"King of the Hill"

A simple random selection seems unfair to me. I think it would be better if players could choose randomly, but within a single round, a player shouldn't be able to be selected a second time. Also, the time for one game should be limited, I think to 3 days, so that there can be as many games as possible. And if you can't play faster, i.e., you can't defend the top of the mountain, then you should drop out.