Continuous-time Discounted Mirror-Descent Dynamics in Monotone Concave Games
This work addresses convergence challenges in game theory for scenarios with limited monotonicity, representing an incremental improvement in dynamics design.
The paper tackles the problem of convergence in concave continuous-kernel games with monotonicity properties by proposing discounted mirror descent-type dynamics, showing that these dynamics can converge asymptotically in games with monotone pseudo-gradients and even in those with hypo-monotone pseudo-gradients when using strongly convex regularizers.
In this paper, we consider concave continuous-kernel games characterized by monotonicity properties and propose discounted mirror descent-type dynamics. We introduce two classes of dynamics whereby the associated mirror map is constructed based on a strongly convex or a Legendre regularizer. Depending on the properties of the regularizer we show that these new dynamics can converge asymptotically in concave games with monotone (negative) pseudo-gradient. Furthermore, we show that when the regularizer enjoys strong convexity, the resulting dynamics can converge even in games with hypo-monotone (negative) pseudo-gradient, which corresponds to a shortage of monotonicity.