Restricted Choice
(Page of 3)

Consider this familiar layout of a suit. You must play this suit without loss of a trick. We will try two experiments to illustrate how the correct play changes when different information is available.

Axxx

K10xxx

Restricted Choice Experiments

Experiment 1
You are blindfolded. When you play a card, your partner will announce whether your opponent has followed, but not which card he played;

T1. You cash the ace and partner announces that both opponents have followed.

T2. You lead towards your K10 and partner announces that RHO has followed.

How should you continue? You could go up with the king hoping for a 2-2 split, or you could finesse the ten hoping your LHO started with precisely singleton queen or singleton jack. It turns out that a 2-2 split occurs more than 3X as often as a singleton honor with LHO (remember the old adage: "eight ever, nine never"?).Playing the K is a huge favorite over finessing with this information. (Since you are blind, you cannot see whether RHO has followed with an honor, so playing for the drop is a big favorite.)

Axxx

K10xxx

Experiment 2
This time, you play this suit with your eyes uncovered.

T1. You cash the A in dummy. RHO plays low. LHO plays Q.

Should you finesse or play king? You might think the correct play should be the same. After all, the card gods don't care whether you wear a blindfold or a tutu.While your eyes did not affect the layout of the cards, they did provide additional information you can use to recalculate the odds.

Previously, we knew only that RHO held 2-3 spades and that LHO held 1-2 spades. We had no information about whether an honor had been played. With this limited information, the best we could do was to go back to the a prioriprobabilities.

In our second experiment, we have more information. We can rule out many holdings for the opponents which are inconsistent with the cards they have actually played so far. For example, If RHO had stared with Jx, we would have seen him play the jack on the first or second trick. Since he actually played two small spades, he must have started with either xx or Jxx. Meanwhile, LHO was originally dealt either:

• QJ
• Q

These holdings are nearly equally likely to be dealt. Given our new information, it would appear the odds are even and finessing the ten succeeds about as often as playing the king.Unfortunately, this is not correct.Our analysis has ignored a second meaningful event that may have occurred and which impacts our odds.

If LHO had been dealt QJ, he had a choice of equal cards to play at T1. He might have chosen to play the jack instead of the queen.His choice to play the queen gives us additional information--we now know that either:

• LHO was dealt the spade queen alone or
• LHO was dealt the Q-J tight AND he played the queen instead of the jack.

If the second possibility is what transpired, and LHO is perfectly random in his plays, we have experienced a parlaywhere two independent events took place.

• Let's say the chance LHO was dealt Q-J tight is X.
• The chance LHO would play the queen instead of the jack at T1 is 1/2.
• The chance he was dealt Q-J tight AND played the queen is 1/2X.

Since singleton queen and Q-J are dealt with similar frequency, after we discount Q-J tight by half, it is now twice as likely that LHO started with singleton queen as Q-J tight and hence the odds in favor of the finesse are 2-to-1!

This notion was dubbed theprinciple of restricted choice by Terrence Reese and has long been known to expert players.

Other Restricted Choice Card Combinations

AJ10

xxxx

Restricted choice can be seen at work in many card combinations.Assume you have plentiful hand entries and no information about how the spade suit is divided in the opponent's hands. On the first round of spades you play a spade the the jack, losing to the K. On the second round of the suit, what are the chances that a spade to the ten will successfully finesse against the Q?

The second finesse is a 2-1 favorite to succeed. Itsucceeds any time LHO started with one honor (two chances, K or Q). It loses whenever RHO started with both honors (1 chance). Thus the odds in favor of the second finesse are2-to-1. But once RHO wins the K on our right, doesn't that mean we can ignore the chance that LHO was dealtthe king? And therefore aren't we comparing only the layouts where LHO started with the queen to layouts where RHOstarted with the queen which are equally likely? E.g.,

1. LHO Qxx RHO Kxx versus
2. LHO xxx RHO KQx
Yes, we are. Howeverin the second case, RHO had a choice of equal honors with which to win the first spade trick. So we must divide the chance RHO was dealt KQby two to account for the fact that half the time hewould have won the first trick with the Q. Hence LHO is twice as likely to hold the queen as RHO is.
KQ9x

