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. 

-Brett


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[-]cata30

Reminiscent of Frozen Synapse.

[-]TrE20

The same applies to Diplomacy and many other games in which one can enhance their performance by thinking probabilistically.

I've played diplomacy, what other examples were you thinking of?

Magic: the Gathering comes to mind, since the community often focuses explicitly on thinking-about-probabilities due to both the hidden information (what cards does your opponent have?) and the random factor (what cards will I draw later in the game?)

One of my favorite stories is about a high-level competitor who had a tendency to win since he could work out "I will ONLY win if I draw this one card" and would then craft his entire game around the expectation of that card - after all, if he failed to draw it, he was dead anyway :)

[-]TrE10

When I wrote the comment, I explicitly thought of poker, chess, the Settlers of Catan and Wizard, all of these games where thinking with probabilities (w.r.t. the opponents' cards / the opponents' actions and the state of the board / the opponents' actions and the roll of the dice / the opponents' cards, respectively) can be more or less helpful. In the end, I chose diplomacy because additionally, it shares the characteristic of simultaneous moves.

There are a large number of games where the probabilistic nature is more readily apparent. Bridge is one of my favorites, but poker and backgammon both come to mind.