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Regret has been established as a foundational concept in online learning, andlikewise has important applications in the analysis of learning dynamics ingames. Regret quantifies the difference between a learner's performance againsta baseline in hindsight. It is well-known that regret-minimizing algorithmsconverge to certain classes of equilibria in games; however, traditional formsof regret used in game theory predominantly consider baselines that permitdeviations to deterministic actions or strategies. In this paper, we revisitour understanding of regret from the perspective of deviations over partitionsof the full \emph{mixed} strategy space (i.e., probability distributions overpure strategies), under the lens of the previously-established $\Phi$-regretframework, which provides a continuum of stronger regret measures. Importantly,$\Phi$-regret enables learning agents to consider deviations from and to mixedstrategies, generalizing several existing notions of regret such as external,internal, and swap regret, and thus broadening the insights gained fromregret-based analysis of learning algorithms. We prove here that thewell-studied evolutionary learning algorithm of replicator dynamics (RD)seamlessly minimizes the strongest possible form of $\Phi$-regret in generic $2\times 2$ games, without any modification of the underlying algorithm itself.We subsequently conduct experiments validating our theoretical results in asuite of 144 $2 \times 2$ games wherein RD exhibits a diverse set of behaviors.We conclude by providing empirical evidence of $\Phi$-regret minimization by RDin some larger games, hinting at further opportunity for $\Phi$-regret basedstudy of such algorithms from both a theoretical and empirical perspective.

Evolutionary Dynamics and $\Phi$-Regret Minimization in Games