Indirect reciprocity with trinary reputations

TL;DR

This paper extends the analysis of indirect reciprocity to a trinary reputation system, revealing conditions under which cooperation can be stable, especially under image scoring, which was previously considered unstable in binary models.

Contribution

It introduces a trinary reputation model for indirect reciprocity, analyzing stability and cooperation, and demonstrates that cooperation under image scoring is possible in this new framework.

Findings

Cooperation can be stable under image scoring with trinary reputations.

The trinary model extends binary reputation results, allowing cooperation in more scenarios.

Stable equilibria include homogeneous and heterogeneous populations.

Abstract

Indirect reciprocity is a reputation-based mechanism for cooperation in social dilemma situations when individuals do not repeatedly meet. The conditions under which cooperation based on indirect reciprocity occurs have been examined in great details. Most previous theoretical analysis assumed for mathematical tractability that an individual possesses a binary reputation value, i.e., good or bad, which depends on their past actions and other factors. However, in real situations, reputations of individuals may be multiple valued. Another puzzling discrepancy between the theory and experiments is the status of the so-called image scoring, in which cooperation and defection are judged to be good and bad, respectively, independent of other factors. Such an assessment rule is found in behavioral experiments, whereas it is known to be unstable in theory. In the present study, we fill both gaps by analyzing a trinary reputation model. By an exhaustive search, we identify all the cooperative and stable equilibria composed of a homogeneous population or a heterogeneous population containing two types of players. Some results derived for the trinary reputation model are direct extensions of those for the binary model. However, we find that the trinary model allows cooperation under image scoring under some mild conditions.