$i$’s value for the item, and ${\mathscr{F}}\subseteq \mathbb R_+^N$ is the all zero vector (representing not selling) and the $N$ standard bases (each representing selling to a corresponding agent). As another example, in a $m$-unit auction with unit-demand buyers, ${\mathscr{F}}\subseteq \mathbb R_+^{N}$ is the set of all integral points in $\{(\alloci[1], \ldots, \alloci[N]) \in \mathbb R_+^{mN} \mid \sum_{i = 1}^N \alloci[ij] \leq 1, j = 1, 2, \ldots, m\}$.

#### Mechanisms.

A (direct revelation) mechanism consists of an *allocation rule* $\allocs: \typespaces \to {\mathscr{F}}$ and a *payment rule* $\pays: \typespaces \to \mathbb R_+^N$. The mechanism elicits type reports from the agents, and on reported type profile $\types$, decides on an allocation $\allocs(\types) \in {\mathscr{F}}$, with each agent $i$ making a payment of $\payi(\types)$. In general, allocation rules can be randomized, in which case $\allocs(\types)$ is a randomized variable supported on ${\mathscr{F}}$. $\allocs(\cdot)$ induces allocation rule $\alloci(\cdot)$ for each agent $i$: for all $\types$, $\alloci(\types) \in \mathbb R_+^m$ is the vector consisting of the $[(i-1)m + 1]$-st to the $im$-th coordinates in $\allocs(\types)$. When $\allocs(\types)$ is a random variable, so are $\alloci(\types)$’s.

When $\allocs(\types)$ is deterministic, we write $\allocs(\types) = \mathbf{y} \in {\mathscr{F}}$ as a shorthand for ${\operatorname{\mathbf{Pr}}_{}{{\mathchoice{ \left [ \allocs(\types) 