  The dynamic behavior of agents in games, which captures how their strategies
evolve over time based on past interactions, can lead to a spectrum of
undesirable behaviors, ranging from non-convergence to Nash equilibria to the
emergence of limit cycles and chaos. To mitigate the effects of selfish
behavior, central planners can use dynamic payments to guide strategic
multi-agent systems toward stability and socially optimal outcomes. However,
the effectiveness of such interventions critically relies on accurately
predicting agents' responses to incentives and dynamically adjusting payments
so that the system is guided towards the desired outcomes. These challenges are
further amplified in real-time applications where the dynamics are unknown and
only scarce data is available. To tackle this challenge, in this work we
introduce the SIAR-MPC method, combining the recently introduced Side
Information Assisted Regression (SIAR) method for system identification with
Model Predictive Control (MPC). SIAR utilizes side-information constraints
inherent to game theoretic applications to model agent responses to payments
from scarce data, while MPC uses this model to facilitate dynamic payment
adjustments. Our experiments demonstrate the efficiency of SIAR-MPC in guiding
the system towards socially optimal equilibria, stabilizing chaotic behaviors,
and avoiding specified regions of the state space. Comparative analyses in
data-scarce settings show SIAR-MPC's superior performance over pairing MPC with
Physics Informed Neural Networks (PINNs), a powerful system identification
method that finds models satisfying specific constraints.
