I feel stupid missing such obivious case. Actually I was thinking of strategies (where players will strategize optimally or atleast don't leave any box unopened) that might backfire instead or more precisely if the system can be modified to make some or all seemingly optimal strategies backfire...
There are 4 players numbered 1 to 4. There is a room with an entrance door to one side and exit door to opposite side. Inside the room, there are 4 boxes numbered 1 to 4. Inside each box, there is a chit containing a number (with equal probability) from 1 to 4 (inclusive). No two chits can have...