No-regret learning and mixed Nash equilibria: They do not mix
This addresses a fundamental question in online learning and game theory, providing a negative result that clarifies the limitations of no-regret learning in predicting game outcomes, which is incremental but important for the field.
The paper tackles the problem of understanding how no-regret learning dynamics, specifically follow-the-regularized-leader (FTRL), correlate with Nash equilibria in games, and shows that only strict (pure) Nash equilibria can be stable and attracting under these dynamics, while non-strict mixed equilibria cannot.
Understanding the behavior of no-regret dynamics in general $N$-player games is a fundamental question in online learning and game theory. A folk result in the field states that, in finite games, the empirical frequency of play under no-regret learning converges to the game's set of coarse correlated equilibria. By contrast, our understanding of how the day-to-day behavior of the dynamics correlates to the game's Nash equilibria is much more limited, and only partial results are known for certain classes of games (such as zero-sum or congestion games). In this paper, we study the dynamics of "follow-the-regularized-leader" (FTRL), arguably the most well-studied class of no-regret dynamics, and we establish a sweeping negative result showing that the notion of mixed Nash equilibrium is antithetical to no-regret learning. Specifically, we show that any Nash equilibrium which is not strict (in that every player has a unique best response) cannot be stable and attracting under the dynamics of FTRL. This result has significant implications for predicting the outcome of a learning process as it shows unequivocally that only strict (and hence, pure) Nash equilibria can emerge as stable limit points thereof.