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O(T^{-3/4}) O(T^{-1}) O(T^{-1/2}) \tilde{O}(T^{-1/2}) We show that natural classes of regularized learning algorithms with a form of recency bias achieve faster convergence rates to approximate efficiency and to coarse correlated equilibria in multiplayer normal form games. When each player in a game uses an algorithm from our class, their individual regret decays at O(T −3/4 ) , while the sum of utilities converges to an approximate optimum at O(T −1 ) --an improvement upon the worst case O(T −1/2 ) rates. We show a black-box reduction for any algorithm in the class to achieve O ~ (T −1/2 ) rates against an adversary, while maintaining the faster rates against algorithms in the class. Our results extend those of Rakhlin and Shridharan~\cite{Rakhlin2013} and Daskalakis et al.~\cite{Daskalakis2014}, who only analyzed two-player zero-sum games for specific algorithms.