For a first date, it is better for the person who had done the asking to pay for most, if not all, the expenses. Some other answers you find on Musing would tell you the same.
So, stop subscribing to the traditional idea of dating. Better yet, let's not stack responsibilities solely on one side or the other.
In your subsequent outings with the same person, you may consider splitting the bill. Some may prefer to split equally, and some may volunteer to cover a specific part. In fact, some may even prefer to add in their own twists as offer.
When two persons are in a committed relationship, the issue of who is paying more should subside. I am not saying to not keep track of your own finances, but cautioning against being too overt about it.
Too often, what started out as an expectation becomes a scoreboard. You don't want to keep scores in a committed relationship because it is a recipe for disaster. You want equity in a partnership, not absolute equality.
Outside of the cost of the date, make sure it is something you both would enjoy. The amount of money you spent is insignificant when you do not enjoy each other's company, or the activity.
Once again, as your relationship progresses, who is paying should not even be an issue. The only limiting factor should be financial constraints and time once you are to that point.
Enjoying your time together should be top priority. No amount of money can mend an empty relationship.