Starting with the empty graph on $n$ vertices, two players alternately
add edges until the graph contains a $p$-path.  The last player to
move wins.  Assuming both players play optimally, the winner will depend
only on $p$ and $n$.  We analyse the game for $p\leq6$ and arbitrary
$n$, determining the winner and providing a winning strategy. 
The results illustrate how deceptive computing the results of small 
cases can be.