Sink equilibria and the attractors of learning in games

Abstract: Characterizing the limit behavior -- that is, the attractors -- of learning dynamics is one of the most fundamental open questions in game theory. In recent work in this front, it was conjectured that the attractors of the replicator dynamic are in one-to-one correspondence with the sink equilibria of the game -- the sink strongly connected components of a game's preference graph -- , and it was established that they do stand in at least one-to-many correspondence with them. We make threefold progress on the problem of characterizing attractors. First, we show through a topological construction that the one-to-one conjecture is false. The counterexamples derive from objects called local sources -- fixed points which lie within the sink equilibrium yet are locally repelling. Second, we make progress on the attractor characterization problem for two-player games by establishing that the one-to-one conjecture is true when a local property called pseudoconvexity holds. Pseudoconvexity prevents the existence of local sources, and generalizes the existing cases -- such as zero-sum games and potential games -- where the conjecture was known to be true.