"You vote with your choice, one through six, or ten, or however many people is (sic) on the ballot paper. If the person you vote one for is eliminated then your number two vote becomes a number one vote for the second person that you chose...and thus your votes may be distributed until the fourth or fifth or sixth count until somebody is finally elected"

We use this system in Australia, where we call it preferential voting, and I reckon it's an excellent system. It rarely causes the election of a minority candidate, but it makes the major parties pay attention to the minority interests in order to win their preferences. This, of course, can be good or bad, depending on your opinion of the minority concerned.