I have a friend who till today would always say any guy taking her out on a date has to be the one to pay the bills. That is her opinion and I am so sure some girls are of that school of thought, well, I am not a saint either. I remember going on a date sometime ago with this guy for lunch and he was going to pay the bills when we were done but I offered to pay part instead. Not because he didn't have enough but because I wanted to show that I could pay bill...that mindset lol...but he didn't allow it.
Well, that was back in the day. However, I am of the opinion that it should not be a tradition for any lady or should I say it would be wrong of you to always expect the guy to pay the bill each time you go on a date. Now, not because he doesn't have the money to pay but because it shows that you actually care....that's a nice gesture from you and it shows that you can also give. You should offer to pay or split the bill.
Personally, I like to pay the bill sometimes...it may mean nothing, but it makes me feel involved. Having the thought of "the guy must pay the bills" is an old way of thinking. It is an amazing thing to show that you are not only a receiver but you are a giver.
However, if a guy ask a lady out on a date, especially the first date...the guy should be ready to pay the bill. Since you invited her to dinner, the responsibility to take care of her falls on you. Except the lady insist on being a part of it which I would strongly advice you reject the offer, especially on the first date. I know of some ladies who just throws out a bait of offering to pay to see your reaction. I have even heard of those who declined going on another date just because you didn't insist but gave in to the offer...there you begin to wonder what exactly women want.
Sometimes, it may not be a matter of who should pay.... it could be a matter of do you want to see him again. Although, it could be a deal breaker for some people and it may not even mean much to another. But whichever it may be, the intentions and a nice gesture from both sides matters more.