Drop a gratuitous anonymous object for storing an array of games

Drop a gratuitous anonymous object for storing an array of games

Recent commits have trimmed down this object until it has a single
property, so we can drop the object and store the value of that
property directly.

This makes a bunch of code much easier to read by replacing all
occurences of "game.game" with just "game". The old code was really
confusing because it was so unclear what the top-level game container
even was.