ؒ઀ޓܙੑʹΑΔڠྗͷਐԽ ଜ੉༸հ RIKEN R-CCS ICETηϛφʔ 2025/1/16 — ؒ઀ޓܙੑͷ਺ཧ —

agenda • طଘཧ࿦ݚڀͷϨϏϡʔ • ݚڀ঺հ̍ɿҙݟಉظԽͷؒ઀ޓܙੑ • ݚڀ঺հ̎ɿࣾձنൣ͕Ͳ͏΍ͬͯਐԽ͢Δ͔ • ࣍ʹղ͘΂͖՝୊͸ʁ • ʢଜ੉ͷओ؍ͱภݟ͕ଟ෼ʹؚ·Ε·͢ʣ • ໨త • ཧ࿦Λઐ໳ͱ͠ͳ͍ํ͕ɺؒ઀ޓܙੑͷཧ࿦ݚڀͷ֓ཁΛ͔ͭΊΔΑ͏ʹ • ʢҰॹʹݚڀ͢ΔਓΛ୳͍ͨ͠ʣ

agenda of the review part • public assessment models • the leading eight • how to solve the public assessment models • private assessment models • why private assessment matters? • three types of models Check out the SI of Murase & Hilbe PNAS (2024). We wrote a brief 5-page review on theoretical studies.

direct & indirect reciprocity • cooperation : an act that bene fi ts others by paying cost (b > c) -c +b AAACzXichVE9T9xAEH0YEsCQcJAGKc2JE4ginOYQX0qFQkMXPnKAhNHJXvaOFf6SvT5BLqSNlBqJgopIKaL8A1oo+AMp+AlRSpBoKBj7rKAEhezK3tk382bfzDihq2JNdNlhdHY9edrd02v29T97PlAYHFqLgyQSsioCN4g2HDuWrvJlVSvtyo0wkrbnuHLd2V1I/etNGcUq8N/p/VBueXbDV3UlbM1QrVC2HNlQfiv0bB2pvQPTmRDFseKEsCzTYYNMS/rbv921QonKlK3iQ6OSGyXkaykonMPCNgIIJPAg4UOz7cJGzHsTFRBCxrbQYixiS2V+iQOYzE04SnKEzegu/xt828xRn+9pzjhjC37F5S9iZhGj9IO+0RVd0Hf6Sbf/zNXKcqRa9vl02lwZ1gY+D6/e/Jfl8amxc896VLNGHXOZVsXawwxJqxBtfvP90dXq65XR1hh9oV+s/4Qu6Ywr8JvX4uuyXDl+RI/gXefq0/7GeMVYknUxQpNVci95dJW/B/XQWJssV2bK08tTpfk3+RB78BIjGOdJzWIei1hCld86xCnOcG68NRLjg/GxHWp05JwX+GMZn+4AYdam9A== ✓ b - c -c b ◆ cooperate(C) defect(D) C D • direct reciprocity • indirect reciprocity

a standard model of indirect reciprocity -c +b A large enough population. Every player has reputation of either “Good” or “Bad”. (public assessment) Population play a repeated donation game. 1. randomly pick two players as a donor and a recipient 2. The donor decides C or D. 3. The donor’s reputation is updated. B G C → G G B Ohtsuki & Iwasa (2004)

action & assessment rules action rule (how to behave) + assessment rule (how to assign reputations) assess Good or Bad? • when "Good" cooperated with “Good”, • when "Good" defected against “Good”, • when "Good" cooperated with “Bad” • when “Good" defected against “Bad” • … • when “Bad” defected against “Bad” Cooperation or Defection? • when “Good” meets “Good” • when “Good” meets “Bad” • when “Bad” meets “Good” • when “Bad” meets “Bad”, action rule assessment rule Under appropriate social norms, cooperation is promoted. Which norms promote stable cooperation? 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 R(X Y AXY ) ! X 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 P(X Y) ! AXY X Y Aij social norm

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 Cooperation level is high if everyone follows the norm. Defector earns a higher payo ff . Not a Nash eq. How can we achieve evolutionarily stable cooperation? 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 P(⇤, B) ! D 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 R(⇤, ⇤, ⇤) ! G 👿

the leading-eight Ohtsuki & Iwasa (2004) (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 successful norms to sustain cooperation & Nash equilibrium (ESS) (i) maintenance of cooperation G G cooperate G G

complexity of social norms defect B defect G B defect fi rst-order norms second-order norms third-order norms 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 R(A) 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 R(Y A) 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 R(X Y A) image scoring generous scoring Simple Standing (L3) Stern Judging (L6) L1, L2, L4, L5, L7, L8 secondary sixteen cooperative ESS exist

how to analyze the public assessment models for a given social norm {P, R}, we calculate the equilibrium fraction of good reptuation h* G G G G G G G G B B B B 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 h 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 ! h 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 h(t) = h(t) RP (G G)+h(t)[ !h(t)][RP (G B)+RP (B G)]+[ !h(t)] RP (B B) with h*, we calculate the self-cooperation probability pc and the payo ff πres 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 ω = (b ! c)pc the social norm is self-cooperative i f 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 µ→ pc =

G G G G G G G B B B B for each action rule P’ ≠ P, we introduce the mutant with P’ and calculate its average reputation H 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 H → 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 h→ 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 ! h→ AAADKnichVFNT9tAEB2btny1JdADlXrZEpEmJERrhCiqVCkKh+QYQgNIcYhsZ5NYOLax15HAyh9of0APPbVSD1X5F+2hZyQOEVcuiFNFpV44MHYs8SVgV/bOvpn39u2Oahu6yyntC+LQo8dPhkdGx8afPns+EZucWnctz9FYRbMMy9lUFZcZuskqXOcG27QdpnRUg22o2ytBfqPLHFe3zA9812a1jtIy9aauKRyheuyT3LC4X+wleYok3pP2lj/XKwabct0vveklCxlSSJH0IEGq0nyQrF3J5lNElhNE3vGURpokpfmwMkWuiuQjkcvsTaF8IFSPxWmWhoPcDqQoiEM0SlbsN8jQAAs08KADDEzgGBuggIuzChJQsBGrgY+Yg5Ee5hn0YAy5HlYxrFAQ3cZ/C3fVCDVxH2i6IVvDUwz8HGQSmKWH9Ac9o3/oT3pCz+/U8kONwMsuruqAy+z6xMeXa/8fZHVw5dC+ZN3rmUMTlkOvOnq3QyS4hTbgd/c+n629K8/6CfqNnqL/r7RPf+ENzO4/7fsqK3+5x4+Gs4m3D97XhQxiXviKDnTRJb4ltk662ajbwfpCVlrKLq0uxnP5qIkj8ApmIImdegs5KEIJKnjWX2FaIMJrcV88EPvi0aBUFCLOC7g2xOMLnzm93Q== H(t) = h → H(t)RP→ (G G) + h →[ ! H(t)]RP→ (G B) + ( ! h →)H(t)RP→ (B G) + ( ! h →)[ ! H(t)]RP→ (B B) from h* and H*, we calculate the cooperation probabilities pres→mut & pmut→res 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 ω = bp → ! cp → the social norm is ESS i f 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 →P→ ↑= P ω > ω the social norm is cooperative ESS (CESS) if it is self-cooperative & ESS.

subsequent studies costly punishment in-group favoritism non-binary reputations dual reputation updates Ohtsuki et al. Nature (2009) P AAAB7nicbVBNS8NAEJ3Ur1q/qh69BIvgxZKIVI9FLx4r2A9oQ5lsN+3SzWbZ3Qgl9Ed48aCIV3+PN/+N2zYHbX0w8Hhvhpl5oeRMG8/7dgpr6xubW8Xt0s7u3v5B+fCopZNUEdokCU9UJ0RNORO0aZjhtCMVxTjktB2O72Z++4kqzRLxaCaSBjEOBYsYQWOl9kUPuRxhv1zxqt4c7irxc1KBHI1++as3SEgaU2EIR627vidNkKEyjHA6LfVSTSWSMQ5p11KBMdVBNj936p5ZZeBGibIljDtXf09kGGs9iUPbGaMZ6WVvJv7ndVMT3QQZEzI1VJDFoijlrknc2e/ugClKDJ9YgkQxe6tLRqiQGJtQyYbgL7+8SlqXVb9WrT1cVeq3eRxFOIFTOAcfrqEO99CAJhAYwzO8wpsjnRfn3flYtBacfOYY/sD5/AH5Q49a ↵ AAAB7XicbVBNS8NAEJ34WetX1aOXYBG8WBKR6rHoxWMF+wFtKJvtpF272Q27G6GE/gcvHhTx6v/x5r9x2+agrQ8GHu/NMDMvTDjTxvO+nZXVtfWNzcJWcXtnd2+/dHDY1DJVFBtUcqnaIdHImcCGYYZjO1FI4pBjKxzdTv3WEyrNpHgw4wSDmAwEixglxkrN826IhvRKZa/izeAuEz8nZchR75W+un1J0xiFoZxo3fG9xAQZUYZRjpNiN9WYEDoiA+xYKkiMOshm107cU6v03UgqW8K4M/X3REZircdxaDtjYoZ60ZuK/3md1ETXQcZEkhoUdL4oSrlrpDt93e0zhdTwsSWEKmZvdemQKEKNDahoQ/AXX14mzYuKX61U7y/LtZs8jgIcwwmcgQ9XUIM7qEMDKDzCM7zCmyOdF+fd+Zi3rjj5zBH8gfP5AzF4juY= X donor recipient N. Masuda, J. Theor. Biol. (2012) Y. Murase et al., Sci.Rep. (2022) Y. Murase et al. PLOS Comp.Biol. (2023) P(G, B) = 0.7 G B R 1 (G, B, D) D R 2 (G, B, D) Alice Bob Alice’s cooperation probability Alice’s realized action Alice’s next reputation Bob’s next reputation Y. Murase to appear in J. Theor. Biol.

