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 SpadesAJ10, and West follows suit with the SpadesK. With SpadesKQ, West could select either the SpadesK or the SpadesQ. But with the SpadesK only, West had no choice: if he were to play an honor, he had to play the SpadesK. That makes it twice as likely that West had the SpadesK but no SpadesQ than that he held both the SpadesK and the SpadesQ.

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 "qu"een – j"ack". 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

BridgeSuitNS
AJ1096|8754Consider the suit combination as in the diagram.

South leads a small spade to dummy's (North's) SpadesJ, and East wins with the SpadesK. 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 SpadesA in an attempt to pin East's now-singleton SpadesQ, or to take another finesse with the Spades10, playing West for an original holding of SpadesQ32? 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.

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

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

Put another way: although the doubleton SpadesKQ is a bit more likely (6.78%) than a singleton SpadesK (6.22%), and a bit more likely than a singleton SpadesQ (6.22%), it is far less likely than a singleton quack (6.78% for the SpadesKQ 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.

External links

* [http://www.acbl-district13.org/artic003.htm Monty Hall problem and the principle of restricted choice]
* [http://www.rpbridge.net/4b73.htm Bridge paradoxes by Richard Pavlicek]
* [http://my.execpc.com/~fkommrus/Bridge/PLAY3.html An advanced example]