Yes, they want men to pay for dinner, call for dates—a writer at the popular dating website The Frisky titled a recent piece “Call me and ask me out for a damn date!”—and open doors for them. A lot of men wonder: “WTF??!” Why should they do the asking? Why should they pay for dinner? After all, they are equals and in any case, the woman a guy is asking out probably has more cash in her pocket than he does; recent female graduates are making more than males in most large cities.
Sure, girls can—and do—ask guys out for dinner and pick up the check without missing a beat. Women can make that choice. Men say they have no choice. If they want a life, they have to ask women out on dates; they have to initiate conversations at bars and parties, they have to take the lead on sex. Women can take a Chinese menu approach to gender roles. They can be all “Let me pay for the movie tickets” on Friday nights, and “A single rose? That’s it?” on Valentine’s Day.
Ahem... woman here.

This is my opinion on this. Some women (especially feminists) may disagree, but I was raised in the South, and this is what I believe.
I EXPECT a man to pursue me. I am not calling, initiating any contact or asking men out (or should I say boys because they are who I encounter while in college). As a women, I should not have to pursue a man. It makes me feel un-feminine and less desirable if I am chasing a man, and it makes me feel desired if the man is taking the initiative to contact me. It is a woman's job to express interest and a man's duty to pursue. Men ought to be ashamed of themselves if they expect a woman to pursue them.

It is ingrained in the male DNA (basically) to pursue and "hunt". I would surmise that when a man sees a woman he's attracted to and he is able to "get" her, it makes him feel like THE man.

If I'm doing all the convo-initiating, I assume the guy just isn't that into me.

Here's a recent example. This guy asked me out and we met at a coffee shop and both had chai lattes.

After the date, he told me
I could call him if I wanted to hang out. He never heard from me. If this man really was interested in me and wanted to do something, he'd call, which he didn't. I'm not chasing any man. I don't believe in this "21st century woman" and the gender role reversals. I expect a man to be a man and a woman to be a woman.
I don't mind paying for dates from time to time.