I'm of the opinion that you should complain "up" and compliment "down".

For complaints at a hotel, start with the first employee you encounter. She or he might be empowered to fix the problem. If not, you can escalate to the next level. If all else fails, you can contact the chain/corporate level.

For compliments, you usually want to have the most impact (without having to do extra work). So, contact the chain/corporate via email, website, or phone. I guarantee that the compliment will be sent down the line. It might take a few days, but the employee will be recognized.

While I don't have any knowledge of Fairmont's system, when I worked for Radisson, part of the hotel's annual review consisted of the number of complaints/compliments received through corporate. So, complimenting "down" did help acknowledge the entire hotel.