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I was checking on this paper where they define some rule-mechanism that is called cutoff rule-mechanism. I do not really understand why is this useful and how is this defined. Any help would be appreciated.

$\textbf{Definition:}$ (Cutoff rule and cutoff mechanism). A mechanism $(x,\pi)$ is a cutoff rule if $x$ is non-decreasing, and the signal $\pi$ can be represented as

$$\tag{1} \pi(s|\theta) x(\theta) = \sum_{c\leq \theta}\gamma(s|c)dx(c)$$

for each $\theta\in \Theta$ and $s\in S$ for some signal function $\gamma: C\to \Delta(s)$. A mechanism $(x,\pi, t)$ is a cutoff mechanism if $(x,\pi)$ is a cutoff rule. (by $t$ refers to transfers, $x$ a non-decreasing allocation rule in $\theta\in \Theta$ where $\theta$ is a value for an object that belongs to a finite sub-set of non-negative real numbers, that is $\Theta$.

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