The Bounds of Algorithmic Collusion; $Q$-learning, Gradient Learning, and the Folk Theorem

We explore the behaviour emerging from learning agents repeatedly interacting strategically for a wide range of learning dynamics, including $Q$-learning, projected gradient, replicator and log-barrier dynamics. Going beyond the better understood classes of potential games and zero-sum games, we consider the setting of a general repeated game with finite recall under different forms of monitoring. We obtain a Folk Theorem-style result and characterise the set of payoff vectors that can be obtained by these dynamics, discovering a wide range of possibilities for the emergence of algorithmic collusion. Achieving this requires a novel technical approach, which, to the best of our knowledge, yields the first convergence result for multi-agent $Q$-learning algorithms in repeated games.