Well ,you had to know that Odds and Ends (Parts 1 and 2) were laying the groundwork for more serious discussion to follow. I would be content to let well enough alone and have my readership bask in the comfort of reviewing stuff they already know, but since part of my mission is to drive those bridge intermediates up a notch or two, I can’t let everybody get too comfortable. Actually, the principle of Restricted Choice is not all that complicated, since the hard part of any sophisticated bridge play is recognizing when it confronts you. In Restricted Choice, it really hits you over the head since you unexpectedly see an opponent drop an honor in the first play in the suit.
Playing Rule for Restricted (Free) Choice Situations
The application of this playing rule often presents itself in a 9 card suit where declarer holds A9xxx in one hand opposite KTxx in the other. With QJxx outstanding, the initial plan is to split the suit 2-2 and drop both honors with the second round played. You start with the Ace from your hand, and, lo and behold, East drops the Q or J under the Ace (it does not matter which, since the math solution ignores the possibility of a false card by East). Now you have to decide whether East holds QJ doubleton or a single honor.
If he was dealt a doubleton honor, then you have to continue your plan to drop the other honor. If East was dealt a singleton honor, that means that West was dealt Jxx or Qxx, as the case may be, and the correct play is to play toward the K10 in the dummy planning to finesse against West for the other honor (Q or J). While the rule with a 9 card suit generally favors a “drop strategy,” this combination of events is an exception to this rule. It is called the Rule of Restricted Choice, although more modern authors generally refer to it as “Free Choice.”
In my example, on the second lead toward the K10x, you should insert the 10, finessing West for the missing honor. This seeming anomaly is mathematically correct and supported by the laws of probability. The correctness of the rule, often disputed in the past, is now conceded by all experts.
Here are the Rules and Odds for Restricted Choice Situations:
(a) In a nine card suit when there are two missing equivalent honor cards in a suit (e.g. Q, J) along with 2 small cards, if on the first lead one of the opponents plays either one of the honors (Q or J), it is almost 2:1 odds that the honor played is a singleton and you should finesse the other opponent for the remaining honor if you can.
(b) If you have an eight card suit (e.g. K765 opposite A1098) and there are two equivalent honor cards outstanding (Q and J, along with 3 small ones), if on the first lead one of the opponents plays either one of the missing honors (Q or J), the odds are approximately 5:3 that the honor played is a singleton and you should finesse the other opponent for the remaining honor if you can.
(c) If you have a seven card suit (e.g. K76 opposite A1098) and there are two equivalent honor cards outstanding (Q and J, along with 4 small ones), if on the first lead one of the opponents plays either one of the honors (Q or J), the odds are approximately 3:2 that the honor played is a singleton and you should finesse for the other opponent for the remaining honor if you can.
Other restricted choice positions are AKQ9 opposite xxx. If on the lead of the Ace you see the 10 played, lead a small heart from your hand and insert the 9. The J and Ten are equivalent honors.
Here is a little different twist. Dummy has J9x and you hold Qxx. Unfortunately, neither opponent led this suit for you, so you have some work to do. Lead small from the dummy toward the Queen, assume this is taken with the Ace by West. Since the Ace and King are equivalent honors, it is unlikely that West holds both of them. The correct play is to come to your hand and play toward the J9. If West plays a small card, insert the 9, since East is a favorite to have the King. If he also has the 10 there is nothing you can do about it, but if West started out with A10x, you will make a trick in this suit.
One final example. K109 opposite xxx. You play small and insert the 9 which loses to either the J or Q. For the purpose of Restricted Choice applications, the Q and J are equivalent honors. We know that it is more likely that any honor played is not likely to be accompanied by the adjacent honor. We need to lead again toward the K10, and if West plays a small card we put in the 10, not the King. East may have the Ace, but the odds favor West holding the Jack and we hope that the 10 will force the Ace in the East hand. If so, we get a trick in the suit.
What is the common denominator? It is that you are missing two equivalent honors in the suit and on the first lead you see one of them, bet that it is a single and take the finesse (if available) against the other opponent.
In Part 4, I will wrap up all this high math with a discussion of the “Deletion Principle”, demonstrating that middle cards can make a difference and that some 7 cards suits can actually be favorites to break 3-3.