Communication complexity with a referee Assume a framework in communication complexity where we have two players A(lice) and B(ob) and a R(eferee). A and B don't communicate directly with each other. In each round of communication, each of them sends a message ($m_A$, $m_B$) to the R. R computes two functions $f_A(m_A,m_B)$ and $f_B(m_A,m_B)$ and sends the results to them. The functions are fixed. The idea is that the communication between the players is restricted. Moreover the referee might do some processing on the messages.

Example:

A and B send two (arbitrary large) numbers to R, R checks which of them is greater and informs the players. 

In this framework, we can design a simple protocol that easily computes the following function using a single round. A and B send $x$ and $y$ to R, R returns the answer to them, and they output the answer.

$$f(x,y)= \begin{cases}0 &amp; x\leq y\\ 1 &amp; ow \end{cases}$$

Obviously this is not an interesting case, since the function we are computing is the same as the referee functions. A more interesting case is when we have a fixed linear inequality $\vec{a} \cdot \vec{x} \leq \vec{b} \cdot \vec{y}$ and the values for the variables are partitioned between players (A has $\vec{x}$ and B has $\vec{y}$). The task is to decide if the inequality is correct. The protocol in this case is that players compute their part and then send them to the referee. 

Question:

Has this kind of communication complexity been studied? If yes where can I find more about this?



note 1: on page 49 Kushilevitz and Nisan mention a framework which involves a referee but seems very different from what I am asking. 

note 2: I am not sure if calling R a referee is the right thing, please comment if you have a better suggestion.
I'm sure you know the following paper, but I put a link to it because other readers may be interested: Interpolation by Games

This paper is an attempt to use the communication complexity framework to show lower bounds for cutting planes.  The protocol is used to produce an interpolant circuit for unsatisfiable CNF: 
$$
A(x,y)\lor B(x,z).
$$

Player $A$ gets input $a$ and $y^a$, player $B$ gets $b$ and $z^b$.  If there is a shallow tree-like proof in cutting planes then the two players have a communication protocol such that 


any communication is mediated by the referee, which helps in evaluating the inequalities in the proof;
the amount of communication is small (the tree is shallow);
the two players either decide which of $A$ or $B$ is falsified;
they find a position $i$ in which $a_i \not= b_i$.


The referee is turned into a probabilistic protocol for inequalities.  In this way it is possible to turn lower bound for tree-like probabilistic protocols in the communication complexity framework into lower bound for tree-like cutting planes proofs.  

If we had lower bound for communication protocol of the form of a PLS, then we would get lower bound for dag-like cutting planes proofs.

Notice that this technique does not depend on the actual inference rules of cutting planes. If we assume the inference rules to be (1) positive combination (2) integer division with floor we can build the monotone interpolant circuit using Pavel Pudlák argument.