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
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
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
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
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
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
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
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
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.
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)
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
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.
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
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
(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
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)
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