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.

Yesterday, we mentioned Occam’s Razor, a hypothesis dating from medieval times. It states that when comparing two explanations, one should assume the truth of the one with the fewer assumptions. This applies to bridge in the form of the Theory of Restricted Choice — and, as we shall see later this week, to what is popularly known as the Monty Hall Problem.

In bridge terms, when comparing two possibilities, we must reduce the probability of an event if a player had previously had a choice of equals to play; this is because he might have played either of them at that turn. But enough theory — let’s look at a deal and see how it works in practice.

In three no-trump, you win the spade lead and drive out the heart ace, then win the spade return and cash the hearts, both defenders pitching small diamonds. With no clue as to who has the fifth spade, you need to bring in the clubs now.

You cash the club ace, then cross to the club queen, bringing down the 10 from West. Should you finesse or play for the drop on the third round? The appropriate percentages to measure up are jack-fourth or 10-fourth of clubs in East against J-10-x in West. You should not look at just the chance of jack-fourth against J-10-x (where the odds would be very close), because West would have had a choice of high spot-cards to play from that holding at his second turn. That makes the finesse the clearly indicated play.

Sixty years ago, Terence Reese produced his seminal work, “The Expert Game,” published in the U.S. as “Master Play.” This book introduced a variety of plays that are now part of every top player’s armory. And the most important idea that the book promulgated was the Theory of Restricted Choice.

This theory borrows from William of Occam, who invented Occam’s Razor. This states that when faced with a choice of competing hypotheses, one should select the simpler option.

How does this apply in bridge terms? Consider today’s deal, where against your contract of five clubs, West leads the heart jack. East takes his ace and returns the suit, letting you ruff. How should you play the trump suit?

Clearly, East is more likely to be short in clubs than West, not only because East has the long hearts, but also because if East had three clubs, he might have been able to shift to a singleton in spades or diamonds at trick two. So lead out the club king; when East follows with the jack, you play a second trump, West following with two small cards, leaving you to decide whether to finesse or play for the drop.

The percentages here might be misleading: A singleton jack is less likely than the doubleton queen-jack, but if East had doubleton honors, he might have followed with either card. So the true percentages to compare are the singleton honor against half the percentage associated with Q-J doubleton. Playing for the finesse is therefore clearly right.

Declarer might easily have relaxed in six no-trump here when West led the heart 10. There were 11 top tricks, with 13 tricks available on a 3-2 spade break, and 12 tricks even on a 4-1 break.

But declarer carefully won the first trick with the heart ace, then carefully cashed the club ace-king followed by the heart king. It was only then that he led a low spade to dummy’s queen. West’s discard turned a potential 13 tricks into 11.

South continued with a low diamond from the table and took the jack with the king, then ran the diamond 10 to East’s queen. East exited with the spade jack, taken by dummy’s ace. Now declarer cashed the diamond ace, pitching the heart queen from hand. He followed up with the heart jack and discarded the club queen from hand, bringing everyone down to three cards.

East was reduced to the spade 10-9 and the diamond nine, and when declarer called for dummy’s club jack, East had no winning discard. He threw the diamond nine, and now dummy’s eight was high.

If East had followed low to the first diamond, declarer would have put in the 10. Had this lost to West, then on any return, declarer would have cashed the heart, diamond and club winners, then played a spade to the ace to cash the diamond ace. This would execute a simple spade-diamond squeeze whenever East had started with four diamonds. It would also work fine when East had begun with at most three diamonds including the nine and at least one diamond honor.

Liam Milne was able to report a fine play by Barbara Travis from the semifinals of this year’s Australian Women’s Playoff. Travis declared six spades on the friendly lead of the club eight to the jack and queen.

If clubs behaved, 12 tricks would be easy; but West’s decision to lead a club instead of a heart argued strongly that the lead was from shortage — and that West must not have the heart ace, or a club lead would be almost pointless.

With the general idea of playing East for the heart ace, Travis ruffed a diamond in the short trump hand, then drew four rounds of trumps. When West followed all the way, she provided additional weight to the theory of club shortness in that hand. Dummy discarded a heart and a club, while East discarded a low heart, a low club and the diamond jack.

Travis now played the last trump and the diamond king, coming down to three clubs and the bare heart king in dummy. East, holding the doubleton heart ace and three clubs, had no choice but to come down to the bare heart ace. Trusting her judgment, Travis cashed the club 10 before exiting with a heart. East had to win and, with only clubs left, was forced to bring dummy back to life. From declarer’s perspective, the only thing that could have made this hand any more spectacular would have been if both heart honors were off-side!

