  Given an increasing graph property $\cal F$, the strong Avoider-Avoider $\cal
F$ game is played on the edge set of a complete graph. Two players, Red and
Blue, take turns in claiming previously unclaimed edges with Red going first,
and the player whose graph possesses $\cal F$ first loses the game. If the
property $\cal F$ is "containing a fixed graph $H$", we refer to the game as
the $H$ game.
  We prove that Blue has a winning strategy in two strong Avoider-Avoider
games, $P_4$ game and ${\cal CC}_{>3}$ game, where ${\cal CC}_{>3}$ is the
property of having at least one connected component on more than three
vertices.
  We also study a variant, the strong CAvoider-CAvoider games, with additional
requirement that the graph of each of the players must stay connected
throughout the game. We prove that Blue has a winning strategy in the strong
CAvoider-CAvoider games $S_3$ and $P_4$, as well as in the $Cycle$ game, where
the players aim at avoiding all cycles.
