Palindromic Game

There are two players A, B playing a game. Player A has a string s with him, and player B has string t with him. Both s and t consist only of lower case English letters and are of equal length. A makes the first move, then B, then A, and so on alternatively. Before the start of the game, the players know the content of both the strings s and t. These players are building one other string w durin