game theory models of bridge

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?

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Here is a link to a paper about double-dummy bridge but it does not involve game theory I believe: ftp.cs.nyu.edu/pub/tech-reports/tr725.ps.gz – Joseph Malkevitch Feb 8 '11 at 22:50
I read an amusing discussion of game theory applied to bidding in four-card bridge once. If I dig it up I'll post it. – Peter Taylor Feb 9 '11 at 8:15

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.

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2/3 odds comes if RHO choose 50-50 probability. Game theory comes in when RHO actually chooses to vary the probability! – Aryabhata Feb 9 '11 at 0:22
@Moron: As I said: "The game-theoretical aspect of this is..." – TonyK Feb 9 '11 at 7:48
So what if your sentence started that way? "Must decide at random what to play" has an implication that the probability is 50-50. For game theory to come into the picture, RHO has to play random alright, but the probability with which the K is chosen needs to vary. If RHO always chooses K or Q with equal probability, declarer has no other correct line, but to finesse! Of course the equilibrium point might well be the 50-50 point, but that is not clear from what you wrote. – Aryabhata Feb 9 '11 at 8:00
@Moron: Yes, I was just stating a result. I didn't claim to have proved it. – TonyK Feb 9 '11 at 8:45
I would be curious about the situations when you need to play randomly. Either in the bidding stage or the playing stage. – Johan Feb 19 '11 at 17:24

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.

• Mixing It Up, published in December 2005.

• Mixing It Up, II, published in January 2006.

Hope that helps.

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I was wondering whether there was a bridge analogue of en.wikipedia.org/wiki/Kuhn_poker. – Johan Feb 9 '11 at 15:20
@Johan: I am not aware of any, but Peter Taylor mentioned something about 4 card bridge. So perhaps there is something... – Aryabhata Feb 9 '11 at 17:06