Given $n$. Two players in turn write different real numbers $x\_1,x\_2,x\_3,\dots$ 
The player after whose turn there is a monotone subsequence of length $n$ loses.
I guess that the question 'who wins' may be hopeless (nice if not so), but possibly there exists a better estimate of the number of turns than Erdős–Szekeres bound $(n-1)^2+1$ (which works not only for optimal, but for all strategies)?



Yet another Erdős–Szekeres game
As noted in the comments (but with not quite the right reference) the game is a first player win for $n \geq 4$. The question here is about the misere form, so this is a combination of Proposition 7, Theorem 10 and a bit of Proposition 9 in the paper [Monotonic Sequence Games](http://arxiv.org/pdf/math/0602630.pdf) by Albert (who he?), Aldred, Atkinson, Handley, Holton, McCaughan and Sagan. This also appeared in [Games of no chance 3](http://library.msri.org/books/Book56/). We did not address the question of the length of the game when one player is playing using a winning strategy, and the other to lose as slowly as possible (obviously, if the players cooperate to lengthen the game then it reaches the Erdos-Szekeres bound). The proof uses a "strategy stealing" argument, but one which is not entirely clear cut. In common with most such arguments it does not actually yield a strategy for the first player to win, only a proof that such a strategy exists.


