I am surprised no one has ever conducted exit polls at the gates after halftime. If you want to uncover the problem, why not address the issues head on and see if there are any patterns?
Talking with some in the athletic department, they don't view students leaving early as a real problem. The goal is to get them to the game, and if it is a close/interesting game they'll generally stay. Doesn't seem to be a priority as much as getting students there in the first place (with giveaways, food, promotions, etc). Don't know if this is the right or wrong approach (I tend to agree - how long someone stays is their choice), but interesting to see what the goal is.