A Penny Game

Two players take turns placing pennies on a rectangular table, one penny
at a time.  The pennies must be placed flat, not overlapping any other
penny, and no penny which has already been placed may be moved.  The
game continues until no more pennies can be placed on the table without
falling off.  The last player to be able to place a (non-falling-off)
penny wins.

The table is empty of pennies at the beginning of the game.  You have
the choice to be the first or the second player to place a penny.  Which
do you choose, and what strategy do you pursue to win the game, or is
there no winning strategy?  Does the game change if the table is round?