Given $n$. Two players in turn mark points on the plane. No three may be collinear, no $n$ may form a convex $n$gon. The player who does not have legal move loses. Who has a winning strategy?

1$\begingroup$ For $n=3,4$ the second player has a winning strategy. Have you checked any bigger $n$ by hand? $\endgroup$– Joonas IlmavirtaMar 9, 2015 at 1:26

2$\begingroup$ The game is parametrized by $n$, presumably, as in the theorem. $\endgroup$– Noah SchweberMar 9, 2015 at 1:56

2$\begingroup$ Unless there is a generic argument like strategy stealing (which doesn't work in this case), this is likely to be a difficult question. It might be more fruitful to consider how long the game lasts. The ErdősSzekeres theorem gives some bound but it might be possible to improve it. $\endgroup$– David ConlonMar 9, 2015 at 5:22
1 Answer
The problem was raised and discussed in Parikshit Kolipaka and Sathish Govindarajan, Two player game variant of the ErdősSzekeres problem, Discrete Math. Theor. Comput. Sci. 15 (2013), no. 3, 73–100, MR3141828. It says the second player wins for $n=5$.