Perfect Bidding

by Don Kersey

[Editor's note: Matt Franklin points out that the problem addressed here and some variants are considered in E. Berlekamp, "Cooperative Bridge Bidding," IEEE Transactions on Information Theory, November 1976, pp. 753-756.]

Many bridge players have seen the very large number of possible bridge deals:

53,644,737,765,488,792,839,237,440,000.

Less familiar is the much larger number of possible bridge auctions:

128,745,650,347,030,683,120,231,926,111,609,371,363,122,697,557.

Years ago, a letter to The Bridge World raised the question of whether a cooperative four-player bidding system could be devised that would allow the complete determination of all four hands. In this note, we construct such a system.

It will be terminologically useful to recall that a bid is any call except pass, double, or redouble.

Arrange the 52 cards of the deck in some order (for the purposes of this note, we will list the suits in descending order, so that the first card is the spade ace, the second the spade king, . . . , the fourteenth the heart ace, . . . , the fifty-second the club deuce). Now, with the bid of 1 club, we associate the first card in our list, the ace of spades. With each subsequent step in the bidding (up to 6 clubs), we associate two cards; so the 1 diamond bid is associated with the king and queen of spades, 1 heart with the jack and ten of spades, and so on. In this way, the bid of 6 clubs is associated with the 4 and 3 of clubs; we don't associate the 2 of clubs with any bid.

The opening bid in our system will always be made by the owner of the ace of spades, and the bid chosen will indicate information about subsequent consecutive cards in our ordering that player holds as well; whichever bid opener makes, the message is "I have all cards associated with this bid and all lower bids, but I don't have both cards associated with the next bidding step." For example, a player holding AKQ107 of spades will open the bidding with 1 diamond, indicating the A, the K and Q, but not both the J and 10.

All calls between bids (passes, doubles, and redoubles) are used to determine the location of the two cards associated with the next available bidding step (let's call that next bid "A"). Once these two cards are located, the bid that is actually chosen (call it "B"; B may be the same bid as A, or it may be higher) resembles the opening bid, in that it says "Now we all know where the 2 cards associated with level A are located; furthermore, I have all the cards associated with any bids higher than A up to B, but I don't have both cards associated with the next bid after B." For example, if South has A9 of spades and West has KQJ1085 of spades, then South will open the bidding with 1 club (showing the A, denying the KQ combination) and West will bid 1 heart (showing the KQ combination, and the J10 as well, but denying the 98 combination).

It remains to see how to use the calls between bids to locate the two cards associated with the next bidding step--of these two cards, call the one that is first in our list F and the one that is second S. Suppose that South, for example, has just made a bid, announcing the location of certain cards, but denying holding both F and S. Then the pattern of the subsequent auction is:

West will bid with both key cards, double with one, or pass with neither.

After West's double, North will bid with F (then West has S), redouble with S (then West has F, and East can bid, as both cards are located), or pass with neither. Over North's pass, East will bid with F (then West has S), or pass otherwise. Over East's pass, South will bid with S (then West has F), or redouble otherwise. Over South's redouble, West will bid with F (then East has S), or pass with S (then South has F, and North can bid, as both cards are located).

After West's pass (showing neither F nor S), North will bid with both, or pass otherwise. Over North's pass, East will bid with both, or double otherwise. Over East's double, South will redouble with F (then whoever has S will bid), or pass otherwise. Over South's pass, West will always pass, and North will bid with S (then East has F), or redouble otherwise. Over North's redouble, East will bid with F (then South has S) or pass otherwise. Over East's pass, South will bid with S (then North has F), or pass otherwise. Finally, over South's pass, West can bid, since it is now known that North has F and East has S.

ESOTERICA

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