it on $\operatorname{Conv}(\typespaces)$, by restricting the “bidding language”, so that a type in $\operatorname{Conv}(\typespaces)$ but not in $\typespaces$ has to report a type in $\typespaces$. Such mechanisms, however, may lose the incentive guarantee which makes DSIC mechanisms attractive in the first place. Unless one can show that agents with types not in $\typespaces$ have a dominant strategy in such mechanisms, such agents need to strategize over which types in $\typespaces$ to report, depending on the types and strategies of their opponents. In the scenario where $\typespaces$ is a set of samples from a continuous distribution, the vast majority of types may not be in $\typespaces$ and have no incentive guarantee, which is clearly undesirable. In settings where the agents’ types are single dimensional, @DFK15 showed that “restricting the bidding language” does turn any mechanism DSIC on $\typespaces$ into a mechanism DSIC on any superset of $\typespaces$: each type in the superset has a dominant strategy, and by the revelation principle this gives rise to a DSIC mechanism that extends the given mechanism’s behavior on $\typespaces$. To the best of our knowledge, no such guarantees are known beyond single dimensional settings.

#### Our Results.

For agents with multi-dimensional types, we first 