Title: On the Variational Interpretation of Mirror Play in Monotone Games

Abstract: Mirror play (MP) is a well-accepted primal-dual multi-agent learning algorithm where all agents simultaneously implement mirror descent in a distributed fashion. The advantage of MP over vanilla gradient play lies in its usage of mirror maps that better exploit the geometry of decision domains. Despite extensive literature dedicated to the asymptotic convergence of MP to equilibrium, the understanding of the finite-time behavior of MP before reaching equilibrium is still rudimentary. To facilitate the study of MP's non-equilibrium performance, this work establishes an equivalence between MP's finite-time primal-dual path (mirror path) in monotone games and the closed-loop Nash equilibrium path of a finite-horizon differential game, referred to as mirror differential game (MDG). Our construction of MDG rests on the Brezis-Ekeland variational principle, and the stage cost functional for MDG is Fenchel coupling between MP's iterates and associated gradient updates. The variational interpretation of mirror path in static games as the equilibrium path in MDG holds in deterministic and stochastic cases. Such a variational interpretation translates the non-equilibrium studies of learning dynamics into a more tractable equilibrium analysis of dynamic games, as demonstrated in a case study on the Cournot game, where MP dynamics corresponds to a linear quadratic game.