# Combinatorial Game Theory and Contract Bridge

__Keywords__: Theory

It turns out that Combinatorial Game Theory can be applied to Double Dummy^{[1]} Bridge as well as go.

We count the value of a position to be net tricks for West. (To keep the diagrams simpler, we will label the two players West and East, even though this isn't standard Bridge notation.)

Consider the end position where West has AQ and East has KJ. Obviously, whoever plays first gives up a trick. So, this position has a value of 1, because East is forced to move to 2 and West is forced to move to 0.

Consider now where West has AJT and East has KQ9. Both are guaranteed one trick, but the third trick is undecided. If East leads first, he gives up a trick. If West leads the J first, East must cover, and now we're left with AT and K9, so East is on lead and gives up a trick.

It turns out that AJT-KQ9 has a Combinatorial Game Theory value of 1/2 (ignoring the sure trick each side is guaranteed)! We "add" positions by assuming there is more than one suit.

Give West S AJT H AJT and give East S KQ9 H KQ9. Again, each side is guaranteed 2 tricks and 2 tricks are undecided. If West is on lead, he leads the S J. East wins, and plays the HQ. West ducks, and now we have West: S AT H AJ and East: S K9 H K9. East has taken 2 tricks and is guaranteed one more. So, if West moves first, the value of the position is zero.

Suppose East was on lead. He would lead the SK. West covers, and plays a low spade. East wins and exits with the S9. Now, West starts hearts, but comes to 2 tricks by the analysis given above. So, when West has S AJT H AJT and East has S KQ9 H KQ9, this position has a value of 1, because whoever is on lead gives up a trick. West has a move to 0 and East has a move to 2.

Similarly, consider West: S AT98 East: KQJ7. West has 1 sure trick and East has 2 sure tricks. East can only gain a 3rd trick if he has 2 safe exits in another suit. This position has a value of -1/4 for West.

As a final example, consider West: S A and East: S K. West is guaranteed a trick, but the value is not 1. Consider West: S AQ H A and East: S KJ H K. The spade position has a value of 1, but when we add the heart suit, East is guaranteed one trick. If West is on lead, he can cash his HA but is then forced to give a spade trick to East. If East is on lead, he can exit safely with his HK.

[1] Combinatorial games have perfect information. With concealed cards, regular bridge does not.

Also, in a combinatorial game the players alternate play. In bridge the player who wins the trick leads to the next trick.

-- Bill Spight

Players do alternate play in this game, if you count "meaningful" plays, which may consist of more than one trick.

- dnerra: On a pedantic note, you should be talking about West's and North's cards, assuming that the other two players only hold irrelevant cards. (West and East are in the same camp :-) )
- I believe immediately that you can fully describe situations where only West's and North's cards are relevant, and where all their suit lenghts are the same. But how about: West QJT AK - - and North AK QJT - - ? And how about if 3 or 4 hands are relevant? That would be extremely cool if CGT could handle that!

Actually, this does seem interesting. Why does CGT have to have only 2 players? There must be some sort of extenstion to allow 3 players. (Bridge only has 2 real players, but each player does not have perfect information on his own position.) For example, double dummy "spades" would be a 3 player game.

- I don't think it can be extended to more than two players, because of the way games are defined, and the the fact that the Class of games includes the Class of surreal numbers. If you tried extending it to three players, the new system would not contain the surreal numbers and the whole thing would break down.

Side comment: Does anybody know of a Contract Bridge wikipedia? It would be nice to see one with specialized notation like the Go notation found here.

- dnerra: Not that I know. Actually I find ASCII-bridge discussion good to follow (and rec.games.bridge is pretty lively and well worth a read). But yes, of course nice hand diagrams would be nice, and maybe even a way to have full boards replayed...hmm, maybe a double dummy solver integrated...