[SS19] Social evolution theory in humans

[SS19-01] Cluster dynamics of indirect reciprocity in complete graphs

*Seung Ki Baek¹ (1. Pukyong National University (Korea))

Keywords:

indirect reciprocity,balance theory,cluster dynamics

Heider's balance theory emphasizes cognitive consistency in assessing others, as expressed by the phrase “The enemy of my enemy is my friend.” At the same time, the theory of indirect reciprocity provides us with a dynamical framework to study how to assess others based on their actions as well as how to act toward them based on the assessments. Well known are the “leading eight” from L1 to L8, the eight norms for assessment and action to foster cooperation in social dilemmas while resisting the invasion of mutant norms prescribing alternative actions.In the first part of this talk, we show that balance is equivalent to stationarity of dynamics only for L4 and L6 (stern judging) among the leading eight. Stern judging reflects an intuitive idea that good merits reward, whereas evil warrants punishment. By analyzing the dynamics of stern judging in complete graphs, we prove that this norm almost always segregates the graph into two mutually hostile groups as the graph size grows. We then compare L4 with stern judging: The only difference of L4 is that a good player's cooperative action toward a bad one is regarded as good. This subtle difference transforms large populations governed by L4 to a “paradise” where cooperation prevails and positive assessments abound. In the second part of this talk, we consider a social network divided into many factions. People are friends within each faction, while they are enemies of the other factions, and even my enemy's enemy is not necessarily my friend. This configuration can be described in terms of a weak form of structural balance. Although weak balance explains a number of real social networks, which dynamical rule achieves it has remained unexplored. We show that the answer can again be found in indirect reciprocity. Weak structural balance is equivalent to stationarity when the rule is given by L8 (judging). By analyzing its cluster dynamics of merging, division, and migration induced by assessment error in complete graphs, we obtain the cluster size distribution in a steady state, and these results are compared with empirical data.We stress that our approach can provide insight into the interplay between a norm that individuals abide by and the macroscopic group structure in society. Our study helps us understand the relationship between individual norms and their emergent consequences at a population level, shedding light on the nuanced interplay between cognitive consistency and segregation dynamics.