  Obvious strategyproofness (OSP) is an appealing concept as it allows to
maintain incentive compatibility even in the presence of agents that are not
fully rational, e.g., those who struggle with contingent reasoning [Li, 2015].
However, it has been shown to impose some limitations, e.g., no OSP mechanism
can return a stable matching [Ashlagi and Gonczarowski, 2015].
  We here deepen the study of the limitations of OSP mechanisms by looking at
their approximation guarantees for basic optimization problems paradigmatic of
the area, i.e., machine scheduling and facility location. We prove a number of
bounds on the approximation guarantee of OSP mechanisms, which show that OSP
can come at a significant cost. However, rather surprisingly, we prove that OSP
mechanisms can return optimal solutions when they use monitoring -- a novel
mechanism design paradigm that introduces a mild level of scrutiny on agents'
declarations [Kovacs et al., 2015].
