Two-player games usually give each player their own controller space. But sometimes you want to fight over control, like in Michael Brough's iPad game O, which looks like air hockey with many colored pucks.
Balls appear on the screen in three colours. You score points by collecting a sequence of balls of the same colour - the first is worth 0, then each after that is worth 1 more. Triangular numbers. This score system ensures that it's not simply a matter of grabbing balls indiscriminately; you get a much higher score by sticking to one colour than by taking anything you can get.
Both players share the entire touchscreen as their input space, with no restrictions on where each player may touch. As above, this invites physical interference. You will have to work out for yourselves what level of physical interaction is appropriate; there's a lot of depth on the digital side of the game that can be trampled right over and missed out on if you get too violent, but some level of physical push and shove is unavoidable when you're both trying to grab the same balls.
Does it sound confusing? Watch this video to get a feel for the subtle finger-fighting O encourages.