Suit Composition as Information

Experts of the old school recognized there was a difference between a raise on 3 small cards and a raise on 3 to the jack-ten. They defined a 5-card suit as being ‘biddable’ if the suit was headed by an honor. In this regard they were saying that suits worthy of bid had to have an honor content that was consistent with their advertised length. That is, a bid was informative to a high degree as the minimum honor content was in accordance with expectations, that is, with what was most probable.

Today’s experts tend to ignore the honor content of a suit when opening the bidding or when raising partner in competition. The Law of Total Tricks maintains that it is the degree of fit that is most important feature. Most of the time this makes little difference as good fits usually produce a requisite number of honor cards merely on the basis of probability, however, the exceptional cases can cause problems. One may think solely in terms of the Law for which adjustments must be made for poor trump quality, but additionally one may think in terms of information. Without restriction on the honor content of suit bids, those bids become less informative than they were traditionally.

Uncertainty as to the quality of the suits being bid may be of benefit in competitive auctions as the opponents may misjudge the situation. On the other hand when it comes to defending a contract, the opening lead is often critical. The more reliable the bidding of one’s partner the more likely one will find the killing lead. Observing on BBO hands played by experts, I conclude that one of the most common sources of failure on defence is due to the bidding of poor suits by defenders who above all wanted to disrupt their opponents’ auction.

The following hand from the Slava Cup is the most recent evidence of a situation where experts bid in an atmosphere of uncertainty for its own sake hoping that the opposition will get it wrong in the end, but in which they are the ones who make the final wrong guess and suffer most from their own misinformation.
 

Dealer: South Both Vulnerable		Welland
		♠	Q105
		♥	93
		♦	AQ954
		♣	K102
Von Arnim				Auken
♠	AKJ987			♠	6432
♥	AJ865			♥	K72
♦	—			♦	82
♣	95			♣	Q643
		Zia
		♠	—
		♥	Q104
		♦	KJ10763
		♣	AJ87

 
At the other table after a Michaels Cue Bid by West, East played in 5♠* off 1 for -200. The double was applied by North whose values appear defensive in nature. Roy Welland thought otherwise, and put pressure on the opposition by taking a 2-way shot at slam, perhaps imagining the aggressive Zia would hold the ♥ A.

Most observers felt they would have found the killing lead of the ♥A. The question that is most relevant is this: could the defenders have bid in a more informative manner? If they had set up the correct defence, Welland would not have bid 6♦ , but would have had to be content with the par result of 5♠* off 1, as was achieved at the other table. As it turned out he and Zia benefited to the tune of 15 IMPs when Von Arnim led the ♠K.

One can see that the East-West defense should be centered on the heart suit. If they declare, they would prefer to play in spades. So the trick is to distinguish between the 2 situations. If over 3NT Auken had bid 4♦, that would have brought the hearts into play. Even a pass (shudder) would have been more informative than the bid of a suit that had no defensive value whatsoever. Occasionally defending against 3NT is not necessarily a bad idea. Down 3 would have been quite satisfactory. Von Arnim for her part might have simply overcalled 1♠. No need to panic, surely, and a more informative auction might develop allowing hearts to be introduced later. We cannot imagine a worse result than was achieved by her phony preemptive style.

Last week I was reminded of this hand when I arrived at a table where the North-South pair were discussing a previous misdefence. It seems North had led an unsupported Ace against a 5-level contract hoping for an attitude signal, which South could not acknowledge as such, since they always lead Ace from Ace-King. North maintained that against a high level contract their agreement no longer applied. Well, we can see Von Arnim had a similar problem. If she could have got an attitude signal on a lead of the ♥A, all would have gone well. It is a matter of being informed.

As it so happened when we finally got around to playing a hand against this pair, another high-level decision arose. As East I was dealt ♠ 543 ♥ QT942 ♦ — ♣ QT862, none vulnerable. North opened 1♦ on my right and I passed rather than bid a ratty unusual 2NT. Not holding spades, I do not expect to outbid the opponents. The bidding proceeded: 1♦ (Pass) 1♠ ( 2♥); 4♠ (???) What is your bid? Of course, 5♦ appears to be obvious, but it seems this is considered by many to be a stroke of genius. My 5♦ was doubled, and I bid 5♥, also doubled. Minus 100 was a clear top against 4 NS scores of 480 and 4 of 300. 5♠ would have been down 1 on a diamond lead, so the informative, lead-directing 5♦ bid was a sure way to achieve a top score. To be able to excel by merely using common sense does not constitute an entirely satisfactory state of affairs, but let’s maintain a clear perspective: it is better than not excelling.

The British Approach
A favorite form of British entertainment is the public inquiry in which political leaders are put to a friendly test with regard to the degree to which they can mislead the public without actually telling a lie under the legal definition. There were the Butler Inquiry, the Hutton Inquiry, the Baker Inquest, and, currently, the Iraqi War Inquiry. Like parliamentary question periods, these shed little light on the true nature of affairs, and whatever information is considered dangerous to the authorities is kept secret on pain of imprisonment and locked away for 70 years. In the same vein there was years ago the unconvincing Foster Inquiry concerning the accusation of cheating by Terrence Reese and Boris Schapiro in the 1965 world championships.

In the game of bridge ‘the British approach’ refers to the use of bids that are not intended to be informative while staying within the legal limit of their definition. Overcalling on a poor suit in a poor hand is a common means of achieving this end, as in this recent example from my club:
 

 
The overcall of 1♠ was made with a topless suit, a speculative toe-in-the-water effort made in the hope of eliciting a raise. The 2♥ bid on inadequate values was intended to show little support for spades. Despite these misleading bids the opposition proceeded to 3NT. The opening leader believed their bidding rather than his own or his partner’s, and led the suit in which he hoped his partner held complementary values. His feeling might have been that he had gained a great deal of information with little cost to himself, but in fact he had drawn a road map for declarer to follow. A tempo had been lost. The most probable outcome is a costly endplay after his minor suit exit cards have been removed and he is required to make a belated lead in a major suit. So it transpired.

We have further evidence to consider taken from Sally Brock’s feature in Bridge Magazine, Leading Questions: Quiz, a particular favorite of mine. Feisty Sally is a world’s champion, so we assume that she represents what is best in British bridge. In her quiz, experts are given an auction and a hand from which they are to choose their opening lead. A pattern has developed in which most often the normal opening lead gets the top (consensus) score, but it turns out not to have been the killing lead. Thus for the reader the problem becomes a 2-part question: what is the normal lead in theory and what is the killing lead in practice? (the Brits actually call it ‘the winning lead’.)

The most entertaining aspect of the feature is that for each problem she gives all 4 hands as they were dealt at the table. In the cases were the obvious lead is not the killing lead there is something fishy about the auction, something that invalidates the normal choice. The panelists are expected to draw correct inferences from the auction, then Sally Brock shows us why they are wrong. The message she is sending is that one cannot trust the auction as the protagonists’ bidding is unreliable. Distrust extends to one’s partner.

The first installment in the May 2009 issue set the tone. On the 5 problems presented, logicians Eric Kokish and Justin Hackett obtained a perfect score of 50, but David Bakhshi found all the killing leads, while finishing last in the consensus department with a score of 36. On Problem 4 he was the only panelist to find the killing lead of the ♣J from ♠J94 ♥J9 ♦8632 ♣JT72, after the sequence: 1♥ (Pass) 2♥ (3♦); 3♥ All Pass. Partner had interjected 3♦ on: ♠KQ6 ♥76 ♦AK974 ♣AQ9. To me the real question is whether East, rather than overcalling 3♦ after NS have established a fit, should have doubled on a balanced hand with 19 HCP thereby showing values in 3 suits at once.

Misleading bidding is disadvantageous in many ways when one ends up on defence. For that reason I prefer overcalls to be informative, that is, made on good suits, the exceptions being in the spade suit when there is a good chance of stealing the hand. I prefer competitive raises that imply that leading the suit is safe. Furthermore, I expect partner to lead the suit I have freely bid. Sally Brock doesn’t agree. She writes, ‘If partners are going to lead my suit it means I can’t bid unless I have a good one!’ Under such a constraint the overcall loses some of its disruptive potential, but this approach is pretty much a one-shot effort that puts a partnership at risk.

