The Principle of Restricted Choice

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 10 to East's jack and South's king. Declarer then played the A and Q. West, however, held up her king. This prevenedt declarer from rectifying the count for a potential double squeeze.

South, then, proceeded by cashing his A and running all the diamonds. This was the position before the last one:

		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 10. (Note that if she had discarded the 9, declarer would have led a spade from the dummy to endplay West to lead away from the club king.)

"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."

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