DHV = Demonstration of Higher Value
Basically what Jurupa said, but I'd like to point out a few minor things.
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I do not respond.
I'm seeing this too much in your post.. Just relax, be laid back in what you do. It's not like responding to a girl is going to blow it. It's responding in a wrong manner.
Now I'm not suggesting to respond all the time. Just try to act like a human would, otherwise it will make you look like an ignorant dickhead. Calibrate it
And now to your questions:
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What is the difference between offering to take someone out and inviting them on a date?
The difference between offering a date and inviting to a date is massive

Heres a few examples:
A date offer = "Hey, I thought that maybe we could go out to the town blahblah.."
An invite to a date = "Hey, I'm going out to the town to have some fun. Tag along!"
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As for flexibility, does it not put me in the Beta position to act as though I'm cool with someone not showing up when they said they would? I assumed these were shit tests and I was supposed to call them on it.
My rule for a flaker is to either show that your time is valuable and you won't hang out with them if they waste it like this.
The other option is to just to forget about it, freeze-out the girl and don't even mention it the next time you might speak to her.
If she asks about it, just say something like you went over to Rita's place who lived nearby and had fun with her. Funny shit ain't it
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I decline. I'm pissed that she's made me an option and I do not want to go on a group date.
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I tell her I am pissed with her for flaking.
A guy got flaked and starts crying about it to the girls who flaked him, while another guy who gets flaked just calls another girl out to satisfy him.

Which one is you?