Last-Iterate Convergence of Optimistic Gradient Descent-Ascent in Behavioral Form for Extensive-Form Games

This paper shows that a simple learning algorithm in the behavioral-form representation of extensive-form games (EFGs) achieves last-iterate convergence, without modifying the game or enforcing exploration. Existing last-iterate guarantees for EFGs are developed mainly in sequence form, while behavioral-form guarantees often rely on regularization or payoff perturbation. Our method applies simultaneous optimistic gradient descent-ascent (OGDA) with independent Euclidean projections at each information set, using ordinary counterfactual values. We show that, for every perfect-recall, two-player zero-sum EFG, any fixed step size below a positive threshold yields last-iterate convergence. Because this guarantee does not follow from the sequence-form analysis, our analysis instead compares the behavioral-form dynamics with sequence-form dilated OGDA and controls the additional correction term that ordinary counterfactual values introduce. We also provide a parameter-free variant that halves the step size until it is small enough and achieves last-iterate convergence for any initial step size. Experiments on benchmark games are consistent with last-iterate convergence.