Evolution of indirect reciprocity in groups of various sizes and comparison with direct reciprocity.

Recently many studies have investigated the evolution of indirect reciprocity through which cooperative action is returned by a third individual, e.g. individual A helped B and then receives help from C. Most studies on indirect reciprocity have presumed that only two individuals take part in a single interaction (group), e.g. A helps B and C helps A. In this paper, we investigate the evolution of indirect reciprocity when more than two individuals take part in a single group, and compare the result with direct reciprocity through which cooperative action is directly returned by the recipient. Our analyses show the following. In the population with discriminating cooperators and unconditional defectors, whether implementation error is included or not, (i) both strategies are evolutionarily stable and the evolution of indirect reciprocity becomes more difficult as group size increases, and (ii) the condition for the evolution of indirect reciprocity under standing reputation criterion where the third individuals distinguish between justified and unjustified defections is more relaxed than that under image scoring reputation criterion in which the third individuals do not distinguish with. Furthermore, in the population that also includes unconditional cooperators, (iii) in the presence of errors in implementation, the discriminating strategy is evolutionarily stable not only under standing but also under image scoring if group size is larger than two. Finally, (iv) in the absence of errors in implementation, the condition for the evolution of direct reciprocity is equivalent to that for the evolution of indirect reciprocity under standing, and, in the presence of errors, the condition for the evolution of direct reciprocity is very close to that for the evolution of indirect reciprocity under image scoring.

[1]  M. Milinski,et al.  Cooperation through indirect reciprocity: image scoring or standing strategy? , 2001, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[2]  P. Taylor,et al.  Evolutionarily Stable Strategies and Game Dynamics , 1978 .

[3]  Yoh Iwasa,et al.  The leading eight: social norms that can maintain cooperation by indirect reciprocity. , 2006, Journal of theoretical biology.

[4]  N. V. Joshi,et al.  Evolution of cooperation by reciprocation within structured demes , 1987, Journal of Genetics.

[5]  博 太郎丸.  ＜書評＞ Robert Sugden, "The economics of rights, co-operation and welfare", Oxford : Basil Blackwell, 1986 , 1992 .

[6]  C. Hauert,et al.  Volunteering as Red Queen Mechanism for Cooperation in Public Goods Games , 2002, Science.

[7]  Karl Sigmund,et al.  Indirect reciprocity, image scoring, and moral hazard. , 2005, Proceedings of the National Academy of Sciences of the United States of America.

[8]  R. Boyd,et al.  Indirect reciprocity can stabilize cooperation without the second-order free rider problem , 2004, Nature.

[9]  C. Hauert,et al.  Replicator dynamics for optional public good games. , 2002, Journal of theoretical biology.

[10]  Arnon Lotem,et al.  Evolution of cooperation between individuals , 1999, Nature.

[11]  O. Leimar,et al.  Evolution of cooperation through indirect reciprocity , 2001, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[12]  M. Milinski,et al.  Reputation helps solve the ‘tragedy of the commons’ , 2002, Nature.

[13]  Hisashi Ohtsuki,et al.  Reactive strategies in indirect reciprocity. , 2004, Journal of theoretical biology.

[14]  Per Molander,et al.  The Prevalence of Free Riding , 1992 .

[15]  Nobuyuki Takahashi,et al.  The importance of subjectivity in perceptual errors on the emergence of indirect reciprocity. , 2006, Journal of theoretical biology.

[16]  M A Nowak,et al.  The logic of contrition. , 1997, Journal of theoretical biology.

[17]  M. Milinski,et al.  Cooperation through image scoring in humans. , 2000, Science.

[18]  Eizo Akiyama,et al.  Reputation and the evolution of cooperation in sizable groups , 2005, Proceedings of the Royal Society B: Biological Sciences.

[19]  M. Nowak,et al.  Tit for tat in heterogeneous populations , 1992, Nature.

[20]  M. Milinski,et al.  Volunteering leads to rock–paper–scissors dynamics in a public goods game , 2003, Nature.

[21]  R. May.  More evolution of cooperation , 1987, Nature.

[22]  M A Nowak,et al.  The dynamics of indirect reciprocity. , 1998, Journal of theoretical biology.

[23]  R. Trivers.  The Evolution of Reciprocal Altruism , 1971, The Quarterly Review of Biology.

[24]  M. Nowak,et al.  Evolution of indirect reciprocity by image scoring , 1998, Nature.

[25]  Karl Sigmund,et al.  The logic of reprobation: assessment and action rules for indirect reciprocation. , 2004, Journal of theoretical biology.

[26]  Lik Mui,et al.  Evolution of indirect reciprocity by social information: the role of trust and reputation in evolution of altruism. , 2003, Journal of theoretical biology.

[27]  R. D. Alexander.  The biology of moral systems , 1989 .

[28]  E. Fehr.  A Theory of Fairness, Competition and Cooperation , 1998 .

[29]  Dirk Semmann,et al.  Donors to charity gain in both indirect reciprocity and political reputation , 2002, Proceedings of the Royal Society of London. Series B: Biological Sciences.

[30]  Karthik Panchanathan,et al.  A tale of two defectors: the importance of standing for evolution of indirect reciprocity. , 2003, Journal of theoretical biology.

[31]  Gary E. Bolton,et al.  Cooperation Among Strangers with Limited Information About Reputation , 2003 .

[32]  W. Hamilton,et al.  The Evolution of Cooperation , 1984 .

[33]  E. Fehr,et al.  Cooperation and Punishment in Public Goods Experiments , 1999, SSRN Electronic Journal.

[34]  P. Richerson,et al.  Punishment allows the evolution of cooperation (or anything else) in sizable groups , 1992 .

[35]  P. Richerson,et al.  The evolution of reciprocity in sizable groups. , 1988, Journal of theoretical biology.

[36]  Yoh Iwasa,et al.  How should we define goodness?--reputation dynamics in indirect reciprocity. , 2004, Journal of theoretical biology.

[37]  Y. Toquenaga,et al.  Effects of information and group structure on evolution of altruism: analysis of two-score model by covariance and contextual analyses. , 2005, Journal of theoretical biology.

[38]  Michael A Fishman,et al.  Indirect reciprocity among imperfect individuals. , 2003, Journal of theoretical biology.

[39]  M. Nowak,et al.  Evolution of indirect reciprocity , 2005, Nature.

[40]  Karl Sigmund,et al.  The good, the bad and the discriminator--errors in direct and indirect reciprocity. , 2006, Journal of theoretical biology.