There have been several descriptions of the principle of restricted choice – for example by Terence Reese in his splendid work ‘The Expert Game’, in the ‘Bedside Book of Bridge’ and more recently in a ’Better Bridge’ publication called ‘the Principle of Restricted Choice’ by Andrew Robson.The most common example of what this principle is all about usually revolves around a combination of cards in one suit like –
Dummy

A10xxx








Declarer

K9xx


Say you play the K from the South hand and the Queen or Jack appears on your right (East’s hand), with West following suit.You then play a small card from South’s hand and again West plays small – now what?Do you play for the drop of the outstanding honour, or finesse the 10?The favourite guideline with four cards outstanding, lets say from AKJxx opposite xxxx, is to play the Ace and King – the old saying ‘Eight ever, Nine never’, implying that with eight cards between the two hands, you finesse, but with nine cards, you play for the drop.This, however, is a slightly different scenario, you have two honours missing, but one has shown up.
Well? I guess you all know that the experts tell you to finesse the 10, presenting various long and convoluted quasi mathematical arguments, culminating in saying that because East has played one honour, he is assumed to have no choice in the matter (hence the ‘restricted choice’ bit!) and therefore cannot have the other.Not very convincing is it?At least, I have never found it so.
What sparked my interest in this principle again, was when I was sitting in Dummy’s chair recently, at a local Bridge Club, with one of Oxfordshire’s esteemed Selectors opposite and this collection of cards in the trump suit turned up
North

Qx








South

AK9xxx


Now you as Declarer with the South cards play a low one to the Queen, with the Jack appearing on your left.When you play dummy’s remaining low card and RHO follows with a small card, do you finesse the 9, playing East to have the 10, or play the ace, to (presumably) drop the 10 on your left?Needless to say, my Partner finessed the 9, lost to the stiff 10 and went down in his contract, with everyone else in the room making theirs.We smiled to ourselves, thinking that this was one of the ‘unlucky experts’ hands and weren’t we hard done by!!However, during the play of the next couple of hands, I found myself thinking about what had happened. Was it really a ‘restricted choice’ situation – or not?In fact, I was so perturbed by this that I mangled the defence on the next two hands.Well, that is my excuse anyhow, for a what is a fairly common occurrence.
Having set the scene, lets go back to the first card combination, with only four cards missing.
Dummy

A10764








Declarer

K985


Going back several years, following the publication of the booklet by Andy Robson, and failing miserably to comprehend the reasoning given there, I investigated further and came up with the following explanation, based upon the a priori expectation of the hand distribution: 
Let us look at all possible hand combinations of the outstanding four cards (assumed to be QJ32 for simplicity):
Combination

West


East





1
2
3
4 5 6 7 8 9 10 11 12 13 14 15 16 
QJ32
QJ3
QJ2
Q32 J32 QJ Q2 Q3 J2 J3 32 Q J 3 2  


2
3
J Q 32 J3 J2 Q3 Q2 QJ J32 Q32 QJ2 QJ3 QJ32 




This means that there are 16 combinations of the four cards in the defender’s hands.Let us go back to the actual deal, where we played the King from the South Hand, and an honour (Jack or Queen) appeared from East and it is a true card – in other words, East has not falsecarded.Then the only hand types of interest are combinations 4, 5 &, 11; i.e. singleton honours, or doubleton QJ.It can be seen immediately therefore, that there are twice the number of combinations where there is a singleton honour in East’s hand than the doubleton QJ, so you should finesse, not play for the drop.None of this business about if West had both, he could have played either, so you assume he didn’t have a choice and therefore he is more likely to have a singleton!You don’t need any of that!It is about twice more likely  simple isn’t it!Terence Reese, in his ‘Expert Game’ very briefly discussed this approach too.
What if East has falsecarded from combinations 7 to 10 inclusive?It doesn’t matter, the finesse still works and cannot cost.
Right, having sorted that, what about the hand that revived my interest?Here it is
North

Qx








South

AK9xxx


As Declarer you lead a small card to the Q and either the Jack or 10 appears from West.Should you finesse the 9 on the way back when you lead from Dummy and East plays a small card?Lets look at all the hand combinations again (the missing cards assumed to be J10432)
Combination

West


East





1
2
3
4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 
J10432
J1043
J1042
J1032 J432 10432 J104 J103 J43 1043 J102 J42 1042 J32 1032 432 J10 J4 J2 J3 104 102 103 43 42 32 J 10 3 2 4  


2
3
4 10 J 23 24 102 J2 34 103 J3 104 J4 J10 432 1032 1043 1024 J32 J43 J42 J102 J103 J104 10432 J432 J1042 J1043 J1032 J10432 




There are 32 combinations in all.What are the possibilities when I lead small to the Q and the J or 10 appears on my left?If it is a genuine card, then the only possible combinations of cards are 17, 27 & 28.Combinations 27 & 28 are the singleton honours and there are two of these compared with the J10 doubleton, so, once more it makes sense to finesse.However, what if West has falsecarded from the J10x combinations – i.e. combinations 7, 8, & 11?It is perfectly safe to do so isn’t it?This means that if you include those J10x combinations in the analysis, there are now six combinations where the J and 10 could have been dealt to the West hand and could reasonably be played.Only two of these six contain the singleton honours, so now it would seem that it is against the odds to finesse, and the cards should be played from the top.
The conclusion from all this?If, with the hand above and you are in your local club and either the Jack or Ten appears, then finesse the nine.If it is Andy Robson sitting West, then play them from the top.
In general therefore, if there are four cards missing, including the QJ and an honour appears, then you should finesse:If there are 5 cards missing to the J10 and one of them appears on the first round, then you should think about it and look at the quality of the opposition!It is not a genuine ‘restricted choice’ situation.
One interesting thing too that I have observed over the years, is that most club players holding J10 doubleton will play the jack, as the obvious false card………something else to take on board!This means that at my local club, if the Jack appears I will play for the drop, if it is the 10, I will finesse.
The sort of analysis described above can be applied to all other ‘restricted choice’ situations, for example
North

A973








South

KQ5


You play off the K and Q from hand and on the second round either the J or 10 appears on your right.You lead the 5, with East playing low – again the odds favour you to finesse the 9.
This is an interesting hand, as there are five cards missing –
North

Q97642








South

A5


South’s lead of the Ace drops the Jack or 10 from East.What should you do when you play the 5 and East plays low?Finesse the 9 or go up with the queen?
Again, if you write down all the hand combinations, you will be able to see that the finesse of the 9 will ‘bring home the bacon’ most of the time.
Any others you can think of?Write them down and analyse the distribution and the frequency of the various combinations – you will be surprised at what you learn.It will make the ‘Principle of Restricted Choice’ much easier to understand, or at least make you realise that you don’t actually need it at all.
It is simply a case of probabilities.
September 2002