  We study the following game version of the generalized graph Tur\'an problem.
For two fixed graphs $F$ and $H$, two players, Constructor and Blocker,
alternately claim unclaimed edges of the complete graph $K_n$. Constructor can
only claim edges so that he never claims all edges of any copy of $F$, i.e. his
graph must remain $F$-free, while Blocker can claim unclaimed edges without
restrictions. The game ends when Constructor cannot claim further edges or when
all edges have been claimed. The score of the game is the number of copies of
$H$ with all edges claimed by Constructor. Constructor's aim is to maximize the
score, while Blocker tries to keep the score as low as possible. We denote by
$g(n,H,F)$ the score of the game when both players play optimally and
Constructor starts the game.
  In this paper, we obtain the exact value of $g(n,H,F)$ when both $F$ and $H$
are stars and when $F=P_4$, $H=P_3$. We determine the asymptotics of $g(n,H,F)$
when $F$ is a star and $H$ is a tree and when $F=P_5$, $H=K_3$, and we derive
upper and lower bounds on $g(n,P_4,P_5)$.
