I friend of mine recently recommended a small independent game to me called Clairvoyance (www.gameofclairvoyance.com), that I thought might be of interest to the LW community. It's effectively a competitive turn-based strategy game, but it's distinguishing feature is that each player places 5 moves at a time, which are then interlaced and executed with the opponent's 5 moves. Thus requiring the clairvoyance of the game's name - you need to anticipate your opponent's moves in order plan your own, included anticipating their anticipations of your actions, etc.
It's clearly not intended as a rationality aid, and I originally did not approach it as such. But after getting trounced in my first few games, I found it very helpful to approach each turn probabilistically - thinking of each round in terms of a probability cloud of possible positions for various pieces, and planning around attacking swathes of that cloud rather than specific squares. Though early on, I can also foresee an element of theory-of-mind practice emerging if you play one opponent repeatedly and begin to learn their particular style.
If any one cares to give it a whirl, the game is just $5 while still in beta. It's cheap, it's fun and easy to learn, and games are relatively fast. If anyone wants to rumble, my handle in-game is also Erdrick - I have a feeling the crowd here would provided for some excellent opponents.