Before revision:  Our method -- Online Double Oracle (ODO) -- achieves the  regret bound of \mathcal{O}(\sqrt{T k \log(k)}) in self-play setting where k is NOT the size of the game , but rather the size of \emph{effective strategy set} that is linearly dependent on the support size of the Nash equilibrium .
After revision:  Our method -- Online Double Oracle (ODO) -- achieves the in the sense that each agent in ODO can exploit strategic adversary with a regret bound of \mathcal{O}(\sqrt{T k \log(k)}) in self-play setting where k is NOT the size of the game , but rather the size of \emph{effective strategy set} that is linearly dependent on the support size of the Nash equilibrium .