[jaclaz]

>Do these chains have some kind of written agreement with the websites where they are NOT allowed to undercut those prices, ever?

It depends on where (which country) and when.

In Europe there were contracts with that "no lower rate" provisions, which have been (in some countries) deemed illegal.

To give you some loose data points, that was 2015 for France and Germany, August 2017 for Italy, here is a map:

http://hotelmanager-blog.trivago.com/rate-parity-hotel-indus...