@Patricia2151
third party bookings are prohibited on Airbnb, however, there is catch 22 - Airbnb allows profile photo and profile name to be changed any time so there is no way the host can know if the booker is indeed the person who showed up on the door.
The only way you will know who have the keys to your home is to ask your guest to show you his ID at arrival. In some countries, including Croatia, Italy, etc... this is mandatory, in others, it is not but it is allowed as long as you have it written in your house rules.
Wherever I stayed in the world, camps, hotels, motels, b&b-s...I was always asked to show my ID and I was registered. I don't see any reason and logic why Airbnb hosts shouldn't do the same.