Second-order effects of mutation in a continuous model of indirect reciprocity
Department of Physics, Pukyong National University, Yongso-ro 45, 48513, Busan, Korea
2 Department of Scientific Computing, Pukyong National University, Yongso-ro 45, 48513, Busan, Korea
Accepted: 19 July 2023
Published online: 16 August 2023
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.
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