Definitions

# Principle of restricted choice (bridge)

In contract bridge, the principle of restricted choice states that the play of a particular card increases the likelihood that the player doesn't have another equivalent one. It is used to help a player find the best line of play in certain situations. It is closely related to the Monty Hall problem.

There are several different ways to express the Principle. One of them is:

The play of a particular card (one that might have been selected from two or more equals) increases the likelihood that the player doesn't have the other one.

If the player "doesn't have the other one," his choice was restricted.

Suppose that declarer leads small toward dummy’s AJ10, and West follows suit with the K. With KQ, West could select either the K or the Q. But with the K only, West had no choice: if he were to play an honor, he had to play the K. That makes it twice as likely that West had the K but no Q than that he held both the K and the Q.

The combination of cards that the player might select from need not be touching: it could be the ♣Q and the ♣10, if it is known that the ♣J has, for example, already been played. But the majority of examples of the Principle show the cards as touching – that is, the QJ, for example, or the KQ. The Principle of Restricted Choice is a somewhat elusive concept, and most people find it necessary to see it discussed several different ways before things start to become clear.

In English language discussions of Restricted Choice, the combination of touching cards is often termed the quack, a contraction of queen – jack. Besides being a convenient way to refer to the combination, it underscores the assumption that the player would choose one of the cards at random, and that it doesn't matter which he selects. As Reese put it, selecting one of the cards affords a presumption that he doesn't hold the other.

## Example

Consider the suit combination as in the diagram.

South leads a small spade to dummy's (North's) J, and East wins with the K. Later in the hand, South leads another small spade, and West again plays low. In the absence of other information, is it better to play the A in an attempt to pin East's now-singleton Q, or to take another finesse with the 10, playing West for an original holding of Q32? According to the Principle of Restricted Choice, the finesse is nearly twice as likely to succeed.

The initial possibilities, prior to any play in the suit, are shown in the following table.

 • KQ 32 • 32 KQ • K3 Q2 • K2 Q3 • Q3 K2 • Q2 K3 • K32 Q • Q32 K • K Q32 • Q K32 • KQ3 2 • KQ2 3 • 3 KQ2 • 2 KQ3 • - KQ32 • KQ32 -

However, before West follows to the second trick in the suit, the remaining possibilities include only those cases where East could play the K the first time:

 • Q3 K2 • Q2 K3 • Q32 K • 32 KQ • 3 KQ2 • 2 KQ3

And the only combinations where it matters which card is played from dummy are:

 • Q32 K • 32 KQ

A priori, four outstanding cards divide as shown in the following table:

Split Probability Number of combinations Probability of a specific combination

3-1 49.74% 8 6.22%
2-2 40.70% 6 6.78%
4-0 9.57% 2 4.78%

Notice that a specific 3-1 split occurs 6.22% of the time, and a specific 2-2 split occurs 6.78% of the time.

So, on the second trick in this suit, should declarer play to drop the remaining honor card from East or finesse West for it? The Principle shows that the finesse works almost twice as often as playing for the drop.

If East had both the K and the Q he had a choice of card to play to the first trick: sometimes he would play the K, and sometimes he would play the Q. The Principle assumes that half the time he would play the K (it turns out that to do so is his best approach). Therefore the probability that East held the doubleton KQ is halved, because he did in fact play the K. With the K but not the Q his choice was restricted (that is, he had no choice) and was forced to play the K.

Put another way: although the doubleton KQ is a bit more likely (6.78%) than a singleton K (6.22%), and a bit more likely than a singleton Q (6.22%), it is far less likely than a singleton quack (6.78% for the KQ versus 12.44% for the singleton quack).

Restricted choice applies in many situations in bridge in addition to the example described above, which appears frequently in the literature.

## References

Alan Truscott wrote of this principle in Contract Bridge Journal, and Terence Reese expanded on Truscott's early writings in The Expert Game (American title: Master Play).

## Math theory

The Principle of Restricted Choice is an application of Bayes' theorem.