Name of game in which remaing win

American Idol is one example.

In general, a game in which those remaining win is an elimination game. Where only one winner remains after all contestants are eliminated, it's called winner take(s) all.

Examples are: beggar-my-neighbor, where the object is to win everybody else's cards; musical chairs, pass-the-parcel, Russian roulette, tennis, Monopoly, and so on.