I'm not saying it doesn't matter - when I'm dating or in a relationship there's always some give or take on who is paying. But there are determining factors, the least of which is how many dates we've been on thus far.

IMO, who pays is some balancing/weighing of:
1. who asked whom out (not determinant but highly influential)
2. what the activity is and how much it costs (you recommend dinner in a helicopter ride, I hope you got helicopter money)
3. each party's financial state (and whether they've made any hardships known to the other person)
4. special occasions (this is obvious - I just got a promotion, you're paying)
5. convenience (they only take cash and I'm the one with cash so I'm going to pay because, easier)
6. culture/belief (some men INSIST they get the bill even when you offer, to the point that offering is offensive to them)
THEN
6. how many times one person has gotten the bill vs. the other