Those joysticks also have a push button action - that'd be another 18 digital inputs if used.
Some more random thoughts after reading this, some more info on the actual project would probably make things a lot clearer.
Oddly it all doesn't really add up: 18 joysticks * 2 directions = 36 movements, but there are only 25 servos and two motors.
Or if using only one direction on the joysticks, that'd leave lots of servos without controls, unless you have a few where two servos are controlled together.
Also how is the control planned to? The joysticks will always fall back to the mid position. Push joystick one way and the servo starts to slowly turn that way?
18 joysticks for one arts installation, when there are four people playing with it that's going to require some good control/design to prevent stuff running into one another and breaking or simply getting entangled or whatnot.
Also it's going to some way for the people to know easily what their joystick controls. Having to figure out what your joystick controls when there are another 8 or 10 things moving at the same time within what sounds like a quite complex installatino is not easy.