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I'm wandering if majority, if not all, of today's gamepads have analog sticks that can be pressed/pushed down?
Well...that´s a hard question but I´d say.....yes
I would focus on the major gamepads with unctionalities and perhaps make the keys changeable in your game to avoid problems.
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I would have to agree with Beaverlicious. Most gamepads I see today have analog sticks, Even the WII had one in a fashion.