If the defenders are in the habit of bidding bad suits, a stopperless 3NT can escape the normal, killing lead. On the other hand leading to partner’s bad suit may give away the contract, a common theme in the quiz. So the game becomes one of bluff and counter-bluff. To reduce the guessing element, one should aim to play for what is most probable, and that involves providing partner with reliable information where it is most needed.

In the quiz of January 2010 Barry Rigal found all 5 killing leads, but scored a lowly 39, good for 10th place in a field of 13. He was the only one got this one right. You are West. South has opened 3♥ at game all, IMPs scoring, has been raised to 4♥ by North, which your partner, East, doubles. What is you lead from: ♠764 ♥Q5 ♦765 ♣KT876 ? The logical answer is ♠6, but the killing lead is the eccentric ♥5. South had opened 3♥, vulnerable, on a 6-card suit: ♠2 ♥AJT732 ♦KJ ♣J532. Although this bid is not illegal, one may say that it does not conform to what one usually expects of such a bid in the given circumstances, thus it is unusual, therefore, improbable.

The lesson I have drawn from these and similar examples is that it is difficult to find the killing lead when the auction is misinformative, therefore, a partnership will benefit if they misinform, yet, in spite of this, reach a playable contract that will make on a lead based on an assumption of normal, most probable conditions. A bidding system that uses limited opening bids has the advantage as the users know early in the auction whether they are aiming for game or for slam. If game is the limit, uninformative bidding may be the best option. There is nothing unethical in the sequence 1NT – 3NT, even if both partners are stretching the limits of definition. Also, one may take advantage of an opponent’s self-inflicted uncertainty. It may be ugly, but unusual is not illegal. There are risks as well as benefits to be derived from such an approach. The benefits are not so obvious when one bids speculatively only to end up on lead against an opponent who has not been intimidated. When in doubt one should ask oneself, ‘if I had not bid, what would I have led?’ To bid one suit tentatively and guess to lead another is a costly bad habit.

Two Chances Are Better than One

One of the conceptual difficulties one encounters in the bridge literature is that one suit is often treated as being independent of the other suits. So we encounter lists of percentage plays in a suit taken in isolation. This can be helpful, but it gives the wrong impression from a theoretical perspective. Suits are intertwined. Here is an example in which an expert asked, ’is there a clear-cut solution?’
 

 
The bidding is unknown. We assume responder evaluated his hand according to the losing trick count as 6 losers opposite a strong 1NT opening bid will often produce a slam. The 5 controls are worth an equivalent of 17 HCP, provided opener can come up with 3 spades to the king. As things turned out the 1NT bidder was happy to bid the slam after an encouraging 4NT RKCB. The lead was a not unexpected ♦K. Declarer won and played 2 rounds of spades hoping for a 2-2 split. The opening leader showed out on the second round, so the RHO had a sure trump trick. Unlucky? The problem was how to arrange a diamond discard on a club or heart winner before the RHO won his trump trick.

There are 3 possible plays that achieve the happy result: 1) play the ♣K and finesse the ♣J, discarding a diamond on the ♣A; 2) play 3 rounds of hearts and discard a diamond on a winning heart; 3) play 3 rounds of clubs hoping the queen falls and the RHO can’t ruff. Viewed in this light one might be inclined to hope the hearts are favorably split, but that is the wrong approach. Well, perhaps the club finesse is a reasonable shot, as the LHO has already shown up with the ♦KQ. Wrong again. Let’s see why #3 is the correct approach, as noted by Tim Bourke.
 

The less often you lose, the more often you win.

Rather than search for the winning percentage, it is often easier to look at the losing percentage. The location of the ♣Q depends on the number of clubs held in the defenders’ hands, and there is no indication that the RHO has many more clubs that the LHO. We conclude the finesse has roughly a 50% chance of losing.

Playing off the hearts will work as long as the RHO has 3 or more hearts. Let’s look at the combinations available in hearts taken in isolation:
 

 
The 3-3 split stands right in the middle and is included in the winning category, so the LHO will hold 3 or more hearts more than 50% of the time. That observation alone makes the heart play superior to the club finesse. The LHO will follow to 3 rounds of hearts for 37 out of 64 combinations, which translates to roughly a 40% chance of failure. The playdown of the hearts clearly has a lesser chance of failure than the club finesse.

The chance of failure by playing 3 rounds of clubs hoping to drop the ♣Q can be calculated in the same way. A full calculation is burdensome at the table, so we consider only a few of the more even splits to get a rough estimate. Whereas the even-numbered heart distribution was shaped like Mt Fuji with one peak in the middle, the odd-numbered club distribution is shaped like Table Mountain with an extensive central plateau.
 

 
The third round of clubs will be ruffed in the ratio of 60 to 320, roughly 19% of the time. This represents clearly a lesser risk than playing on hearts. Success may depend on what transpires during the next phase of play.

The Chance of Success
With regard to the first 2 strategies, either one wins or one loses on the completion of the play in the suit. However, with regard to playing 3 rounds of clubs, there are card combinations for which declarer neither wins nor loses: the queen doesn’t drop, but the RHO doesn’t ruff. Failure has been avoided, so there is now an additional chance to win, namely the hearts may be played down with the hope that the RHO holds at least 3 hearts. It has become a case of 2 chances are better than 1.

A represents the number of combinations where the initial club play works.

B represents the number of combinations where the initial club play fails.

C represents the number of combinations without a resolution.

A + B + C represents the total of all combinations allowed.

We start with a division of sides for the defenders of 4=6=9=7. The lead is the ♦K. Early in the play West shows out of spades so we assume he began with ♦KQ and a singleton spade, whereas East has 3 spades and at least 1 diamond. This leaves us with 10 vacant places in the West hand and 9 in the East hand. However, West had to discard something on the second round of spades. What information has that sent us, and by how much does it affect the odds?

If West were to discard a heart, declarer would be happy enough to play on clubs as planned. If West discards a low club, that might raise concerns that West is long in clubs (5) and, therefore, that East has only 2 clubs. If West discards a diamond, declarer proceeds happily in his plan in a neutral state of mind. In theory one assumes the discard was a random selection from the unseen cards in West’s hand. The appearance of a diamond, heart, or club doesn’t materially affect the odds, so we stick with the vacant places 9 and 10 for the sake of convenience. On that basis we do our probability calculations.
 

 
The club plays have left us with combinations where the club splits of 3-4 and 4-3 represent over 80% of the remnant combinations. We are now working on the high plateau of card distributions in accordance with the general tendency of proportionally more even splits as the cards are played out without incident. It is matter of calculating with high expectations for how many of these combinations are the hearts evenly split as well. This is tedious to do by hand, but I did it. The probability that the heart play will succeed is 67.4%. So the probability of success overall is given by the following:

Probability of success = 0.365 + .466 x .674 = .679
Probability of failure = .321

A rough approximation is suggested:
Probability of success = 1/3 + (1/2) x (2/3) = 2/3
Probability of failure = 1/6 + 1/6 = 1/3

If one employs the a priori odds for the club splits, the probability of failure in the first phase is less, close to 1 in 7.

The full deal was as follows:
 

		♠	A86532
		♥	AQ9
		♦	72
		♣	K9
♠	4			♠	QJ10
♥	8532			♥	107
♦	KQ1093			♦	J854
♣	Q94			♣	10872
		♠	K97
		♥	KJ64
		♦	A9
		♣	AJ53

 
Again thanks go to Tim Bourke for bringing an interesting problem to my attention along with the correct solution. He is developing a computer program that should prove very useful for calculating compound probabilities for complex situations. In the end one hopes to develop from such results insights which will serve as good guidance at the table.

