A guy approaches a girl and they go out on a date, he should pay. Its not more complicated than that. If you want sex only approach girls who want the same.cause you see it as a fight or a war. the scenario i described if the girl is actually trying to avoid awkwardness but the guy is just aloof. had she not paid, he would have with no hesitation but it's no psychological war. you're scenario is just a war of who's gonna come out on top and 1up the other.
You are not going to get a relationship material girl who is going to accept being treated as disposable.