  How can the system operator learn an incentive mechanism that achieves social
optimality based on limited information about the agents' behavior, who are
dynamically updating their strategies? To answer this question, we propose an
\emph{adaptive} incentive mechanism. This mechanism updates the incentives of
agents based on the feedback of each agent's externality, evaluated as the
difference between the player's marginal cost and society's marginal cost at
each time step. The proposed mechanism updates the incentives on a slower
timescale compared to the agents' learning dynamics, resulting in a
two-timescale coupled dynamical system. Notably, this mechanism is agnostic to
the specific learning dynamics used by agents to update their strategies. We
show that any fixed point of this adaptive incentive mechanism corresponds to
the optimal incentive mechanism, ensuring that the Nash equilibrium coincides
with the socially optimal strategy. Additionally, we provide sufficient
conditions that guarantee the convergence of the adaptive incentive mechanism
to a fixed point. Our results apply to both atomic and non-atomic games. To
demonstrate the effectiveness of our proposed mechanism, we verify the
convergence conditions in two practically relevant games: atomic networked
quadratic aggregative games and non-atomic network routing games.
