score:21
Put simply, it doesn't. Satan will not be the punisher in hell, but among the punished. Satan is not any any sense the king of hell
And when the thousand years are ended, Satan will be released from his prison and will come out to deceive the nations that are at the four corners of the earth, Gog and Magog, to gather them for battle; their number is like the sand of the sea. And they marched up over the broad plain of the earth and surrounded the camp of the saints and the beloved city, but fire came down from heaven[b] and consumed them, and the devil who had deceived them was thrown into the lake of fire and sulfur where the beast and the false prophet were, and they will be tormented day and night forever and ever.
So, the idea is a misconception.
Upvote:1
To be "punished by Satan" would imply that Satan has some sort of authority over the inhabitants of Hell.
It would seem to me that Mathew 23:12 (KJV), which states:
would equally apply to the inhabitants of Hell as it would those of Heaven.
Therefore, since Satan's greatest sin was exalting himself, he would actually be the LOWEST inhabitant of Hell.
Upvote:4
In reference to your comment on Narnian's answer, my guess is that since Satan is referred to as the "prince of this world" (John 12:31, 14:30), it's been misconstrued over the years to mean "ruler" &/or "king". Along the same lines, since we understand he'll be in Hell for eternity, people have associated the "prince of this world" and "will be in hell" together and, voila, you have a "King."
Wikipedia states (emphasis added):
God therefore grants [Satan] the chance to test Job. Due to this, it has been interpreted that [Satan] is under God's control and cannot act without God's permission. This is further shown in the epilogue of Job in which God is speaking to Job, [Satan] is absent from these dialogues. "For Job, for [Job's] friends, and for the narrator, it is ultimately [God] himself who is responsible for Job's suffering; as [God] says to the 'satan', 'You have incited me against him, to destroy him for no reason.'".
I'd say from observation that a King usually does not need permission to do such things.
This is mostly an educated guess, though :)