$\allocs$ is implemented by payment rule $\pays$.

#### Extensions.

Given a subset $S \subseteq \mathbb R^n$, we denote by $\operatorname{Conv}(S)$ the convex hull of $S$.

\[def:extension\] An allocation rule $\extallocs: \operatorname{Conv}(\typespaces) \to {\mathscr{F}}$ is an *extension* of an allocation rule $\allocs: \typespaces \to {\mathscr{F}}$ if for all $\types \in \typespaces$, $\extallocs(\types)$ has the same distribution as $\allocs(\types)$. Similarly, a payment rule $\extpays: \operatorname{Conv}(\typespaces) \to \mathbb R_+^N$ is an extension of payment rule $\pays: \typespaces \to \mathbb R_+^N$ if for all $\types \in \typespaces$, $\extpays(\types) = \pays(\types)$.

In Definition \[def:extension\], if $\allocs(\cdot)$ is deterministic, then $\extallocs(\cdot)$ being an extension simply means $\extallocs(\types) = \allocs(\types)$ for all $\types \in \typespaces$.

#### Downward closed settings.

The feasible allocation set ${\mathscr{F}}$ is *downward closed* if $\mathbf y \in {\mathscr{F}}$ entails $\allocs \in {\mathscr{F}}$ for all $\allocs \preceq \mathbf y$, where $\allocs \preceq \mathbf y$ denotes $\alloci[j] \leq y_j$ for $j = 1, \cdots, mN$.

#### Weak monotonicity.

A well-known necessary condition for an allocation rule to be DSIC implementable is weak monotonicity:

An allocation rule $\allocs: \typespaces \to {\mathscr{F}}$ is *weakly monotone* if for each agent $i$, any $\typei, \typei' \in \typespacei$ and $\typesmi \in \typespacesmi$, $$\begin{aligned}
        {\operatorname{\mathbf E}_{}{{\mathchoice{ \left [ \langle \typei - 