In this paper, we propose and study the utilization of the
Dirichlet-to-Neumann (DN) map to uniquely identify the discount functions r,k and cost function F in a stationary mean field game (MFG) system. This
study features several technical novelties that make it highly intriguing and
challenging. Firstly, it involves a coupling of two nonlinear elliptic partial
differential equations. Secondly, the simultaneous recovery of multiple
parameters poses a significant implementation challenge. Thirdly, there is the
probability measure constraint of the coupled equations to consider. Finally,
the limited information available from partial boundary measurements adds
another layer of complexity to the problem. Considering these challenges and
problems, we present an enhanced higher-order linearization method to tackle
the inverse problem related to the MFG system. Our proposed approach involves
linearizing around a pair of zero solutions and fulfilling the probability
measurement constraint by adjusting the positive input at the boundary. It is
worth emphasizing that this technique is not only applicable for uniquely
identifying multiple parameters using full-boundary measurements but also
highly effective for utilizing partial-boundary measurements