The person who asked the other person out pays. Simple as that.
If a girl asks me out, we set a date and she picks a place and then as the bill comes around she's expecting me to foot all of it? I'd probably lose interest right there. It would come off as incredibly spoiled and arrogant. It doesn't really have anything to do with being cheap either, I don't mind paying for a dinner or two. Quite the contrary, I enjoy feeling generous and tend to insist once or twice that I pay the entire thing if I took her out. However paying for someone else when they take you out is just awkward.
Splitting is always fine and a safe default for girls. Guys might, sometimes get some grief for it. Honestly if you expect someone to pay the bill for you you're ruining the good gesture it is meant to be.
I can see girls are gonna be all 'guys pay and girls do their hair and their make-up and wear push-up bras' and whatever, but honestly I don't care about that. Show up with patted down hair, casual clothes, bare-faced and colloquial and I'll feel much more able to relax with you and connect than if you look like a model. When it comes down to money, who was there for who? You were there for each other? Oh, what a shock. You both got something out of it so you both put something into it, go halfsies for first date every time.
Since it's the first date, the one who invited should pay. You imagine a girl asks you out to this 3 star Michelin restaurant, all fancy and expensive, and then she expects you to pay for the $200 bill. Like, the fuck? You invited, you pay, or we split, I'm not paying for your shit when YOU invited.