If the first player starts in a corner (4/9 odds) and the second player doesn't answer in the center (7/8 odds), the first one will win; otherwise, it will be a draw.
If the first player starts in the center (1/9 odds) and the second doesn't answer in a corner (4/8 odds) the first will win; otherwise, it will be a draw.
Finally, if the first player starts in one of the four other possible squares (4/9 odds), if the second plays next to his (3/8 odds) or opposite (1/8 odds) it will be a draw; otherwise, the first player will win.
Thus, the odds of the first player winning are 4/9x7/8 + 1/9x4/8 + 4/9x4/8 = 2/3, and the odds of a draw are 1/3.
If a player goes first (1/2 odds) he will win with 2/3 odds, and tie with 1/3 odds. If he goes second (1/2 odds) he will tie with 1/3 odds, and lose with 2/3 odds. So, the odds of winning are 1/2x2/3=1/3; tying, 1/2x1/3+1/2x1/3=1/3, and losing, logically also 1/3.