Second-order effects of mutation in continuous indirect reciprocity

TL;DR

This paper develops a second-order analytical model to understand how small mutations affect cooperative social norms in continuous indirect reciprocity, revealing key features that promote norm stability.

Contribution

It introduces a second-order perturbative analysis of mutation effects in continuous models of indirect reciprocity, addressing limitations of linear approximations for cooperative norms.

Findings

Second-order analysis reveals norm characteristics that resist mutation.

Norms should allow refusal to cooperate with the ill-reputed.

Cooperation among ill-reputed players can help reduce mutation impact.

Abstract

We have developed a continuous model of indirect reciprocity and thereby investigated effects of mutation in assessment rules. Within this continuous framework, the difference between the resident and mutant norms is treated as a small parameter for perturbative expansion. Unfortunately, the linear-order expansion leads to singularity when applied to the leading eight, the cooperative norms that resist invasion of another norm having a different behavioral rule. For this reason, this study aims at a second-order analysis for the effects of mutation when the resident norm is one of the leading eight. We approximately solve a set of coupled nonlinear equations using Newton's method, and the solution is compared with Monte Carlo calculations. The solution indicates how the characteristics of a social norm can shape the response to its close variants appearing through mutation. Specifically, it shows that the resident norm should allow one to refuse to cooperate toward the ill-reputed, while regarding cooperation between two ill-reputed players as good, so as to reduce the impact of mutation.This study enhances our analytic understanding on the organizing principles of successful social norms.