  In the tournament game two players, called Maker and Breaker, alternately
take turns in claiming an unclaimed edge of the complete graph on n vertices
and selecting one of the two possible orientations. Before the game starts,
Breaker fixes an arbitrary tournament T_k on k vertices. Maker wins if, at the
end of the game, her digraph contains a copy of T_k; otherwise Breaker wins. In
our main result, we show that Maker has a winning strategy for k =
(2-o(1))log_2 n, improving the constant factor in previous results of Beck and
the second author. This is asymptotically tight since it is known that for k =
(2-o(1))log_2 n Breaker can prevent that the underlying graph of Maker's graph
contains a k-clique. Moreover the precise value of our lower bound differs from
the upper bound only by an additive constant of 12.
  We also discuss the question whether the random graph intuition, which
suggests that the threshold for k is asymptotically the same for the game
played by two "clever" players and the game played by two "random" players, is
supported by the tournament game: It will turn out that, while a
straightforward application of this intuition fails, a more subtle version of
it is still valid.
  Finally, we consider the orientation-game version of the tournament game,
where Maker wins the game if the final digraph -- containing also the edges
directed by Breaker -- possesses a copy of T_k. We prove that in that game
Breaker has a winning strategy for k = (4+o(1))log_2 n.
