  We derive the rate of convergence to Nash equilibria for the payoff-based
algorithm proposed in \cite{tat_kam_TAC}. These rates are achieved under the
standard assumption of convexity of the game, strong monotonicity and
differentiability of the pseudo-gradient. In particular, we show the algorithm
achieves $O(\frac{1}{T})$ in the two-point function evaluating setting and
$O(\frac{1}{\sqrt{T}})$ in the one-point function evaluation under additional
requirement of Lipschitz continuity of the pseudo-gradient. These rates are to
our knowledge the best known rates for the corresponding problem classes.
