Communication between Cooperating Companies..Got an equation?

There are two teams working on a single project.
Each team has a distinct and separate part in the project,
both are required for completion.

The Ralph Shaw Law of Cooperating Companies (which I just made up and named for the guy from whom I learned to live it) says that in order to have the least opportunity for miscommunication, misdirection, or misunderstanding, one and only one person from team A represents official conversation to one and only one person from team B.

I propose that for each added person to either team the opportunity for mistake or misunderstanding expands by a pretty big number. But I don’t know how to figure out what that number is. If you can, I’d love to see it.

Here’s the back story:
Ralph Shaw owns Shaw & Sons Amusements. I was the volunteer chair of the Baltimore City Fair, a giant three day celebration of life in the city of Baltimore. Ralph’s organization was responsible for the rides and midway set up to compliment the neighborhood and other exhibits that were the showcase of the event.

The Fair needed the Midway to entertain the huge number of people for longer than the time to walk thru the exhibits. The Midway need the Fair as a reason to block streets and set up in downtown Baltimore.

Each organziaiton had its own team of workers who knew their own jobs. But as you can imagine, it’s important that not just anyone from the Fair’s team could go to any one on Ralph’s team, for example, on the day before the event and say.. “OH, gee, we don’t like where you set the merry go round. Could you move it over there?”

So there exists one person from the Fair — the chair — who talks to one person at Shaw & Sons — Ralph — about any major decisions affecting the the event.

This is not to imply that other people in the organizations can’t talk together or work together. I think the referenced conversations are sort of contract level discusssions. We will do this; we won’t do that.

My hypothesis:
In any conversation where instructions are given there are three options for out come.. Do it, dont do it, or do something else.

So if there are two people in the matrix, then there are six options (each person could choose any of the answers) ? Or is it only three?.. because it’s only one instruction?
And if you add a person to one side, how does that number increase?
And what if there are two people on each side? Then what?
What about more?

Do you get the math? Could you send me an equation? I’d really like to know.

PS.. Don’t tell my old math professor, Marv Brubaker, that I can’t do this on my own.