A LOT goes into game design. You need to give more information as to what element you want to participate in. Do you want to do the writing (as in dialog and such), the music, the framework (such as the physics and combat engines), the art, the animation, the concept (just the creativity part that gets an idea rolling, which would require more of a surface knowledge of existing video games and a spirit of innovation than anything; you could just sell your ideas*), the graphics, the debugging, the testing, the storyline, etc?
I’ve wanted to be a game designer myself, but haven’t quite answered this question yet, although considering I’m going to major in software engineering whether or not I go into game design, I’m probably going to go with the framework. Seeing as how game design is actually one of my possible elective core classes (as it is for Computer Engineering or Computer Science as well; the three majors are pretty close to one another) I think such a major is probably the way to go for multiple facets of the industry, but again, make sure you have it straight.
*revolutionary ideas in gaming are kind of complicated to apply, however, as told in the “innovation” video in this web series, which I strongly urge you to follow either way. His unfailing insight and ideas have certainly interested me so far.