Bridge As Opera

Sometimes bridge glides along as smoothly as an adagio movement from a Beethoven string quartet. When the four participants sustain a harmonious tension over a long period of time, it can be lovely, but it can get boring for all but the most refined connoisseurs, like myself. At other times the bidding becomes highly competitive, emotions rise and bridge becomes like an Italian opera with lots of chaos, shouting, and entertaining suicides by the foolish protagonists. That is the part most love: it is exciting. The trick to achieving success in an operatic setting is to get oneself into the proper frame of mind.

Imagine you are a member of the chorus of a famous opera house. The performance this night is Il Trovatore, a barn burner. Suddenly the famous mezzo-soprano and her understudy are taken ill with food poisoning (oysters), and you are thrust into the role of Azucena, the crazed gypsy woman who years previously had mistakenly thrown her own baby into a bonfire and has regretted it ever since. Let’s set the scene. The all-too-familiar Anvil Chorus has pounded its way to conclusion; you envy your erstwhile colleagues as they wend their way off stage for a much deserved cup of tea leaving you crouched beside the stage campfire awaiting your cue. The stage darkens, and the spotlight shines in your eyes but not before you manage to catch a glimpse of the bald-headed conductor as his lifts his baton and gives you an encouraging wink. Quietly you clear your throat. The maestro is hoping for accuracy, but you know that loudness is the key. You take a deep breath and let it fly.

So it is in competitive bidding – sometimes you have to let it fly. You are a veteran, and this is what you have been playing for all these years, a chance to shine. This is not a quartet or a duet. It is not a time for looking around for someone else to share the responsibility. The difference from opera is that you don’t have a conductor there to signal that it’s your call. You have to recognize the moment yourself and take charge.

This is what to look for. First, in the partnership your hand is by far the stronger. Second, your hand is short in the opponents’ suit. If those conditions exist, it is up to you to make the decisions for the partnership. It is well known that one should never suggest a penalty when holding a void in their trump suit.

Generally in a preemptive auction there is not enough information available for anyone to make a reasoned decision. One operates on instinct and general principles. No length of thinking about the circumstances is going to further one’s cause, and, in fact, may prove counter-productive. Last week there was a deal against an aggressive pair on which my partner made a weak jump overcall of 3♥ over 1♠ . Unhesitatingly my RHO raised to 4♠. I held ♠xxx ♥ QTxx ♦Axx ♣xxx, and just as quickly bid 5♥.

The opener said, ‘This goes against my rule, so sorry partner if this doesn’t work,’ and he bid 5♠. It turned out to be a top for us as 5♠ went down 1 on a diamond ruff. Yes, dummy held 6 diamonds which she had not deigned to show, preferring an uninformative ‘pressure’ bid. This is a characteristic of the operatic approach: loudness, not subtlety reigns. If the opponents are going to put on pressure, it behooves one to do the same. If I had hesitated before bidding 5♥ do you think the opener would not have doubled instead of taking the push to a losing contract? Under the circumstances, there was not enough information available to make a reasoned judgment, and I was not likely to gather more. So I went with what I had. It could be right, it probably was wrong, but if I didn’t know, neither did they. As with any stage performance, the trick is to appear credible at all times.

Linda Lee’s Question
In this light consider the problem discussed by Linda Lee in her blog of Feb 17, 2010. Should West double 5♣ ? The answer to me is clear cut.

After the preemptive 3NT opening bid, West must recognize the spotlight is going to be on her: a void, 4 losers, 8 controls. She knows the preemptor has long clubs, but her partner does not, so she has to mark time with a double and await developments. North cooperates by making a takeout bid in the minors, a vital piece of information being transmitted around the table, and South’s pass completes the picture. At this point West should take a deep breath and bid a conservative 6♣ . (Remember Rozzy’s Rule: Never bid a grand slam in competition.)

Yes, you might on a very bad day not make slam, but do the opponents know that? If one were to double and partner bids 5♠, say, can one judge that this is the right spot? The preempting side will be happy enough to let you play there on the grounds you may have missed your slam or grand slam. Even if you achieve par and make only 11 tricks they will be happy enough to have driven you to the 5-level. Why make them happy?

More important than what you think you can make is what the opponents think you can make. Suppose that West bids 6♣ and partner bids 6♠. How confident can the preemptors be now? Why shouldn’t they believe you can make 6♠? Might they not bid 7♣ as a sacrifice against a slam that is going down? So there is a second way to win when one boldly goes for the top. The greater your uncertainty, the greater theirs, so don’t advertise your doubts with a dubious double that really is not as wise as it seems.

I hate interviews of inarticulate jocks coached to give out mind numbing answers to asinine questions. In the Vancouver Winter Olympics snowboarder Maelle Ricker did Canada proud by being the first Canadian woman to win a gold medal on home ice and snow. In her after-run interview she kept repeating, ‘it was lots of fun!’ This was simple, genuine, and informative, as it demonstrated vividly and succintly the winning spirit: to win one must have fun and not be overly concerned about ending ass-up in a snow bank.

The Four Tenors
Over 60 years ago S.J.Simon wisely cautioned that in competition one should strive to achieve the best result possible rather than the best possible result, yet time and again we observe even experts making ridiculous overbids trying for an imaginary maximum. What has happened to the concept that good bridge is never having to say ‘I’m sorry’? To tell the truth they sometimes make their silly contracts such as 6NT missing 2 aces. The top level competition in the 2010 NEC Cup gave several examples of this tendency to take the bait and bid on and on. Here is my favorite operatic hand in which Italians faced Italians.

Dealer: East EW vulnerable		Bocchi
		♠	Q
		♥	AQ754
		♦	9
		♣	KJ9643
Nunes				Fantoni
♠	KJ9642			♠	A10853
♥	J96			♥	—
♦	J54			♦	8762
♣	Q			♣	A1082
		Madala
		♠	7
		♥	K10832
		♦	AKQ73
		♣	75

A critic once claimed that a good writer never uses the exclamation point, but my view is that sometimes one is not enough. Because information, or the lack of it, is often the key, the question one asks in these auctions that get too high is: who knew what? If Agustin Madala, the latest addition to the Lavazza team, had opened a sensible, informative 1♦ there may have been no story to tell, for drama often turns on ignorance and misunderstanding, as Jane Austen and Charles Dickens well knew. The recent introduction of transfers in competition hasn’t appeared to improve the situation. Eventually Madala did show his diamonds at the 5-level, so his partner, Noberto Bocchi, was in a good position to double 5♠ which Claudio Nunes had interjected before the correction to 5♥ , which, by the way, would have been down 1. But it didn’t end there.

Madala must have thought he had more than so far promised, and he became the first of 3 players on this deal to freely bid a slam missing 2 aces. Or was he just guessing? We know the answer, don’t we? Nunes, emulating the style of his countryman, the late Luciano Pavorotti, overdid it immensely when he bid his mediocre suit a 3rd time, as if he feared that his partner may not have been paying proper attention. Bocchi passed having already expressed an opinion that had been ignored. Fluvio Fantoni didn’t have enough to redouble, but he must have smiled when he put down a dummy with a useable void and 2 aces. It was not nearly enough, as 6♠* went down 2 for a loss of 14 IMPs.

At the other table where 2 Poles faced 2 Italians, there were variations on the same themes but with greater discord and cue bids galore. Adam Zmudzinski was strangely silent over Sementa’s 4♥ although one can imagine he could have chosen with reason from 6 alternatives: double, 4♠, 4NT, 5♦ , 5♥, or 5♠. (Make that 7 if 5♣ is a transfer.)

Balicki’s 4NT appears to have been an attempt to gather more information, although he was the first to jump the bidding and take away bidding room. He had jumped but not high enough. Sementa’s first double is dubious. Without the libretto it remains a mystery as to what this was meant to convey. Zmudzinski’s 5♣ might or might not have shown diamonds, but, regardless, Balicki knew enough to stop in 5♥. Sementa’s push to 5♠ was understandable, however, Zmudzinski’s pass held an unexpected meaning for Balicki.

