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Successful social norms with ternary reputation for indirect reciprocity

Yohsuke Murase
November 26, 2022

Successful social norms with ternary reputation for indirect reciprocity

presentation slide for the paper https://www.nature.com/articles/s41598-021-04033-w

Ref: Yohsuke Murase, Minjae Kim, Seung Ki Baek
“Social norms in indirect reciprocity with ternary reputations”
Scientific Reports, 12, 455 (2022)

Yohsuke Murase

November 26, 2022
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  1. Successful social norms with ternary reputation for indirect reciprocity Yohsuke

    Murase, Minjae Kim, Seung Ki Baek RIKEN R-CCS, Pukyong National Univ.
  2. indirect reciprocity If we have Good/Bad reputations, we might be

    cooperated/retaliated against by someone else. => incentive cooperation a mechanism to sustain cooperation by means of reputation
  3. model • In a large population, randomly selected two players

    in population play donation game repeatedly. • One is selected as 'donor', the other one is 'recipient'. • The donor decides either to cooperate (C) or defect (D). • Cooperation costs the donor a payo ff loss c, while the recipients receive the bene fi t b. (b>c) • The process is repeated su ff i ciently long to reach a stationary state. • Players are given "reputations" based on their actions and the reputations are shared by the community (public reputation). • Players fail to cooperate with mu_e (implementation error) and are assigned a wrong reputation with mu_a (assignment error). donor: X recipient: Y Cooperate: Defect: donation -c +b 0 0 b>c
  4. social norm Norm: Assignment rule + Action rule Assignment rule

    : Which reputation is given to the donor Action rule : Which action is taken <latexit sha1_base64="fraYy70EMQrAQwS6ySJOW0oJZ1U=">AAAB9XicbVBNS8NAEJ34WetX1aOXxSJUKCWRoh6rXjxWsR/axrLZbtqlm03Y3Sgl9H948aCIV/+LN/+N2zYHbX0w8Hhvhpl5XsSZ0rb9bS0sLi2vrGbWsusbm1vbuZ3dugpjSWiNhDyUTQ8rypmgNc00p81IUhx4nDa8weXYbzxSqVgobvUwom6Ae4L5jGBtpIebQrN4Vzw/Qm0dovtOLm+X7AnQPHFSkocU1U7uq90NSRxQoQnHSrUcO9JugqVmhNNRth0rGmEywD3aMlTggCo3mVw9QodG6SI/lKaERhP190SCA6WGgWc6A6z7atYbi/95rVj7Z27CRBRrKsh0kR9zZF4cR4C6TFKi+dAQTCQztyLSxxITbYLKmhCc2ZfnSf245JyUytflfOUijSMD+3AABXDgFCpwBVWoAQEJz/AKb9aT9WK9Wx/T1gUrndmDP7A+fwDjKpDV</latexit> R(X, Y, A) ! Z <latexit sha1_base64="rW75HBlUH/juK3PC0I2L8EC+GxA=">AAAB83icbVBNS8NAEJ3Ur1q/oh69LBahgpREinqsevFYwX5IE8pmu2mXbjZhdyOU0L/hxYMiXv0z3vw3btsctPXBwOO9GWbmBQlnSjvOt1VYWV1b3yhulra2d3b37P2DlopTSWiTxDyWnQArypmgTc00p51EUhwFnLaD0e3Ubz9RqVgsHvQ4oX6EB4KFjGBtJK9R6Zw9niJPx+i6Z5edqjMDWiZuTsqQo9Gzv7x+TNKICk04VqrrOon2Myw1I5xOSl6qaILJCA9o11CBI6r8bHbzBJ0YpY/CWJoSGs3U3xMZjpQaR4HpjLAeqkVvKv7ndVMdXvkZE0mqqSDzRWHKkXlxGgDqM0mJ5mNDMJHM3IrIEEtMtImpZEJwF19eJq3zqntRrd3XyvWbPI4iHMExVMCFS6jDHTSgCQQSeIZXeLNS68V6tz7mrQUrnzmEP7A+fwDFGpA5</latexit> P(X, Y ) ! A donor: X recipient: Y C: D: donation -c +b 0 0 b>c action: A reputation of donor reputation of recipient donor's action new reputation of donor reputation of donor reputation of recipient donor's action
  5. examples <latexit sha1_base64="2w4jAVE2FQHM3ykoBGn7lQ91qzU=">AAAB+XicbVBNS8NAEJ34WetX1KOXxSJUkJJIUY/FHvRYwX5AE8pmu2mXbjZhd1Moof/EiwdFvPpPvPlv3LY5aOuDgcd7M8zMCxLOlHacb2ttfWNza7uwU9zd2z84tI+OWypOJaFNEvNYdgKsKGeCNjXTnHYSSXEUcNoORvWZ3x5TqVgsnvQkoX6EB4KFjGBtpJ5tN8oeVvoS3V8gT8eo3rNLTsWZA60SNyclyNHo2V9ePyZpRIUmHCvVdZ1E+xmWmhFOp0UvVTTBZIQHtGuowBFVfja/fIrOjdJHYSxNCY3m6u+JDEdKTaLAdEZYD9WyNxP/87qpDm/9jIkk1VSQxaIw5ci8OIsB9ZmkRPOJIZhIZm5FZIglJtqEVTQhuMsvr5LWVcW9rlQfq6XaXR5HAU7hDMrgwg3U4AEa0AQCY3iGV3izMuvFerc+Fq1rVj5zAn9gff4A8qaR7g==</latexit> P(⇤, G) ! C <latexit sha1_base64="UwJGvobklAWS5N9wQLgT6bMv2Ds=">AAAB+XicbVBNS8NAEJ34WetX1KOXxSJUkJJIUY+levBYwX5AE8pmu2mXbjZhd1Moof/EiwdFvPpPvPlv3LY5aOuDgcd7M8zMCxLOlHacb2ttfWNza7uwU9zd2z84tI+OWypOJaFNEvNYdgKsKGeCNjXTnHYSSXEUcNoORnczvz2mUrFYPOlJQv0IDwQLGcHaSD3bbpQ9rPQlql8gT8fovmeXnIozB1olbk5KkKPRs7+8fkzSiApNOFaq6zqJ9jMsNSOcToteqmiCyQgPaNdQgSOq/Gx++RSdG6WPwliaEhrN1d8TGY6UmkSB6YywHqplbyb+53VTHd76GRNJqqkgi0VhypF5cRYD6jNJieYTQzCRzNyKyBBLTLQJq2hCcJdfXiWtq4p7Xak+Vku1eh5HAU7hDMrgwg3U4AEa0AQCY3iGV3izMuvFerc+Fq1rVj5zAn9gff4A7HOR6g==</latexit> P(⇤,

