HI Nikola
If you have done nothing wrong, then you have nothing to worry about. IF the guests stays one night, and decides to leave tomorrow, then you need to have HIM cancel his reservation for the remaining nights by agreeing to your cancellation policy which is clearly stated on your profile or you can also go into the actual booking and ALTER the reservation to end early. You are the host, its entirely up to you what you want to do.
My suggestion is to call Airbnb as this is a situation that they need to be made aware of. They will help guide you thru this process to ensure you get paid properly and to possbly ask them to help the guest find alternate accommodations. But please remind the guests, that its HIS reposponsiblity to also contact Airbnb customer service to inform them that he wants to leave after spending one night. This is your business, run your business the way you want, and if you choose to refund him, you can, but depending on YOUR cancellation policy he may or may not be entitled to anything back. Its entirely up to you.
I want to recommend that you go through your profile setion by section and make sure you understand what you have chosen for your settings on cancellation policy as you are the one who set it. Look at everything....quite a few hosts have not properly gone though their entire profiles to make sure all settings are what they want. I know this because I tried to book a place myself with another host, and even that host didnt' know they had chosen the arrival time as FLEXIBLE when I informed them that I was arriving at Noon, only to be told that check in was 3p.m. but that's not what their profile said. Just some helpful advice. Good Luck.....
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Aloha, Momi
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