Back in my day, this was a sign of desperation. I know it doesn't happen a whole lot but it happens a lot more than it ever has (women asking guys out.)
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I never did understand the reason for gender roles in the first place.
Why should gender determine who asks who out or who pays for dates? It's no big deal. I don't see how it automatically makes you "desperate" either. As I stated below, most people don't understand the vocabulary they use.
I like the idea of a woman asking guys out, since most girls find it "creepy" when a guy approaches them anyway.1