My feedback is mostly generic in the context of career hunting in the scope of Computer Science. I've never worked a job outside of this umbrella of an industry so I can't rightly speak to much of anything else.
Personally (I may catch some heat about this) I believe there is a huge discrepancy from "going to college", and "Going to College to earn a degree" as opposed to "Going to College to earn a degree, and get a job in my Major".
Do what you think is right. A degree is a good thing, but viewing college as a meal ticket to any career is a dangerous mentality, and it is healthy to be weary of joining the college bubble! College imho should be treated as a place to make connections, and to undergo self enlightenment in some skill or trade. Which indirectly may land you a job, but nothing is for certain.
A brief digression in the context of choosing your major, the last thing you want to do is type-cast your credentials in a market segment that some employers, honest to god, don't even believe constitutes a separate career path.
I can't speak to much about Game Programming as a profession, but I can't imagine it being that much different than Software Development in general. Obviously you have to know your stuff when it comes to whatever technology, APIs, and languages the development house is using (Or demonstrate an ability to pick them up quickly). Which leads to my point.
I think a degree is good for showing you can stick things through; It can be beneficial as well if you feel you learn best in a more structured environment, and also for passing any bots the HR department may be using to filter out any resume that doesn't have a key word of "Bachelors" in it. But, conversely I believe a decent portfolio showing your prowess in the technologies the development house is using, and critical thinking skills, and a natural ability to speak well and give thought out articulated responses during an interview (Naturally, being able to pass the technical portion helps as well ) speaks larger volumes than a degree.
To play devils advocate to my earlier assertion of the evil HR bots; you will always have certain postings stating, "5 Years Experience, Bachelors Degree Required". But, having been a interviewer before, I find semantics to be a harsh lady indeed, as most should say "5 Years Experience, Bachelors Degree Preferred" as the exception to the rule always seems to occur more often than most people think.