I think the best thing you'll get out of coming to us for help is finding just how unfeasible the idea is. No disrespect, sounds good in theory, but...
You say "work your way across the country" implying people can get on game shows in certain parts of the country. Of the few remaining regional lottery shows one can go on, for example, potential contestants must first complete some sort of qualifying exercise based on lottery tickets. In Ohio, it involves getting a lucky scratch followed by winning a drawing - back when Michigan had one, it was winning a drawing whose entry was three losing scratch-off tickets.
Winnings on game shows typically take time to process as well, so the idea of only spending the money won on game shows would be a tough one to enact. People don't just stand in line, get on a show, and walk out in the same day. If I'm insulting your intelligence, my apologies, but that's what your idea seems to flesh out.
Even PlayMania (if you chose to count it) who give away small (usually) tax-free sums take a week for their check to be delivered and all of that.
-Jason