Optimality of the square-root player dependence and sharp lower bounds

Determine whether the \(\sqrt n\) dependence in the individual regret bound for uncoupled learning in finite general-sum games is optimal, and establish sharp lower bounds for uncoupled learning in general games.

Background

Multiplicatively Optimistic Regret Matching (MORM) achieves individual external regret of order $\mathcal{O}(\sqrt n\log d)$ in finite general-sum self-play, where $n$ is the number of players and $d$ is the maximum number of actions per player. The conclusion explicitly identifies the optimality of the $\sqrt n$ dependence as unresolved.

The paper further notes that sharp lower bounds for uncoupled learning in general games are largely absent. Establishing such lower bounds would clarify whether the player dependence achieved by MORM reflects a fundamental limitation or can be improved by other uncoupled learning rules.