[vikarti]

Chess is special case. Women are allowed to play in men torunaments. Not reverse. There are even special rules (since 1976) which specifically says that women cannot be prevented from participating based on gender (it wasn't clear to Indians at this time). Some (Judit Polgár,Vera Menchik) almost never played in woman ones. Chess rating system didn't take take gender into account but in 1986 active woman chess players excpt Susan Polgár got small bonus to rating.

There is research from 2007 ( https://onlinelibrary.wiley.com/doi/epdf/10.1002/ejsp.440 ) which shows that if is via Internet and women thinks their opponent are men - they show worse results. If gender is not known to both sides - results are mostly same.

It's not currently clear if M2F transgenders should be allowed to play in women competitions.