Aces on Bridge — Daily Columns

The Aces on Bridge: Wednesday, June 12th, 2019

There is little friendship in the world, and least of all between equals.

Francis Bacon

N North
N-S ♠ A K J
 Q 9 5 2
 9 2
♣ K 7 6 4
West East
♠ 9 8 4
 J 8 7
 K J 10 6
♣ 10 8 2
♠ 10 7 6 3 2
 K 10
 Q 5 4 3
♣ A 5
♠ Q 5
 A 6 4 3
 A 8 7
♣ Q J 9 3
South West North East
    1 ♣ Pass
1 Pass 2 Pass
4 All pass    


This week we are addressing the thorny problem of Restricted Choice in bridge. This dictum says that if (and only if) a player had a choice of equal cards to play, then the probability that he had one of those cards singleton should be compared to half the probability that he had both of those cards — because with the doubleton he might have played the other card. Granted, this does assume that he was equally likely to play the queen or jack from queen-jack doubleton, but unless you know to the contrary, you should indeed assume that.

A deal may make the point more clearly than words. Declaring four hearts, you win the diamond lead for fear of a club shift, after which the defenders might maneuver a ruff. You cash the heart ace, dropping the 10 from East. Should you now lead a heart to the nine or to the queen?

As indicated above, it is correct to play East for the doubleton K-10 rather than for the J-10 doubleton. That is because, with the former holding, he had no choice but to play the 10 at his first turn, whereas with the J-10 doubleton he might have played either of those cards. Thus, one should not compare the initial probabilities of each doubleton holding, which are equally likely, but instead assume that the K-10 is twice as likely as J-10 doubleton.

How does that relate to the Monty Hall problem? We will find out tomorrow.

Despite your heart support, it may be wrong to raise hearts directly. Your partner could be worried that the opponents have a spade fit. On the other hand, responding one spade may not work well if you finish up there instead of in hearts. Still, I would bid one spade, expecting to be able to raise hearts at my next turn (assuming I get another one).


♠ 10 7 6 3 2
 K 10
 Q 5 4 3
♣ A 5
South West North East
  1 ♣ 1 Pass

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Bob LiptonJune 26th, 2019 at 3:00 pm

I would express it a little differently. Restricted choice tells us that when east plays the H10, we should play him fas not having the HJ, because if he did, he might have played that. So for the second round of hearts, we should play the H9. West might have 4 hearts, including the King. Playing the 9 helps hold the defense to one trick in hearts regardless.

bobbywolffJune 26th, 2019 at 3:28 pm

Hi Bob,

Yes, your point is well taken since by playing the nine in this case after leading a second low heart from hand toward the Q95 not only is it more likely that West has the jack of hearts, but also he may still have the remaining three hearts, namely the KJ8.

However then it should probably also be mentioned that if West wins the king, next after declarer returns to his hand to lead another heart that West can then rise and then initiate the first of two straight diamonds which would then might develop the setting trick.

Not that the declarer can do anything about it, but only that would then take more explaining, too many words used, etc. Of course, and before playing the 2nd heart toward dummy, the declarer might lead three rounds of spades, throwing a diamond away from hand, but then it gets complicated later, since declarer will still be badly placed when and if the hearts turn out to be a singleton 10 with East with more forces by the defense together with the declarer still having work to do in clubs, etc.

However, none of the above in any way denies what could and if allowed, should be said, but attempting to describe the rest of the hand and its options will likely be just too difficult to get across and not the space necessary to attempt to do so.

In any event thanks for your input, which does generally apply, but when other problems emerge, it is difficult to impossible for the column writer to cover all those bases.

Bob LiptonJune 26th, 2019 at 4:23 pm

Thank you, Bobby. To expand, the play of card on the first round informs us that — assuming the defenders are rational — east does not hold the H8 (or H7 is west played the 8 on the first round of Hearts). East ‘s possible holdings are stiff Ten, JT, KT, or KJT. The chance of holding the stiff to holding one of the KT combinations is unchanged from before the first Heart is played, while his holding of the Jack is reduced by Restricted Choice. Therefore the most likely holding is KT, probably followed by the stiff 10. We choose the play for the holding that benefits us.

All of this argues, along with your statement about west rising with the King on the second round of hearts, that perhaps we should have begun by winning the DA on the first round, followed by three rounds of Spades, then a Heart to the Ace, followed by a Heart, intending to play the 9 if West follows low. It’s too complicated for me to calculate the deadly hands, when West has no more than a doubleton Spade or doubleton Club. Or we might rely on West not finding the optimal line of play, should he hold HKJ87, which is what he would do were I East. In life, we rely on the kindness of strangers, and in Bridge, that of opponents.


Iain ClimieJune 26th, 2019 at 4:30 pm

Hi Bob, Bobby,

Any case for 3 rounds of spades dumping a diamond before playing A and another heart? I accept that if West improbably has 6 spades and (say) HKJx or Kxx this could look a bit silly, but there is a rather higher risk that spades are 5-3 and there is a possible trump promotion. Against aggressive players, these concerns may not apply but it would be just typical if West had HKx and East had Sxxx and HJ10x or similar. Partner is then best advised not to say “Wouldn’t it have been easier just to play HAx first….” until the end of the session at least, and preferably not at all.



Iain ClimieJune 26th, 2019 at 5:16 pm

Hi Bob,

Looks like your 2nd post and mine crossed!