Local stability of cooperation in a continuous model of indirect reciprocity

TL;DR

This paper models the stability of cooperation in indirect reciprocity using a continuous reputation framework, revealing conditions under which cooperation persists despite noise and disagreement.

Contribution

It introduces a continuous model of reputation and behavior, deriving stability conditions for cooperation that extend beyond binary reputation systems.

Findings

A derived condition for social norm stability under continuous reputations

Identification of a threshold for benefit-cost ratio based on reputation sensitivities

Numerical simulations supporting the theoretical stability conditions

Abstract

Reputation is a powerful mechanism to enforce cooperation among unrelated individuals through indirect reciprocity, but it suffers from disagreement originating from private assessment, noise, and incomplete information. In this work, we investigate stability of cooperation in the donation game by regarding each player's reputation and behaviour as continuous variables. Through perturbative calculation, we derive a condition that a social norm should satisfy to give penalties to its close variants, provided that everyone initially cooperates with a good reputation, and this result is supported by numerical simulation. A crucial factor of the condition is whether a well-reputed player's donation to an ill-reputed co-player is appreciated by other members of the society, and the condition can be reduced to a threshold for the benefit-cost ratio of cooperation which depends on the reputational sensitivity to a donor's behaviour as well as on the behavioural sensitivity to a recipient's reputation. Our continuum formulation suggests how indirect reciprocity can work beyond the dichotomy between good and bad even in the presence of inhomogeneity, noise, and incomplete information.