The Principle of Restricted Choice is a guideline for locating particular cards held by the opponents.

Imagine you are declarer with the following suit combination:

Dummy | ||||

A J 10 9 | ||||

Declarer | ||||

8 7 6 5 4 |

You lead low to dummy, losing to East's queen. Upon regaining control, you lead toward dummy a second time.

West plays low. Should you put up the ace, or finesse again?

The answer can be found within the brain-teaser known as "The Monty Hall Problem."

Monty Hall was a Canadian-American television personality, best known as the host of the television game show "Let's Make a Deal."

On the TV show, contestants often encountered the challenge of choosing one of three doors, with the hope of revealing a grand prize like a new car.

Imagine you are a contestant who must pick between Door #1, #2, and #3. Each door has an equal 1/3 probability of hiding a car. The other two doors contain booby prizes.

Say you choose Door #1.

Monty, who knows the locations of each prize, then teases you by opening Door #2, revealing a booby prize. Next, he gives you the opportunity to *switch your choice* to Door #3.

Should you switch?

The answer is yes, you should change doors.

Here's why:

- When you initially picked Door #1, your chance of choosing the new car was 1/3.
- That means there is a 2/3 probability that Door #2 or #3 contain the car instead.
- If Door #2 or #3 do indeed contain the car, Monty will always reveal the one that has the booby prize. In other words, he is prevented — i.e. "restricted" — from opening the door with the car.
- Thus, if you switch doors, you are effectively betting that Monty was forced to keep Door #2 OR Door #3 hidden from you. This gives you a 2/3 chance to win.

The crux of the Monty Hall Problem is that the odds change based on Monty's action, even though it may seem counterintuitive.

Let's return to the card combination at the beginning of this article.

Dummy | ||||

A J 10 9 | ||||

Declarer | ||||

8 7 6 5 4 |

After losing one finesse to East's queen, should you attempt a second finesse or try to drop the king?

The answer is to finesse again. The odds are closer to 2/3 that West holds the king. Why? Because East may have been "restricted" to playing the queen *because she doesn't hold the king.*

This is analogous to Monty's disclosure of a booby prize. In the game show scenario, Monty may have been forced to open Door #2 because he was restricted from revealing Door #3. Or vice-versa.

Restricted choice does not only concern double finesses for K-Q or Q-J. It can also apply to discards, as once shown by Philip Alder in The New York Times: ^{1}

North | ||||

J 8 5 4 2 | ||||

3 | ||||

West | A Q J 9 6 5 | East | ||

K 9 6 3 | 4 | 10 7 | ||

10 4 | Q J 9 7 6 4 2 | |||

7 2 | South | 10 8 3 | ||

K 10 8 7 3 | A Q | J | ||

A K 8 | ||||

K 4 | ||||

A Q 9 6 5 2 |

North Pass 4 |
East 3 Pass |
South Dbl 6NT |
West Pass All Pass |

Opening lead: 10

West led the

South, then, proceeded by cashing his

North | ||||

J 8 | ||||

— | ||||

West | 5 | East | ||

K 9 | 4 | — | ||

— | Q 9 7 | |||

— | South | — | ||

K 10 | — | J | ||

8 | ||||

— | ||||

A Q 9 |

To quote Alder:

"On the last diamond, East and South pitched hearts, then West did very well, smoothly throwing the

"Now declarer had to guess. Were spades 3-3 all along, when a spade lead would endplay West? Or had West come down to a singleton club?

"Mathematically, it is much more likely that spades will be 4-2 and clubs 5-1 than spades 3-3 and clubs 6-0. Also, if East had started with 10-9-7 of spades, he might have played the nine on the second round instead of the ten. The Principle of Restricted Choice advises that in these situations, when an opponent might have played either of two equivalent cards, assume he had no choice, that he had only one card left.

"So Stayman played a club to his ace, dropping West’s king, and took the last two tricks with the queen and nine of clubs."

^{1}Alder, P. (2010, August 18). A Principle Is Put to the Test. *The New York Times.*

- Probabilities: Suit Distributions

The probabilities of suit splits when two hands are hidden (e.g. the defenders). - Refinements to Restricted Choice

Discussion whether restricted choice is always a valid guideline.

2023 © Jeff Tang. All Rights Reserved.