The problem is that I need to be able to quickly fetch which games a user is participating in, as well as which users are participating in any individual game.
Then what you really want is a simple two- or three-column table for the associations.
| user_game_id (primary key) | user_id (key) | game_id (key) |
Unless you have hundreds of thousands of entries in this table, queries against it will be very fast. There is probably no reason to cache this info in the users table... such systems can become hard to work with.
You have a second table for the actual game data (game title, etc.)