competition between norms by group-level selection Pacheco et al., PLOS Comp. Biol. (2011) Santos et al., Nature (2018) Summary of the public assessment models: • Mathematically, public assessment models are easier to work with. • The leading-eight have four simple principles in common. Similar principles also explain the cooperative ESS in extended models. • Several studies conclude that L6 (Stern Judging) is the most successful.

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private assessment disagreement can escalate even the leading eight does not work Uchida PRE (2010) Hilbe et al. PNAS (2018) B G A B C D D observers donor recipient )FNVTUCFCBE *NVTUQVOJTI #JTTVDIBOJDFQFSTPO *DBO`UGPSHJWF" #JTBOBTUZHVZ )FEFTFSWFTQVOJTINFOU G B

several types of private assessment models 3 6 2 1 4 5 C or D observers donor recipient solitary observation model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer Okada et al. Sci.Rep (2020) 1 2 3 4 5 6 1 2 3 4 5 6 3 6 2 1 4 5 C or D observers donor recipient simultaneous observation model the donor is assessed by multiple observers Fujimoto et al. PNAS (2023) Murase et al. PNAS (2024) 3 6 2 1 4 5 C or D observers donor recipient gossiping model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer opinions are exchanged by gossiping Kawakatsu et al. PNAS (2024)

1 2 3 4 5 6 1 2 3 4 5 6 3 6 2 1 4 5 C or D observers donor recipient simultaneous observation model the donor is assessed by multiple observers Fujimoto et al. PNAS (2023) Murase et al. PNAS (2024) •most basic model •analytically intractable •Harder to maintain cooperation. L6 fails terribly.

3 6 2 1 4 5 C or D observers donor recipient solitary observation model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer Okada et al. Sci.Rep (2020) •only one observer updates their opinion •analytically tractable •L6 fails

3 6 2 1 4 5 C or D observers donor recipient gossiping model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer opinions are exchanged by gossiping Kawakatsu et al. PNAS (2024) •exchange opinions between players •analytically tractable to some extent •L6 works

ϨϏϡʔύʔτ·ͱΊ • େ͖͘෼͚Δͱ public assessment ͱ private assessment model ͕͋Δ • public assessment model ͸ཧ࿦తʹѻ͍΍͍͕͢ɺධ൑͕׬શʹಉظ͍ͯ͠Δͱཧ ૝Խ͍ͯ͠Δ • private assessment model ͸ҙݟͷ৯͍ҧ͍Λڐ͕͢ɺཧ࿦తͳѻ͍͸جຊతʹࠔ೉ • ͞Βʹ̏छྨͷΫϥεʹ͓͓·͔ʹ෼ྨͰ͖Δ

agenda • طଘཧ࿦ݚڀͷϨϏϡʔ • ݚڀ঺հ̍ɿҙݟಉظԽͷؒ઀ޓܙੑ • ݚڀ঺հ̎ɿࣾձنൣ͕Ͳ͏΍ͬͯਐԽ͢Δ͔ • ࣍ʹղ͘΂͖՝୊͸ʁ • ʢଜ੉ͷओ؍ͱภݟ͕ଟ෼ʹؚ·Ε·͢ʣ • ໨త • ཧ࿦Λઐ໳ͱ͠ͳ͍ํ͕ɺؒ઀ޓܙੑͷཧ࿦ݚڀͷ֓ཁΛ͔ͭΊΔΑ͏ʹ • ʢҰॹʹݚڀ͢ΔਓΛ୳͍ͨ͠ʣ

indirect reciprocity models 3 6 2 1 4 5 C or D observers donor recipient solitary observation model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer 1 2 3 4 5 6 1 2 3 4 5 6 3 6 2 1 4 5 C or D observers donor recipient simultaneous observation model the donor is assessed by multiple observers 3 6 2 1 4 5 C or D observers donor recipient gossiping model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer opinions are exchanged by gossiping 3 6 2 1 4 5 C or D observers donor recipient public assessment model 1 2 3 4 5 6 1 2 3 4 5 6 opinions are fully synchronized by model definition AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqs5IqS6LblxJBfuQdiiZNNOGJpkxyQhl6Fe4caGIWz/HnX9j2s5CWw8EDufcS+45QcyZNq777eRWVtfWN/Kbha3tnd294v5BU0eJIrRBIh6pdoA15UzShmGG03asKBYBp61gdD31W09UaRbJezOOqS/wQLKQEWys9PCIuiZC3tltr1hyy+4MaJl4GSlBhnqv+NXtRyQRVBrCsdYdz42Nn2JlGOF0UugmmsaYjPCAdiyVWFDtp7ODJ+jEKn0URso+adBM/b2RYqH1WAR2UmAz1IveVPzP6yQmvPRTJuPEUEnmH4UJRzbkND3qM0WJ4WNLMFHM3orIECtMjO2oYEvwFiMvk+Z52auWq3eVUu0qqyMPR3AMp+DBBdTgBurQAAICnuEV3hzlvDjvzsd8NOdkO4fwB87nD16Hj30= q ! 1/N AAAB8XicbVDLSgMxFL1TX7W+qi7dBIvgqsyIVJdFNy4r2Ad2hpJJM21oJjMkd4RS+hduXCji1r9x59+YtrPQ1gOBwzn3kntOmEph0HW/ncLa+sbmVnG7tLO7t39QPjxqmSTTjDdZIhPdCanhUijeRIGSd1LNaRxK3g5HtzO//cS1EYl6wHHKg5gOlIgEo2ilRx9pRnxMiNsrV9yqOwdZJV5OKpCj0St/+f2EZTFXyCQ1puu5KQYTqlEwyaclPzM8pWxEB7xrqaIxN8FkfvGUnFmlT6JE26eQzNXfGxMaGzOOQzsZUxyaZW8m/ud1M4yug4lQaYZcscVHUSaJjTiLT/pCc4ZybAllWthbCRtSTRnakkq2BG858ippXVS9WrV2f1mp3+R1FOEETuEcPLiCOtxBA5rAQMEzvMKbY5wX5935WIwWnHznGP7A+fwBnhCQPg== ⌧ ! 0 AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkoiUj0WvXisYGuhCWWz3bRLN5uwOyvE0l/ixYMiXv0p3vw3btsctPXBwOO9GWbmRZngGjzv2ymtrW9sbpW3Kzu7e/tV9+Cwo1OjKGvTVKSqGxHNBJesDRwE62aKkSQS7CEa38z8h0emNE/lPeQZCxMylDzmlICV+m41AGJwACkOuIwh77s1r+7NgVeJX5AaKtDqu1/BIKUmYRKoIFr3fC+DcEIUcCrYtBIYzTJCx2TIepZKkjAdTuaHT/GpVQY4TpUtCXiu/p6YkETrPIlsZ0JgpJe9mfif1zMQX4UTLjMDTNLFotgIbP+cpYAHXDEKIreEUMXtrZiOiCIUbFYVG4K//PIq6ZzX/Ua9cXdRa14XcZTRMTpBZ8hHl6iJblELtRFFBj2jV/TmPDkvzrvzsWgtOcXMEfoD5/MHbv2S9w== ⌧ ! 1 AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KolI9SIUPeixgv2ANpTNdtMs3WzC7kQopT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMC1IpDLrut7Oyura+sVnYKm7v7O7tlw4OmybJNOMNlshEtwNquBSKN1Cg5O1UcxoHkreC4e3Ubz1xbUSiHnGUcj+mAyVCwShaqRX17sg1iXqlsltxZyDLxMtJGXLUe6Wvbj9hWcwVMkmN6Xhuiv6YahRM8kmxmxmeUjakA96xVNGYG388O3dCTq3SJ2GibSkkM/X3xJjGxoziwHbGFCOz6E3F/7xOhuGVPxYqzZArNl8UZpJgQqa/k77QnKEcWUKZFvZWwiKqKUObUNGG4C2+vEya5xWvWqk+XJRrN3kcBTiGEzgDDy6hBvdQhwYwGMIzvMKbkzovzrvzMW9dcfKZI/gD5/MHDEqOvg== hG = h AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KolI9SIUPeixgv2ANpTNdtMu3WzC7kQooT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMCxIpDLrut7Oyura+sVnYKm7v7O7tlw4OmyZONeMNFstYtwNquBSKN1Cg5O1EcxoFkreC0e3Ubz1xbUSsHnGccD+iAyVCwShaqTXs3ZFr4vVKZbfizkCWiZeTMuSo90pf3X7M0ogrZJIa0/HcBP2MahRM8kmxmxqeUDaiA96xVNGIGz+bnTshp1bpkzDWthSSmfp7IqORMeMosJ0RxaFZ9Kbif14nxfDKz4RKUuSKzReFqSQYk+nvpC80ZyjHllCmhb2VsCHVlKFNqGhD8BZfXibN84pXrVQfLsq1mzyOAhzDCZyBB5dQg3uoQwMYjOAZXuHNSZwX5935mLeuOPnMEfyB8/kDuN+Ohw== hG = 1 (no synchronization) (complete synchronization) Fujimoto et al. PNAS (2023) Murase et al. PNAS (2024) Murase et al. PlosCompBiol (2023) Kawakatsu et al. PNAS (2024) Okada et al. Sci.Rep (2020) private assessment models uni fi ed understanding is lacking

