  In this short note, we consider an inverse problem to a mean-field games
system where we are interested in reconstructing the state-independent running
cost function from observed value-function data. We provide an elementary proof
of a uniqueness result for the inverse problem using the standard
multilinearization technique. One of the main features of our work is that we
insist that the population distribution be a probability measure, a requirement
that is not enforced in some of the existing literature on theoretical inverse
mean-field games.
