Kings are meant to destroy 1 enemy driller for every 3 of yours left AFTER the specialist phase, but I'm in a game now where I'm surrounded by 4 players with kings, and for some of those players the king acts after the specialist phase and for some it acts before.
Also, I'm not sure exactly how the the number of drillers to be destroyed by the king is calculated, but I have seen some pretty random numbers turn up, like the king destroying 13 when the king's side has 30.
Finally, and possibly not a bug, the player whose king acts before the specialist phase also sometimes appears to have two kings in the combat preview but then the combat happens like they only have one. Could be they're just promoting and cancelling the hypnotist though, I guess.