I got offered scholarships to a few schools (Duke, Arizona State, Univ. of Arizona), but I can't say anyone was entertaining any sort of bidding...heheh. I suppose it happens, though.
One thing I hear happening more and more for those who can't quite get all the funding together is folks using a "full ride" honors scholarship at the local community college of your choice (here in my city, they pay for all your courses and give you an allowance for books) for two years and then transfer your credits to a major university. It drastically cuts the cost of attending college for those families who find themselves a bit strapped.
The way it works here is that you secure the scholarship, enroll in the CC's honors program, maintain a certain average and go to school for four semesters for free. As long as you're careful about what you take, all of your credits will transfer if they fit your major when you go to the larger university. You should make every attempt to ensure you're taking courses relevant to your expected major, of course.
These days, unless you're lucky enough to get a full ride somewhere, most of us are stick with combinations of scholarships, grants and loans. Financial aid can be a hard thing for a white male to qualify for these days, especially if your parents claimed you on their tax return the year before your application year.