The institutionalized office of a Game Master

This question has come up in my mind in connection with pbem Diplomacy. I wondered what the office of the Game Master would be called if you consider it an institution. You know, like a president’s office is the Presidency, or a monarch’s office the Crown. Like in the sentence: “XY is currently holding the Presidency.” What is a GM holding?
Similarly, if the Game Master were a board of several members, what would it be called? “Board of Game Masters, headed by the Game Master-General” is a bit bulky. I’ve thought about Game Mastery, but this seems to apply to skilled play, and not to the institution.

What would you say? And feel free to comment on the uselessness of this question and any reply it might generate, if you wish to do so. I agree completely.