Have you ever played
Praetorian? It's another abstract with a strong focus on deduction and shared pieces. Your description of Mr. Jack really reminded me of this one.
In Praetorian, one player plays an assassin and the other plays the police, trying to capture the assassin before they kill their targets. There are two police pieces that only the police player can move, and 24 common pieces that either player can move ("the crowd"). The assassin piece is one of the common pieces, but only the assassin player knows which one, and two of the other common pieces are targets. Again, only the assassin player knows which pieces are the targets. The assassin's goal is to kill both targets before the police player interrogates the assassin piece.
On each turn, the assassin player can ether move one of the crowd pieces or have the assassin kill a target who is adjacent to the assassin piece. The police player can move a cop, move a crowd member, or have each cop interrogate an adjacent crowd member to see if they're the assassin.
It's a pretty cool game, and can be played online at
Super Duper Games.