Poincar\'{e}-Bendixson Limit Sets in Multi-Agent Learning

Aleksander Czechowski, Georgios Piliouras

A key challenge of evolutionary game theory and multi-agent learning is to characterize the limit behaviour of game dynamics. Whereas convergence is often a property of learning algorithms in games satisfying a particular reward structure (e.g. zero-sum), it is well known, that for general payoffs even basic learning models, such as the replicator dynamics, are not guaranteed to converge. Worse yet, chaotic behavior is possible even in rather simple games, such as variants of Rock-Paper-Scissors games (Sato et al., 2002). Although chaotic behavior in learning dynamics can be precluded by the celebrated Poincar\'e-Bendixson theorem, it is only applicable to low-dimensional settings. Are there other characteristics of a game, which can force regularity in the limit sets of learning? In this paper, we show that behaviors consistent with the Poincar\'e-Bendixson theorem (limit cycles, but no chaotic attractor) follows purely based on the topological structure of the interaction graph, even for high-dimensional settings with 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 games where each player has two strategies at disposal, and for interaction graphs where payoffs of each agent are only affected by one other agent (i.e. interaction graphs of indegree one). Since chaos has been observed in a game with only two players and three strategies, this class of non-chaotic games is in a sense maximal. Moreover, we provide simple conditions under which such behavior translates to social welfare guarantees, implying that FoReL learning achieves time average social welfare which is at least as good as that of a Nash equilibrium; and connecting the topology of the dynamics to the Price of Anarchy analysis.

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