  We analyze the duration of the unbiased Avoider-Enforcer game for three basic
positional games. All the games are played on the edges of the complete graph
on $n$ vertices, and Avoider's goal is to keep his graph outerplanar,
diamond-free and $k$-degenerate, respectively. It is clear that all three games
are Enforcer's wins, and our main interest lies in determining the largest
number of moves Avoider can play before losing.
  Extremal graph theory offers a general upper bound for the number of
Avoider's moves. As it turns out, for all three games we manage to obtain a
lower bound that is just an additive constant away from that upper bound. In
particular, we exhibit a strategy for Avoider to keep his graph outerplanar for
at least $2n-8$ moves, being just 6 short of the maximum possible. A
diamond-free graph can have at most $d(n)=\lceil\frac{3n-5}{2}\rceil$ edges,
and we prove that Avoider can play for at least $d(n)-3$ moves. Finally, if $k$
is small compared to $n$, we show that Avoider can keep his graph
$k$-degenerate for as many as $e(n)$ moves, where $e(n)$ is the maximum number
of edges a $k$-degenerate graph can have.
