arXiv:2610.13448v1 Announce Type: new Abstract: Incentives can align private interests with collective goals, but their design becomes intertwined with learning when agents' preferences are unknown. This paper develops an adaptive incentive-design framework for multi-agent linear-quadratic dynamic games, where a system planner can simultaneously learn agents' private preferences and steer their strategic behavior towards the social optimum. To this end, we first characterize the complete family of affine incentives that implement the social optimum as the unique open-loop Nash equilibrium while ensuring exact budget balance. When agents' preferences are unknown, we develop an online algorithm to estimate their preference parameters and establish strong consistency of the estimates under a weak trajectory-dependent excitation condition, without requiring persistent excitation or i.i.d. trajectories. We then integrate this estimator into an active-learning incentive mechanism that combines certainty-equivalent coordination with decaying random probing. We prove that the probing generates sufficient information through the agents' equilibrium responses and the system dynamics, ensuring consistent preference learning from the resulting closed-loop trajectories. We show that the induced trajectories converge almost surely to their socially optimal counterparts, while cumulative social-cost regret relative to the full-information oracle and cumulative net-budget imbalance grow sublinearly. Simulations of a transboundary pollution game demonstrate the effectiveness of the proposed framework.