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Understanding the behavior of no-regret dynamics in general $N$-player gamesis a fundamental question in online learning and game theory. A folk result inthe field states that, in finite games, the empirical frequency of play underno-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 dynamicscorrelates to the game's Nash equilibria is much more limited, and only partialresults are known for certain classes of games (such as zero-sum or congestiongames). In this paper, we study the dynamics of "follow-the-regularized-leader"(FTRL), arguably the most well-studied class of no-regret dynamics, and weestablish a sweeping negative result showing that the notion of mixed Nashequilibrium is antithetical to no-regret learning. Specifically, we show thatany Nash equilibrium which is not strict (in that every player has a uniquebest response) cannot be stable and attracting under the dynamics of FTRL. Thisresult has significant implications for predicting the outcome of a learningprocess as it shows unequivocally that only strict (and hence, pure) Nashequilibria can emerge as stable limit points thereof.

No-regret learning and mixed Nash equilibria: They do not mix