Restricted Revisited
What is the principle of restricted choice? How should we apply it? There are two questions to be answered. The first question deals in generalities, the second with specifics. The Principle arises from Bayes’ Theorem, a mathematically precise statement relating to the calculation of probabilities after cards have been played. It can be difficult to translate from the language of mathematics into plain English as something may get lost in the translation. In his masterwork, Master Play (1960), Terence Reese put it this way:
It comes to this: that a defender should be assumed not to have had a choice rather than to have exercised a choice in a particular way.
From this we may gather that if in a suit the JT are missing and one defender plays the jack, it affords the assumption he does not also hold the ten. It is not always correct to act on this assumption. Reese knew that, but he was trying to be helpful and may have inadvertently put the wrong idea in some readers’ minds. When making decisions one should take into account all possible combinations remaining.
Here is my statement on the application of Bayes’ Theorem to card play.
After one observes a sequence of plays in a suit, the probability that the observed sequence arose from a particular card combination is in inverse proportion to the number of equally plausible plays available with that combination. The greater the number of plausible alternatives, the less likely it is that the observed sequence was chosen from that combination.
The Encyclopedia of Bridge gives the following cautionary example in the section on restricted choice: North holds A2, South, KQ9843 and EW hold JT765. Declarer plays the Ace and then the 2. East follows with 2 low cards, but West has followed with the ten on the first round. Should declarer finesse the 9 on that assumption that West had no choice but to play the ten? No. That is not what Reese meant. Let’s look at the combinations remaining after East follows to the second round. When looking at probabilities one should specify which cards have appeared, and not state vaguely that East has followed twice with low cards. Let’s say East has played the 5 followed by the 7.
Here are the combinations remaining along with their probabilities:
West 
East 
Plausible Plays 
A Priori % 
A Posteriori % 
T 
J765 
6 
2.83 
29 
JT 
765 
12 
3.39 
18 
JT6 
75 
4 
3.39 
53 
We assume that West would habitually play the ‘obligatory’ false card from JTx. The single most likely combination is the 32 split. In total JT in the West is more likely than the singleton T in the ratio of 12:5, so it’s not even close: declarer should play for the drop of the Jack in 3 rounds.
The question arises: does the same logic apply if West follows first with the Jack? That depends on how often a player would play the Jack from JT. If he always plays the lower of two touching honors, then JT is impossible. Always playing low from touching honors is more informative than playing either honor at random. In the above example partner need not be informed, so the random approach is best. More on this later.
This scenario is unrealistic as some bids have to be made and some cards have to played before declarer gets around to leading his ace. If the vacant places are equal, as they are when the cards are unseen, the results based on the a priori odds will provide reasonably sound guidance. However, if there has been preemptive bidding by the opponents, the vacant places may be unbalanced, and this significantly affects the odds. All suits may have to be included in the calculation, as in this example from the 2013 Spingold Final.
Brad Moss led the ♣9, covered by the ♣Q, ducked by Joe Grue (♣2). Jacek Kalita led the ♥T from dummy, covered by the ♥K and ♥A, Moss following with the ♥2. The ♠4 was played towards dummy, Moss playing the ♠9, a significant card. The ♠J won the trick, Grue playing the ♠7. On the lead of the ♠2 from dummy Grue followed with the ♠5, and Kalita was faced with the decision as to whether or not to finesse the ♠8.
If one blindly follows the guidance of Reese as quoted as above, one finesses, playing for the ♠T and ♠9 to have been dealt to separate hands. The Encyclopedia of Bridge has warned us that such simplistic thinking can be wrong. One has to examine the remaining combinations before deciding.
Let’s simplify and assume on the bidding that Moss holds 7 diamonds, Grue 2, and that Moss has led passively from a tripleton club. The vacant places are 3 in the South and 8 in the North. The probabilities before a major suit has been played relate to the number of combinations of hearts and spades possible on the deal. The hearts and spades may be split in the following manner to fill the vacant places.

North – South 
North – South 
Heart Splits 
3 – 2 
4 – 1 
Spade Splits 
5 – 1 
4 – 2 
Combinations 
60 
75 
If there were no restrictions on the play of the cards, one round of hearts and one round of spades would not affect the relative probabilities. Thus, the probability of South being dealt a doubleton spade would still be more likely than his being dealt a singleton spade in the ratio of 5:4. If there are restrictions on the play, probabilities can change drastically as an adjustment must be made for the number of plausible plays for each remaining combination. Below are given the number of plausible plays at the time of decision when the ♠2 is led from dummy and Grue follows with the ♠5.

