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 [Kovács et al., 2015].Li [2015] has recently formalized the aforementioned idea of simple mechanisms, by introducing the concept of Obviously Strategy-Proof (OSP) mechanisms. This notion stems from the observation that the very same mechanism can be more or less truthful in practice depending on the implementation details. For example, in lab experiments, Vickrey's famous second-price mechanism results to be "less" truthful when implemented via a sealed-bid auction, and "more" truthful when run via an ascending auction. The quite technical definition of OSP formally captures how implementation details matter by looking at a mechanism as an extensive-form game; roughly speaking, OSP demands that strategyproofness holds among subtrees of the game (see below for a formal definition). An important validation for the 'obviousness' is further provided by Li [2015] via a characterization of these mechanisms in terms of agents with limited cognitive abilities (i.e., agents with limited skills in contingent reasoning). Specifically, Li shows that a strategy is obviously dominant if and only if these "limited" agents can recognize it as such. OSP is consequently a very appealing notion as in many cases rationality has been seen as the main obstacle to concrete applications of mechanism design paradigms, cf. e.g. ; such a relaxation might be a panacea in these cases.Nevertheless, for all its significant aspects, there appear to be hints that the notion of OSP mechanisms might be too restrictive. Ashlagi and Gonczarowski [2015] prove, for example, that no OSP mechanism can return a stable matching -thus implying that the Gale-Shapley matching algorithm is not OSP despite its apparent simplicity.