Symmetry in tag-based games & invariants under replicator dynamics

Mathematicians and physicists love finding symmetries. The reason is simple: symmetries make life easier. The situation is no different when studying the evolutionary dynamics of life. If the fitness functions of your organisms have some symmetry or other nice structure then you can usually exploit it to make analyzing your replicator equations easier. In this post, I want to show an example of this in tag-based models. This analysis is an essential base case when building more complicated models of ethnocentrism — like our work in the Hammond and Axelrod model — and I have been meaning to write about it for a while. This will give me a chance to show a concrete example where my method for factoring the replicator equation is useful, and how observing a straighforward symmetry can reduce the dimensionality of a problem. Maybe this exercise will also teach us something about the evolution of ethnocentrism.
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