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Sometimes they do, sometimes they don't, at least in Europe. If you're price sensitive, always check both before booking it.

Some examples: I've booked directly with the place to pay half price for one night in an amazing farm in Iceland; I've got the opposite last time I booked a night in a cheap hotel in Portugal. I usually check the Booking.com price and then check directly with the hotel then "Oh, but I've seen this cheaper in Booking.com", "Yes, please do the reservation via Booking.com then".

For most hotel chains if you join the loyalty programs then when you log in and book directly you get the member price which is usually $10-20/night lower.

The hotels are not allowed to offer a lower price than booking.com. To undercut, or offer bonuses, risks then getting pulled from one of their largest sales channels.

It went to court in the EU, this is not the case anymore