Non-chaotic limit sets in multi-agent learning

Non-convergence is an inherent aspect of adaptive multi-agent systems, and even basic learning models, such as the replicator dynamics, are not guaranteed to equilibriate. Limit cycles, and even more complicated chaotic sets are in fact possible even in rather simple games, including variants of the Rock-Paper-Scissors game. A key challenge of multi-agent learning theory lies in characterization of these limit sets, based on qualitative features of the underlying game. Although chaotic behavior in learning dynamics can be precluded by the celebrated Poincaré–Bendixson theorem, it is only applicable directly to low-dimensional settings. In this work, we attempt to find other characteristics of a game that can force regularity in the limit sets of learning. We show that behavior consistent with the Poincaré–Bendixson theorem (limit cycles, but no chaotic attractor) follows purely from the topological structure of interactions, even for high-dimensional settings with an arbitrary number of players, and arbitrary payoff matrices. We prove our result for a wide class of follow-the-regularized leader (FoReL) dynamics, which generalize replicator dynamics, for binary games characterized interaction graphs where the payoffs of each player are only affected by one other player (i.e., interaction graphs of indegree one). Moreover, for cyclic games we provide further insight into the planar structure of limit sets, and in particular limit cycles. We propose simple conditions under which learning comes with efficiency guarantees, implying that FoReL learning achieves time-averaged sum of payoffs at least as good as that of a Nash equilibrium, thereby connecting the topology of the dynamics to social-welfare analysis.