impossibility of cooperative ESS when opinions are independent When opinions are independent, we only need to keep track of h Introduce a small fraction of mutants. Their average reputation H is H and the mutant's payo ff depend on pmut→res only. AAAB/HicbVBNS8NAEN3Ur1q/oj16WSyCp5KIVI9FLx4r2FpoQthsJ+3S3STsboQQ6l/x4kERr/4Qb/4bt20O2vpg4PHeDDPzwpQzpR3n26qsrW9sblW3azu7e/sH9uFRTyWZpNClCU9kPyQKOIuhq5nm0E8lEBFyeAgnNzP/4RGkYkl8r/MUfEFGMYsYJdpIgV1Pg8KTAotMY08nWIKaBnbDaTpz4FXilqSBSnQC+8sbJjQTEGvKiVID10m1XxCpGeUwrXmZgpTQCRnBwNCYCFB+MT9+ik+NMsRRIk3FGs/V3xMFEUrlIjSdguixWvZm4n/eINPRlV+wOM00xHSxKMo4Nl/OksBDJoFqnhtCqGTmVkzHRBKqTV41E4K7/PIq6Z033VazdXfRaF+XcVTRMTpBZ8hFl6iNblEHdRFFOXpGr+jNerJerHfrY9FascqZOvoD6/MHhgOUtQ== pmut→res AAAB/HicbVBNS8NAEN3Ur1q/oj16WSyCp5KIVI9FLx4r2FpoQthsJ+3S3STsboQQ6l/x4kERr/4Qb/4bt20O2vpg4PHeDDPzwpQzpR3n26qsrW9sblW3azu7e/sH9uFRTyWZpNClCU9kPyQKOIuhq5nm0E8lEBFyeAgnNzP/4RGkYkl8r/MUfEFGMYsYJdpIgV1Pg8KTAktQ2NMJFpmeBnbDaTpz4FXilqSBSnQC+8sbJjQTEGvKiVID10m1XxCpGeUwrXmZgpTQCRnBwNCYCFB+MT9+ik+NMsRRIk3FGs/V3xMFEUrlIjSdguixWvZm4n/eINPRlV+wOM00xHSxKMo4Nl/OksBDJoFqnhtCqGTmVkzHRBKqTV41E4K7/PIq6Z033VazdXfRaF+XcVTRMTpBZ8hFl6iNblEHdRFFOXpGr+jNerJerHfrY9FascqZOvoD6/MHhaOUtQ== pres→mut AAAB6HicbVBNS8NAEJ34WetX1aOXxSJ4KolI9Vj04rEF+wFtKJvtpF272YTdjVBCf4EXD4p49Sd589+4bXPQ1gcDj/dmmJkXJIJr47rfztr6xubWdmGnuLu3f3BYOjpu6ThVDJssFrHqBFSj4BKbhhuBnUQhjQKB7WB8N/PbT6g0j+WDmSToR3QoecgZNVZqjPqlsltx5yCrxMtJGXLU+6Wv3iBmaYTSMEG17npuYvyMKsOZwGmxl2pMKBvTIXYtlTRC7WfzQ6fk3CoDEsbKljRkrv6eyGik9SQKbGdEzUgvezPxP6+bmvDGz7hMUoOSLRaFqSAmJrOvyYArZEZMLKFMcXsrYSOqKDM2m6INwVt+eZW0LitetVJtXJVrt3kcBTiFM7gAD66hBvdQhyYwQHiGV3hzHp0X5935WLSuOfnMCfyB8/kD0YuM9w== h AAAB6HicbVDLTgJBEOzFF+IL9ehlIjHxRHaNQY9ELxwhkUcCGzI7NDAyO7uZmTUhG77AiweN8eonefNvHGAPClbSSaWqO91dQSy4Nq777eQ2Nre2d/K7hb39g8Oj4vFJS0eJYthkkYhUJ6AaBZfYNNwI7MQKaRgIbAeT+7nffkKleSQfzDRGP6QjyYecUWOlRq1fLLlldwGyTryMlCBDvV/86g0iloQoDRNU667nxsZPqTKcCZwVeonGmLIJHWHXUklD1H66OHRGLqwyIMNI2ZKGLNTfEykNtZ6Gge0MqRnrVW8u/ud1EzO89VMu48SgZMtFw0QQE5H512TAFTIjppZQpri9lbAxVZQZm03BhuCtvrxOWldlr1KuNK5L1bssjjycwTlcggc3UIUa1KEJDBCe4RXenEfnxXl3PpatOSebOYU/cD5/AKELjNc= H linear function of pmut→res

AAAB/HicbVBNS8NAEN3Ur1q/oj16WSyCp5KIVI9FLx4r2FpoQthsJ+3S3STsboQQ6l/x4kERr/4Qb/4bt20O2vpg4PHeDDPzwpQzpR3n26qsrW9sblW3azu7e/sH9uFRTyWZpNClCU9kPyQKOIuhq5nm0E8lEBFyeAgnNzP/4RGkYkl8r/MUfEFGMYsYJdpIgV1Pg8KTAotMY08nWIKaBnbDaTpz4FXilqSBSnQC+8sbJjQTEGvKiVID10m1XxCpGeUwrXmZgpTQCRnBwNCYCFB+MT9+ik+NMsRRIk3FGs/V3xMFEUrlIjSdguixWvZm4n/eINPRlV+wOM00xHSxKMo4Nl/OksBDJoFqnhtCqGTmVkzHRBKqTV41E4K7/PIq6Z033VazdXfRaF+XcVTRMTpBZ8hFl6iNblEHdRFFOXpGr+jNerJerHfrY9FascqZOvoD6/MHhgOUtQ== pmut→res AAAB/HicbVBNS8NAEN3Ur1q/oj16WSyCp5KIVI9FLx4r2FpoQthsJ+3S3STsboQQ6l/x4kERr/4Qb/4bt20O2vpg4PHeDDPzwpQzpR3n26qsrW9sblW3azu7e/sH9uFRTyWZpNClCU9kPyQKOIuhq5nm0E8lEBFyeAgnNzP/4RGkYkl8r/MUfEFGMYsYJdpIgV1Pg8KTAktQ2NMJFpmeBnbDaTpz4FXilqSBSnQC+8sbJjQTEGvKiVID10m1XxCpGeUwrXmZgpTQCRnBwNCYCFB+MT9+ik+NMsRRIk3FGs/V3xMFEUrlIjSdguixWvZm4n/eINPRlV+wOM00xHSxKMo4Nl/OksBDJoFqnhtCqGTmVkzHRBKqTV41E4K7/PIq6Z033VazdXfRaF+XcVTRMTpBZ8hFl6iNblEHdRFFOXpGr+jNerJerHfrY9FascqZOvoD6/MHhaOUtQ== pres→mut ALLC or ALLD is the best action rule! Conditional cooperation cannot be optimal.

intuition behind the impossibility C A B Even if A determines her action according to her opinion about B, A looks as if she is cooperating randomly from C's perspective. C has no idea about A's opinion because of the statistical independence. No social norm can prevent (neutral) invasion by random cooperators. Cooperation cannot be stable.

