explicit description with samples from the agents type distribution [@likhodedov2004methods; @likhodedov2005approximating; @sandholm2015automated]. Our work highlights how interpolating between discrete samples can effect not only the objective but the implementablility of the mechanism itself. Luckily, the research on sample based automated mechanism design is able to avoid the pitfalls of only having discrete samples. They do this either by optimizing over parameterized families of mechanisms which are guaranteed to be implementable on the entire typespace or by working in settings where the addition of new types has no effect on the objective (i.e. downward closed settings, see )[@sandholm2015automated; @guo2010computationally; @balcan2018; @morgenstern2016learning]. However, our work points out some difficulties that might arise if one wishes to take a more general, non-parameterized approach to automated mechanism design in settings which are not downward closed.

There are a variety of well-studied settings that are not downward closed in which extending a type space to its the convex hull could cause problems. One commonly studied not downward closed setting arises from the job scheduling problem introduced in @nisan2001algorithmic and later built upon by @schedule2 and @ashlagi2012optimal. since in this problem every job must eventually be scheduled and the set of feasible solutions is not downward 