Local reputation dynamics: Stable tripartite relationships under Kandori and Sugden assessment rules

In the theories of indirect reciprocity, Kandori assessment rule is renowned to be one of the best moral systems that secures mutual cooperation under the challenges by cheating. Previous models of indirect reciprocity often assumed that the assessment by the third party is public or global, i.e., the updated reputation of a donor of action is immediately shared by all the members of the community. Here I introduce a local/private assessment rule and modeled how the reputation in each tripartite relationship in the community develops with time. I found that, among 17 possible patterns for the mutual trust/distrust structure in a tripartite relationship, only two are stable. One is just a mutual-trusts between all three. The other more interesting pattern is classified as a common-enemy-mediated mutual bond. I also found that there remains no one-way trust in a stable tripartite relationship. With Sugden rule, by contrast, only stable tripartite relationship is mutual-trusts between all three.