indirect reciprocity models 3 6 2 1 4 5 C or D observers donor recipient solitary observation model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer 1 2 3 4 5 6 1 2 3 4 5 6 3 6 2 1 4 5 C or D observers donor recipient simultaneous observation model the donor is assessed by multiple observers 3 6 2 1 4 5 C or D observers donor recipient gossiping model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer opinions are exchanged by gossiping 3 6 2 1 4 5 C or D observers donor recipient public assessment model 1 2 3 4 5 6 1 2 3 4 5 6 opinions are fully synchronized by model definition AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqs5IqS6LblxJBfuQdiiZNNOGJpkxyQhl6Fe4caGIWz/HnX9j2s5CWw8EDufcS+45QcyZNq777eRWVtfWN/Kbha3tnd294v5BU0eJIrRBIh6pdoA15UzShmGG03asKBYBp61gdD31W09UaRbJezOOqS/wQLKQEWys9PCIuiZC3tltr1hyy+4MaJl4GSlBhnqv+NXtRyQRVBrCsdYdz42Nn2JlGOF0UugmmsaYjPCAdiyVWFDtp7ODJ+jEKn0URso+adBM/b2RYqH1WAR2UmAz1IveVPzP6yQmvPRTJuPEUEnmH4UJRzbkND3qM0WJ4WNLMFHM3orIECtMjO2oYEvwFiMvk+Z52auWq3eVUu0qqyMPR3AMp+DBBdTgBurQAAICnuEV3hzlvDjvzsd8NOdkO4fwB87nD16Hj30= q ! 1/N AAAB8XicbVDLSgMxFL1TX7W+qi7dBIvgqsyIVJdFNy4r2Ad2hpJJM21oJjMkd4RS+hduXCji1r9x59+YtrPQ1gOBwzn3kntOmEph0HW/ncLa+sbmVnG7tLO7t39QPjxqmSTTjDdZIhPdCanhUijeRIGSd1LNaRxK3g5HtzO//cS1EYl6wHHKg5gOlIgEo2ilRx9pRnxMiNsrV9yqOwdZJV5OKpCj0St/+f2EZTFXyCQ1puu5KQYTqlEwyaclPzM8pWxEB7xrqaIxN8FkfvGUnFmlT6JE26eQzNXfGxMaGzOOQzsZUxyaZW8m/ud1M4yug4lQaYZcscVHUSaJjTiLT/pCc4ZybAllWthbCRtSTRnakkq2BG858ippXVS9WrV2f1mp3+R1FOEETuEcPLiCOtxBA5rAQMEzvMKbY5wX5935WIwWnHznGP7A+fwBnhCQPg== ⌧ ! 0 AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkoiUj0WvXisYGuhCWWz3bRLN5uwOyvE0l/ixYMiXv0p3vw3btsctPXBwOO9GWbmRZngGjzv2ymtrW9sbpW3Kzu7e/tV9+Cwo1OjKGvTVKSqGxHNBJesDRwE62aKkSQS7CEa38z8h0emNE/lPeQZCxMylDzmlICV+m41AGJwACkOuIwh77s1r+7NgVeJX5AaKtDqu1/BIKUmYRKoIFr3fC+DcEIUcCrYtBIYzTJCx2TIepZKkjAdTuaHT/GpVQY4TpUtCXiu/p6YkETrPIlsZ0JgpJe9mfif1zMQX4UTLjMDTNLFotgIbP+cpYAHXDEKIreEUMXtrZiOiCIUbFYVG4K//PIq6ZzX/Ua9cXdRa14XcZTRMTpBZ8hHl6iJblELtRFFBj2jV/TmPDkvzrvzsWgtOcXMEfoD5/MHbv2S9w== ⌧ ! 1 AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KolI9SIUPeixgv2ANpTNdtMs3WzC7kQopT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMC1IpDLrut7Oyura+sVnYKm7v7O7tlw4OmybJNOMNlshEtwNquBSKN1Cg5O1UcxoHkreC4e3Ubz1xbUSiHnGUcj+mAyVCwShaqRX17sg1iXqlsltxZyDLxMtJGXLUe6Wvbj9hWcwVMkmN6Xhuiv6YahRM8kmxmxmeUjakA96xVNGYG388O3dCTq3SJ2GibSkkM/X3xJjGxoziwHbGFCOz6E3F/7xOhuGVPxYqzZArNl8UZpJgQqa/k77QnKEcWUKZFvZWwiKqKUObUNGG4C2+vEya5xWvWqk+XJRrN3kcBTiGEzgDDy6hBvdQhwYwGMIzvMKbkzovzrvzMW9dcfKZI/gD5/MHDEqOvg== hG = h AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KolI9SIUPeixgv2ANpTNdtMu3WzC7kQooT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMCxIpDLrut7Oyura+sVnYKm7v7O7tlw4OmyZONeMNFstYtwNquBSKN1Cg5O1EcxoFkreC0e3Ubz1xbUSsHnGccD+iAyVCwShaqTXs3ZFr4vVKZbfizkCWiZeTMuSo90pf3X7M0ogrZJIa0/HcBP2MahRM8kmxmxqeUDaiA96xVNGIGz+bnTshp1bpkzDWthSSmfp7IqORMeMosJ0RxaFZ9Kbif14nxfDKz4RKUuSKzReFqSQYk+nvpC80ZyjHllCmhb2VsCHVlKFNqGhD8BZfXibN84pXrVQfLsq1mzyOAhzDCZyBB5dQg3uoQwMYjOAZXuHNSZwX5935mLeuOPnMEfyB8/kDuN+Ohw== hG = 1 (no synchronization) (complete synchronization) Fujimoto et al. PNAS (2023) Murase et al. PNAS (2024) Murase et al. PlosCompBiol (2023) Kawakatsu et al. PNAS (2024) Okada et al. Sci.Rep (2020) private assessment models

theoretical framework 1 2 3 4 5 6 1 2 3 4 5 6 h: probability of G when an opinion is randomly chosen. hG: conditional probability that Alice assigns a good reputation to Bob as well, given that Charlie does. hB: conditional probability that Alice assigns a good reputation to Bob given that Charlie does not. AAAB/HicbZDLSsNAFIYn9VbrLdqlm8EitAtLIlLdCKUudFnBXqANYTKdNkMnkzAzEUKor+LGhSJufRB3vo2TNgtt/WHg4z/ncM78XsSoVJb1bRTW1jc2t4rbpZ3dvf0D8/CoK8NYYNLBIQtF30OSMMpJR1HFSD8SBAUeIz1vepPVe49ESBryB5VExAnQhNMxxUhpyzXLftU+893bGryGGdV8t+WaFatuzQVXwc6hAnK1XfNrOApxHBCuMENSDmwrUk6KhKKYkVlpGEsSITxFEzLQyFFApJPOj5/BU+2M4DgU+nEF5+7viRQFUiaBpzsDpHy5XMvM/2qDWI2vnJTyKFaE48WiccygCmGWBBxRQbBiiQaEBdW3QuwjgbDSeZV0CPbyl1ehe163G/XG/UWl2crjKIJjcAKqwAaXoAnuQBt0AAYJeAav4M14Ml6Md+Nj0Vow8pky+CPj8wemB5I2 h(1 → hG) = (1 → h)hB 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 1 2 3 4 5 6 AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KolI9SIUPeixgv2ANpTNdtMu3WzC7kQooT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMCxIpDLrut7Oyura+sVnYKm7v7O7tlw4OmyZONeMNFstYtwNquBSKN1Cg5O1EcxoFkreC0e3Ubz1xbUSsHnGccD+iAyVCwShaqTXs3ZFr4vVKZbfizkCWiZeTMuSo90pf3X7M0ogrZJIa0/HcBP2MahRM8kmxmxqeUDaiA96xVNGIGz+bnTshp1bpkzDWthSSmfp7IqORMeMosJ0RxaFZ9Kbif14nxfDKz4RKUuSKzReFqSQYk+nvpC80ZyjHllCmhb2VsCHVlKFNqGhD8BZfXibN84pXrVQfLsq1mzyOAhzDCZyBB5dQg3uoQwMYjOAZXuHNSZwX5935mLeuOPnMEfyB8/kDuN+Ohw== hG = 1 AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KolI9SIUPeixgv2ANpTNdtMs3WzC7kQopT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMC1IpDLrut7Oyura+sVnYKm7v7O7tlw4OmybJNOMNlshEtwNquBSKN1Cg5O1UcxoHkreC4e3Ubz1xbUSiHnGUcj+mAyVCwShaqRX17sg1iXqlsltxZyDLxMtJGXLUe6Wvbj9hWcwVMkmN6Xhuiv6YahRM8kmxmxmeUjakA96xVNGYG388O3dCTq3SJ2GibSkkM/X3xJjGxoziwHbGFCOz6E3F/7xOhuGVPxYqzZArNl8UZpJgQqa/k77QnKEcWUKZFvZWwiKqKUObUNGG4C2+vEya5xWvWqk+XJRrN3kcBTiGEzgDDy6hBvdQhwYwGMIzvMKbkzovzrvzMW9dcfKZI/gD5/MHDEqOvg== hG = h

At this moment, we do not make any further assumptions on how exactly opinions get correlated. (Our results are independent of how correlations are achieved.) Consider the dynamics of h for a given social norm: Introduce a small amount of mutants with another action rule P’. The relation between h and hG. We can calculate pc for a given norm and hG.

