There is a well known fact that we can use the existence of external regret minimization algorithms to prove the minimax theorem of two-player zero-sum games.
The proof can be found in the survey Learning, Regret minimization, and Equilibria A. Blum and Y. Mansour.
Immediately arises question whether can we use external regret minimization to find mixed Nash equilibrium in finite non-zero two-player game, in other words does zero external regret leads to find mixed Nash equilibrium, in per round sense.
Intuitively it doesn't seem so, I am not sure that we can get external regret equals zero in general case, but there are polynomial time algorithm for good approximation (see the above paper), on the other hand approximation of Nash equilibrium seems like more difficult task.
I would appreciate if someone could help to answer the question.