Robust equilibria in continuous games: From strategic to dynamic robustness
This work addresses robustness in game theory for applications requiring stable equilibria under uncertainty, but it appears incremental as it builds on existing concepts like FTRL and regularization.
The paper tackles the problem of Nash equilibria robustness in continuous games under strategic and dynamic uncertainty, establishing a structural correspondence between these notions and showing that entropically regularized learning converges at a geometric rate in affinely constrained games.
In this paper, we examine the robustness of Nash equilibria in continuous games, under both strategic and dynamic uncertainty. Starting with the former, we introduce the notion of a robust equilibrium as those equilibria that remain invariant to small -- but otherwise arbitrary -- perturbations to the game's payoff structure, and we provide a crisp geometric characterization thereof. Subsequently, we turn to the question of dynamic robustness, and we examine which equilibria may arise as stable limit points of the dynamics of "follow the regularized leader" (FTRL) in the presence of randomness and uncertainty. Despite their very distinct origins, we establish a structural correspondence between these two notions of robustness: strategic robustness implies dynamic robustness, and, conversely, the requirement of strategic robustness cannot be relaxed if dynamic robustness is to be maintained. Finally, we examine the rate of convergence to robust equilibria as a function of the underlying regularizer, and we show that entropically regularized learning converges at a geometric rate in games with affinely constrained action spaces.