Does anyone know if there exists any small-scale game-theoretical models of the game of contract bridge, similar to the many simplified versions of poker that have been analysed?
There are many applications of game theory to bridge. Perhaps the simplest is the Principle of Restricted Choice:
Your trump suit is AJ1098 in dummy and four small in hand. You play small to the Jack, which loses to the Queen. On regaining the lead, you play small towards dummy, and left-hand opponent plays small. Do you finesse again, or play the Ace hoping to drop the King?
This is a kind of pons asinorum for bridge players. The probability is nearly 2/3 that the finesse will succeed, but some players never grasp it. The reason is that if right-hand opponent held both honours, they might have won with the King on the first round. Conditional probability does the rest. The game-theoretical aspect of this is that right-hand opponent, holding both the King and the Queen, must decide at random which one to play on the first round, otherwise declarer can improve the odds by remembering how this particular defender played on previous occasions.
There are more complicated situations, where in principle both declarer and defender must play randomly. But in practice they are so rare that gathering statistics on a player's habits is impossible, so the best play in a particular situation is more likely to depend on your estimation of your opponents' technical skill. I once (thirty years ago) wrote a letter to an internationally popular bridge magazine, explaining how game theory could be used to resolve a supposedly intractable problem discussed in their pages. I was gently mocked for my pains.
Also, surprisingly, there are situations in the bidding that (in theory) call for randomised choices. Such situations are easiest to construct in the context of a head-to-head teams-of-four match.
If any of this interests anybody, let me know and I will try to construct some concrete examples.
If you are looking for "solving completely" the game of bridge (like some variants of poker), I would say you definitely won't find any current literature.
If you are just looking for applications of game theory, I believe Bridge World did have a couple of articles by Jeff Rubens published, which talked about game theory.
Hope that helps.