On the thread page, right at the top of the first post -- look to the top right. Just below the star ratings/votes there should be a button marked "...v" (or looks a little like that). Just to the right of the [Watch]
button.
If you click it, you should get a menu marked "More options".
For normal threads, there's an option for "Edit Thread" and within that there's an option to edit the thread prefixes.
I'm guessing (I don't own a game thread) that the game tags are either there or there abouts.
If it's not there, it might be that you have to report your own thread to the moderators and ask them to edit the tags for you. But I'm guessing you won't need to do that.