The robustness of robust implementation
We show that a mechanism that robustly implements optimal outcomes in a one-dimensional supermodular environment continues to robustly implement e-optimal outcomes in all close-by environments. Robust implementation of e-optimal outcomes is thus robust to small perturbations of the environment. This is in contrast to ex-post implementation which is not robust in this sense as only trivial social choice functions are ex-post implementable in generic environments.