Yay! I have an answer for this one, unlike the first puzzle..
The player who goes first will win if they follow this strategy:
First turn: Player one should put their penny in the
middle of the round table.
Second turn:Player two can then go anywhere.
Third turn:Player one should then go directly
opposite of where player two has gone. This way, as long as player two can
still put a penny down, player one can always put a penny down.