Men and women play in chess tournaments against each other all the time. Most tournaments are open class meaning everyone can play, or rather they are classed by rating.
The problem that chess and a lot of other sports have is that there are less women joining the sport and women are more likely to leave the sport as well. So there are far more men then women playing competitive chess. In an effort to promote top female chess players the top tournaments have separate female classes in addition to the open class. When signing up women can chose which class to compete in. But if they sign up for the open class they are going to face much stronger competitors and get much lower scores which means less exposure and less prize money.
This have helped get more women into chess as it have made several top raking female chess players famous. Something their equally strong male competitors have not been. However this system have gotten some criticism, even among these top female players. Some have been facing a dilemma when signing up to tournaments as they can do pretty well against the top competitors in the open class but will be guaranteed to win the female class.
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