The payo ff of the mutant is 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 ωG > nd ωB < Conditional cooperation (P’(G) = 1 and P’(B) = 0) is stable i f We can check if a given norm is stable or not.

L3 (Simple Standing) L6 (Stern Judging) average reputation stable b/c range A B C D Conditional Cooperatoin Conditional Cooperatoin ALLC ALLD ALLD Higher correlation hG ↓ higher cooperation level h (=pc) Higher correlation hG ↓ CC is stable for a broader b/c range

comparison with the speci fi c models (1) 3 6 2 1 4 5 C or D observers donor recipient solitary observation model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer 1 2 3 4 5 6 1 2 3 4 5 6 3 6 2 1 4 5 C or D observers donor recipient simultaneous observation model the donor is assessed by multiple observers 3 6 2 1 4 5 C or D observers donor recipient gossiping model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer opinions are exchanged by gossiping 3 6 2 1 4 5 C or D observers donor recipient public assessment model 1 2 3 4 5 6 1 2 3 4 5 6 opinions are fully synchronized by model definition AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqs5IqS6LblxJBfuQdiiZNNOGJpkxyQhl6Fe4caGIWz/HnX9j2s5CWw8EDufcS+45QcyZNq777eRWVtfWN/Kbha3tnd294v5BU0eJIrRBIh6pdoA15UzShmGG03asKBYBp61gdD31W09UaRbJezOOqS/wQLKQEWys9PCIuiZC3tltr1hyy+4MaJl4GSlBhnqv+NXtRyQRVBrCsdYdz42Nn2JlGOF0UugmmsaYjPCAdiyVWFDtp7ODJ+jEKn0URso+adBM/b2RYqH1WAR2UmAz1IveVPzP6yQmvPRTJuPEUEnmH4UJRzbkND3qM0WJ4WNLMFHM3orIECtMjO2oYEvwFiMvk+Z52auWq3eVUu0qqyMPR3AMp+DBBdTgBurQAAICnuEV3hzlvDjvzsd8NOdkO4fwB87nD16Hj30= q ! 1/N AAAB8XicbVDLSgMxFL1TX7W+qi7dBIvgqsyIVJdFNy4r2Ad2hpJJM21oJjMkd4RS+hduXCji1r9x59+YtrPQ1gOBwzn3kntOmEph0HW/ncLa+sbmVnG7tLO7t39QPjxqmSTTjDdZIhPdCanhUijeRIGSd1LNaRxK3g5HtzO//cS1EYl6wHHKg5gOlIgEo2ilRx9pRnxMiNsrV9yqOwdZJV5OKpCj0St/+f2EZTFXyCQ1puu5KQYTqlEwyaclPzM8pWxEB7xrqaIxN8FkfvGUnFmlT6JE26eQzNXfGxMaGzOOQzsZUxyaZW8m/ud1M4yug4lQaYZcscVHUSaJjTiLT/pCc4ZybAllWthbCRtSTRnakkq2BG858ippXVS9WrV2f1mp3+R1FOEETuEcPLiCOtxBA5rAQMEzvMKbY5wX5935WIwWnHznGP7A+fwBnhCQPg== ⌧ ! 0 AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkoiUj0WvXisYGuhCWWz3bRLN5uwOyvE0l/ixYMiXv0p3vw3btsctPXBwOO9GWbmRZngGjzv2ymtrW9sbpW3Kzu7e/tV9+Cwo1OjKGvTVKSqGxHNBJesDRwE62aKkSQS7CEa38z8h0emNE/lPeQZCxMylDzmlICV+m41AGJwACkOuIwh77s1r+7NgVeJX5AaKtDqu1/BIKUmYRKoIFr3fC+DcEIUcCrYtBIYzTJCx2TIepZKkjAdTuaHT/GpVQY4TpUtCXiu/p6YkETrPIlsZ0JgpJe9mfif1zMQX4UTLjMDTNLFotgIbP+cpYAHXDEKIreEUMXtrZiOiCIUbFYVG4K//PIq6ZzX/Ua9cXdRa14XcZTRMTpBZ8hHl6iJblELtRFFBj2jV/TmPDkvzrvzsWgtOcXMEfoD5/MHbv2S9w== ⌧ ! 1 AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KolI9SIUPeixgv2ANpTNdtMs3WzC7kQopT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMC1IpDLrut7Oyura+sVnYKm7v7O7tlw4OmybJNOMNlshEtwNquBSKN1Cg5O1UcxoHkreC4e3Ubz1xbUSiHnGUcj+mAyVCwShaqRX17sg1iXqlsltxZyDLxMtJGXLUe6Wvbj9hWcwVMkmN6Xhuiv6YahRM8kmxmxmeUjakA96xVNGYG388O3dCTq3SJ2GibSkkM/X3xJjGxoziwHbGFCOz6E3F/7xOhuGVPxYqzZArNl8UZpJgQqa/k77QnKEcWUKZFvZWwiKqKUObUNGG4C2+vEya5xWvWqk+XJRrN3kcBTiGEzgDDy6hBvdQhwYwGMIzvMKbkzovzrvzMW9dcfKZI/gD5/MHDEqOvg== hG = h AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KolI9SIUPeixgv2ANpTNdtMu3WzC7kQooT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMCxIpDLrut7Oyura+sVnYKm7v7O7tlw4OmyZONeMNFstYtwNquBSKN1Cg5O1EcxoFkreC0e3Ubz1xbUSsHnGccD+iAyVCwShaqTXs3ZFr4vVKZbfizkCWiZeTMuSo90pf3X7M0ogrZJIa0/HcBP2MahRM8kmxmxqeUDaiA96xVNGIGz+bnTshp1bpkzDWthSSmfp7IqORMeMosJ0RxaFZ9Kbif14nxfDKz4RKUuSKzReFqSQYk+nvpC80ZyjHllCmhb2VsCHVlKFNqGhD8BZfXibN84pXrVQfLsq1mzyOAhzDCZyBB5dQg3uoQwMYjOAZXuHNSZwX5935mLeuOPnMEfyB8/kDuN+Ohw== hG = 1 (no synchronization) (complete synchronization) Fujimoto et al. PNAS (2023) Murase et al. PNAS (2024) Murase et al. PlosCompBiol (2023) Kawakatsu et al. PNAS (2024) Okada et al. Sci.Rep (2020) private assessment models hG = 1 by model assumption The previous study is reproduced. Y. Murase et al. PLOS Comp.Biol. (2023)

