Social norms in indirect reciprocity with ternary reputations

TL;DR

This paper explores the use of ternary reputations in indirect reciprocity, identifying stable norms that promote cooperation and analyzing their behaviors through extensive computational analysis.

Contribution

It introduces a comprehensive classification of cooperation-promoting norms with three reputation levels and derives rules for their success, extending beyond binary reputation models.

Findings

Identified all stable norms with ternary reputations through supercomputing.

Found four key rules for successful norms in ternary reputation systems.

Demonstrated that leading eight norms are special cases of these rules.

Abstract

Indirect reciprocity is a key mechanism that promotes cooperation in social dilemmas by means of reputation. Although it has been a common practice to represent reputations by binary values, either `good' or `bad', such a dichotomy is a crude approximation considering the complexity of reality. In this work, we studied norms with three different reputations, i.e., `good', `neutral', and `bad'. Through massive supercomputing for handling more than thirty billion possibilities, we fully identified which norms achieve cooperation and possess evolutionary stability against behavioural mutants. By systematically categorizing all these norms according to their behaviours, we found similarities and dissimilarities to their binary-reputation counterpart, the leading eight. We obtained four rules that should be satisfied by the successful norms, and the behaviour of the leading eight can be understood as a special case of these rules. A couple of norms that show counter-intuitive behaviours are also presented. We believe the findings are also useful for designing successful norms with more general reputation systems.