\typei', \alloci(\typei,\typesmi)  - \alloci(\typei', \typesmi) \rangle   \right ]}{[ \langle \typei - \typei', \alloci(\typei,\typesmi)  - \alloci(\typei', \typesmi) \rangle  ]}{[ \langle \typei - \typei', \alloci(\typei,\typesmi)  - \alloci(\typei', \typesmi) \rangle  ]}{[ \langle \typei - \typei', \alloci(\typei,\typesmi)  - \alloci(\typei', \typesmi) \rangle  ]} }}} \geq 0. 
    \end{aligned}$$

An allocation rule is implementable only if it is weakly monotone.

In fact, @SY05 showed that, if $\typespaces$ is convex, then weak monotonicity is also a sufficient condition for DSIC implementability.

#### Revenue.

A mechanism $(\allocs, \pays)$ is ex post *individually rational* (IR) if for each agent $i$ and for every $\types \in \typespaces$, $\langle \typei, \alloc_{i}(\types) \rangle - \payi(\types) \geq 0$.

Given a distribution $D$ on $\typespaces$ and a mechanism $(\allocs, \pays)$ that is DSIC and ex post IR, the *expected revenue* of the mechanism is ${\operatorname{\mathbf E}_{\types \sim D}{{\mathchoice{ \left [ \sum_i \payi(\types)  \right ]}{[ \sum_i \payi(\types) ]}{[ \sum_i \payi(\types) ]}{[ \sum_i \payi(\types) ]} }}}$. The *optimal* revenue is the maximum expected revenue achievable among all DSIC, ex post IR mechanisms.

DSIC Convex Extensions {#sec:map}
======================

\[sec:mapping\] Before presenting our main result on the impossibility to extend DSIC allocation rules, we first complement @DFK15’s result in single-dimensional setting with a simple observation in multi-dimensional preference settings: 