comparison with the speci fi c models (2) 3 6 2 1 4 5 C or D observers donor recipient solitary observation model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer 1 2 3 4 5 6 1 2 3 4 5 6 3 6 2 1 4 5 C or D observers donor recipient simultaneous observation model the donor is assessed by multiple observers 3 6 2 1 4 5 C or D observers donor recipient gossiping model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer opinions are exchanged by gossiping 3 6 2 1 4 5 C or D observers donor recipient public assessment model 1 2 3 4 5 6 1 2 3 4 5 6 opinions are fully synchronized by model definition AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqs5IqS6LblxJBfuQdiiZNNOGJpkxyQhl6Fe4caGIWz/HnX9j2s5CWw8EDufcS+45QcyZNq777eRWVtfWN/Kbha3tnd294v5BU0eJIrRBIh6pdoA15UzShmGG03asKBYBp61gdD31W09UaRbJezOOqS/wQLKQEWys9PCIuiZC3tltr1hyy+4MaJl4GSlBhnqv+NXtRyQRVBrCsdYdz42Nn2JlGOF0UugmmsaYjPCAdiyVWFDtp7ODJ+jEKn0URso+adBM/b2RYqH1WAR2UmAz1IveVPzP6yQmvPRTJuPEUEnmH4UJRzbkND3qM0WJ4WNLMFHM3orIECtMjO2oYEvwFiMvk+Z52auWq3eVUu0qqyMPR3AMp+DBBdTgBurQAAICnuEV3hzlvDjvzsd8NOdkO4fwB87nD16Hj30= q ! 1/N AAAB8XicbVDLSgMxFL1TX7W+qi7dBIvgqsyIVJdFNy4r2Ad2hpJJM21oJjMkd4RS+hduXCji1r9x59+YtrPQ1gOBwzn3kntOmEph0HW/ncLa+sbmVnG7tLO7t39QPjxqmSTTjDdZIhPdCanhUijeRIGSd1LNaRxK3g5HtzO//cS1EYl6wHHKg5gOlIgEo2ilRx9pRnxMiNsrV9yqOwdZJV5OKpCj0St/+f2EZTFXyCQ1puu5KQYTqlEwyaclPzM8pWxEB7xrqaIxN8FkfvGUnFmlT6JE26eQzNXfGxMaGzOOQzsZUxyaZW8m/ud1M4yug4lQaYZcscVHUSaJjTiLT/pCc4ZybAllWthbCRtSTRnakkq2BG858ippXVS9WrV2f1mp3+R1FOEETuEcPLiCOtxBA5rAQMEzvMKbY5wX5935WIwWnHznGP7A+fwBnhCQPg== ⌧ ! 0 AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkoiUj0WvXisYGuhCWWz3bRLN5uwOyvE0l/ixYMiXv0p3vw3btsctPXBwOO9GWbmRZngGjzv2ymtrW9sbpW3Kzu7e/tV9+Cwo1OjKGvTVKSqGxHNBJesDRwE62aKkSQS7CEa38z8h0emNE/lPeQZCxMylDzmlICV+m41AGJwACkOuIwh77s1r+7NgVeJX5AaKtDqu1/BIKUmYRKoIFr3fC+DcEIUcCrYtBIYzTJCx2TIepZKkjAdTuaHT/GpVQY4TpUtCXiu/p6YkETrPIlsZ0JgpJe9mfif1zMQX4UTLjMDTNLFotgIbP+cpYAHXDEKIreEUMXtrZiOiCIUbFYVG4K//PIq6ZzX/Ua9cXdRa14XcZTRMTpBZ8hHl6iJblELtRFFBj2jV/TmPDkvzrvzsWgtOcXMEfoD5/MHbv2S9w== ⌧ ! 1 AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KolI9SIUPeixgv2ANpTNdtMs3WzC7kQopT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMC1IpDLrut7Oyura+sVnYKm7v7O7tlw4OmybJNOMNlshEtwNquBSKN1Cg5O1UcxoHkreC4e3Ubz1xbUSiHnGUcj+mAyVCwShaqRX17sg1iXqlsltxZyDLxMtJGXLUe6Wvbj9hWcwVMkmN6Xhuiv6YahRM8kmxmxmeUjakA96xVNGYG388O3dCTq3SJ2GibSkkM/X3xJjGxoziwHbGFCOz6E3F/7xOhuGVPxYqzZArNl8UZpJgQqa/k77QnKEcWUKZFvZWwiKqKUObUNGG4C2+vEya5xWvWqk+XJRrN3kcBTiGEzgDDy6hBvdQhwYwGMIzvMKbkzovzrvzMW9dcfKZI/gD5/MHDEqOvg== hG = h AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KolI9SIUPeixgv2ANpTNdtMu3WzC7kQooT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMCxIpDLrut7Oyura+sVnYKm7v7O7tlw4OmyZONeMNFstYtwNquBSKN1Cg5O1EcxoFkreC0e3Ubz1xbUSsHnGccD+iAyVCwShaqTXs3ZFr4vVKZbfizkCWiZeTMuSo90pf3X7M0ogrZJIa0/HcBP2MahRM8kmxmxqeUDaiA96xVNGIGz+bnTshp1bpkzDWthSSmfp7IqORMeMosJ0RxaFZ9Kbif14nxfDKz4RKUuSKzReFqSQYk+nvpC80ZyjHllCmhb2VsCHVlKFNqGhD8BZfXibN84pXrVQfLsq1mzyOAhzDCZyBB5dQg3uoQwMYjOAZXuHNSZwX5935mLeuOPnMEfyB8/kDuN+Ohw== hG = 1 (no synchronization) (complete synchronization) Fujimoto et al. PNAS (2023) Murase et al. PNAS (2024) Murase et al. PlosCompBiol (2023) Kawakatsu et al. PNAS (2024) Okada et al. Sci.Rep (2020) private assessment models The previous study is reproduced. Fujimoto et al. derived “goodness” distribution h and hG are calculated by the fi rst- and the second-order moments of P(g). By plugging those into our theory, Y. Fujimoto et al. PNAS (2023)

comparison with the speci fi c models (3) 3 6 2 1 4 5 C or D observers donor recipient solitary observation model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer 1 2 3 4 5 6 1 2 3 4 5 6 3 6 2 1 4 5 C or D observers donor recipient simultaneous observation model the donor is assessed by multiple observers 3 6 2 1 4 5 C or D observers donor recipient gossiping model 1 2 3 4 5 6 1 2 3 4 5 6 the donor is assessed by a single observer opinions are exchanged by gossiping 3 6 2 1 4 5 C or D observers donor recipient public assessment model 1 2 3 4 5 6 1 2 3 4 5 6 opinions are fully synchronized by model definition AAAB8HicbVDLSgMxFL1TX7W+qi7dBIvgqs5IqS6LblxJBfuQdiiZNNOGJpkxyQhl6Fe4caGIWz/HnX9j2s5CWw8EDufcS+45QcyZNq777eRWVtfWN/Kbha3tnd294v5BU0eJIrRBIh6pdoA15UzShmGG03asKBYBp61gdD31W09UaRbJezOOqS/wQLKQEWys9PCIuiZC3tltr1hyy+4MaJl4GSlBhnqv+NXtRyQRVBrCsdYdz42Nn2JlGOF0UugmmsaYjPCAdiyVWFDtp7ODJ+jEKn0URso+adBM/b2RYqH1WAR2UmAz1IveVPzP6yQmvPRTJuPEUEnmH4UJRzbkND3qM0WJ4WNLMFHM3orIECtMjO2oYEvwFiMvk+Z52auWq3eVUu0qqyMPR3AMp+DBBdTgBurQAAICnuEV3hzlvDjvzsd8NOdkO4fwB87nD16Hj30= q ! 1/N AAAB8XicbVDLSgMxFL1TX7W+qi7dBIvgqsyIVJdFNy4r2Ad2hpJJM21oJjMkd4RS+hduXCji1r9x59+YtrPQ1gOBwzn3kntOmEph0HW/ncLa+sbmVnG7tLO7t39QPjxqmSTTjDdZIhPdCanhUijeRIGSd1LNaRxK3g5HtzO//cS1EYl6wHHKg5gOlIgEo2ilRx9pRnxMiNsrV9yqOwdZJV5OKpCj0St/+f2EZTFXyCQ1puu5KQYTqlEwyaclPzM8pWxEB7xrqaIxN8FkfvGUnFmlT6JE26eQzNXfGxMaGzOOQzsZUxyaZW8m/ud1M4yug4lQaYZcscVHUSaJjTiLT/pCc4ZybAllWthbCRtSTRnakkq2BG858ippXVS9WrV2f1mp3+R1FOEETuEcPLiCOtxBA5rAQMEzvMKbY5wX5935WIwWnHznGP7A+fwBnhCQPg== ⌧ ! 0 AAAB+HicbVBNS8NAEN3Ur1o/GvXoZbEInkoiUj0WvXisYGuhCWWz3bRLN5uwOyvE0l/ixYMiXv0p3vw3btsctPXBwOO9GWbmRZngGjzv2ymtrW9sbpW3Kzu7e/tV9+Cwo1OjKGvTVKSqGxHNBJesDRwE62aKkSQS7CEa38z8h0emNE/lPeQZCxMylDzmlICV+m41AGJwACkOuIwh77s1r+7NgVeJX5AaKtDqu1/BIKUmYRKoIFr3fC+DcEIUcCrYtBIYzTJCx2TIepZKkjAdTuaHT/GpVQY4TpUtCXiu/p6YkETrPIlsZ0JgpJe9mfif1zMQX4UTLjMDTNLFotgIbP+cpYAHXDEKIreEUMXtrZiOiCIUbFYVG4K//PIq6ZzX/Ua9cXdRa14XcZTRMTpBZ8hHl6iJblELtRFFBj2jV/TmPDkvzrvzsWgtOcXMEfoD5/MHbv2S9w== ⌧ ! 1 AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KolI9SIUPeixgv2ANpTNdtMs3WzC7kQopT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMC1IpDLrut7Oyura+sVnYKm7v7O7tlw4OmybJNOMNlshEtwNquBSKN1Cg5O1UcxoHkreC4e3Ubz1xbUSiHnGUcj+mAyVCwShaqRX17sg1iXqlsltxZyDLxMtJGXLUe6Wvbj9hWcwVMkmN6Xhuiv6YahRM8kmxmxmeUjakA96xVNGYG388O3dCTq3SJ2GibSkkM/X3xJjGxoziwHbGFCOz6E3F/7xOhuGVPxYqzZArNl8UZpJgQqa/k77QnKEcWUKZFvZWwiKqKUObUNGG4C2+vEya5xWvWqk+XJRrN3kcBTiGEzgDDy6hBvdQhwYwGMIzvMKbkzovzrvzMW9dcfKZI/gD5/MHDEqOvg== hG = h AAAB7nicbVBNS8NAEJ34WetX1aOXxSJ4KolI9SIUPeixgv2ANpTNdtMu3WzC7kQooT/CiwdFvPp7vPlv3LY5aOuDgcd7M8zMCxIpDLrut7Oyura+sVnYKm7v7O7tlw4OmyZONeMNFstYtwNquBSKN1Cg5O1EcxoFkreC0e3Ubz1xbUSsHnGccD+iAyVCwShaqTXs3ZFr4vVKZbfizkCWiZeTMuSo90pf3X7M0ogrZJIa0/HcBP2MahRM8kmxmxqeUDaiA96xVNGIGz+bnTshp1bpkzDWthSSmfp7IqORMeMosJ0RxaFZ9Kbif14nxfDKz4RKUuSKzReFqSQYk+nvpC80ZyjHllCmhb2VsCHVlKFNqGhD8BZfXibN84pXrVQfLsq1mzyOAhzDCZyBB5dQg3uoQwMYjOAZXuHNSZwX5935mLeuOPnMEfyB8/kDuN+Ohw== hG = 1 (no synchronization) (complete synchronization) Fujimoto et al. PNAS (2023) Murase et al. PNAS (2024) Murase et al. PlosCompBiol (2023) Kawakatsu et al. PNAS (2024) Okada et al. Sci.Rep (2020) private assessment models The previous study is reproduced. Kawakatsu et al. derived By plugging those into our theory, M. Kawakatsu et al. PNAS (2024) 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 ωG > nd ωB < 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 ω > ω→

