Last-Iterate Convergence of Parameter-Free and Scale-Invariant Regret Matching${}^{+}$

Regret Matching${}^{+}$ (RM${}^{+}$) is widely used as a local regret minimizer in counterfactual regret minimization (CFR) for two-player zero-sum extensive-form games. RM${}^{+}$ and its predictive variants are parameter-free and invariant to positive rescaling of the payoffs, and existing guarantees control their average or best iterates. Their last-iterate behavior, however, has remained theoretically unresolved even in two-player zero-sum matrix games, where their self-play dynamics can be studied in isolation. We study two predictive RM${}^{+}$ dynamics in this setting. The first is the Increasing Regret Extra-Gradient Predictive Regret Matching${}^{+}$ (IREG-PRM${}^{+}$) dynamics of zhang2026, and the second is the Increasing-Regret Optimistic PRM${}^{+}$ (IRO-PRM${}^{+}$) dynamics that we introduce. We prove that the played strategy profiles of both dynamics converge to the Nash equilibrium set for every two-player zero-sum matrix game. For IREG-PRM${}^{+}$, we further bound the cumulative duality gap by $\mathcal{O}(\sqrt{T})$. To the best of our knowledge, these are the first last-iterate convergence guarantees for a parameter-free and payoff-scale-invariant RM${}^{+}$-type algorithm. Experiments on structured and random matrix games support the theoretical results.