# Comparative communication complexity?

I was reading the book "Communication Complexity" by Kuschilevitz and Nisan and in Exercise 1.18 they introduce a variant of the normal vanilla 2-person deterministic communication complexity protocol which they call a comparative protocol.

I'll deal just with functions of the form $$f: \{0,1\}^ \times \{0,1\}^n \to \{0,1\}$$. The definition of a comparison protocol is identical to a classic definition of a communication protocol, except the vertex functions $$\theta_v$$ are restricted to a very special form. Namely, for each vertex $$v$$ in a binary tree describing a protocol $$\mathcal{P}$$, the vertex function $$\theta_v$$ must be 0 for all inputs less than a given $$z \in \{0,1\}^n$$ and 1 for all inputs greater than or equal to $$z$$ (where here I am using the usual lexicographical ordering on $$\{0,1\}^n$$. We can then of course define the communication complexity of a function $$f$$ with respect to these comparison protocols and in fact the aforementioned exercise gives a place to start.

Is there any literature on this communication complexity model? I tried some googling but I didn't have any luck.