Stackelberg game has attracted recent interest modeling an asymmetric, general sum game between leaders and followers. Understanding of Stackelberg game is limited especially when there are multiple Nash equilibria among the followers. We will start the talk by some simple examples to illustrate the issue of robustness and sensitivity of the game with respect to multiple Nash equilibria. We will then present a discrete-time Stackelberg mean field game framework to analyze this robustness issue, where the followers have bounded rationality and only aim at achieving some -Nash equilibrium while the leader wants to maximize her objective in the worst case scenario among all -NEs. We formulate this problem into an explicit minimax optimization problem when the leader has a finite number of actions. Finally, we also study the sensitivity analysis when the leader only has access to some perturbed model and our results suggest that one needs to be more pessimistic in this case and solve a relaxed problem to obtain a near optimal solution.