Thanks, writerguyfl, for the insights into hotel revenue management. You raised some issues I hadn't considered.

I certainly don't expect Hilton's revenue management system to account for third parties. I mentioned Citi only because it affected my overall total, after accounting for the 4th night free. Without that benefit from Citi, I'd pay the extra $4, just to avoid the possibility of my key not working for the entire stay. A few years ago, in a stay that joined consecutive reservations at one of the mammoth Vegas hotels, I had to go to the front desk at 3 am because the keys were programmed to stop working after the first part of the stay. I would have gladly paid more than $4 to avoid the long slog to get new keys (walk to the elevator, take it down 45 floors, go through the casino, wait in line 15 minutes to reach the front desk, then reverse the steps). All in all, it took almost 30 minutes (and seemed longer at that time of night).