  (Note. The results of this manuscript has been merged and published with
another paper of the same authors: A new approach to nonrepetitve sequences.)
  A repetition of size $h$ ($h\geqslant1$) in a given sequence is a subsequence
of consecutive terms of the form: $xx=x_1... x_hx_1... x_h$. A sequence is
nonrepetitive if it does not contain a repetition of any size. The remarkable
construction of Thue asserts that 3 different symbols are enough to build an
arbitrarily long nonrepetitive sequence. We consider game-theoretic versions of
results on nonrepetitive sequences. A nonrepetitive game is played by two
players who pick, one by one, consecutive terms of a sequence over a given set
of symbols. The first player tries to avoid repetitions, while the second
player, in contrast, wants to create them. Of course, by simple imitation, the
second player can force lots of repetitions of size 1. However, as proved by
Pegden, there is a strategy for the first player to build an arbitrarily long
sequence over 37 symbols with no repetitions of size $>1$. Our techniques allow
to reduce 37 to 6. Another game we consider is an erase-repetition game. Here,
whenever a repetition occurs, the repeated block is immediately erased and the
next player to move continues the play. We prove that there is a strategy for
the first player to build an arbitrarily long nonrepetitive sequence over 8
symbols. Our approach is inspired by a new algorithmic proof of the Lov\'asz
Local Lemma due to Moser and Tardos and previous work of Moser (his so called
entropy compression argument).