North – South 
North – South 
North – South 
Heart Division 
KQJ 72 
KQ7 J2 
KQJ7 2 
Spade Division 
T7653 9 
T7653 9 
7653 T9 
Plausible Plays 
72 
24 
72 
Probability (%) 
20 
60 
20 
Many possible combinations have been reduced to a mere three. We have assumed North would cover randomly with an honor on the lead of the ♥T from dummy. The results show that a singleton spade in the South is 4 times as likely as a doubleton, so the finesse is the correct play.
It is said that only a genius or a fool leads from a worthless doubleton and it doesn’t pay to assume your opponent is a genius. Let’s suppose the lead was from a singleton, as was the case. If the clubs were dealt 15, the vacant places are 6 in the North and 5 in the South. The hearts and spades may be split in the following manner to fill the vacant places.

North – South 
North – South 
North – South 
Heart Splits 
1 – 4 
2 – 3 
3 – 2 
Spade Splits 
5 – 1 
4 – 2 
3 – 3 
Combinations 
30 
150 
200 
On an a priori basis the chances of a singleton spade in the South hand are slim indeed, but let’s again look at the situation at the time the ♠2 is led from dummy and Grue follows with the ♠5. We assume that North would split his heart honors randomly from ♥KQx on the lead of the ♥T from dummy. Under those assumptions there are just 8 combinations left to consider.

North 
South 
Plausible Plays 
Probability (%) 
I 




Hearts 
K 
QJ72 


Spades 
T7653 
9 
24 
33 
II 




Hearts 
KQ 
J72 


Spades 
7653 
T9 
96 
7 
III 




Hearts 
KQJ 
72 


Spades 
76x 
T9x 
108 
15 
IV 




Hearts 
KQ7 
J2 


Spades 
75x 
T9x 
36 
44 
There are 2 possibilities included in Cases III and IV. The odds are 2:1 in favor of playing for the drop. In order to justify the right decision and finesse for the ♠T, as Kalita did, one would have to assume that the ♥K would be played from ♥KQ(x) much less often than half the time, in particular, at trick 2 Grue would play ♥Q about 7 times out of 8. The ♥Q would be the informative play, as normally the King would deny the Queen. If the lower honor would always be played in Cases IIIV, the only remaining possibility is Case I with the singleton ♥K.
We have observed on BBO that the experts usually strive to play the informative card, probably because they trust their partners to make good use of the information they transmit. So, it is quite possible that, early in the play especially, the ♥K would not often be chosen over the ♥Q. This would reinforce Reese’s advice and lead to the conclusion that if North plays the ♥K, one should assume tentatively he had no equivalent card.
This isn’t the whole story, but this report is long enough and as a twofinger typist we were happy that Nowosadzki didn’t play a larger role. At least one can see how the calculations should proceed – even computers have been known to make mistakes. (That’s a comforting thought!) Bayes’ Theorem is correct but one must assign realistic probabilities to the play options. As Voltaire noted, a logical conclusion is only as valid as the assumptions that went into it. Detractors of restricted choice will be glad to discover that in the end judgment is the determining factor, and mathematics acts merely as the Handmaiden to Success (or the MopLady to Failure.) But …
Don’t throw out the baby with the bath water – German proverb (1512)
Hi Bob,
Enjoyed your great description complete with mathematics entwined with bridge and its psychology.
Jacek Kalita made a great intuitive play against Joe Grue and particularly Brad Moss when he played Brad for the singleton 9 of spades, If he would have known that Moss had only a singleton 9 of clubs (the opening lead) he may have had second thoughts about finessing, but sometimes the smaller the knowledge the more effective the play.
In any event, with me having strong beliefs about the law of averages over time evening out the luck element in all forms of competition, the next time his intuition reacts he may be losing that spade trick to an original 109xx, with of course Grue having the KQJ of hearts. Of course, with that layout the declarer would only have been increasing his down tricks from 1 to perhaps 5, but in retrospect his gamble was justified since by so doing, taking advantage of the actual count of the hand, enabled the contract to be scored up.
A wise bridge player once advised me when I was very young, “Bobby, always let the WINNER explain”.
In your first chart, the bottom row should read JTx opposite xx, with the same 12 plausible plays as the row above it. Unless you believe your opponents defend differently with 76 and 75, you need to cater to original cases. Another way to think about this is a restricted choice argument, with the same six options from an original 765 counting the other way against that holding compared to 75.
In your last chart, you over counted your plausible plays by 50%, presumably from counting T9x as three plausible cards on the first round, when you only have two (you need to drop a T or 9 to give declarer a losing option).
Thanks for the comments. Yes, we don’t know for sure what went on in Jacek’s mind,and if he told us, could we believe him? There may have been psychological reasons he prefers to keep to himself. As the Billie Holliday song goes, ‘Don’t Explain’.
As for plausible plays, the situation is complex, and one is free to assume what a player might do in any given circumstance and carry on from there. This was my simplification.
If you take the remaining 3 hearts, 3 spades, 4 clubs and split them into 3 and 7 don’t you get that ratio for the location of the spade ten?
Greg, the method you propose is valid when one is considering the location of a card in a suit that has not been broached. It is the probability you would get if the remaining cards were reshuffed so the memory of what happened before was erased. Of course, that is where the a priori odds come from.