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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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Transcript

  1. Successful social norms with ternary
    reputation for indirect reciprocity
    Yohsuke Murase, Minjae Kim, Seung Ki Baek

    RIKEN R-CCS, Pukyong National Univ.

    View Slide

  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

    View Slide

  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

    View Slide

  4. social norm
    Norm: Assignment rule + Action rule

    Assignment rule : Which reputation is given to the donor



    Action rule : Which action is taken
    AAAB9XicbVBNS8NAEJ34WetX1aOXxSJUKCWRoh6rXjxWsR/axrLZbtqlm03Y3Sgl9H948aCIV/+LN/+N2zYHbX0w8Hhvhpl5XsSZ0rb9bS0sLi2vrGbWsusbm1vbuZ3dugpjSWiNhDyUTQ8rypmgNc00p81IUhx4nDa8weXYbzxSqVgobvUwom6Ae4L5jGBtpIebQrN4Vzw/Qm0dovtOLm+X7AnQPHFSkocU1U7uq90NSRxQoQnHSrUcO9JugqVmhNNRth0rGmEywD3aMlTggCo3mVw9QodG6SI/lKaERhP190SCA6WGgWc6A6z7atYbi/95rVj7Z27CRBRrKsh0kR9zZF4cR4C6TFKi+dAQTCQztyLSxxITbYLKmhCc2ZfnSf245JyUytflfOUijSMD+3AABXDgFCpwBVWoAQEJz/AKb9aT9WK9Wx/T1gUrndmDP7A+fwDjKpDV
    R(X, Y, A) ! Z
    AAAB83icbVBNS8NAEJ3Ur1q/oh69LBahgpREinqsevFYwX5IE8pmu2mXbjZhdyOU0L/hxYMiXv0z3vw3btsctPXBwOO9GWbmBQlnSjvOt1VYWV1b3yhulra2d3b37P2DlopTSWiTxDyWnQArypmgTc00p51EUhwFnLaD0e3Ubz9RqVgsHvQ4oX6EB4KFjGBtJK9R6Zw9niJPx+i6Z5edqjMDWiZuTsqQo9Gzv7x+TNKICk04VqrrOon2Myw1I5xOSl6qaILJCA9o11CBI6r8bHbzBJ0YpY/CWJoSGs3U3xMZjpQaR4HpjLAeqkVvKv7ndVMdXvkZE0mqqSDzRWHKkXlxGgDqM0mJ5mNDMJHM3IrIEEtMtImpZEJwF19eJq3zqntRrd3XyvWbPI4iHMExVMCFS6jDHTSgCQQSeIZXeLNS68V6tz7mrQUrnzmEP7A+fwDFGpA5
    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

