I think the host should initiate, plan, arrange, and entirely pay for dates, and then swap roles for the next date. CMV.

In the early stages of a romantic relationship, dating works best on a host-guest model, the host being the person who asks the guest on a date. The host should be responsible for the work of planning and arranging the date, perhaps consulting the guest as appropriate. The host should take care of transportation. And the host should pay for the entire thing: a date is a gift that one person gives another, and offering to pay half for a gift is rude.

For the next date, the former guest should become the host and ask out the other.