We consider the super-hedging price of an American option in a discrete-time market in which stocks are available for dynamic trading and European options are available for static trading. We show that the super-hedging price $π$ is given by the supremum over the prices of the American option under randomized models. That is, $π=\sup_{(c_i,Q_i)_i}\sum_ic_iφ^{Q_i}$, where $c_i \in \mathbb{R}_+$ and the martingale measure $Q^i$ are chosen such that $\sum_i c_i=1$ and $\sum_i c_iQ_i$ prices the European options correctly, and $φ^{Q_i}$ is the price of the American option under the model $Q_i$. Our result generalizes the example given in ArXiv:1604.02274 that the highest model based price can be considered as a randomization over models.