I would normally expect the application itself to appear in the folder named "Applications" at the root of my system drive, not in the "Application Support" folder (which would contain files the support the application).
Not all apps use Application Support files. Some do, some don't. It sounds as if your game doesn't. If the game is in your root Applications folder and is working, I wouldn't worry about finding it in ~/Library/Application Support.
There is also a system-wide applications support folder in the Library folder at the root level of your hard disk (as opposed to the Library folder in your home directory). Maybe the file you seek is there.