A "Complete" Bridge Bidding System

People sometimes point out that the number of possible bidding sequences in contract bridge exceeds the number of possible deals. Someone has used this fact to create a bidding system which, if used by all four players, allows anyone hearing the bidding sequence to construct the entire deal from it. I have seen the notes for such a system, though I cannot now find them: it always ended with a contract of 7HXX.

In fact, the number of possible bidding sequences does not merely exceed the number of ways that a hand of thirteen cards can be dealt validly. It also exceeds the number of possible misdeals. (I define a misdeal as a way of allocating each card of a valid bridge pack to one of the four players. I exclude cases in which the pack is defective, and those in which one or more cards end up held by none of the players.) The purpose of this page is to present a bidding system which, if followed by all four players, even when there has been a misdeal, allows anyone hearing the bidding sequence to construct the entire deal or misdeal from it.

How it works

The system assigns an order to the cards, and finds the cards in this order. Any order will do, but we shall use the order C2, D2, H2, S2, C3, ... SA.

Initially, a player bids 1C if he holds the C2, otherwise passes. Once the C2 has been placed, the bidding proceeds in "steps". Each step takes the bidding up two levels (e.g. from 2C to 2H, or from 4NT to 5D), and places the next three cards. Thus there are 17 steps, which are just sufficient to place the remaining 51 cards.

A step starts with the player after the final bid of the previous step, or after the 1C bid; and ends with the second bid of the step. For the duration of the step, the first player is labelled "1", the second player "2", etc. They bid the step according to the diagram below.

In diagram, black numbers like indicate the player who is next to act. Red numbers like indicate the player who is known to hold some card. Thus means that player 1 holds the first card (first to be placed in this step) and player 4 holds the second. The lines connecting the circles are bids, passes, doubles, and redoubles, as shown in the key at the bottom left of the diagram.

The circles indicate the player who is next to call: means that player 3 is to call. Terminal nodes show, in red, where the three cards placed in the step are: places the first card with player 4, the second with player 2, and the third with player 3.

Some terminal nodes are tinted pink like this: . These indicate that only two cards have been placed, but there is a level of bidding left to locate the third one. This is then easily done:
The next player bids if he holds the card; else he passes and
the next player bids if he holds the card; else he passes and
the next player bids if he holds the card; else he doubles and
the next player, who must hold the card, bids.

Comments

The bidding diagram has (I hope) these properties:

• Each path through it involves no more than two bids
• Each double is preceded by a bid (not a double or redouble)
• Each redouble is preceded by a double
• There are no three consecutive passes
• Whenever a player has to make a decision, his own cards decide how it should be made

There is plenty of slack in this system. I suspect it is possible to devise a system like this which can also cope with cards misdealt to no player. But arranging to place three cards (each in some player's hand or on the floor) in each two levels of bidding won't quite do; we cannot guarantee to place the 52nd card in the initial 1C bid and any final doubles and redoubles.

Some Numbers