Continuous-Time Convergence Rates in Potential and Monotone Games

In this paper, we provide exponential rates of convergence to the Nash equilibrium of continuous-time game dynamics such as mirror descent (MD) and actor-critic (AC) in $N$-player continuous games that are either potential games or monotone games but possibly potential-free. In the first part of this paper, under the assumption the game admits a relatively strongly concave potential, we show that MD and AC converge in $\mathcal{O}(e^{-\beta t})$. In the second part of this paper, using relative concavity, we provide a novel relative characterization of monotone games and show that MD and its discounted version converge with $\mathcal{O}(e^{-\beta t})$ in relatively strongly and relatively hypo-monotone games. Moreover, these rates extend their known convergence conditions and also improve the results in the potential game setup. Simulations are performed which empirically back up our results.