White Moves First by Albert Awol

The reason for the stalemate in the light vs dark game of Lost was the refusal of a player to make the next move. In games like Chess making a move is compulsory, you are not able to skip your turn. If the rules do not dictate a time limit, you could end up waiting a very long time.

And in chess, a King may not checkmate another King alone. There are a minimum number of pieces required to prevent stalemate. Near the end of the game, killing off too many opponent pieces prevents you from winning. You've got to keep some around.

It is reasonable to assume that Jacob's death resulted from a "move" being made in the game. A move that Black made, thus making it White's turn. Since the game isn't over, Jacob obviously wasn't the King piece. Perhaps he was positioned on the field of play in such a manner as to be indispensable to the opposing player. Perhaps positioned to prevent stalemate. The only loophole then would be to travel back in time and change a move. (or take an extra turn)

Considering Black MADE A MOVE, Black must also have changed the conditions leading up to the stalemate, thus negating the need to halt play. It's almost as if so many candidate pieces are now alive as to allow him to move again without fear of stalemate. This is the ALT world. Black's extra or switched move was then Jughead.

Now it's White's turn. Perhaps White has the ability to change time as well (and Black doesn't know). This could bring our story to times even prior to 1954, blackrock or older era's. If Jughead was Black's move, maybe White can throw a big enough boulder into the river to prevent Jughead. Sink the ship in route with the military folks and time's got some serious course correcting to do. Or sink the island so they never test there to begin with.

Off the wall suspicion:
I *suspect* that the dead Jacob we've been seeing is actually not dead. It's a time traveling Jacob from before his own death. (Think time-travelers wife)

TL:DR

Black's Jughead triggered the ability for the game to recommence, putting enough pieces in play as to allow a move without stalemate. White's turn will be to prevent the US government from every trying to get to the island in the first place, by putting the island underwater and winning the game. Checkmate.