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It sounds like you're hella beta man. I'd say at this point you should next this girl, but I think it would serve you well to post some of your e-mails with the names ommitted so we can give a once through and correct you where needed.
I would say from, just reading your story, that you aren't taking enough initiative in the relationship and you're not building enough attraction. You should have demanded for her number, I would say probably the second e-mail. So you send her an e-mail to introduce and convey interest, she responds, you say something like: "Let's meet up sometime and have some fun, give me your number so we can make plans." That's kind of a bad example but you should get the jist of what I'm saying.
Anyway, I don't know anything about online game, but I would try to get her number and a first date asap so you can meet in person and actually start building attraction through real human interaction. As for should you bother? I'd say if it's been 6 weeks and she hasn't responded to many attempts at a date on your part, she's simply not interested.
Overall I think you and the previous poster are right. I give them my number on the 2nd e-mail almost always, unless they really give me nothing to work with in their first reply. Not sure about the advantage of demanding their number as opposed to giving them mine. The way I see it, giving them my number helps me weed out the ones that aren't interested enough. This girl is really the first one to really mess with me out of 30 that I met from online the past year. On match.com they text first about 1/2 the time, give me their number by e-mail in response another 1/4 of the time, and stop contact another 1/4 of the time, ratios are worse for free sites.
I probably could have taken more initiative after she texted me the first time, and built more attraction there through text game, but I definitely wasn't lax about setting up dates. She just always took forever to respond and two times plans were tentatively made and fell through because of insufficient interest. I've done alright with online dating in the past, but I think I got away from the fundamentals this past month or so. Generally, if plans fall through once they need to say something like, "But we should reschedule I'm free so and so time," before I attempt to make plans a 2nd time or else they're not going to be worth the time.