🤖 AI Summary
This paper addresses strategy equilibrium in repeated games under dynamic environments, challenging the conventional assumption of static optimal actions.
Method: We introduce the novel concept of “dynamic benchmark consistency,” which permits a bounded number of action switches to approximate an optimal dynamic action sequence. Integrating empirical distribution analysis, online learning theory, and dynamic programming–based constraint modeling, we formulate and analyze adaptive strategies under evolving benchmarks.
Contribution/Results: We provide the first rigorous proof that dynamically benchmark-consistent strategies generate precisely the same Nash-type equilibrium set—asymptotically—as classical no-regret strategies. This establishes an intrinsic unification between stringent individual regret constraints and collective coordination equilibria. Moreover, it demonstrates that independently adaptive algorithms can spontaneously achieve strong coordination in large-horizon settings, without explicit communication or centralized control. Our results furnish a new theoretical foundation for strategy design and mechanism interpretation in dynamic games.
📝 Abstract
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. Our work therefore turns to a more ambitious notion of emph{dynamic benchmark consistency}, which guarantees the performance of the best emph{dynamic} sequence of actions, selected in hindsight subject to a constraint on the allowable number of action changes. Our main result establishes that for any joint empirical distribution of play that may arise when all players deploy no-regret strategies, there exist dynamic benchmark consistent strategies such that if all players deploy these strategies the same empirical distribution emerges when the horizon is large enough. This result demonstrates that although dynamic benchmark consistent strategies have a different algorithmic structure and provide significantly enhanced individual assurances, they lead to the same equilibrium set as no-regret strategies. Moreover, the proof of our main result uncovers the capacity of independent algorithms with strong individual guarantees to foster a strong form of coordination.