Inverse Noncooperative Games With Indistinguishable Observations

In this article, the inverse problem is studied for a class of nonzero-sum linear quadratic (LQ) games with indistinguishable observations. The goal is to reconstruct the unknown individual objectives from permuted observations of the Nash equilibrium trajectories, where the main challenge lies in the unknown and time-varying permutations of agent states. Taking advantage of the symmetry in the game structure, the identifiability of the underlying game model is established and the well-posedness of the inverse game problem is guaranteed. The identification problem is solved by a least squares program that admits a unique global minimum at the true value of the unknown parameter. To address the mixed-integer constraints, an efficient subdifferential algorithm is designed by incorporating the linear sum assignment substructures. In addition, robust estimators are further designed in the stochastic setup, whose statistical consistency can be guaranteed in the presence of noisy observations. Finally, extensions to the infinite time inverse game problem are discussed and its solution set is derived analytically. Efficiency and effectiveness of the proposed method are demonstrated by numerical examples.