    View Slide

  5. examples
    AAAB+XicbVBNS8NAEJ34WetX1KOXxSJUkJJIUY/FHvRYwX5AE8pmu2mXbjZhd1Moof/EiwdFvPpPvPlv3LY5aOuDgcd7M8zMCxLOlHacb2ttfWNza7uwU9zd2z84tI+OWypOJaFNEvNYdgKsKGeCNjXTnHYSSXEUcNoORvWZ3x5TqVgsnvQkoX6EB4KFjGBtpJ5tN8oeVvoS3V8gT8eo3rNLTsWZA60SNyclyNHo2V9ePyZpRIUmHCvVdZ1E+xmWmhFOp0UvVTTBZIQHtGuowBFVfja/fIrOjdJHYSxNCY3m6u+JDEdKTaLAdEZYD9WyNxP/87qpDm/9jIkk1VSQxaIw5ci8OIsB9ZmkRPOJIZhIZm5FZIglJtqEVTQhuMsvr5LWVcW9rlQfq6XaXR5HAU7hDMrgwg3U4AEa0AQCY3iGV3izMuvFerc+Fq1rVj5zAn9gff4A8qaR7g==
    P(⇤, G) ! C
    AAAB+XicbVBNS8NAEJ34WetX1KOXxSJUkJJIUY+levBYwX5AE8pmu2mXbjZhd1Moof/EiwdFvPpPvPlv3LY5aOuDgcd7M8zMCxLOlHacb2ttfWNza7uwU9zd2z84tI+OWypOJaFNEvNYdgKsKGeCNjXTnHYSSXEUcNoORnczvz2mUrFYPOlJQv0IDwQLGcHaSD3bbpQ9rPQlql8gT8fovmeXnIozB1olbk5KkKPRs7+8fkzSiApNOFaq6zqJ9jMsNSOcToteqmiCyQgPaNdQgSOq/Gx++RSdG6WPwliaEhrN1d8TGY6UmkSB6YywHqplbyb+53VTHd76GRNJqqkgi0VhypF5cRYD6jNJieYTQzCRzNyKyBBLTLQJq2hCcJdfXiWtq4p7Xak+Vku1eh5HAU7hDMrgwg3U4AEa0AQCY3iGV3izMuvFerc+Fq1rVj5zAn9gff4A7HOR6g==
    P(⇤, B) ! D AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARKkiZkaIuS3Xhsop9QGcomTTThmYeJHeEMnbhr7hxoYhbf8Odf2PazkJbDyQczrmXe+/xYsEVWNa3kVtaXlldy68XNja3tnfM3b2mihJJWYNGIpJtjygmeMgawEGwdiwZCTzBWt7wauK3HphUPArvYRQzNyD9kPucEtBS1zy4KzlEwSme/dcn2IEI17pm0SpbU+BFYmekiDLUu+aX04toErAQqCBKdWwrBjclEjgVbFxwEsViQoekzzqahiRgyk2n+4/xsVZ62I+kfiHgqfq7IyWBUqPA05UBgYGa9ybif14nAf/STXkYJ8BCOhvkJwLrEydh4B6XjIIYaUKo5HpXTAdEEgo6soIOwZ4/eZE0z8r2eblyWylWa1kceXSIjlAJ2egCVdENqqMGougRPaNX9GY8GS/Gu/ExK80ZWc8++gPj8wfiQJQY
    R(⇤, ⇤, D) ! B
    AAAB/3icbVDLSgMxFM3UV62vUcGNm2ARKkiZkaIui13osop9QGcomTTThmYeJHeEMnbhr7hxoYhbf8Odf2PazkJbDyQczrmXe+/xYsEVWNa3kVtaXlldy68XNja3tnfM3b2mihJJWYNGIpJtjygmeMgawEGwdiwZCTzBWt6wNvFbD0wqHoX3MIqZG5B+yH1OCWipax7clRyi4BTP/toJdiDC112zaJWtKfAisTNSRBnqXfPL6UU0CVgIVBClOrYVg5sSCZwKNi44iWIxoUPSZx1NQxIw5abT/cf4WCs97EdSvxDwVP3dkZJAqVHg6cqAwEDNexPxP6+TgH/ppjyME2AhnQ3yE4H1iZMwcI9LRkGMNCFUcr0rpgMiCQUdWUGHYM+fvEiaZ2X7vFy5rRSrV1kceXSIjlAJ2egCVdENqqMGougRPaNX9GY8GS/Gu/ExK80ZWc8++gPj8wfoSZQc
    R(⇤, ⇤, C) ! G
    G B
    defect
    B
    B
    Bad players increase.

    Cooperation level drops.
    Image scoring All-Good
    AAAB+XicbVBNS8NAEJ34WetX1KOXxSJUkJJIUY/FHvRYwX5AE8pmu2mXbjZhd1Moof/EiwdFvPpPvPlv3LY5aOuDgcd7M8zMCxLOlHacb2ttfWNza7uwU9zd2z84tI+OWypOJaFNEvNYdgKsKGeCNjXTnHYSSXEUcNoORvWZ3x5TqVgsnvQkoX6EB4KFjGBtpJ5tN8oeVvoS3V8gT8eo3rNLTsWZA60SNyclyNHo2V9ePyZpRIUmHCvVdZ1E+xmWmhFOp0UvVTTBZIQHtGuowBFVfja/fIrOjdJHYSxNCY3m6u+JDEdKTaLAdEZYD9WyNxP/87qpDm/9jIkk1VSQxaIw5ci8OIsB9ZmkRPOJIZhIZm5FZIglJtqEVTQhuMsvr5LWVcW9rlQfq6XaXR5HAU7hDMrgwg3U4AEa0AQCY3iGV3izMuvFerc+Fq1rVj5zAn9gff4A8qaR7g==
    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??
    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
    P(⇤, B) ! D
    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
    R(⇤, ⇤, ⇤) ! G

    View Slide

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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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.

