As Scott said, in general you don't change your price or rules a lot. If you can get 5 clients with your rules, don't change them for the 6th client.

Now, there are some situations that you WANT to get that job. It's either interesting or you see more work or you don't have much work currently etc. No matter what, if you really want that job, then soften your rules while still earning money.

For example, offer lower price than Ann by offering discount. Everyone likes discount. Make sure he knows your real price. If Ann can meet him each week, offer him meeting once a month, over the weekend maybe, if you can afford it. If convenient for you, offer him work on Saturdays, not only on weekdays, etc.

My point is that yourself being better than Ann just by 1 point will not make him change his mind. But if he sees 3,4,5,... points better than Ann, then you will get the job.