I can understand Business, English, and Graphic design, but why put mathematics at the top of the list? Roleplaying games definitely seem more of a literary occupation, requiring good storytelling skills. Of course, the ability to lay out a product to make it have good appeal is important, as is the ability to run a company that produces these things, but unless you're thinking just about general game-theory, I don't see how mathematics really fits in.
And no, actually I wouldn't want to restrict it to people of particular majors. People change their majors all the time, and who knows how a brilliant gamer who is majoring in Latin or Psychology might find a way to bring his or her knowledge into the game world. Heck, I took a Sociology class this year because I thought it would help me understand group roleplaying dynamics. Practically any field of study could be applied directly to RPGs, except maybe the hard sciences. I know that biology, physics, and chemistry might inspire some interesting and accurate rules for RPGs, but you can write rules for RPG-science with just a layman's understanding. I don't think you really need to know the exact specifics of DNA gene recombination in order to write a monster manual.
Still, though, gamers come from all walks of life. Why restrict who can apply. I like your idea about requiring a writing sample, but only for some people. We could have a few different categories--fiction, illustration, rules mechanics, adventure-writing, and possibly graphic design. Then maybe a general 'other' category, for people who want to do stuff like write music that can be played during game sessions, or people who believe they could run a gaming business well. There'd have to be a pretty flexible way to enter the competition, but I'm sure we can think of something.