Inheritance Game

A couple delegates a person, D, to divide a cake (their inheritance) of unit length among their children, players A and B. Players are in separate rooms and each have half of the cake on the table in front of them. Each chooses (e.g., by cutting) a piece from the cake, [0, 1/2]. The rules are as follows: (1) If some of the players disagree with the rules, nobody will receive anything. The choice 0 expresses disagreement. (2) Otherwise, players receive their own piece. And, if there is some piece left from either player, D will pay each an extra 1 unit of money (as he'd like to taste the cake and convince them to agree with this rule)