Total: 1
We examine the interplay between ordinal, preference-based solution concepts in games and the outcomes of payoff-driven learning dynamics, asking to what extent the combinatorial data of a game—its *preference graph*—can predict the long-run behavior of no-regret dynamics such as *follow-the-regularized-leader* (FTRL). In one direction, we show that the skeleton of every *dynamically stable* set (i.e., the set of pure profiles it contains) must also be *preferentially stable*, that is, it must be closed under profitable deviations. We then ask the converse question: when are preferences sufficient to describe the long-run behavior of the players' learning dynamics? We begin by showing that preferences are indeed enough to fully characterize asymptotic stability in the case of *subgames*—i.e., subsets of pure profiles obtained by restricting players' action sets. Beyond this case however, the equivalence between dynamic and preferential stability breaks down: in particular, we construct a three-player game with a preferentially stable set whose span is dynamically *unstable*, showing that preferences are *not sufficient* to describe dynamically stable behavior in general. To restore stability, we introduce the notion of *leaklessness*, a measure of aggregate payoff drift away from a set of pure profiles, and we use it to identify a payoff-based condition guaranteeing that the span of a set of pure profiles is stable and attracting.