Most of the bridge world likes to have a late pass available to convey the meaning, ‘nothing to add at this time.’ It is sometimes apologetic in nature, a sly abrogation of responsibility. Poles are notorious for adopting an alternative approach. Copernicus was such a Pole, so they are not always wrong. Maybe confusion is natural in a country overrun repeatedly from both East and West. Balicki, yet another famous Polish émigré, remembers the Polish Pass as meaning ‘Bid on! I’ve got a void and several other features I neglected to mention.’ To non-Poles, that doesn’t appear consistent with Zmudzinski’s previous actions, but if that’s what you think, bidding 6♥ becomes automatic.

Antonio Sementa’s final double doesn’t make sense, either, unless Giorgio Duboin is an idiot, but rumor has it that he is not. Let’s be fair. Maybe double says, ’I’m void, so consider bidding on.’ I admit I don’t know what these bids mean, but on the evidence of the auction I am not sure any one at the table knew what their partners were divulging, so why should I know? Uncertainty is the essence of the modern competitive style.

As reported on BBO the appearance of the ♠7 from declarer’s hand provoked operatic shouting and arm waving that must have shocked the ever-polite Japanese audience. However, Japan is a nation that has developed the act of self-destruction from its humble beginnings as a sincere apology for misdeeds to the ultimate means of self expression. They would have appreciated fully the suicidal aspects of the subsequent events. The Zimmerman team, which had built up a huge lead, continued its downward path to self-destruction, losing the final on a 3 IMP swing on the very last board. As Maelle Ricker might comment on behalf of the Lavazza team, ‘It was lots of fun!’

Tim Bourke The Master Carver

Once upon a time two craftsmen sat at a dimly lit workbench earning a scant wage from the rich-pilgrim trade by carving lucky turtles from wood, lucky turtle after lucky turtle, day after day. Maybe they should have kept some of that luck for themselves. One turned to the other and said

‘Look at this piece of wood, all knurled and knotted. How can anyone make a turtle out of that?’
‘How can they send us wood like that? There should be a law against it,’ commented his companion. “Throw it in the trash bin.’

Just then the master carver entered the workshop.
‘Let me have a look at that. Interesting. Let’s see what I can make of it.’

The master took away the defective specimen and in a few days everyone was marveling at an impressive figurine of a fierce-eyed eagle with wings outspread, sharp talons clutching a forlorn fish, and looking altogether as un-turtlelike as could be.

Tim Bourke is the master carver of bridge hands. Give him a contract that appears hopelessly misconceived and he can turn something ugly into something that everyone admires. He reads the grain of the hand and turns its odd features into something aptly beautiful. Here is an example that the detractors of Precision might cherish, as the auction has given away information that will allow the opponents to defend double dummy.
 

 
Critics of the losing trick count will note that it predicts 13 tricks, but that it is off the mark as there is an inevitable spade loser. They chortle too soon. He who chortles last chortles best. The opening lead is the ♣K. Do you see how one might make 7♦?

Tim, the master carver, looks at this configuration and wonders what can be made of it. There is a chance to utilize the clubs and spades if the player holding the ♠K is subjected to a squeeze in the black suits. How to create an ending that produces that effect? That is where the master excels: he can envision the patterns that others miss.

The ♣A is won and a club is ruffed immediately, an essential move. Both defenders follow with low cards. Six diamonds are cashed to see if the defenders give away something by their discards, but they have been well informed by the auction so there is nothing to be read from their selection. Here is Tim’s visualization of the 4-card ending after the ♥A and ♥K have been cashed and all hearts have been accounted for:
 

		North
		♠	Q3
		♥	3
		♦	3
		♣	—
West				East
♠	K9			♠	J107
♥	—			♥	—
♦	—			♦	—
♣	J3			♣	10
		South
		♠	A
		♥	Q
		♦	—
		♣	98

 
The ♥3 is led from dummy and West is caught in a trump squeeze, and he knows it! Interchange the East-West hands and East gets squeezed. The squeeze will be effective when the defender with the ♠K also was dealt 4 or more clubs. The same ending will be effective if the lead were a passive heart or a trump.

Tim has moved a stage further in analysis, as he has written a program that calculates the probability of success of alternative strategies, although probabilities are not critical when there is just one winning option. There are 3 major influences on the probabilities once play commences: 1) the opening lead, (2) the order of play, and (3) the discards. A passive diamond lead can be thought of as the standard trump lead against a grand slam. As it is expected, there is no surprise in that lead and little to be gathered from the implications of that choice. The lead of the ♣K is relevant, especially when declarer has admitted to holding 5 cards in the suit including the ace. The gift of knowledge revealed during the auction is being repaid in part by information from the opening lead.

Let’s assume both defenders have followed to 2 rounds of clubs. Suppose that on the first 2 rounds West has played the ♣K followed by the ♣3 and East has played ♣5 followed by ♣6. We must rule out a lead from ♣K3. Here are the remaining possibilities.
 

 
The plausible plays are the same for each remaining combination, provided we assume that East will choose randomly between his 2 low cards, and West will always lead the ♣K and follow on the second round with his low card. There is no need to signal as thanks to the Precision auction both defenders know the distribution fully. In cases where the plausible plays reach equality, vacant place analysis can be used to calculate the probabilities exactly. The situation in the club suit is such that the clubs have been reduced by 2 rounds of play from 7 to 3. Here are the resulting possible distributions:

 
The number of combinations remaining would be the normal weights to be applied to the club splits when there has been no restriction on the way the clubs were dealt or played. However, in the hand being considered, the opening lead has provided information that drastically changes the probability weighting of the various club splits. There is 1 combination remaining from a possible 5-2 split, 2 from a 4-3 split and 1 from a 3-4 split. As the plausible plays are equal, one may apply probability weights equal to the number of combinations remaining in each category.

The club weights can be applied to Tim’s 4-card end position where the heart split has been revealed and the spade split is unknown. The diamonds have split 2-3 and the hearts have been seen to split 4-3, so the red suits taken together were dealt 6 and 6.
 

 
The expected number of clubs in the West is 3.89 and the expected number of spades is 3.11, for a total of 7. As the red suits are known to split 6-6, the blacks must split 7-7.

A Vacant Place Interpretation
In the 4-card ending the odds that the ♠K is in the West is exactly 4 out 9. It is as if there are 9 black cards missing, 2 clubs and 7 spades, yet there are only 8 cards still outstanding. Why? The implications on the opening lead have placed a restriction on the West hand to having been dealt 3, 4, or 5 clubs, and, as we know the reds are split 6-6, which places a corresponding restriction on the spades to having been dealt 4, 3, or 2 in the West. The West player has followed to 2 rounds of clubs, so the lower bound on the number of clubs in the West has to be satisfied by one of the 3 missing clubs leaving the 2 remaining clubs free to be dealt to either defender in a random fashion. We have filled a vacant place with one of the missing clubs.

Once the clubs are placed, the spades fall into line. The 2 spade discards are not relevant directly. One has seen the heart discards which have enabled an exact count in that suit. The assumption is that the spade discards are ‘free’ and have not altered the probability of the holdings in that suit, because they have not transmitted any significant information, or, rather, because we have chosen to ignore whatever information there was transmitted. There are still enough insignificant spades remaining to satisfy the indirect constraint imposed by way of the club suit.

The Plausibility of Plays
The vacant place interpretation is a consequence of Bayes’ Theorem applied to the play in the club suit wherein one took account of the number of plausible plays. Because after 2 rounds the plausible plays were equal for all remaining possible club combinations, one can use the vacant place argument. The two methods are exactly equivalent: advancing a vacant place argument is the same as claiming the plausible plays are equal.

If we were to alter the probabilities of the club combinations by changing the number of plausible plays, then this nice interpretation would no longer be strictly valid. Consider the permutations in play where West has been dealt: ♣KQJT3. He decides on the second round to play one of the touching honors, giving him a choice of 3 equally plausible plays, and the partnership 6 in all on the 2 rounds. One might also claim that West could have lead a deceptive ♣Q, ♣J or ♣10, giving him 4 possible equivalent leads and 4 possible plays on the second round. Considerations of this kind greatly reduce the probability of the 5-2 split. We no longer achieve the vacant place probability weights of 1-2-1.