Axx
T1. You cash the K and RHO follows small.
T2. You cash the A and RHO follows with the J
T3. You play a small spade towards the dummy and LHO plays low.
Which card do you play and why? Finesse the 9. RHO started with either Jx or J10x which occur with similar frequency.However, we must discount the chance RHO started with J10x by half because in that case,he had an equal choice on the second round of spades and half the time he would play the ten instead of the jack.
Qxx

AK8x
T1. You cash the A and LHO follows with the 9.
T2. You play a small spade to the Q and LHO follows with the 10.
T3. You play the finalsmall spade from dummy and RHO follows small.
Do you finesse or play for an original 3-3 break? Why? What are the odds? Restricted choice does not always lead to 2-1 odds. In this case, a finesse is a favorite by 3-to-1. The possible original holdings for LHO are:
• 109
• J109
If LHO started with the JT9, he had two opportunities to choose a card to play from equals. On the first two tricks he could have played: J-10, J-9, or 10-9. In fact he chose to play 10-9 whichoccursonly 1/3 of the time.Thus wedivide the chance LHO was dealt J109 by 3 to figure out how often LHO was dealt J109 AND we would have seen the 109 on the first two plays.
Q9xxxx

Ax
T1. You cash the A and RHO follows with the T.
T2. You play a small spade to the dummy and LHO plays low.
Do you go up queen or finesse the 9? And what are the odds? Once again this is a restricted choice situation. RHO began with one of the following holdings:
1. J10
2. K10
3. KJ10(your play is irrelevant so we ignore it)
4. 10 (your play is irrelevant so we ignore it)

J10and K10 are equally likely. However, if RHO had been dealt the equivalent cards J and 10, he had a choice of plays, while if he was dealt K10 he did not. Therefore we discount the possibility of J10 by halfand the odds in favor of a finesse are 2-to-1.

Unrestricted Choice Card Combinations

After reading the last page, it's easy to think that the correct play must always be to finesse whenever one honor drops. Not so.

KQ9xx

Ax

T1. You cash the A and RHO follows with the J.

T2. You play a small spade towards the dummy and LHO plays low.Which card do you play?
Play an honor. This looks like arestrictedchoice card combination, but strangely, it is not.For his play of the Jack, RHO could hold any of the following:
1. J
2. J10
3. J10x
4. J10xx (irrelevant since you will always lose a trick in this case)

There is a possible holding we have not had to explore before. RHO might have been dealt an original holding of J10xand chosen to falsecard with an honor to lure us into a finesse that would lose a trick unnecessarily. As in this position:

KQ9x

xx 10x

x

If you finesse the 9, you will lose a trick to the tenthat you would have scored if you just cashed the king and queen.Of course playing the king means you will lose an extra trick if RHO started with a singleton jack. So which original holding for RHO is more likely?

There are fourJ10x holdings (JT2, JT3, JT4 and JT5) plus one J-10 tight holding. There is only one singleton jack holding. So even after adjusting for the fact that RHO might have played the ten on the first round holding both the jack and ten, you are still more than twice as likely to pick up the suit by rising with the king as by finessing.We must play the king to protect ourselves from being fooled by a devious opponent.

K109xxx

Ax

T1. You cash the A and RHO follows with the Q.

T2. You play a small spade to the dummy and LHO plays low.
Which card do you play and why? If you want to score all six tricks in this suit, you play the K. Yes, this is a restricted choice situation and consequently LHO is more than a 2-to-1 favorite to hold the missing J. Unfortunately, a finesse won't help you. If your finesse succeeds, you will still lose a trick in the suit because LHO will still hold Jxafter the finesse and you will have to concedea trick to his jack to establish the suit. Therefore, your only hope is to play for QJtight on your right and rise with the king.

Conclusion

Most intermediate players find restricted choice counter-intuitive when they first encounter it. There are a surprising number of card combinations where restricted choice comes into play and recognizing them is a useful skill to pick up. Equally important is recognizing combinations that merely appear to be restricted choice.

KitWoolseycommented to me recently that at bridge, it is rarely correct to play for a parlay (to make a play that succeeds only when two or more unknowns happen to be true). Kit's observation applies to true restricted choice situations. Finessing plays for one fact to be true--an opponent was dealt a singleton honor. Trying to drop the other honor plays for two facts to be true--an opponent was dealt two touching honors and decided to play the one you have seen.
Lastly, remember that restricted choice is a guideline, not alaw. If your count shows that RHO started with exactly 3 spades, then a restricted choice finesse which plays him to have started with 2 spades is going to lose.