    B) ! D <latexit sha1_base64="GYm+2XNxkVAelWTb9wuTfbRw0Ns=">AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARKkiZkaIuS3Xhsop9QGcomTTThmYeJHeEMnbhr7hxoYhbf8Odf2PazkJbDyQczrmXe+/xYsEVWNa3kVtaXlldy68XNja3tnfM3b2mihJJWYNGIpJtjygmeMgawEGwdiwZCTzBWt7wauK3HphUPArvYRQzNyD9kPucEtBS1zy4KzlEwSme/dcn2IEI17pm0SpbU+BFYmekiDLUu+aX04toErAQqCBKdWwrBjclEjgVbFxwEsViQoekzzqahiRgyk2n+4/xsVZ62I+kfiHgqfq7IyWBUqPA05UBgYGa9ybif14nAf/STXkYJ8BCOhvkJwLrEydh4B6XjIIYaUKo5HpXTAdEEgo6soIOwZ4/eZE0z8r2eblyWylWa1kceXSIjlAJ2egCVdENqqMGougRPaNX9GY8GS/Gu/ExK80ZWc8++gPj8wfiQJQY</latexit> R(⇤, ⇤, D) ! B <latexit sha1_base64="r58JGQ49/QWFWfkNXtADKR70z9k=">AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARKkiZkaIui13osop9QGcomTTThmYeJHeEMnbhr7hxoYhbf8Odf2PazkJbDyQczrmXe+/xYsEVWNa3kVtaXlldy68XNja3tnfM3b2mihJJWYNGIpJtjygmeMgawEGwdiwZCTzBWt6wNvFbD0wqHoX3MIqZG5B+yH1OCWipax7clRyi4BTP/toJdiDC112zaJWtKfAisTNSRBnqXfPL6UU0CVgIVBClOrYVg5sSCZwKNi44iWIxoUPSZx1NQxIw5abT/cf4WCs97EdSvxDwVP3dkZJAqVHg6cqAwEDNexPxP6+TgH/ppjyME2AhnQ3yE4H1iZMwcI9LRkGMNCFUcr0rpgMiCQUdWUGHYM+fvEiaZ2X7vFy5rRSrV1kceXSIjlAJ2egCVdENqqMGougRPaNX9GY8GS/Gu/ExK80ZWc8++gPj8wfoSZQc</latexit> R(⇤, ⇤, C) ! G G B defect B B Bad players increase. Cooperation level drops. Image scoring All-Good <latexit sha1_base64="2w4jAVE2FQHM3ykoBGn7lQ91qzU=">AAAB+XicbVBNS8NAEJ34WetX1KOXxSJUkJJIUY/FHvRYwX5AE8pmu2mXbjZhd1Moof/EiwdFvPpPvPlv3LY5aOuDgcd7M8zMCxLOlHacb2ttfWNza7uwU9zd2z84tI+OWypOJaFNEvNYdgKsKGeCNjXTnHYSSXEUcNoORvWZ3x5TqVgsnvQkoX6EB4KFjGBtpJ5tN8oeVvoS3V8gT8eo3rNLTsWZA60SNyclyNHo2V9ePyZpRIUmHCvVdZ1E+xmWmhFOp0UvVTTBZIQHtGuowBFVfja/fIrOjdJHYSxNCY3m6u+JDEdKTaLAdEZYD9WyNxP/87qpDm/9jIkk1VSQxaIw5ci8OIsB9ZmkRPOJIZhIZm5FZIglJtqEVTQhuMsvr5LWVcW9rlQfq6XaXR5HAU7hDMrgwg3U4AEa0AQCY3iGV3izMuvFerc+Fq1rVj5zAn9gff4A8qaR7g==</latexit> P(⇤, G) ! C G G G G G G G B Cooperation level is high. Defector earns a higher payo ff . Not a Nash eq. How can we achieve a fully-cooperative Nash equilibrium?? <latexit sha1_base64="COHypbRafDNAx3d0RTb59t9cOPk=">AAACdXichVHLSsNAFD2Nr1ofjboRRChWRUHqVETFlagLl9XaB7RSkjhqME1CMi3U4g/4Ay50o6AifoYbf8CFnyAuK3Tjwps0IFrUO8zMmTP33Dkzo9qG7grGXkJSR2dXd0+4N9LXPzAYlYeGs65VcTSe0SzDcvKq4nJDN3lG6MLgedvhSlk1eE493vD2c1XuuLpl7oqazffKyqGpH+iaIogqyXJqpqi4Ym59NlYUVmwzUpLjLMH8iLWDZADiCCJlyXcoYh8WNFRQBocJQdiAApdaAUkw2MTtoU6cQ0j39zlOESFthbI4ZSjEHtN4SKtCwJq09mq6vlqjUwzqDiljmGLP7J412BN7YK/s49dadb+G56VGs9rScrsUPRtNN/9VlWkWOPpS/elZ4AArvledvNs+491Ca+mrJ+eN9OrOVH2aXbM38n/FXtgj3cCsvms323znAt4HJH8+dzvILiSSS4nF7cX42nrwFWGMYQIz9N7LWMMWUsjQuVVc4ga3oaY0Lk1K061UKRRoRvAtpPlP5qKO0A==</latexit> P(⇤, B) ! D <latexit sha1_base64="yx60UR+jkYrpbuj32sg0q1jfX8w=">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</latexit> R(⇤, ⇤, ⇤) ! G
  6. the leading-eight comprehensive search for cooperative evolutionarily stable states (CESS)

