No doubt they do somewhat, but don't kid yourself. No matter how much women want to pretend its all men chasing women and they themselves are above such things, its all a front. They very much DO want men and male attention - at least from men they want.
Women like to say "Men compete and women select". Which is true up to a point. The missing piece, though, is that women can only select from the subset of men who are interested in them.