Communication complexity with a referee
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.