    View Slide

  13. comprehensive enumeration
    binary reputation :
    ternary reputation :
    AAAB+HicbVDLTsJAFL3FF+KDqks3VWKCG2wJUZdENy4xkUcChUyHKUw6nTYzUxNs+BI3LjTGrZ/izr9xgC4UPMm9OTnn3syd48WMSmXb30ZubX1jcyu/XdjZ3dsvmgeHLRklApMmjlgkOh6ShFFOmooqRjqxICj0GGl7we3Mbz8SIWnEH9QkJm6IRpz6FCOlpYFZLFeDfvW8n+o+vQhOBmbJrthzWKvEyUgJMjQG5ldvGOEkJFxhhqTsOnas3BQJRTEj00IvkSRGOEAj0tWUo5BIN50fPrXOtDK0/Ejo4sqaq783UhRKOQk9PRkiNZbL3kz8z+smyr92U8rjRBGOFw/5CbNUZM1SsIZUEKzYRBOEBdW3WniMBMJKZ1XQITjLX14lrWrFuazU7mul+k0WRx6O4RTK4MAV1OEOGtAEDAk8wyu8GU/Gi/FufCxGc0a2cwR/YHz+ALCzkdQ=
    (2k2)k2
    /k!
    # of norms with k-ary reputations:
    AAACBnicbVDLSsNAFJ3UV62vqEsRRotQQWoSgnYjFN24rGAf0KZlMp22QyeTMDMRSujKjb/ixoUibv0Gd/6N0zYLbT1w4XDOvdx7jx8xKpVlfRuZpeWV1bXsem5jc2t7x9zdq8kwFphUcchC0fCRJIxyUlVUMdKIBEGBz0jdH95M/PoDEZKG/F6NIuIFqM9pj2KktNQxDwtOuwRbigZEQqftnsJz6BzBK+icWW6pY+atojUFXCR2SvIgRaVjfrW6IY4DwhVmSMqmbUXKS5BQFDMyzrViSSKEh6hPmppypNd6yfSNMTzRShf2QqGLKzhVf08kKJByFPi6M0BqIOe9ifif14xVr+QllEexIhzPFvViBlUIJ5nALhUEKzbSBGFB9a0QD5BAWOnkcjoEe/7lRVJzivZF0b1z8+XrNI4sOADHoABscAnK4BZUQBVg8AiewSt4M56MF+Pd+Ji1Zox0Zh/8gfH5A31rlKo=
    (28 ⇥ 24)/2! = 2, 048
    AAACEnicbVC7TgJBFJ31ifhCLW1GiQkkBHcBZSlMiDaWmMgjgYXMDrMwYfaRmVkTsuEbbPwVGwuNsbWy828cYAsFT3KTk3Puzb332AGjQur6t7ayura+sZnYSm7v7O7tpw4OG8IPOSZ17DOft2wkCKMeqUsqGWkFnCDXZqRpj26mfvOBcEF9716OA2K5aOBRh2IkldRLZTPFbmSYE9iR1CUCFrqVLDyHxRN4BYvFnH5RyZmmkSsXzF4qref1GeAyMWKSBjFqvdRXp+/j0CWexAwJ0Tb0QFoR4pJiRibJTihIgPAIDUhbUQ+p/VY0e2kCz5TSh47PVXkSztTfExFyhRi7tup0kRyKRW8q/ue1Q+mYVkS9IJTEw/NFTsig9OE0H9innGDJxoogzKm6FeIh4ghLlWJShWAsvrxMGoW8cZkv3ZXS1es4jgQ4BqcgAwxQBlVwC2qgDjB4BM/gFbxpT9qL9q59zFtXtHjmCPyB9vkDOh+YIg==
    (318 ⇥ 29)/3! = 33, 059, 881, 728