In the Seniors, Open and Women’s events, most declarers who received a club lead reduced themselves to guessing hearts — and not all of them did so correctly.

At the Dyspeptics Club, the rivalries are more than about winning and losing, since there is an unspoken contest between North and East, each of whom considers himself far superior to the other.

While neither of them would consider criticizing the other directly (as opposed to eviscerating their hapless partners) when the opportunity arises, a cryptic aside can turn the knife just as sharply as a direct criticism.

Today’s deal gave North the opportunity to add insult to injury after an unsophisticated auction had led South to a marginal six spades. When East competed over North’s transfer bid, you can hardly blame South for joining in, and that led North to something of an overbid when he took control and drove to slam.

West led the heart eight to East’s ace, and when that player returned a trump, declarer could simply draw trumps and claim, disposing of both dummy’s diamonds on the winning hearts.

While South was waiting for his partner to acknowledge the brilliance of his play, North turned sympathetically to East and commented on what a difficult opportunity he had missed. Stung, East asked what North meant. Can you see the answer?

West’s spot-card lead had to be from shortage, so taking the heart ace was virtually conceding defeat. The only real chance was that partner would hold the diamond king, so East should have followed at trick one with the heart jack.

Declarer will not lose a heart trick now, but he will have two inescapable losers in diamonds!

Declarer in today’s deal from the Common Game played three spades competently. Clubs were originally 3-3, so South could eliminate that suit and endplay the defenders with the third trump to hold his diamond losers to one. However, I have changed the layout in the minors to make the task for declarer more challenging.

The defense begins with two top hearts followed by a shift to the club jack. Declarer takes the club in hand and cashes both top spades, then runs the club winners as West discards a heart. When South leads the fourth club, planning to ruff this in dummy, West must pitch a heart. Otherwise, declarer can set up diamonds easily enough for one loser. So West pitches a second heart, and, after ruffing the club in dummy, South exits from the North hand with a trump to East’s queen. What four cards does West come down to now?

If he comes down to one heart and three diamonds, then East can do no better than lead a heart, and declarer discards a diamond to endplay West to lead away from the diamond king or East from the jack.

But if West pitches a diamond, to come down to two diamonds and two hearts, declarer ruffs the third round of hearts and leads a diamond to the 10 and jack. East has only diamonds left, so he leads one, and South plays low from hand. When the king pops up, declarer has the last two tricks.

This line may require playing East for the diamond jack, against perfect defense.

Bidding over pre-empts is more about judgment than science. Here, South should not double a three-diamond pre-empt, since he cannot stand to have partner bid spades at just about any level.

When North raises to four diamonds, suggesting at least a good high-card raise to four hearts, South might stretch just a little and use key-card, hoping his extra shape in the side suits will come in handy, as indeed it will.

After the lead of the diamond 10, declarer should not put up the diamond queen, since doing so might provide the defenders with some communications. He will win the first diamond in hand and lay down the heart ace, then draw trumps and pitch his diamond loser on dummy’s good spade. Now he can ruff a diamond to hand and reach a five-card ending where he has four clubs and a trump in each hand. Can you see what he should do next?

If clubs are 3-2, the hand is cold for 12 tricks. If clubs don’t break, then declarer cannot protect himself against West having begun with all of the four significant spot cards. But he can guard against East having a singleton intermediate or honor in clubs by leading a small club from hand and playing low from dummy, no matter what West plays. In the layout shown, if West plays low, then East will win his 10 and be forced to surrender a ruff-sluff. If West plays high on the first club, he is left on lead; now, whether he plays a high club or a low one, declarer is home.

West leads the club three against four spades, and South can see that he must expect to lose a club. He can discard one of his hearts on a club and ruff his other losing heart with one of dummy’s small trumps. He should then be in good shape to hold his spade losers to two at most.

When East wins the ace, South drops the king from hand to create an entry to dummy. That will allow him to win the next club in dummy rather than in his own hand.

East returns a low spade, and South can afford to put in the jack, since he needs only one trump in dummy to ruff with. When West wins trick two with the spade queen and returns a diamond, South wins dummy’s king rather than risking the finesse. South also cannot afford to take another trump finesse immediately, since West might be able to win and return a third trump to keep South from ruffing his losing heart at all.

To avoid this fate, now is the right moment to take the spade ace, then cash the two club winners in dummy so that South can get rid of one of his losing hearts. Next, declarer cashes the two top hearts and ruffs the fourth heart with dummy’s seven. Though East is out of hearts, he cannot over-ruff, and even if he could, it would be with the master trump.

Only now can South afford to resume the play of trumps. He concedes one trick to the defenders’ master trump but makes his contract.