You wouldn’t patent the “game” you’d patent the various forms of utility or designs within that game. So throwing a sphere at a life form to then capture it could be one patent, but maybe then you’d also file another patent to cover keeping it alive and caring for it inside the ball habitat. You might file the second off of what is called a continuation filling and in combination, as you need both actions to get the full effect, you might get a bit of extended coverage in practice.

But the bigger thing here would probably be trademark law, which is a whole different beast.