    Ohtsuki & Iwasa (2004) Hilbe & Nowak PNAS (2018)
  7. the leading-eight comprehensive search for cooperative evolutionarily stable states (CESS)

    Ohtsuki & Iwasa (2004) Hilbe & Nowak PNAS (2018) (i) maintenance of cooperation G G cooperate
  8. the leading-eight comprehensive search for cooperative evolutionarily stable states (CESS)

    Ohtsuki & Iwasa (2004) Hilbe & Nowak PNAS (2018) (i) maintenance of cooperation G G cooperate (ii) identi fi cation of defectors G G defect G B
  9. the leading-eight comprehensive search for cooperative evolutionarily stable states (CESS)

    Ohtsuki & Iwasa (2004) Hilbe & Nowak PNAS (2018) (i) maintenance of cooperation G G cooperate (ii) identi fi cation of defectors G G defect G B (iii) punishment and its justi fi cation G B defect G B
  10. the leading-eight comprehensive search for cooperative evolutionarily stable states (CESS)

    Ohtsuki & Iwasa (2004) Hilbe & Nowak PNAS (2018) (i) maintenance of cooperation G G cooperate (ii) identi fi cation of defectors G G defect G B (iii) punishment and its justi fi cation G B defect G B (iv) apology and forgiveness B G G cooperate G
  11. the leading-eight comprehensive search for cooperative evolutionarily stable states (CESS)

    Ohtsuki & Iwasa (2004) Hilbe & Nowak PNAS (2018) (i) maintenance of cooperation G G cooperate (ii) identi fi cation of defectors G G defect G B (iii) punishment and its justi fi cation G B defect G B (iv) apology and forgiveness B G G cooperate G Laeding Eight G 0.9980 B 0.0020 GG:CG 0.9960 GB:DG 0.0020 GG:DB 0.0020 BG:CG 0.0020
  12. Reputations are often represented by a binary value, "Good/Bad". What

