Discrete Mathematics, Volume 344, Issue 3, March 2021, 112256 https://doi.org/10.1016/j.disc.2020.112256 arXiv
I’ve previously written about the Namer-Claimer game. I can now prove that the length of the game is 
with optimal play from each side, matching the greedy lower bound. The upper bound makes use of randomness, but in a very controlled way. Analysing a truly random strategy still seems like it will be very difficult.
The proof brings up a surprising connection to the Ramsey theory of Hilbert cubes.