I've asked men out on first dates, and when I've done that, I've paid. To be fair, I've been asked out by men a lot more than I've asked them, but I have done it. However, whoever asks, I don't expect a first date to be expensive. I prefer something simple like a walk in the woods or a cup of coffee in a cozy local cafe.
One of the most romantic first dates a guy ever took me on was almost free - he picked me up and took me to the beach. We walked on the sand and talked for a while, then he spread a towel for me to sit on and serenaded me with the guitar tucked in his trunk. After, we went up the boardwalk for ice cream cones. He did pay for the ice-cream, but it was just a few bucks.