Explaining the “Harmonic†Signature in a Tag-Based Model of the Evolution of Altruism

This paper offers an explanation for the anomalous and unexplained behavior of an agent-based model previously described in JASSS in 2015. In their model of tag-based altruism, Shutters and Hales described a curious and unexpected “harmonic†or quantized signature observed in the model's output. This manuscript synthesizes two lines of analysis in an attempt to explain these results. First, it provides a conceptual explanation for the presence of sharp thresholds in the evolved donation rate when the number of interactions per generation $P$ crosses integer multiples of the benefit–cost ratio $b/c$. Second, it presents a minimal analytic toy model that reproduces quantized cooperation levels with thresholds at $P approx n(b/c)$. Throughout, this study interprets these results in the broader context of graph-structured cooperation and tag-based altruism in evolutionary game theory