chrisseaton specifically mentioned "every major hotel chain", which is true. See my other comment listing all the websites for all the big (American) brands showing they promise the cheapest prices on their own websites.
Edit: I'm going to have to retract my comments about cheapest price being on official hotel brand websites. I did a cursory search and find many cheaper options on Hotel Tonight (and expedia) for same day reservations than the brands, and while I haven't looked at all the brand's best price guarantees, at least IHG's exempts them from having to provide the lowest price within 24 hours of checkin, so it seems like a loophole to let them dump rooms for cheaper on third party websites day of arrival.