A Maximally Likely Guess
Early in the play declarer should consider what is most likely on the basis of the play so far. This approach helps in the visualization and the planning of the play. The initial crude estimate may not be far off if the suits split as evenly as one hopes. Here are the final 3 choices of distribution shown in full with their initial weightings and their true weightings in the 4-card ending:

 
After the trumps are seen to split 2-3 declarer could take note that Conditions II and III have equal probability greater than Condition I in the ratio of 100:36 Therefore, it is more probable that the ♠K doesn’t sit with the longer clubs.
 

 
On the basis of the opening lead, declarer might judge that Condition II is more probable than Condition III. If declarer were to consider the maximally likely distribution, Condition II, by itself the probability of the ♠K being with 4 clubs would be 42.9%, not a favorable percentage, but one not without hope. The crude initial guess of 3 out of 7 is not far removed from the 4-card ending estimate of 4 out of 9 (44.4%).

The Tiny Voices, Yours and Mine

With regard to bidding, everyone knows from experience artificial is best because it makes the transmission of information easier and more efficient. No one would willingly give up Stayman, Jacoby, and Blackwood. The only question is, how much is too much? Through the medium of its bulletin, the ACBL attempts to smooth ruffled feathers by providing a forum for disgruntled members. Therein we read letters about aged parents and grandparents, frozen in the past before all this artificiality took root, who can still play a mean game merely by exercising their common sense. Frank Stewart is their adopted champion. In his column he writes favorably of traditional rights and wrongs.

In the January 2010 Bulletin he writes, ‘discipline is having a reason to bid and being willing to pass when no bid is just right.’ The problem with that advice is that the nature of the bidding mechanism is such that one may not be confident at to what is the right bid, or even if there is a right bid. No system is capable of allowing one always to bid comfortably within strictly defined limits. Too many hands, not enough bids. So if you pass because no bid is just right, you’ll pass a lot more than you should. This advice caters to the conservative mindset that impedes action during our senior years.

Stewart’s predisposition is towards wait-and-see. He writes, ‘discipline is taking the action you know is best when a tiny voice in your head is urging you to take a flyer.’ This is a justification for passing when in doubt, but we all know that good advice is, ‘when in doubt bid one more.’ So, his advice might be reworded to the following, ‘discipline is taking the action you feel is best when a tiny voice in your head is urging you to underbid.’ Competent players should not imagine they will get doubled every time they overbid because their opponents can see through the cards. Incompetent players may not realize that what they are doing is highly dangerous. To underbid in competition planning to bid again if necessary is a perilous procedure that should be employed only when one wants to get doubled. Passing may not be safer than bidding but more hazardous.

The most dangerous actions are related to not bidding according to one’s prior agreements. Take for instance, a preemptive raise of partner’s suit defined as 0-6 HCP. The conservative player may decide to make a preemptive jump raise to 3♠ on 4 spades and some outside stuffing, for example, with ♠ QT76 ♥ K764 ♦ Q98 ♣ T9. This may seem safer than a bare 4 HCP, but it is not, because the opposition may pass and defeat 3♠. The red suit honors and the length in hearts make it less likely that the opponents will be tempted to indiscreet action, and it makes it less likely they can make a contract at the 3-level. A bad player will raise to 2♠, then bid 3♠ after the opponents compete to 3♣, even though the losing trick count is 9. So, another case of more is less.

The Human Factor

It is somewhat odd that the simple-is-best readers of the ACBL Bulletin have come to look at Stewart as their champion, as his methods are quite idiosyncratic. We’ll go back to the July 2007 issue of the ACBL Bulletin in which Stewart goes through the bidding of a successful slam hand giving us his thoughts along the way. His honesty is refreshing and much appreciated. In typical fashion Stewart passes a hand that others would open, then overbids to a slam that is made by astute declarer play. Strange as it may seem, his approach is based less on logic than on the tiny voice that tempts him into distorting his bidding practices to conform to his prejudices. It would be very difficult to deduce a general approach that his readers could adopt, although many bid in a similar fashion.

He admits his style requires having a bold partner, ‘as odd-couple partnerships work because the players exert a moderating influence on each other.’ Notice that the emphasis is again put on the suppression of natural impulses. This is objectionable to the logical mind, because the success of the bidding will depend on the character of the players rather than on the character of the cards dealt them. Exchange the hands and the auction would be entirely different, which Stewart admits. Yes, we know that in practice there are those who bid one way with Jane and another way with John. Adjusting to a partner’s personality is a practical approach down at the club, but there should be a general structure of constraints within which one can safely maneuver with any good player.

Convention Free Bidding

We shall study what Stewart’s tiny voice was whispering on the hand shown below from a regional pairs event. Looking at both hands one concludes that any reasonable method should allow the partnership to explore for a small slam in spades. We begin by considering bid by bid the natural auction on the right. Most players prefer to ask rather than tell, so to such players a natural system which promotes the free exchange of information between partners treated on an equal basis is an anathema. In theory a bidding sequence is an exercise in constrained optimization, the bidders revealing only what is necessary to reach the best contract. What the best destination may be depends on the route taken to get there. Some players obsessively hide their tracks.

With the hand shown below the HCP content is not a major factor, as the partners hold a mere 26 HCP between them. Distribution is the key. The losing trick count is the proper means by which to measure potential with shapely hands, and the bidding provides information with regard to the degree of fit. The losing trick counts and number of controls are given below the hands as a reminder that these are the critical factors.

If there is a fit, all should proceed smoothly, but if there isn’t, there must be means by which one can put on the brakes and bring the process to a grinding halt. The honor combination of ♣KQxx is a critical factor, so it is East who should have the final say on whether or not to bid slam. Now we shall examine each bid in turn to demonstrate that bidding this slam is quite easy when each player simply bids what he has.

 
1 ♦ Often the most important call is the first one. The question is this: can one open the West hand without misleading partner as to its strength? That is a matter of agreement as to what constitute an opening bid. The high card content, 10 HCP, is dead average, but the hand satisfies the Rule of 20. (HCPs + the lengths of the 2 longest suit = 20). The hand has 4 controls (Ace=2, King=1), 1 more than normal, and they sit in the long suits. What power it possesses is offensive, and that power is considerable as expressed in the loser count of 6. By losing trick evaluation an opening bid in a minor should have 7 or fewer losers, so in that respect West’s hand is quite promising. How good it is will depend on the degree of fit with East’s hand.

There is an 84% chance that West will find at least one 8-card fit with partner’s hand. West should not worry about the opponents entering the auction as one doesn’t anticipate being outbid. If the opponents have a super heart fit, 4♠ may be a profitable sacrifice. The greatest worry is that the 8-card fit is in diamonds, and that partner with a flat 13 HCP hand will insist on playing in a contract of 3NT. During the auction West hopes to reveal that his opening bid was based on shape, not power. No trump bidding has to be different from good-fit bidding, and this difference must be recognized.

Trying to avert playing in 3NT on minimal HCP strength, some would prefer to open 1♠ to put the emphasis on a major suit game, perhaps in a good 5-2 fit with diamonds functioning as surrogate trumps. This prejudice towards the majors is often observed at the table. Some live by it, to the detriment of their slam bidding in the minor suits. There is no need to panic. The primary need is to identify the different hand types.

3♠ The auction has proceeded smoothly and the spade fit has been uncovered. Most Easts have available the Fourth Suit Forcing convention by which they can bid 2♣ to ask a partner to provide more information without revealing their own intent. In that way one can subjugate a partner and take charge of the auction. It is a popular and sometimes necessary treatment in NT bidding. A major virtue is that it saves bidding space.

Slam bidding proceeds more smoothly when a partner can show a fit with a game forcing raise. ‘Support with support’ is the means by which a partnership switches priorities from high card to loser count evaluation. This one promises 4 spades and 7 losers or less. To use 3♠ as merely invitational means there must be an artificial bid (such as 2♣) available by which East can force to game. This makes for confusion.

4♦ With a 9-card fit in a 6-loser hand, West sees that slam is a distinct possibility. He can hope that East has something good in hearts to keep the defenders at bay. Because the major value of his hand lies in its shape rather that his HCPs, West can’t make a descriptive jump to 5♣ to show his void in clubs. He is short the ♥K which would bring the loser count down to 5. The solution is simple enough: reveal the good diamond suit and await developments. That encourages slam, but doesn’t insist on it.

