  Algorithms based on regret matching, specifically regret matching$^+$
(RM$^+$), and its variants are the most popular approaches for solving
large-scale two-player zero-sum games in practice. Unlike algorithms such as
optimistic gradient descent ascent, which have strong last-iterate and ergodic
convergence properties for zero-sum games, virtually nothing is known about the
last-iterate properties of regret-matching algorithms. Given the importance of
last-iterate convergence for numerical optimization reasons and relevance as
modeling real-word learning in games, in this paper, we study the last-iterate
convergence properties of various popular variants of RM$^+$. First, we show
numerically that several practical variants such as simultaneous RM$^+$,
alternating RM$^+$, and simultaneous predictive RM$^+$, all lack last-iterate
convergence guarantees even on a simple $3\times 3$ game. We then prove that
recent variants of these algorithms based on a smoothing technique do enjoy
last-iterate convergence: we prove that extragradient RM$^{+}$ and smooth
Predictive RM$^+$ enjoy asymptotic last-iterate convergence (without a rate)
and $1/\sqrt{t}$ best-iterate convergence. Finally, we introduce restarted
variants of these algorithms, and show that they enjoy linear-rate last-iterate
convergence.
