The Omission Stone

The stone he put in its place looked exactly like Jon's stone, except that it wasn't. It was the Omission Stone. It worked just like the normal stone, but after every wish, someone who was considered "close" to the wisher (relationship-wise) would be erased from existence. Reality would change as if they never existed in the first place. Only the wisher would notice that the person is missing. But these "omissions" weren't permanent. Once a person had been erased, someone could wish them back, but since that's another wish, another person would be erased from existence.