The central question in mechanism design is how to implement a given social choice function. One of the most studied concepts is that of
implementations in which truth-telling is always the best response of the players. The Revelation Principle says that one can focus on truthful implementations without loss of generality (if there is no truthful implementation then there is no implementation at all). Green and Laffont  showed that, in the scenario in which players’ responses can be
, the revelation principle holds only in some particular cases.
When the Revelation Principle does not hold, non-truthful implementations become interesting since they might be the only way to implement a social choice function of interest. In this work we show that, although non-truthful implementations may exist, they are hard to find. Namely, it is NP-hard to decide if a given social choice function can be implemented in a non-truthful manner, or even if it can be implemented at all. This is in contrast to the fact that truthful implementability can be recognized efficiently, even when partial verification of the agents is allowed. Our results also show that there is no “simple” characterization of those social choice functions for which it is worth looking for non-truthful implementations.