Principle of restricted choice
The Principle of restricted choice is used in bridge to guide a player (usually the declarer) into finding the best line of play in certain situations. It is closely tied to the Monty Hall problem.
The principle can be expressed in several different ways; one of them is:
When a defender freely plays an important card (over declarer's lead), assume that it had to be played rather than it was result of a particular choice; adjust the subsequent play accordingly.
In other words, if an opponent unprovoked plays a honor card (e.g. a king) on declarer's or dummy's lead, it should be assumed that he had to play it (i.e. it was a singleton) rather than it was played from a combination of equal-rank cards (e.g. king-queen). With KQ, he could select either king or queen, but with bare king he had no choice. That makes singleton king twice as possible as bare KQ, so it should be assumed that it was singleton.
Example
| AJT9x |
| xxxx |
Consider the situation as in the diagram (with "x" denoting insignificant cards with a small face value).
South leads a small card to dummy's (North's) Jack, but East wins with the King. Later in the hand, South leads a small card again, and West plays low. In the absence of other information, is it better to play the Ace in an attempt to crush East's Queen, or to take another finesse by playing the ten, playing West for three cards? The Principle of Restricted Choice explains that finessing is roughly twice as likely.
The initial possibilities were (ignoring 4-0 breaks):
|
xx | |
|
KQ | |
|
Qx | (?, as the small cards can be swapped around) |
|
Kx | (?) |
|
Q | (and Q Kxx) |
|
K | (and K Qxx) |
|
x | (and x KQx) (?) |
However, the remaining possibilities are:
|
Kx | (?) |
|
K | |
|
KQ | |
|
KQx | (?) |
The only combinations where it matters what you do are:
|
K | |
|
KQ |
Which is more likely? Na?ely playing for KQ doubleton seems more likely as a 2-2 split is just over 50% but the Principle of Restricted Choice shows that it is almost twice as likely to be the first combination.
Simply put; if RHO had both the King and the Queen he had a choice over what card to play - half the time he would play the King. Therefore the weighting of the xx KQ possibilty is halved! With stiff King he has a restricted choice (i.e. none) and always plays the king.
Restricted choice applies in many situations in bridge in addition to the frequent occurrence described above.
Math theory
The Principle of Restricted Choice is an application of Bayes' theorem.
