Occupy Living Room.
It's a tale as old as time: Millie has a ball, and Goggles has no ball. Goggles wants Millie's ball, but Millie does not want Goggles to have her ball. This leads to a lot of loud barking, fevered stamping and earnest posturing, but the ball ultimately does not change hands and the status quo is maintained.
I think the only fair way to end this situation is for the person of the house to intervene, cut the ball with a hacksaw and give half to each dog. That way, everybody loses.