Search on your own for that hotel, try extended date searches like 7 or 10-day stays, and if you don't see any availability there isn't any availability and calling isn't going to change anything. And by search on your own I mean direct searches with Hyatt not using a third party scraper that can often be out of date. Games or not, if it shows no availability there isn't any availability that a concierge will be able to book.