    happens when "Good/Bad" assumption is relaxed? → ternary model : "Good / Neutral / Bad" How should we revise the conclusion learned from the binary model? What are the universal mechanisms that work for general cases? previous studies motivation of this study Tanabe et al. "Indirect reciprocity with trinary reputations" J.Theor.Biol. -> assumes ordinal relationships between G/N/B. Only subsets were investigated.
  13. comprehensive enumeration binary reputation : ternary reputation : <latexit sha1_base64="uAeh2X/b0BkBLyre2GDQUvp7CGk=">AAAB+HicbVDLTsJAFL3FF+KDqks3VWKCG2wJUZdENy4xkUcChUyHKUw6nTYzUxNs+BI3LjTGrZ/izr9xgC4UPMm9OTnn3syd48WMSmXb30ZubX1jcyu/XdjZ3dsvmgeHLRklApMmjlgkOh6ShFFOmooqRjqxICj0GGl7we3Mbz8SIWnEH9QkJm6IRpz6FCOlpYFZLFeDfvW8n+o+vQhOBmbJrthzWKvEyUgJMjQG5ldvGOEkJFxhhqTsOnas3BQJRTEj00IvkSRGOEAj0tWUo5BIN50fPrXOtDK0/Ejo4sqaq783UhRKOQk9PRkiNZbL3kz8z+smyr92U8rjRBGOFw/5CbNUZM1SsIZUEKzYRBOEBdW3WniMBMJKZ1XQITjLX14lrWrFuazU7mul+k0WRx6O4RTK4MAV1OEOGtAEDAk8wyu8GU/Gi/FufCxGc0a2cwR/YHz+ALCzkdQ=</latexit>

    (2k2)k2 /k! # of norms with k-ary reputations: <latexit sha1_base64="VUBWYmqM4g+aXda1v58NmDraM2I=">AAACBnicbVDLSsNAFJ3UV62vqEsRRotQQWoSgnYjFN24rGAf0KZlMp22QyeTMDMRSujKjb/ixoUibv0Gd/6N0zYLbT1w4XDOvdx7jx8xKpVlfRuZpeWV1bXsem5jc2t7x9zdq8kwFphUcchC0fCRJIxyUlVUMdKIBEGBz0jdH95M/PoDEZKG/F6NIuIFqM9pj2KktNQxDwtOuwRbigZEQqftnsJz6BzBK+icWW6pY+atojUFXCR2SvIgRaVjfrW6IY4DwhVmSMqmbUXKS5BQFDMyzrViSSKEh6hPmppypNd6yfSNMTzRShf2QqGLKzhVf08kKJByFPi6M0BqIOe9ifif14xVr+QllEexIhzPFvViBlUIJ5nALhUEKzbSBGFB9a0QD5BAWOnkcjoEe/7lRVJzivZF0b1z8+XrNI4sOADHoABscAnK4BZUQBVg8AiewSt4M56MF+Pd+Ji1Zox0Zh/8gfH5A31rlKo=</latexit> (28 ⇥ 24)/2! = 2, 048 <latexit sha1_base64="WUm9od7Zn6L1+JeX+4Zh+q4LDJo=">AAACEnicbVC7TgJBFJ31ifhCLW1GiQkkBHcBZSlMiDaWmMgjgYXMDrMwYfaRmVkTsuEbbPwVGwuNsbWy828cYAsFT3KTk3Puzb332AGjQur6t7ayura+sZnYSm7v7O7tpw4OG8IPOSZ17DOft2wkCKMeqUsqGWkFnCDXZqRpj26mfvOBcEF9716OA2K5aOBRh2IkldRLZTPFbmSYE9iR1CUCFrqVLDyHxRN4BYvFnH5RyZmmkSsXzF4qref1GeAyMWKSBjFqvdRXp+/j0CWexAwJ0Tb0QFoR4pJiRibJTihIgPAIDUhbUQ+p/VY0e2kCz5TSh47PVXkSztTfExFyhRi7tup0kRyKRW8q/ue1Q+mYVkS9IJTEw/NFTsig9OE0H9innGDJxoogzKm6FeIh4ghLlWJShWAsvrxMGoW8cZkv3ZXS1es4jgQ4BqcgAwxQBlVwC2qgDjB4BM/gFbxpT9qL9q59zFtXtHjmCPyB9vkDOh+YIg==</latexit> (318 ⇥ 29)/3! = 33, 059, 881, 728
  14. comprehensive enumeration binary reputation : ternary reputation : cooperation level

