Puzzle of the Week for 7 December 1998

Here are two games played on a chess board using dominoes. Each domino can cover exactly two squares on the chess board.

Game 1: Two players take turns placing a domino on the board so that it covers two squares. Dominoes may not overlap, and players may not move dominoes that have been played in previous turns. The last player able to place a domino successfully is the winner.

Game 2: This is played using the same rules as Game 1, except that the dominoes may be placed anywhere on the board, in any orientation (the dominoes do not have to line up with the squares).

Is there a winning strategy for either the first or the second player in Game 1? Is there a winning strategy for either player in Game 2?