We have reached a point in the auction where conventions can be helpful to further describe the opening bid, especially when opening bids can be light on HCP. Modern bidders at willing to give up on playing in 3NT once a major suit fit has been established, so they employ 3NT to investigate slam below the level of 4 of their agreed major. Holding powerhouse 2-suited hands they can employ 5♣ as Exclusion Blackwood. Think of Goldilocks and the 3 Bears. Hands with shape try 3NT (Mamma Bear) and those with power try 5♣ (Papa Bear). Without strong slam interest they may complete a description by cuebidding where their values lie, here 4♦ (Baby Bear), which is ‘just right’.

4♥ East is encouraged even though he has no support for diamonds. The ♦Q would be sooo nice. As West hasn’t given up on slam, and as he holds such good support for spades, East marks time with a cue bid in his previously bid suit. Some play this as ‘Last Train’, a non-descriptive bid showing slam interest – ‘I’ve got my bids, have you?’

4♠ In the natural auction West is content that he has indicated some slam interest and has shown where his values lie. This is a situation where he can legitimately make a minimal ‘just right’ bid at the end of a sequence that has accurately disclosed his values.

5♣ East is not ready to give up. A reflexive 4NT Blackwood won’t provide the material necessary to build a case for a grand slam. If West holds the ♥K, East expects to see a bid of 5♥ next. He needn’t decide about the grand slam (!) until all the information is in.

6♠ West has no new information to add to the mix. He can see that East must have very good spades, because West has promised just 4 on the auction thus far. He expects partner has hearts well covered and he may well have the ♦Q. There is enough encouraging news here to take a well-reasoned flyer at slam.

Frank Stewart’s Tiny Voice

We can take Stewart’s subjective approach as reflective of many who play a 2/1 structure where the limits that define the bids are expressed largely in terms of HCP. One does not assume a fit at the start. Rather one assumes one may have to defend or support partner in his attempt to bid and make a NT contract. One plans to make adjustments for shape as the auction progresses and fits are established.

Within such a system it is often easier to pass initially and bid aggressively once a fit is established than it is to open the bidding, raising expectations, thereafter trying to apply the brakes. The modern trend is better: open light and make provision for that by adding conventions, like Last Train and 3NT Slam Try, to set limits on one’s ambitions. Coming soon to a table near you: Reverse Two-Way Drury by an unpassed hand.

 
Pass or 1♠? Stewart admits he might go along with the current trend and open 1♠ if the spade suit had more stuffing. His partner would open the given hand. He makes a revealing comment on why he would not: if responder bids 2 H as a game force, he would have to raise to 3♥ on ♥542. That might lead to a bad 6 H contract with responder holding just ♥AK963. Exactly! That is why the single raise should be avoided on such a weak holding. It could happen that partner bids 2♥, but is this is likely? It is much more likely that partner has a fit in one of the long suits, or that his long suit is clubs.

Returning to the past glories of American bridge, Goren required at least Qxx and Schenken, J10x for such a raise. So one shouldn’t engage in the sequence that Stewart fears. This is a red herring. A 2-suited hand should be bid as a 2-suited hand. After 1♠ – 2 ♥ opener would have to bid 3♦. This is a poorly defined, strong sequence that has got too high and covers too many possibilities. The case is strengthened for opening 1♦ where the HCP are, thus preparing for a sensible and descriptive rebid. The spade suit is not going to be lost on this hand. So if one decides to open, as one should, the proper plan is to open 1♦ and rebid twice in spades. As we saw in the natural auction, partners can learn a lot without reaching the exalted level of 1NT.

Pass – 1♦? After a pass, Stewart responds 1♠, not 1♦, on the grounds that one bids the higher-ranking of 5-card suits. This is not necessarily so. There is a box on the ACBL convention card for players to check if they frequently bypass a bid of 1♦ over 1♣ when holding 4+diamonds. At one time this was a controversial practice that required an alert. The conservatives argued voraciously against change. Nowadays players follow the Walsh procedure in which one responds 1♦ with a good hand and 1♠ with a lesser hand. The West hand is within the category of a super passed hand that requires at least 2 bids for proper description. The quality of the diamond suit is a prime consideration, as well.

Pass – 1♠? The traditional response of 1♦ is more descriptive, more efficient, and less judgmental than a bid of 1♠. The preference to spades over diamonds is based on the assumption that the contract should most likely be played in spades. This prejudice distorts the auction unnecessarily. Bidding the better suit, diamonds, does not preclude finding a spade fit, it saves bidding space, and it conforms more closely to the expected distribution of HCP between the suits. A one-over-one bid should be forcing, even when made by a passed hand, but not all agree with this simple solution. Again, there is too much allowance being made for rare occurrences. It shouldn’t be necessary to jump around to show a normal hand.

4♦! Well, there you go, East did come up with strong support for spades and almost any system can be used to explore for slam whether the West hand is opened or not. Stewart disagrees with his partner’s splinter bid of 4♦ – it should be much stronger, he maintains. The East hand fulfills the requirement for a splinter raise if West had opened 1♠, namely at least 13 support points. After a pass, the requirements for a splinter raise go from a minimum of 13 to a minimum of 18, a jump of 5 points. This jump may be caused by a difference of 1 HCP in the West hand. The effect on the bidding system is one of instability where a small difference leads to a huge change in methods.

The justification goes that an opening bid has a lower limit of 11 HCP, but the response to 1♣ has a lower limit of 6 HCP, a difference of 5 HCP, so this difference must be reflected in the response structure. This is against the probabilities, as it is more likely the initial pass is closer to the 10 HCP mark than to the miserable 6 HCP mark. The opponents have passed, the opening bidder has 16 HCP, so it is reasonable to assume partner holds 1/3 of the remainder, 8 HCP. This is in the middle of the range 6-10 HCP for a passed hand, not at the bottom. Seen in this light, East should be encouraged. A splinter raise is justified, especially so when the trump holding is this strong. One should bid on the basis of what is most probable, not on what is minimal and least likely.

3♦? East need not stretch to a game forcing splinter bid, because there is available an non-forcing splinter of 3♦ that invites game. The general rule for recognizing a splinter bid is that it is a jump in a suit a level above what would have been a forcing bid in that suit. As a bid of 2♦ would be a forcing reverse on the part of the opening bidder, even opposite a passed hand, bids of 3♦ and 4♦ can be defined as splinters of differing strengths, 3♦ being game invitational and 4♦ being slam invitational. This distinction is a necessary consequence of the wide-ranging 1♣ opening bid.

When in doubt about these matters I go to an authoritative prime source, in this case Max Hardy’s Two Over One Game Forcing, which is the major reference for players in my area who are happy to claim 2/1 as their system. Here is his invitational splinter: ♠KQ84 ♠7 ♦A82 ♣AK975. After opening 1♦ and receiving a 1♠ response, opener may jump to 3♥ intending to pass responder’s 3♠ rebid. This resembles closely East’s hand – good spades, 16 HCP, sufficient controls, 5 losers, shortage in a side suit. Personally, I would be ashamed of myself if I left partner in 3♠ on this hand.

3♠? For Stewart East’s proper reply to the 1♠ response is 3♠, a limit raise. He comments, ‘players often use splinter bids with high-card values that are too skimpy. They succumb to the lure of finding the perfect hand.’ This rather implies that the bidder has made a decision to go to slam regardless. What if the bidder in a cooperative manner is merely describing a hand capable of making game most of the time? What if the bidder is merely revealing a promising loser count? In this case it is controls, aces and kings, and trump quality that are the major factors. A sideboard of queens and jacks may represent nothing of value, like macaroni on a buffet table laden with turkey and smoked salmon.

