I just stepped into the world of game design a few weeks ago, so I'm completely confused about the whole process
That's understandable when you consider that people evolve into a game designer by experience rather than just slide into it one day. A game designer must not only know his own field, but also the fields of his team members and of course a solid background in the game subject matter. It's not something I can really explain in a few words. If you sit down with the various programmers, artists, designers, QA, management and just watch them interact with each other, then a game designer's role is to be in the middle of all that. The centre of the web that connects all the pieces together. He doesn't have to be a an expert in every field, but he should have sufficient technical understanding to collaborate. In a lot of cases, a game designer may even be required to get involved with programming or art development. Often it's not enough just to have someone dictate the result, but also help create it. Resources are often limited, so people have to take on multiple roles.
I can write out the whole plot and game theory documentation first if that would be any help
From a writers perspective, storytelling is all that matters. From a game development perspective, you have to look at everything. Having just a plot and game theory is like reading a short story. It can be good or bad, but it takes more than words to push a project onto the movie screen or into game development. Your first problem is that you need to build a team of specialists to help you make your project a reality. You can either hire them for cash, or attempt to persuade the online community for free help. The only way you're going to persuade is to push this project forward on your own until it reaches a point that you can sell it to the public. Meaning you have to take on the multiple roles and get the game to a state that shows others you mean business. In the TV and movie business, they call these pilots. In the game industry, they are often called prototypes.