I totally feel for you! I've studied Geology in college and gotten a bachelor's degree, but now I have enrolled into an MBA program hoping to learn more about marketing.
My dream is to become a video game marketer and completely change my industry. I think an MBA offers an adequate academical background for most business-related jobs in the video game industry. Most companies I've looked on the internet don't focus that much on your education (mostly asking for just a BA), but your experience and knowledge of the tools of the "trade".
So my advice for you is to get on with that MBA. Follow logistics, marketing, whatever you like and just get yourself out there to gain some experience. Eventually your CV will become rich enough to cover your industry switch and get to work for a gaming company.
Wishing you the best of luck!