Regret Minimization with Dynamic Benchmarks in Repeated Games

In repeated games, strategies are often evaluated by their ability to guarantee the performance of the single best action that is selected in hindsight (a property referred to as \emph{Hannan consistency}, or \emph{no-regret}). However, the effectiveness of the single best action as a yardstick to evaluate strategies is limited, as any static action may perform poorly in common dynamic settings. We propose the notion of \emph{dynamic benchmark consistency}, which requires a strategy to asymptotically guarantee the performance of the best \emph{dynamic} sequence of actions selected in hindsight subject to a constraint on the number of action changes the corresponding dynamic benchmark admits. We show that dynamic benchmark consistent strategies exist if and only if the number of changes in the benchmark scales sublinearly with the horizon length. Further, our main result establishes that the set of empirical joint distributions of play that may emerge, when all players deploy such strategies, asymptotically coincides with the set of \emph{Hannan equilibria} (also referred to as \emph{coarse correlated equilibria}) of the stage game. This general characterization allows one to leverage analyses developed for frameworks that consider static benchmarks, which we demonstrate by bounding the social efficiency of the possible outcomes in our~setting. Together, our results imply that dynamic benchmark consistent strategies introduce the following \emph{Pareto-type} improvement over no-regret strategies: They enable stronger individual guarantees against arbitrary strategies of the other players, while maintaining the same worst-case guarantees on the social welfare, when all players adopt these strategies.