  This paper proposes and studies a class of discrete-time finite-time-horizon
Stackelberg mean-field games, with one leader and an infinite number of
identical and indistinguishable followers. In this game, the objective of the
leader is to maximize her reward considering the worst-case cost over all
possible $\epsilon$-Nash equilibria among followers. A new analytical paradigm
is established by showing the equivalence between this Stackelberg mean-field
game and a minimax optimization problem. This optimization framework
facilitates studying both analytically and numerically the set of Nash
equilibria for the game; and leads to the sensitivity and the robustness
analysis of the game value. In particular, when there is model uncertainty, the
game value for the leader suffers non-vanishing sub-optimality as the perturbed
model converges to the true model. In order to obtain a near-optimal solution,
the leader needs to be more pessimistic with anticipation of model errors and
adopts a relaxed version of the original Stackelberg game.
