@inproceedings{
leme2024convergence,
title={Convergence of No-Swap-Regret Dynamics in Self-Play},
author={Renato Paes Leme and Georgios Piliouras and Jon Schneider},
booktitle={The Thirty-eighth Annual Conference on Neural Information Processing Systems},
year={2024},
url={https://openreview.net/forum?id=CEnoUjEqNx}
}
In this paper, we investigate the question of whether no-swap-regret dynamics have stronger convergence properties in repeated games than regular no-external-regret dynamics. We prove that in almost all symmetric zero-sum games under symmetric initializations of the agents, no-swap-regret dynamics in self-play are guaranteed to converge in a strong ``frequent-iterate'' sense to the Nash equilibrium: in all but a vanishing fraction of the rounds, the players must play a strategy profile close to a symmetric Nash equilibrium.  Remarkably, relaxing any of these three constraints, i.e. by allowing either i) asymmetric initial conditions, or ii) an asymmetric game or iii) no-external regret dynamics suffices to destroy this result and lead to complex non-equilibrating or even chaotic behavior. 

In a dual type of result, we show that the power of no-swap-regret dynamics comes at a cost of imposing a time-asymmetry on its inputs. While no-external-regret dynamics can be completely determined by the cumulative reward vector received by each player, we show there does not exist any general no-swap-regret dynamics defined on the same state space. In fact, we prove that any no-swap-regret learning algorithm must play a time-asymmetric function over the set of previously observed rewards, ruling out any dynamics based on a symmetric function of the current set of rewards.
Convergence of No-Swap-Regret Dynamics in Self-Play
Renato Paes Leme Georgios Piliouras Jon Schneider