    <latexit sha1_base64="sp+Qn+3ECk0VmnRQLYCrrky9ysc=">AAAB9HicbVBNSwMxEJ31s9avqkcvwSJ4Krsi6rHoxWMF+wHtUrJptg3NJjHJFsvS3+HFgyJe/THe/Dem7R609cHA470ZZuZFijNjff/bW1ldW9/YLGwVt3d29/ZLB4cNI1NNaJ1ILnUrwoZyJmjdMstpS2mKk4jTZjS8nfrNEdWGSfFgx4qGCe4LFjOCrZNC1SWog5XS8gkF3VLZr/gzoGUS5KQMOWrd0lenJ0maUGEJx8a0A1/ZMMPaMsLppNhJDVWYDHGfth0VOKEmzGZHT9CpU3ooltqVsGim/p7IcGLMOIlcZ4LtwCx6U/E/r53a+DrMmFCppYLMF8UpR1aiaQKoxzQllo8dwUQzdysiA6wxsS6nogshWHx5mTTOK8FlJbi/KFdv8jgKcAwncAYBXEEV7qAGdSDwCM/wCm/eyHvx3r2PeeuKl88cwR94nz8Cv5Gb</latexit> pc ⇡ 1 AND Evolutionarily Stable State (ESS) Cooperative Evolutionarily Stable State (CESS) <latexit sha1_base64="uAeh2X/b0BkBLyre2GDQUvp7CGk=">AAAB+HicbVDLTsJAFL3FF+KDqks3VWKCG2wJUZdENy4xkUcChUyHKUw6nTYzUxNs+BI3LjTGrZ/izr9xgC4UPMm9OTnn3syd48WMSmXb30ZubX1jcyu/XdjZ3dsvmgeHLRklApMmjlgkOh6ShFFOmooqRjqxICj0GGl7we3Mbz8SIWnEH9QkJm6IRpz6FCOlpYFZLFeDfvW8n+o+vQhOBmbJrthzWKvEyUgJMjQG5ldvGOEkJFxhhqTsOnas3BQJRTEj00IvkSRGOEAj0tWUo5BIN50fPrXOtDK0/Ejo4sqaq783UhRKOQk9PRkiNZbL3kz8z+smyr92U8rjRBGOFw/5CbNUZM1SsIZUEKzYRBOEBdW3WniMBMJKZ1XQITjLX14lrWrFuazU7mul+k0WRx6O4RTK4MAV1OEOGtAEDAk8wyu8GU/Gi/FufCxGc0a2cwR/YHz+ALCzkdQ=</latexit> (2k2)k2 /k! # of norms with k-ary reputations: <latexit sha1_base64="VUBWYmqM4g+aXda1v58NmDraM2I=">AAACBnicbVDLSsNAFJ3UV62vqEsRRotQQWoSgnYjFN24rGAf0KZlMp22QyeTMDMRSujKjb/ixoUibv0Gd/6N0zYLbT1w4XDOvdx7jx8xKpVlfRuZpeWV1bXsem5jc2t7x9zdq8kwFphUcchC0fCRJIxyUlVUMdKIBEGBz0jdH95M/PoDEZKG/F6NIuIFqM9pj2KktNQxDwtOuwRbigZEQqftnsJz6BzBK+icWW6pY+atojUFXCR2SvIgRaVjfrW6IY4DwhVmSMqmbUXKS5BQFDMyzrViSSKEh6hPmppypNd6yfSNMTzRShf2QqGLKzhVf08kKJByFPi6M0BqIOe9ifif14xVr+QllEexIhzPFvViBlUIJ5nALhUEKzbSBGFB9a0QD5BAWOnkcjoEe/7lRVJzivZF0b1z8+XrNI4sOADHoABscAnK4BZUQBVg8AiewSt4M56MF+Pd+Ji1Zox0Zh/8gfH5A31rlKo=</latexit> (28 ⇥ 24)/2! = 2, 048 <latexit sha1_base64="WUm9od7Zn6L1+JeX+4Zh+q4LDJo=">AAACEnicbVC7TgJBFJ31ifhCLW1GiQkkBHcBZSlMiDaWmMgjgYXMDrMwYfaRmVkTsuEbbPwVGwuNsbWy828cYAsFT3KTk3Puzb332AGjQur6t7ayura+sZnYSm7v7O7tpw4OG8IPOSZ17DOft2wkCKMeqUsqGWkFnCDXZqRpj26mfvOBcEF9716OA2K5aOBRh2IkldRLZTPFbmSYE9iR1CUCFrqVLDyHxRN4BYvFnH5RyZmmkSsXzF4qref1GeAyMWKSBjFqvdRXp+/j0CWexAwJ0Tb0QFoR4pJiRibJTihIgPAIDUhbUQ+p/VY0e2kCz5TSh47PVXkSztTfExFyhRi7tup0kRyKRW8q/ue1Q+mYVkS9IJTEw/NFTsig9OE0H9innGDJxoogzKm6FeIh4ghLlWJShWAsvrxMGoW8cZkv3ZXS1es4jgQ4BqcgAwxQBlVwC2qgDjB4BM/gFbxpT9qL9q59zFtXtHjmCPyB9vkDOh+YIg==</latexit> (318 ⇥ 29)/3! = 33, 059, 881, 728
  15. comprehensive enumeration binary reputation : ternary reputation : supercomputer Fugaku

