Hmm... not as simple, this 'un.
The Thing consumes dead people and imitates them. Eric the Red, as a Viking, is officially a Bit of a Dumb Twat and will probably charge straight in and get eaten by The Thing. This leaves The Thing lying around for 15minutes while it transforms enough to be usable again. Either way, The Thing has got the munchies and will ignore the ring, which will then be picked up by Martin Luther King, turning him invisible and most likely scaring the shite out of him. However, MLK was obviously a pretty brave guy to stand up for black people's rights, so lets say he doesn't run away but uses the invisibility to his advantage.
For now, then, we're left with MLK and the Ring versus Judge Dredd and the Evil Dead. Invisibility counts for shit against either of them, as Judge Dredd probably has thermoptics on his helmet, and the Evil Dead is double-hard. Either way, MLK will bite the dust despite The One Ring. Whether Dredd picks up the ring or not isn't too important, but he probably wouldn't. Anyway, The Thing is still trying to transform into Eric The Red so is in a pretty vulnerable state and can be dealt with shortly.
Judge Dredd and the Evil Dead can now celebrate their victory, but there is more yet to come: the use of the One Ring has called to the Nazgul, who arrive on their big flappy Wyvern things. Judge Dredd is only one man, but has a big gun, though it's probably not enough against the undead incarnations of evil kings of yore. The Evil Dead can't possess something just as evil as itself really, so it comes down to a straight fight between the Nazgul, newly arrived on the scene, versus the Evil Dead. All the Nazgul have to do is hang in till sunrise when they can just take the Ring, but we can assume that once they have it they will leave for Mordor, leaving the battlefield empty (since the Evil Dead hides away in day) and the final result of: a draw.
