Memes, compound strategies, and factoring the replicator equation

When you work with evolutionary game theory for a while, you end up accumulating an arsenal of cute tools and tricks. A lot of them are obvious once you’ve seen them, but you usually wouldn’t bother looking for them if you hadn’t know they existed. In particular, you become very good friends with the replicator equation. A trick that I find useful at times — and that has come up recently in my on-going project with Robert Vander Veldge, David Basanta, and Jacob Scott — is nesting replicator dynamics (or the dual notion of factoring the replicator equation). I wanted to share a relatively general version of this trick with you, and provide an interpretation of it that is of interest to people — like me — who care about the interaction of evolution in learning. In particular, we will consider a world of evolving agents where each agent is complex enough to learn through reinforcement and pass its knowledge to its offspring. We will see that in this setting, the dynamics of the basic ideas — or memes — that the agents consider can be studied in a world of selfish memes independent of the agents that host them.
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