    By Hiroko Hama - Own work, CC BY-SA 4.0, https:// commons.wikimedia.org/w/index.php?curid=102551957 cooperation level <latexit sha1_base64="sp+Qn+3ECk0VmnRQLYCrrky9ysc=">AAAB9HicbVBNSwMxEJ31s9avqkcvwSJ4Krsi6rHoxWMF+wHtUrJptg3NJjHJFsvS3+HFgyJe/THe/Dem7R609cHA470ZZuZFijNjff/bW1ldW9/YLGwVt3d29/ZLB4cNI1NNaJ1ILnUrwoZyJmjdMstpS2mKk4jTZjS8nfrNEdWGSfFgx4qGCe4LFjOCrZNC1SWog5XS8gkF3VLZr/gzoGUS5KQMOWrd0lenJ0maUGEJx8a0A1/ZMMPaMsLppNhJDVWYDHGfth0VOKEmzGZHT9CpU3ooltqVsGim/p7IcGLMOIlcZ4LtwCx6U/E/r53a+DrMmFCppYLMF8UpR1aiaQKoxzQllo8dwUQzdysiA6wxsS6nogshWHx5mTTOK8FlJbi/KFdv8jgKcAwncAYBXEEV7qAGdSDwCM/wCm/eyHvx3r2PeeuKl88cwR94nz8Cv5Gb</latexit> pc ⇡ 1 AND Evolutionarily Stable State (ESS) Cooperative Evolutionarily Stable State (CESS) <latexit sha1_base64="uAeh2X/b0BkBLyre2GDQUvp7CGk=">AAAB+HicbVDLTsJAFL3FF+KDqks3VWKCG2wJUZdENy4xkUcChUyHKUw6nTYzUxNs+BI3LjTGrZ/izr9xgC4UPMm9OTnn3syd48WMSmXb30ZubX1jcyu/XdjZ3dsvmgeHLRklApMmjlgkOh6ShFFOmooqRjqxICj0GGl7we3Mbz8SIWnEH9QkJm6IRpz6FCOlpYFZLFeDfvW8n+o+vQhOBmbJrthzWKvEyUgJMjQG5ldvGOEkJFxhhqTsOnas3BQJRTEj00IvkSRGOEAj0tWUo5BIN50fPrXOtDK0/Ejo4sqaq783UhRKOQk9PRkiNZbL3kz8z+smyr92U8rjRBGOFw/5CbNUZM1SsIZUEKzYRBOEBdW3WniMBMJKZ1XQITjLX14lrWrFuazU7mul+k0WRx6O4RTK4MAV1OEOGtAEDAk8wyu8GU/Gi/FufCxGc0a2cwR/YHz+ALCzkdQ=</latexit> (2k2)k2 /k! # of norms with k-ary reputations: <latexit sha1_base64="VUBWYmqM4g+aXda1v58NmDraM2I=">AAACBnicbVDLSsNAFJ3UV62vqEsRRotQQWoSgnYjFN24rGAf0KZlMp22QyeTMDMRSujKjb/ixoUibv0Gd/6N0zYLbT1w4XDOvdx7jx8xKpVlfRuZpeWV1bXsem5jc2t7x9zdq8kwFphUcchC0fCRJIxyUlVUMdKIBEGBz0jdH95M/PoDEZKG/F6NIuIFqM9pj2KktNQxDwtOuwRbigZEQqftnsJz6BzBK+icWW6pY+atojUFXCR2SvIgRaVjfrW6IY4DwhVmSMqmbUXKS5BQFDMyzrViSSKEh6hPmppypNd6yfSNMTzRShf2QqGLKzhVf08kKJByFPi6M0BqIOe9ifif14xVr+QllEexIhzPFvViBlUIJ5nALhUEKzbSBGFB9a0QD5BAWOnkcjoEe/7lRVJzivZF0b1z8+XrNI4sOADHoABscAnK4BZUQBVg8AiewSt4M56MF+Pd+Ji1Zox0Zh/8gfH5A31rlKo=</latexit> (28 ⇥ 24)/2! = 2, 048 <latexit sha1_base64="WUm9od7Zn6L1+JeX+4Zh+q4LDJo=">AAACEnicbVC7TgJBFJ31ifhCLW1GiQkkBHcBZSlMiDaWmMgjgYXMDrMwYfaRmVkTsuEbbPwVGwuNsbWy828cYAsFT3KTk3Puzb332AGjQur6t7ayura+sZnYSm7v7O7tpw4OG8IPOSZ17DOft2wkCKMeqUsqGWkFnCDXZqRpj26mfvOBcEF9716OA2K5aOBRh2IkldRLZTPFbmSYE9iR1CUCFrqVLDyHxRN4BYvFnH5RyZmmkSsXzF4qref1GeAyMWKSBjFqvdRXp+/j0CWexAwJ0Tb0QFoR4pJiRibJTihIgPAIDUhbUQ+p/VY0e2kCz5TSh47PVXkSztTfExFyhRi7tup0kRyKRW8q/ue1Q+mYVkS9IJTEw/NFTsig9OE0H9innGDJxoogzKm6FeIh4ghLlWJShWAsvrxMGoW8cZkv3ZXS1es4jgQ4BqcgAwxQBlVwC2qgDjB4BM/gFbxpT9qL9q59zFtXtHjmCPyB9vkDOh+YIg==</latexit> (318 ⇥ 29)/3! = 33, 059, 881, 728
  16. method: judgement of CESS 1. Calculate the fraction of players

