This question - originally due to Nash himself - has been at the forefront of game-theoretic research ever since the early days of the field. However, despite immense progress and intense scrutiny, a full and complete answer remains elusive.

In this talk, we will examine the long-run behavior of a wide array of algorithms for learning in games – including the multiplicative/exponential weights algorithm, gradient descent/ascent, their optimistic variants, etc. We will consider both finite and continuous games, with different types of feedback (from full information to purely emprical, bandit-type observations), and we will present a unified framework for their analysis through the lens of stochastic approximation. We will subsequently use this viewpoint to survey a range of recent results in the field – both positive and negative – and we will discuss a number of open questions that have attracted considerable interest in machine learning and beyond.