5♦ – 5♥? Stewart expects his partner to bid outside of the normal range, but he is bidding mostly on what East should have, not so much on what he thinks East does have. The auction is getting high, but his status as a passed hand allows for a slam invitation at the 5-level. What happens, I wonder, if East now bids 5♠ thinking, ‘well, I only have 16 HCP, after all, and partner expects 18 at least. My hearts have gaps and the minors don’t appear to fit well. 5♠ most accurately describes my hand in the context of the bidding thus far.’ Will this sudden conversion to honest evaluation keep Stewart out of slam?

6♣? Recall this statement: discipline is taking the action you know is best when a tiny voice in your head is urging you to take a flyer. Can one say that the above auction demonstrates this principle in action? As he has bid against the grain by never showing length or strength in diamonds, Stewart feels he is justified in master-minding the final contract. If slam should fail, who do you call to chase away the gremlins? Slambusters.

Hold that call – slam makes! It is fortunate that a club is led rather than a heart. One of the benefits of bad bidding is that the opponents are in doubt as to the best lead. The ♣K is covered by the ♣A and declarer discards a heart instead of ruffing. A second heart goes on the established ♣Q, so Stewart avoids losing 2 hearts. But note that this slam makes because East holds the ♣KQxx opposite a void. Has the bidding revealed this situation? If not, is the slam justifiable? Well, one could argue that what East doesn’t have in clubs he must have in hearts. East’s bid of 5♥ is a strong indication he holds the ♥AKxx. That may not be so helpful, as declarer may have to ruff a diamond with the ♠ Q to make way for a triple squeeze. The luck in the slam resides in a happy ending to a disorderly bidding sequence rather than in the subsequent, careful play.

The Case For Side-Suit Quacks

We are left with an account how a conservative bidder thinking primarily in terms of points was able to reach a makeable slam on just 26 HCP. Granted, it was only a major suit slam. One might observe that the ♥J is a wasted value. The ♣KQxx could be reduced to the ♣KJ10x and the slam might still make in the same way after a lead to the ♣J and ♣A. We wouldn’t want to reduce the ♠ Q to the ♠ J as trump solidarity is always a key component in bidding slams. With this trimming down the losing trick count would still indicate bidding the slam, although the chances of making it are greatly reduced.

The losing trick count strips the hand down to the bare essentials. It caters to a ‘nothing wasted’ condition. The possession of the ♣Q removes the need for the ♣A to be well placed, provided the lead is a club. However, the chances of a club lead are reduced, if dummy doesn’t have the ♣Q because it would be dangerous to lead away from a holding of ♣Qxxx. Declarer is more likely to receive a trump lead. Dummy’s not holding the ♥J would increase the chances of receiving a threatening heart lead.

A great advantage of holding side-suit quacks is that the opponents don’t have them. That makes their defence more difficult; defenders may not have an obvious lead; they have to envision a particularly favorable distribution of unseen honors. They may lose a tempo by adopting an overly passive approach. Although slam might be made on a double dummy basis without the sideboard quacks, in practice it’s going to be helpful to declarer when he tries to manipulate the defence. Thus, the notion that one needs a considerable number of HCPs to make encouraging raises is a conservative one that provides for extra chances that may not be immediately obvious and that may not be necessary. The system is not geared towards finding the ‘perfect’ result.

The realization that ‘what I have, they don’t have’ is also important in competitive bidding, as noted at the beginning. When deciding whether or not to push on, one has to gauge the effect on the opponents’ bidding. They are less likely to take the push if they hold secondary honors in your suit and you have secondary honors in theirs. It becomes too easy for them to pass and let you play in a bad contract.

My Year So Far

Most think it’s the beginning of a new decade, but they’re wrong. The year 2011 is the beginning of a new decade, in the same way January 1, 2001 was the beginning of a new millennium. The decades run from 1 to 10. As Ibsen said, ‘the majority is always wrong.’ Take bridge as an example: the vast majority prefers 2/1 to Precision, but they’re wrong, dead wrong. Nevertheless, in deference to my partners my primary New Year’s Resolution is to ‘go with the field’. The big question is, ‘where is the field going?’ The answer: downhill. Before we expound on that, I’ll give you my predictions for 2010.

Barack Obama will forecast a budget surplus for the year 2020.
George W. Bush’s as-told-to memoirs will top the hardcover fiction list.
Vancouver will make money on the Winter Olympics.
Investment bankers will agree to cut back on huge, unwarranted bonuses.
The Chinese economic bubble will burst.
Toyota will return to making all its cars in Japan.
Martha Stewart will teach Hillary Clinton how to bake cookies.
Sarah Palin will star in the hit show Desperate Hockey Moms on Fox.
Tiger Woods will appear on the David Letterman Show.
Silvio Berlusconi will go to prison, but only to visit old friends.
Rihanna will crash and burn.
In the Iraqi elections the Peace and Prosperity Party will come in last.
Karzai will be accused of intercepting bribes intended for the Taliban.
The Pope will visit Mecca.

I Pass the First Test My first game of the year was with Dorothy who long ago gave up Precision in favor of SAYC. Behaving myself and going with the field I scored 48%. I caused one bad board because I mistakenly believed with 2NT the field played Puppet Stayman. I had asked Dorothy if she would add it to our card, and she agreed, either because she’s a good sport, or she thought it would never come up. Sorry, Dorothy. Late in the game I♣ was dealt a hand with 20 HCP and a 5-card spade suit ♠ AQT96, 5-3-3-2 shape, a perfect setup for Puppet. Wrong! Partner held 3 spades and zero points. I played too fast – my euphemism for screwing up – when I led out the ♠ A, ending up down 3. I imagined I might have some company, but, no. Everyone with my hand opened 1♠, making 80 or 110, so a well deserved bottom for me. I was buoyed by the hope that the field I was going to emulate was not as dumb as I feared.

On seeing the scores on the travelers the nice old granny on my right cackled, ‘That’s the kind of top I like.’ It was on the tip of my tongue to reply, ‘I prefer the tops I earn myself’, but I didn’t say it. My new rule of life was not, ‘do unto others’, but ‘turn the other cheek’. I had found it doesn’t hurt to thank partner for a totally inappropriate dummy, or to congratulate the opponents on their lucky result. This was the New Me. Being virtuous feels so good it’s a wonder more don’t practice it. I wish there was a pill for men to promote a surge in friendly feelings towards those of female persuasion that lasts up to 3 hours without affecting performance.

The Old Me Returns A week later I thought to pick up more ideas on which direction the field is currently leaning, so I went early to the club to sit in on a lecture given by our local 2/1 expert. There are always attempts to explain why 2/1 should work, but doesn’t. He often speaks of ‘lying about your hand’. It’s nonsense, of course. Don’t make a liar of everyone; if the demands of the system require that you bid that way, it can’t be a lie. Today’s topic was opening 1NT on a 5-card major within a 5-4-2-2 shape. When at the end of his lecture, he asked, ‘any questions?’ I kept mum. I have lots of questions, but they always come out as declarations about the superiority of a Big Club approach.

My last surviving Precision partner, Harry, and I began play against a pair of old dears. We announced happily that we play Precision, upon which the one on the right, festively bedecked with decorations that wouldn’t look out of place on the municipal Christmas tree, turned to me and declared severely, ‘Bridge is a game, and I come here to have fun.’ To which I replied, ‘Good for you; I am sure there are others here who think the same way.’ Harry laughed. So there went my first New Year’s Resolution which had been, ‘Be 100% tolerant of any asinine comment directed your way.’ It might have been one of those I’ll-hate-myself-in-the–morning moments, but it turned out that it wasn’t. Frankly, I was already tired of the New Me. The Old Me returned to score 67%.

I want things to be done the proper way. Does that make me an elitist? Yes. When I look in the mirror I see what could easily be mistaken for a mean, old geezer. So what? At my age no one looks like Howdy Doody. Don’t I have a right to have fun, too?

After 10 straight pluses Harry and I faced the lecturer. He let Harry play in 2♠, plus number 11, after which this hand came up.

