that track ball is digital, it clicks when it rolls and makes a connection in that particular direction.
faster it rolls the faster it clicks. So on that ball there are essentially 5 normally open switches... One for each direction(up, down, left, right) and one for center click.
You may find that the clicks are too short to be very useful in gaming, might want to have a one shot 555 timer circuit on each direction in order to extend the time the switches are closed.
In other words the click will trigger the 555 into sending a signal to the controller that is of the right length rather then short amount of time that the click makes a connection... srry if this is confusing.