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arXiv · 2010.09514

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

Lampros Flokas·Emmanouil-Vasileios Vlatakis-Gkaragkounis·Thanasis Lianeas·Panayotis Mertikopoulos·Georgios Piliouras
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Abstract

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.

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Lampros Flokas, Emmanouil-Vasileios Vlatakis-Gkaragkounis, Thanasis Lianeas, Panayotis Mertikopoulos, Georgios Piliouras. 2020-10-19. No-regret learning and mixed Nash equilibria: They do not mix. https://arxiv.org/abs/2010.09514

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