give a condition under which any DSIC mechanism on $\typespaces$ can be extended to a DSIC mechanism on $\operatorname{Conv}(\typespaces)$, via an argument that is different from @DFK15’s yet still straightforward (Theorem \[thm:swap\]). In particular, the condition is satisfied whenever the set of feasible outcomes is *downward closed* (Theorem \[thm:downward\]).

Our main result, however, is a construction of a set $\typespaces$ of multi-dimensional types and a DSIC mechanism on it, for which we show that no DSIC mechanism on $\operatorname{Conv}(\typespaces)$ can output the same social outcomes on types in $\typespaces$. The impossibility result stands even if the extension mechanism is allowed to be randomized. This shows that, without conditions such as single-dimensional types or downward closed set of feasible outcomes, designing incentive compatible mechanisms by focusing on a discrete subset of types can be a questionable approach to designing mechanisms for the whole type space — there may not be any mechanism DSIC on the whole type space which behave the same way on the subset.

Near the end, we give a multi-dimensional setting where the expected *revenue* of a mechanism with only correct incentives for a set $\typespace$ of types can be unboundedly more than the revenue of a mechanism 