  Games are natural models for multi-agent machine learning settings, such as
generative adversarial networks (GANs). The desirable outcomes from algorithmic
interactions in these games are encoded as game theoretic equilibrium concepts,
e.g. Nash and coarse correlated equilibria. As directly computing an
equilibrium is typically impractical, one often aims to design learning
algorithms that iteratively converge to equilibria. A growing body of negative
results casts doubt on this goal, from non-convergence to chaotic and even
arbitrary behaviour. In this paper we add a strong negative result to this
list: learning in games is Turing complete. Specifically, we prove Turing
completeness of the replicator dynamic on matrix games, one of the simplest
possible settings. Our results imply the undecicability of reachability
problems for learning algorithms in games, a special case of which is
determining equilibrium convergence.