    View Slide

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

    View Slide

  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 AAAB9HicbVBNSwMxEJ31s9avqkcvwSJ4Krsi6rHoxWMF+wHtUrJptg3NJjHJFsvS3+HFgyJe/THe/Dem7R609cHA470ZZuZFijNjff/bW1ldW9/YLGwVt3d29/ZLB4cNI1NNaJ1ILnUrwoZyJmjdMstpS2mKk4jTZjS8nfrNEdWGSfFgx4qGCe4LFjOCrZNC1SWog5XS8gkF3VLZr/gzoGUS5KQMOWrd0lenJ0maUGEJx8a0A1/ZMMPaMsLppNhJDVWYDHGfth0VOKEmzGZHT9CpU3ooltqVsGim/p7IcGLMOIlcZ4LtwCx6U/E/r53a+DrMmFCppYLMF8UpR1aiaQKoxzQllo8dwUQzdysiA6wxsS6nogshWHx5mTTOK8FlJbi/KFdv8jgKcAwncAYBXEEV7qAGdSDwCM/wCm/eyHvx3r2PeeuKl88cwR94nz8Cv5Gb
    pc
    ⇡ 1
    AND
    Evolutionarily Stable State (ESS)
    Cooperative Evolutionarily Stable State (CESS)
    AAAB+HicbVDLTsJAFL3FF+KDqks3VWKCG2wJUZdENy4xkUcChUyHKUw6nTYzUxNs+BI3LjTGrZ/izr9xgC4UPMm9OTnn3syd48WMSmXb30ZubX1jcyu/XdjZ3dsvmgeHLRklApMmjlgkOh6ShFFOmooqRjqxICj0GGl7we3Mbz8SIWnEH9QkJm6IRpz6FCOlpYFZLFeDfvW8n+o+vQhOBmbJrthzWKvEyUgJMjQG5ldvGOEkJFxhhqTsOnas3BQJRTEj00IvkSRGOEAj0tWUo5BIN50fPrXOtDK0/Ejo4sqaq783UhRKOQk9PRkiNZbL3kz8z+smyr92U8rjRBGOFw/5CbNUZM1SsIZUEKzYRBOEBdW3WniMBMJKZ1XQITjLX14lrWrFuazU7mul+k0WRx6O4RTK4MAV1OEOGtAEDAk8wyu8GU/Gi/FufCxGc0a2cwR/YHz+ALCzkdQ=
    (2k2)k2
    /k!
    # of norms with k-ary reputations:
    AAACBnicbVDLSsNAFJ3UV62vqEsRRotQQWoSgnYjFN24rGAf0KZlMp22QyeTMDMRSujKjb/ixoUibv0Gd/6N0zYLbT1w4XDOvdx7jx8xKpVlfRuZpeWV1bXsem5jc2t7x9zdq8kwFphUcchC0fCRJIxyUlVUMdKIBEGBz0jdH95M/PoDEZKG/F6NIuIFqM9pj2KktNQxDwtOuwRbigZEQqftnsJz6BzBK+icWW6pY+atojUFXCR2SvIgRaVjfrW6IY4DwhVmSMqmbUXKS5BQFDMyzrViSSKEh6hPmppypNd6yfSNMTzRShf2QqGLKzhVf08kKJByFPi6M0BqIOe9ifif14xVr+QllEexIhzPFvViBlUIJ5nALhUEKzbSBGFB9a0QD5BAWOnkcjoEe/7lRVJzivZF0b1z8+XrNI4sOADHoABscAnK4BZUQBVg8AiewSt4M56MF+Pd+Ji1Zox0Zh/8gfH5A31rlKo=
    (28 ⇥ 24)/2! = 2, 048
    AAACEnicbVC7TgJBFJ31ifhCLW1GiQkkBHcBZSlMiDaWmMgjgYXMDrMwYfaRmVkTsuEbbPwVGwuNsbWy828cYAsFT3KTk3Puzb332AGjQur6t7ayura+sZnYSm7v7O7tpw4OG8IPOSZ17DOft2wkCKMeqUsqGWkFnCDXZqRpj26mfvOBcEF9716OA2K5aOBRh2IkldRLZTPFbmSYE9iR1CUCFrqVLDyHxRN4BYvFnH5RyZmmkSsXzF4qref1GeAyMWKSBjFqvdRXp+/j0CWexAwJ0Tb0QFoR4pJiRibJTihIgPAIDUhbUQ+p/VY0e2kCz5TSh47PVXkSztTfExFyhRi7tup0kRyKRW8q/ue1Q+mYVkS9IJTEw/NFTsig9OE0H9innGDJxoogzKm6FeIh4ghLlWJShWAsvrxMGoW8cZkv3ZXS1es4jgQ4BqcgAwxQBlVwC2qgDjB4BM/gFbxpT9qL9q59zFtXtHjmCPyB9vkDOh+YIg==
    (318 ⇥ 29)/3! = 33, 059, 881, 728

    View Slide

  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)

    View Slide

  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.

    View Slide

  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

    View Slide

  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
    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
    h

    G
    = O(1)
    h

    N
    = O(µ)
    h

    B
    = O(µ)
    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
    h

    G
    = O(1)
    h

    N
    = O(
    p
    µ)
    h

    B
    = O(µ)
    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
    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

    View Slide

  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.

    View Slide

  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

    View Slide

  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

    View Slide

  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

    View Slide

  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)

    View Slide