    in each reputation in a stationary state, denoted by h∗B , h∗N , and h∗G , respectively, under the assumption that the entire population uses a norm S. 2. Calculate the cooperation level pc, which means the probability that a donor cooperates towards a recipient when both are randomly picked from the resident population. 3. Reject the norm if p < pth, where pth is a threshold for the cooperation level. 4. Otherwise, calculate the payoff of a mutant with a different action rule from the resident one under the assumption that mutants occupy a suf fi ciently small fraction. 5. Repeat the above step for all possible action rules. If the payoff of a resident is higher than that of any possible mutants, S is a CESS. G G B G G G G G G N N N N N B B G G B G G G G G G N N N N N B B player with a di ff erent action rule (mutant)
  17. method: labeling of reputations • We assume no ordinal relationships

    between the three reputations. • Norms remain equivalent by swapping G/N/B labels. • We use the following protocol to remove the trivial multiplicity. 1. The reputation with the largest fraction is labelled as G. • In a CESS, G players cooperates with G. 2. When a G-player defects against another G-player by mistake, the player is assigned a reputation other than G, which we de fi ne as B. 3. The remaining reputation is called N.
  18. diverse social norms # of CESS "core set" CESS for

    any b/c C1P1R1 C1P1R2 C1P2R1 C1P2R2 G 0.9980 B 0.0015 N 0.0005 GG:CG 0.9940 GG:DB 0.0010 BG:CG 0.0015 GB:DG 0.0015 G 0.9965 B 0.0015 N 0.0020 GG:CG 0.9911 GG:DB 0.0010 GN:DG 0.0020 NG:DG 0.0020 GB:DG 0.0015 BG:CN 0.0015 G 0.9965 B 0.0015 N 0.0020 GG:CG 0.9910 GG:DB 0.0010 BG:CG 0.0015 GN:DG 0.0020 GB:DN 0.0015 NG:DG 0.0020 G 0.9950 B 0.0015 N 0.0035 GG:CG 0.9881 GG:DB 0.0010 BG:DN 0.0015 GN:DG 0.0034 GB:DN 0.0015 NG:CG 0.0034 C3P1R1 C3P1R2 C3P2R1 C3P2R2 G 0.6155 B 0.0061 N 0.3784 BG:CG 0.0037 GG:DB GN:DB 0.0006 NG:CG 0.2324 GB:DG 0.0038 NB:DB 0.0023 GN:CG 0.2324 NN:CG 0.1429 GG:CN 0.3781 BN:DB 0.0023 NG:DB NN:DB 0.0003 G 0.4989 B 0.0030 N 0.4981 BG:CN 0.0015 GG:DB GN:DB 0.0004 NN:CG 0.2476 NB:DG 0.0015 GN:CG 0.2480 NG:CN 0.2480 BN:CG 0.0015 GG:CN 0.2484 GB:DB 0.0015 NG:DB NN:DB 0.0004 G 0.4993 B 0.0030 N 0.4978 GN:CN 0.2480 BN:CG 0.0015 NN:CG 0.2480 GB:DN 0.0015 NB:DG 0.0015 GG:CG 0.2488 BG:DB 0.0015 NG:CN 0.2480 GG:DB GN:DB 0.0004 NG:DB NN:DB 0.0004 G 0.5000 B 0.0015 N 0.4985 GG:CN 0.2495 BG:CG BN:CG 0.0014 NG:CG 0.2488 GB:DG 0.0007 NB:DN 0.0007 GN:CG 0.2488 NN:CN 0.2480 GG:DB GN:DB 0.0004 NG:DB NN:DB 0.0004 C2P2R2 C2P2R1 C2P1R2 C2P1R1 G 0.9406 B 0.0016 N 0.0579 GG:CG 0.8829 GN:CG 0.0543 NG:CN 0.0543 GB:DN 0.0015 NN:CG 0.0033 BG:CN 0.0015 GG:DB 0.0009 G 0.9558 B 0.0015 N 0.0427 GG:CG 0.9117 GN:CN 0.0408 NG:CG 0.0408 GB:DN 0.0014 NN:DG 0.0018 BG:CG 0.0014 GG:DB 0.0009 G 0.9565 B 0.0015 N 0.0420 GG:CG 0.9130 GN:CG 0.0401 NG:CN 0.0401 GB:DG 0.0014 NN:DG 0.0018 BG:CN 0.0014 GG:DB 0.0009 G 0.9773 B 0.0015 N 0.0212 GG:CG 0.9531 GN:CN 0.0207 GB:DG 0.0015 BG:CG 0.0015 GG:DB 0.0010 NG:CG 0.0207 NN:CG 0.0005 2,166,764 CESS comprehensive enumerations of 30billion norms
  19. taxonomy of CESS cooperation pattern C1: G occupies the system

