If one of the PD players has a third option of "get two bucks guaranteed and screw everyone else" - if the game structure doesn't allow other players to punish him - then no algorithm at all can punish him. Or did you mean something else?

Yep, I know what the core is, and it does seem relevant. But seeing as my solution is definitely wrong for stability reasons, I'm currently trying to think of any stable solution (continuous under small changes in game payoffs), and failing so far. Will think about the core later.