Continuous-Time Convergence Rates in Potential and Monotone Games

In this paper, we provide exponential rates of convergence to the Nash equilibrium for continuous-time dual-space game dynamics such as mirror descent (MD) and actor-critic (AC). We perform our analysis in $N$-player continuous concave games that are either potential games or monotone games but possibly potential-free. In the first part of this paper, 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, respectively. In the second part of this paper, we specialize our results to games that admit a relatively strongly concave potential and show that MD and AC converge with $\mathcal{O}(e^{-\beta t})$. Moreover, these rates extend their known convergence conditions. Simulations are performed which empirically back up our results.