    C2: a large fraction of G and a small fraction of N C3: both G and N occupies the system recovery pattern R1: instantaneous recovery R2: gradual recovery punishment pattern P1: full justi fi cation of punishment P2: partial justi fi cation of punishment leading eight are special cases of C1-P1-R1 or C3-P1-R1 leading eight C1P1R1 C1P1R2 C1P2R1 C1P2R2 C2P1R1 C2P1R2 C2P2R1 C2P2R2 C3P1R1 C3P1R2 C3P2R1 C3P2R2 C1 C2 C3 <latexit sha1_base64="1v7odxFig9RvQkJToXqk+husUvk=">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</latexit> h ⇤ G = O(1) h ⇤ N = O(µ) h ⇤ B = O(µ) <latexit sha1_base64="8vN7jpJtvMVOuQd9Ru1X+YZmmD0=">AAACWHicbVHPT9swGHUCDMhg6+C4i0XVCS5VMiHgMgmxA5wGkygg1V3luF9aC9sJ9hekKso/OWmH8a9wwS1B6mBPsvT03vfDfk4LJR3G8d8gXFpeebe6th6939j88LH1aevK5aUV0BO5yu1Nyh0oaaCHEhXcFBa4ThVcp7ffZ/71PVgnc3OJ0wIGmo+NzKTg6KVhK+8wBRmyKmIpjKWpnN+KdTQZnv6qGHdYf/l2vpvsUca89mNBY+7OYsV0WTfmyaKpy72IgRm9zOswK8cT7A5b7bgbz0HfkqQhbdLgYtj6zUa5KDUYFIo710/iAgcVtyiFgjpipYOCi1s+hr6nhmtwg2oeTE07XhnRLLf+GKRzdbGj4tq5qU59peY4ca+9mfg/r19idjSopClKBCOeF2WlopjTWcp0JC0IVFNPuLDS35WKCbdcoP+LyIeQvH7yW3L1tZscdPd/7rePT5o41shnskN2SUIOyTE5IxekRwT5Qx6D5WAleAhJuBquP5eGQdOzTf5BuPUEYXax9g==</latexit> h ⇤ G = O(1) h ⇤ N = O( p µ) h ⇤ B = O(µ) <latexit sha1_base64="s0zNS+sih28V2sYx/MGaEqTx/Ao=">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</latexit> h ⇤ G = O(1) h ⇤ N = O(1) h ⇤ B = O(µ) R1: instantaneous recovery R2: gradual recovery P1: full justification of punishment P2: partial justification of punishment error rate fraction of N players
  20. 1: unjusti fi ed punishment (iii) punishment is always justi

    fi ed G B defect G B binary reputation ternary reputation (iii) punishment is not always justi fi ed G B defect N B N is defected by G. This player is defected even though he/she correctly follow the norm.
  21. 2. apology by defecting (iv) show apology by cooperating with

    G players B G G cooperate G (iv) defect fi rst, and then cooperate to make an apology B G G G G N cooperate defect two steps are required to recover the reputation. binary reputation ternary reputation
  22. 3. emergence of inequality binary reputation ternary reputation G G

    G G G G G G G G G G G G ... ... ... G Almost all players are in G. Cooperation is maintained by G players. G G G G G G G G G G G ... ... ... G N N N Players spontaneously split into majority G-players & minority N-players. N-players don't cooperate with each other. B B
  23. Conclusion (i) maintenance of cooperation (ii) identi fi cation of

    defectors (iii) punishment and its justi fi cation (iv) apology and forgiveness (i) maintenance of cooperation by the majority (but not necessarily all) of the population. (ii) identi fi cation of defectors (iii) punishment and its partial or full justi fi cation (iv) apology and forgiveness, either gradual or instantaneous • We exhaustively enumerated Cooperative Evolutionary Stable Strategies (CESS) with ternary reputations. • CESS are categorized into 12 classes. The leading eight are special cases of them. The rules for CESS's are revised: Open Issues: private reputation, incomplete observation, communication, structured populations
  24. reference & source code https://github.com/yohm/sim_game_ternary_reputation source code & the list

    of the CESS norms Y. Murase et al. "Social norms in indirect reciprocity with ternary reputations" Sci.Rep 12, 455 (2022)