. Biol.
**2011**, 277, 48–54. [Google Scholar] [CrossRef] [PubMed] - Sasaki, T.; Okada, I.; Nakai, Y. The evolution of conditional moral assessment in indirect reciprocity. Sci. Rep.
**2017**, 7, 41870. [Google Scholar] [CrossRef] [PubMed] - Uchida, S.; Sigmund, K. The competition of assessment rules for indirect reciprocity. J. Theor. Biol.
**2010**, 263, 13–19. [Google Scholar] [CrossRef] [PubMed][Green Version] - Uchida, S. Effect of private information on indirect reciprocity. Phy. Rev. E
**2010**, 82, 036111. [Google Scholar] [CrossRef] - Sigmund, K. Moral assessment in indirect reciprocity. J. Theor. Biol.
**2012**, 299, 25–30. [Google Scholar] [CrossRef][Green Version] - Martinez-Vaquero, L.A.; Cuesta, J.A. Evolutionary stability and resistance to cheating in an indirect reciprocity model based on reputation. Phys. Review. E
**2013**, 87, 052810. [Google Scholar] [CrossRef][Green Version] - Ohtsuki, H.; Iwasa, Y.; Nowak, M.A. Reputation effects in public and private interactions. PLoS Comput. Biol.
**2015**, 11, e1004527. [Google Scholar] [CrossRef][Green Version] - Olejarz, J.; Ghang, W.; Nowak, M.A. Indirect reciprocity with optional interactions and private information. Games
**2015**, 6, 438–457. [Google Scholar] [CrossRef][Green Version] - Okada, I.; Sasaki, T.; Nakai, Y. Tolerant indirect reciprocity can boost social welfare through solidarity with unconditional cooperators in private monitoring. Sci. Rep.
**2017**, 7, 9737. [Google Scholar] [CrossRef][Green Version] - Okada, I.; Sasaki, T.; Nakai, Y. A solution of private assessment in indirect reciprocity using solitary observation. J. Theor. Biol.
**2018**, 455, 7–15. [Google Scholar] [CrossRef] - Takahashi, N.; Mashima, R. The importance of subjectivity in perceptual errors on the emergence of indirect reciprocity. J. Theor. Biol.
**2006**, 243, 418–436. [Google Scholar] [CrossRef] [PubMed][Green Version] - Inaba, M.; Takahashi, N. Linkage Based on the Kandori Norm Successfully Sustains Cooperation in Social Dilemmas. Games
**2019**, 10, 10. [Google Scholar] [CrossRef][Green Version] - Hilbe, C.; Schmid, L.; Tkadlec, J.; Chatterjee, K.; Nowak, M.A. Indirect reciprocity with private, noisy, and incomplete information. Proc. Natl. Acad. Sci. USA
**2018**, 115, 12241–12246. [Google Scholar] [CrossRef] [PubMed][Green Version] - Sugden, R. The Economics of Rights, Cooperation and Welfare; Basil Blackwell: Oxford, UK, 1986. [Google Scholar]
- Milinski, M.; Semmann, D.; Bakker, T.C.M.; Krambeck, H.J. Cooperation through indirect reciprocity: Image scoring or standing strategy? Pr. R. Soc. B
**2001**, 268, 2495–2501. [Google Scholar] [CrossRef] - Kandori, M. Introduction to repeated games with private monitoring. J. Econ. Theo.
**2002**, 102, 1–15. [Google Scholar] [CrossRef] - Dieckmann, U.; Law, R. The dynamical theory of coevolution: A derivation from stochastic ecological processes. J. Math. Biol.
**1996**, 34, 579–612. [Google Scholar] [CrossRef] - Yamamoto, H.; Okada, I.; Uchida, S.; Sasaki, T. A norm knockout method on indirect reciprocity to reveal indispensable norms. Sci. Rep.
**2017**, 7, 44146. [Google Scholar] [CrossRef][Green Version] - Uchida, S.; Yamamoto, H.; Okada, I.; Sasaki, T. A theoretical approach to norm ecosystems: Two adaptive architectures of indirect reciprocity show different paths to the evolution of cooperation. Front. Phys.
**2018**, 6, 14. [Google Scholar] [CrossRef][Green Version]

**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 |

© 2020 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 (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**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