summary of this study • We propose a general framework to analyze the evolutionary stability of indirect reciprocity. • Previous work can be organized by considering a single key quantity: the degree to which individual opinions are correlated. • The more opinions are correlated, the easier it becomes to sustain cooperation. Conversely, if opinions turn out to be completely uncorrelated, cooperative norms become evolutionarily unstable. • Evolutionary origin of conformity? • How mechanisms for opinion synchronization coevolve with social norms? Open Questions:

agenda • طଘཧ࿦ݚڀͷϨϏϡʔ • ݚڀ঺հ̍ɿҙݟಉظԽͷؒ઀ޓܙੑ • ݚڀ঺հ̎ɿࣾձنൣ͕Ͳ͏΍ͬͯਐԽ͢Δ͔ • ࣍ʹղ͘΂͖՝୊͸ʁ • ʢଜ੉ͷओ؍ͱภݟ͕ଟ෼ʹؚ·Ε·͢ʣ • ໨త • ཧ࿦Λઐ໳ͱ͠ͳ͍ํ͕ɺؒ઀ޓܙੑͷཧ࿦ݚڀͷ֓ཁΛ͔ͭΊΔΑ͏ʹ • ʢҰॹʹݚڀ͢ΔਓΛ୳͍ͨ͠ʣ

limitations on previous studies 2. "local" stability after a norm has been established Fujimoto et al. PRX Life (2024) Fujimoto et al. PNAS (2023) Yamamoto et al. Sci.Rep. (2017) 3. only a handful of norms were considered Hilbe et al. PNAS (2018) Okada Sci.Rep. (2018) Schmid et al. Nat. Comm. (2023) Radzvilavicius et al. eLife (2019) Kessinger et al. PNAS (2023) 1. public / private assessment models AAACBXicbVDLSgMxFM3UV62vqks3wSK4kDIjRV0W3bhswT6gHUomvdOGJpkhyRTK0LULt/oZ7sSt3+FX+Aum7Sxs64HA4Zx7uScniDnTxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ihRFBo04pFqB0QDZxIahhkO7VgBEQGHVjB6mPmtMSjNIvlkJjH4ggwkCxklxkp11iuW3LI7B14nXkZKKEOtV/zp9iOaCJCGcqJ1x3Nj46dEGUY5TAvdRENM6IgMoGOpJAK0n86DTvGFVfo4jJR90uC5+ncjJULriQjspCBmqFe9mfif10lMeOenTMaJAUkXh8KEYxPh2a9xnymghk8sIVQxmxXTIVGEGtvN0hVKZ9H0FSQc1FhMC7Yhb7WPddK8Lns35Uq9UqreZ13l0Rk6R5fIQ7eoih5RDTUQRYBe0Ct6c56dd+fD+VyM5pxs5xQtwfn6BXx3mTc= i AAACB3icbVDLSgMxFM3UV62vqks3wSK4kDIjRV0W3bisaB/QDiWT3mlDk8yQZAql9ANcuNXPcCdu/Qy/wl8w087Cth4IHM65l3tygpgzbVz328mtrW9sbuW3Czu7e/sHxcOjho4SRaFOIx6pVkA0cCahbpjh0IoVEBFwaAbDu9RvjkBpFsknM47BF6QvWcgoMVZ6FF3WLZbcsjsDXiVeRkooQ61b/On0IpoIkIZyonXbc2PjT4gyjHKYFjqJhpjQIelD21JJBGh/Mos6xWdW6eEwUvZJg2fq340JEVqPRWAnBTEDveyl4n9eOzHhjT9hMk4MSDo/FCYcmwin/8Y9poAaPraEUMVsVkwHRBFqbDsLVyhNo+kLSDiokZgWbEPech+rpHFZ9q7KlYdKqXqbdZVHJ+gUnSMPXaMqukc1VEcU9dELekVvzrPz7nw4n/PRnJPtHKMFOF+/FB6aFw== mi AAACBXicbVDLSgMxFM3UV62vqks3wSK4kDIjRV0W3bhswT6gHUomvdOGJpkhyRTK0LULt/oZ7sSt3+FX+Aum7Sxs64HA4Zx7uScniDnTxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ihRFBo04pFqB0QDZxIahhkO7VgBEQGHVjB6mPmtMSjNIvlkJjH4ggwkCxklxkp11iuW3LI7B14nXkZKKEOtV/zp9iOaCJCGcqJ1x3Nj46dEGUY5TAvdRENM6IgMoGOpJAK0n86DTvGFVfo4jJR90uC5+ncjJULriQjspCBmqFe9mfif10lMeOenTMaJAUkXh8KEYxPh2a9xnymghk8sIVQxmxXTIVGEGtvN0hVKZ9H0FSQc1FhMC7Yhb7WPddK8Lns35Uq9UqreZ13l0Rk6R5fIQ7eoih5RDTUQRYBe0Ct6c56dd+fD+VyM5pxs5xQtwfn6BXx3mTc= i AAACBXicbVDLSgMxFM34rPVVdekmWAQXUmakqMuiG5ct2Ae0Q8mkd9rYJDMkmUIZunbhVj/Dnbj1O/wKf8G0nYVtPRA4nHMv9+QEMWfauO63s7a+sbm1ndvJ7+7tHxwWjo4bOkoUhTqNeKRaAdHAmYS6YYZDK1ZARMChGQzvp35zBEqzSD6acQy+IH3JQkaJsVLtqVsouiV3BrxKvIwUUYZqt/DT6UU0ESAN5UTrtufGxk+JMoxymOQ7iYaY0CHpQ9tSSQRoP50FneBzq/RwGCn7pMEz9e9GSoTWYxHYSUHMQC97U/E/r52Y8NZPmYwTA5LOD4UJxybC01/jHlNADR9bQqhiNiumA6IINbabhSuUTqPpS0g4qJGY5G1D3nIfq6RxVfKuS+VauVi5y7rKoVN0hi6Qh25QBT2gKqojigC9oFf05jw7786H8zkfXXOynRO0AOfrF34YmTg= j AAACBXicbVDLSgMxFM3UV62vqks3wSK4kDIjRV0W3bhswT6gHUomvdOGJpkhyRTK0LULt/oZ7sSt3+FX+Aum7Sxs64HA4Zx7uScniDnTxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ihRFBo04pFqB0QDZxIahhkO7VgBEQGHVjB6mPmtMSjNIvlkJjH4ggwkCxklxkr1Ua9YcsvuHHideBkpoQy1XvGn249oIkAayonWHc+NjZ8SZRjlMC10Ew0xoSMygI6lkgjQfjoPOsUXVunjMFL2SYPn6t+NlAitJyKwk4KYoV71ZuJ/Xicx4Z2fMhknBiRdHAoTjk2EZ7/GfaaAGj6xhFDFbFZMh0QRamw3S1conUXTV5BwUGMxLdiGvNU+1knzuuzdlCv1Sql6n3WVR2foHF0iD92iKnpENdRAFAF6Qa/ozXl23p0P53MxmnOynVO0BOfrF3+5mTk= k AAACBXicbVDLSgMxFM3UV62vqks3wSK4kDIjRV0W3bhswT6gHUomvdOGJpkhyRTK0LULt/oZ7sSt3+FX+Aum7Sxs64HA4Zx7uScniDnTxnW/ndzG5tb2Tn63sLd/cHhUPD5p6ihRFBo04pFqB0QDZxIahhkO7VgBEQGHVjB6mPmtMSjNIvlkJjH4ggwkCxklxkp13iuW3LI7B14nXkZKKEOtV/zp9iOaCJCGcqJ1x3Nj46dEGUY5TAvdRENM6IgMoGOpJAK0n86DTvGFVfo4jJR90uC5+ncjJULriQjspCBmqFe9mfif10lMeOenTMaJAUkXh8KEYxPh2a9xnymghk8sIVQxmxXTIVGEGtvN0hVKZ9H0FSQc1FhMC7Yhb7WPddK8Lns35Uq9UqreZ13l0Rk6R5fIQ7eoih5RDTUQRYBe0Ct6c56dd+fD+VyM5pxs5xQtwfn6BYFamTo= l AAACCnicbVDLSsNAFJ34rPVVdelmsAgupCRS1GXRjcsK9gFtKJPpTTvtTBJmJoUS8gcu3OpnuBO3/oRf4S84abOwrQcGDufcyz1zvIgzpW3721pb39jc2i7sFHf39g8OS0fHTRXGkkKDhjyUbY8o4CyAhmaaQzuSQITHoeWN7zO/NQGpWBg86WkEriCDgPmMEm2kpuglbJT2SmW7Ys+AV4mTkzLKUe+Vfrr9kMYCAk05Uarj2JF2EyI1oxzSYjdWEBE6JgPoGBoQAcpNZmlTfG6UPvZDaV6g8Uz9u5EQodRUeGZSED1Uy14m/ud1Yu3fugkLolhDQOeH/JhjHeLs67jPJFDNp4YQKpnJiumQSEK1KWjhCqVZNHUJMQc5EWnRNOQs97FKmlcV57pSfayWa3d5VwV0is7QBXLQDaqhB1RHDUTRCL2gV/RmPVvv1of1OR9ds/KdE7QA6+sXyQSblw== mij AAACCnicbVDLSsNAFJ3UV62vqks3wSK4kJJIUZdFNy4r2Ae0oUymN+3YmUmYmRRKyB+4cKuf4U7c+hN+hb/gpM3Cth4YOJxzL/fM8SNGlXacb6uwtr6xuVXcLu3s7u0flA+PWiqMJYEmCVkoOz5WwKiApqaaQSeSgLnPoO2P7zK/PQGpaCge9TQCj+OhoAElWBupxfsJHaf9csWpOjPYq8TNSQXlaPTLP71BSGIOQhOGleq6TqS9BEtNCYO01IsVRJiM8RC6hgrMQXnJLG1qnxllYAehNE9oe6b+3UgwV2rKfTPJsR6pZS8T//O6sQ5uvISKKNYgyPxQEDNbh3b2dXtAJRDNpoZgIqnJapMRlphoU9DCFUKyaOoCYgZywtOSachd7mOVtC6r7lW19lCr1G/zroroBJ2ic+Sia1RH96iBmoigJ/SCXtGb9Wy9Wx/W53y0YOU7x2gB1tcvyqabmA== mik AAACCnicbVDLSsNAFJ3UV62vqks3wSK4kJJIUZdFNy4r2Ae0oUymN+3YmUmYmRRKyB+4cKuf4U7c+hN+hb/gpM3Cth4YOJxzL/fM8SNGlXacb6uwtr6xuVXcLu3s7u0flA+PWiqMJYEmCVkoOz5WwKiApqaaQSeSgLnPoO2P7zK/PQGpaCge9TQCj+OhoAElWBupxfsJZWm/XHGqzgz2KnFzUkE5Gv3yT28QkpiD0IRhpbquE2kvwVJTwiAt9WIFESZjPISuoQJzUF4yS5vaZ0YZ2EEozRPanql/NxLMlZpy30xyrEdq2cvE/7xurIMbL6EiijUIMj8UxMzWoZ193R5QCUSzqSGYSGqy2mSEJSbaFLRwhZAsmrqAmIGc8LRkGnKX+1glrcuqe1WtPdQq9du8qyI6QafoHLnoGtXRPWqgJiLoCb2gV/RmPVvv1of1OR8tWPnOMVqA9fULzEibmQ== mil public assessment model private assessment model B G A B C D D observers donor recipient )FNVTUCFCBE *NVTUQVOJTI #JTTVDIBOJDFQFSTPO *DBO`UGPSHJWF" #JTBOBTUZHVZ )FEFTFSWFTQVOJTINFOU B G

purpose of this study • A more complete view of the evolution of social norms would require a global analysis that includes a more comprehensive set of norms. • We study evolution with all possible third-order norms (2080 norms), taking into account all possible transitions between them. • Does indirect reciprocity evolve? If yes, which social norm plays a key role?

model + evolutionary dynamics (imitation & mutation)

challenges • We need to calculate the fi xation probabilities between all K x (K-1) strategy pairs. • To calculate each fi xation probability, we need to calculate the payo ff for the residents and mutants for N-1 di ff erent population con fi gurations. • No analytic formula to calculate the long-term payo ff s for each con fi guration. • total amount of computation • K(K-1)(N-1) x 5 samples x 10^6 donation games ʙ 10^15 games

© RIKEN

evolution in well-mixed population In previous studies, some strategies are known to be stable against ALLC and ALLD. However, cooperation does NOT emerge when considering all possible transitions. No strategy is stable enough against a diverse set of possible mutants. ALLD L1, L7 L2 ALLC L5 L3, L4 Type II Type I evolution with the full set of third-order strategies (2080 strategies)

evolution in group-structured population mutation out-group imitation in-group imitation Cooperation does evolve! within group-structured populations, cooperation can emerge, particularly facilitated by one of the leading eight norms, L1. Natural populations are rarely well mixed. Instead, they often have some intrinsic structure. To explore whether cooperation can evolve in principle, we study a group-structured population.

the key to promote cooperation L1 is the most successful norm. Context independency is the key. • Cooperation → Good • Defection → Bad except the justi fi ed punishment -KVEHFT "--%BT( When Bad Defected against Bad, the donor should remain Bad to suppress ALLD Cooperation with Bad should be judged as Good to maintain agreement D C B G G G B G ALLD ALLD

Conclusion • We study evolution of indirect reciprocity with full set of third-order social norms. • Cooperation rarely evolves when populations are well mixed. • The success in the three-species system does not mean the success in the full system. • Despite these negative results, cooperation is still possible in a group-structured population. • L1 is the key to success. It is the most context-insensitive: • Cooperation → Good • Defection → Bad (except justi fi ed punishment) Y. Murase & C. Hilbe PNAS (2024)

agenda • طଘཧ࿦ݚڀͷϨϏϡʔ • ݚڀ঺հ̍ɿҙݟಉظԽͷؒ઀ޓܙੑ • ݚڀ঺հ̎ɿࣾձنൣ͕Ͳ͏΍ͬͯਐԽ͢Δ͔ • ࣍ʹղ͘΂͖՝୊͸ʁ • ʢଜ੉ͷओ؍ͱภݟ͕ଟ෼ʹؚ·Ε·͢ʣ • ໨త • ཧ࿦Λઐ໳ͱ͠ͳ͍ํ͕ɺؒ઀ޓܙੑͷཧ࿦ݚڀͷ֓ཁΛ͔ͭΊΔΑ͏ʹ • ʢҰॹʹݚڀ͢ΔਓΛ୳͍ͨ͠ʣ

1: ؒ઀ޓܙͱ৘ใަ׵ ྑ͍ᷚɺѱ͍ᷚɺͲͪΒʹॏ͖Λ͓͘ʁ Ͳ͏͍͏࣌ʹਖ਼௚ʹҙݟΛ఻͑Δʁ ޡ৘ใΛ཈੍͢Δʹ͸ʁ ίϛϡχέʔγϣϯίετ͕ۃখʹͳ͍ͬͯͬͨ৔߹ʁ gossip ͳͲͷίϛϡχέʔγϣϯ͕ࣾձنൣͱ ͲͷΑ͏ʹڞਐԽ͖ͯͨ͠ͷ͔ʁ #BE (PPE (PPE #BE ίϛϡχ έʔγϣϯ → SNSͳͲͷσδλϧσʔλͱͷൺֱ͕Ͱ͖ͳ͍͔ʁ

2. ύʔτφʔબ୒ • ؒ઀ޓܙϞσϧͰ͸શͯͷਓ͕શͯͷਓΛ஌͍ͬͯΔɺ౳ ͘͠૬ޓ࡞༻͢ΔͱԾఆ͖ͯͨ͠ɻ • ूஂαΠζ͕Dunbar਺ҎԼ • ΑΓେ͖ͳࣾձͰ͸ɺωοτϫʔΫ͸ඇۉҰ͔ͭɺઓུత ʹύʔτφʔΛબ୒͢Δɻύʔτφʔબ୒ͷཧ࿦ݚڀ͸ෆ े෼ • ֊૚͕ࣗൃతʹൃੜ͢Δͷ͔ʁ • ؔ܎ੑ͕ྲྀಈੑ͸ڠྗΛ༰қʹ͢Δʁ • triadic closure, structural balance, community structure ͸ͳͥͰ͖Δʁ mobile phone call data Onnela et al. PNAS (2007)

·ͱΊ • ؒ઀ޓܙͷཧ࿦ݚڀͷϨϏϡʔ • public / private assessment model ʹେผ͞ΕΔ • Indirect reciprocity under opinion synchronization • ҙݟͷಉௐ͕ڠྗͷҡ࣋ʹॏཁ • Computational evolution of social norms • L1 ʢίϯςΩετඇґଘੑʣ͕ॏཁ • ະղܾ໰୊ • ৘ใަ׵΍ύʔτφʔબ୒