 
When Harry bypassed 3NT I had a fair indication that we had little wastage in diamonds. The loser count led me to conclude that 6♣ would be a good contract, and a tiny voice whispered in my ear, ‘bid it.’ Then I remembered my primary New Year’s Resolution, ‘go with the field’. So I counted up my 11 HCPs, added 19 and bid 5♣ praying something might go wrong to hold me to 11 tricks. My prayers went unanswered. I had missed the mark by a long shot as the field was in 3NT going down. Missing a cold minor suit slam was worth 8 out of 11 matchpoints. I blame Harry for not opening 1♣ with his 7 controls. I note once more that the field commonly bids 3NT with a singleton in the suit about to be led. Is that an unforeseen consequence of 2/1?

My First 2/1 Session Despite my studious preparation, or perhaps because of it, my first 2/1 outing with Carl was disappointing. Our local expert was lecturing as usual, and last week’s hand was discussed as our fortunate result against him apparently still rankled. After a roundabout analysis he suggested that I should have bid 4♣ Gerber, then 6♣ when the required 3 aces were revealed. Well, that works on that hand, but what about a possible 4-4 spade fit? Isn’t 2/1 all about reaching spade contracts? My reading suggests they even seek out the juicy 4-3 fits. I kept my peace then, but here are my thoughts now.

The Proliferation of Conventional Bidding has at its source the need for precise information, however, no one wants to give out information, as we have been told repeatedly that giving away too much information is detrimental. So the field goes by guesses. The Losing Trick Count is a superior way of guessing, however, there is this fundamental need that can’t be denied. The 1NT bid has limited partner’s hand. Responder must make a decision. How can he find out about controls? At the table I could have lied about my hand and bid 4♦, but where does the lying stop?

Another convention made to order: the bid of 4♦ should be defined as RKCB-for-Bob. Problem solved. We’d all love to have a convention named after us, but I am probably not in the first thousand Canadians to suggest this. It doesn’t come up very often, so we can double the frequency by extending its application to the diamond suit as follows: 1NT – 2♣; 2♥ or 2♠ – 3♦; 4♣ – 4♦, where 4♣ agrees diamonds and 4♦ is RKCB-for-Bob in diamonds. It’s easy to remember as a natural 4♣ bid is highly unusual.

The Weak Hand Decides I resolved never to comment to Carl after a bidding failure that it would have been easier playing Precision as we used to do, but, like my other New Year’s resolutions, this one didn’t survive the month of January. Most don’t subscribe to the idea that the stronger hand decides. Imagine you are driving carefully through town and the person in the passenger seat is constantly grabbing at the wheel. That’s how I felt all day. One of the features that attracts the masses to 2/1 is that either partner can decide at any time what the final contract should be. Indeed, they may slant the early bidding towards reaching what appears at first glance to be the best spot. If that’s how they behave, how can they imagine they’re playing a system? My resolve to button my lip was sorely tested on the 3rd board:

 
I have read that one shouldn’t open 2♣ with a 2-suited hand. My excuse was that as we were playing control responses, I would immediately obtain useful information that could allow me to place the contract in the best spot. My heart leapt with joy when partner showed a big hand. A tiny voice whispered in my ear, ‘seven clubs’. As Frank Stewart tells us in the ACBL Bulletins, we shouldn’t listen to the tiny voice which tempts us into indiscretions, but here I was again, shamelessly out to beat the field. My 3♥ bid was a firm step forward, and I felt on the brink of a great success. Then I became exposed to one of the most attractive features of 2/1, the take-charge jump to 4NT. This shook me momentarily and I thought of bidding 6♦ to show the void, but we certainly had no agreement, so I kept it simple, declared my key cards, and held my breath.

Void or no void, 7NT, if it had made, it would have been a top, no argument there, but then so would 7♥, a contract that has the great advantage of actually producing 13 tricks. My unrepentant partner argued that he didn’t expect my hearts to be so good. Of course, how could he know? He missed the point entirely. How the field managed to avoid a cold grand slam, I am not sure. I’ll have to study the 2/1 system more carefully before I can state it authoritatively, but I think most began with 1♥ followed by a Jacoby 2NT raise putting the weaker hand in control. Opener showed shortness in diamonds and responder happily bid 4♥, which he was always going to do, regardless. Opener stopped in 6♥ thinking he needed the spade finesse. This is really stupid. Can it be true?

My First 2/1 Top Our first zero came early, so I was only marginally peeved. The chill turned to thrill a few hands later when against my expectations 2/1 produced a good minor suit slam. I initiated a dreaded reverse, opening on shorter clubs planning to bid longer diamonds later to show a strong hand while keeping my shape open to question. One must keep the weaker hand informed about the HCPs, just in case he wants to spring to 4NT. I was surprised when partner responded 1♦, as my diamonds were not only strong but plentiful. This bidding a minor over a minor is very significant in 2/1 as it suggests no interest at all in spades.

 
The old granny cum witch on my right admonished us for not alerting that we no longer played Precision where 1♣-1♦ means something entirely different. Fair enough, but I wonder if such an alert is required under the ACBL rules. After 6♦ was seen to have scored a clear top, partner politely commended me for my successful bid.
‘Pretty standard’ I replied, ‘you had to have a good hand to reply 1♦ like that.’
‘Well, actually, I had only 9 HCP, and was hoping you would bid a major.’
‘That’s the kind of top I like,’ I commented cheerily. I have a long memory.

So here was my introduction to yet another attractive feature of the 2/1 system as played by the field: you don’t have to follow it if you don’t feel like it. Once again a player opted to bid 3NT with a singleton, but this occasion was extraordinary, I think, in that he knew his partner held a singleton, too. I suppose at the other tables the auction went 1♣ – 1♥; 2♦ – 2♠; 2NT – 3NT, illustrating the method by which the weaker hand takes charge through the application of the nebulous Fourth Suit Forcing. Neither partner would know the other held a singleton, but then neither would the defenders. The consensus is that either you get very lucky, or you have lots of company.

Although it was very much in my mind, throughout the game the word ‘Precision’ had not audibly escaped my lips, but a lip reader might have caught it slipping through the cracks once or twice. That changed on the very last hand. I held a very attractive 2♣ bid with 22 HCP. Eight controls represent a very potent holding, so a rebid of 2NT can be a slam killer, but just how to avoid it I wasn’t sure. My concerns were alleviated when partner bid 2NT first, announcing 4+ controls. I could immediately place him with the ♠AK and ♣K. Ah, Science! The time was ripe for the strong hand to take control.

 
Prospects for a grand slam were good, to say the least. I tried to get more information from Carl, but it is like pulling teeth. When all he could contribute were 2 minimal bids showing spades, I cautiously jumped to 6NT feeling that told the story. The doubleton ♦AK detracted somewhat from the hand. If Carl held the ♣Q along with the ♣K he might have supported my phantom clubs. There was no tiny voice urging me to bid 7NT.

Declaring in 6NT holding 2 singletons might upset some, but not Carl. He was upset because I hadn’t supported his spades. Consequently he adopted an unusual line of play under a delusion of wish fulfillment. Winning the diamond lead in dummy, he cashed the second diamond, played to the ♣K and attempted to ruff a diamond, carefully choosing the ♠Q to guard against an over-ruff. Not an optimum line even if spades were trumps. He had played too quickly. It was then I said a bad word followed by what I had resolved never to say, and more. I refused to take responsibility.

Taking 13 tricks off the top 6NT would have scored a shared top with the field stopping in 6♠. I have so much to learn that the thought of bidding 6♠ had never crossed my mind. I suppose others bid 2♣ – 2♦ (waiting); 2NT (22-23 HCP) – 3 (transfer); 3♠ – 6♠. The weak hand would be happy to be in a slam with just 32 HCP.

Competitive Bidding In competition 2 over 1 is out and 3 against 1 is in. Not only do you compete against the opponents, you also compete against your partner. When the hands are approximately of equal strength, and sometimes even when they’re not, the auction becomes a game of musical chairs for sedentary folk, where the trick is not to be left standing when the bidding stops. As everyone bids on nothing there is little danger of being doubled in the early stages, but partner is always a danger. An over-zealous partner may raise twice so you end up in a really bad contract. The principle skill in declarer play lies in keeping the opponents from spotting the killing defense during the first 3 rounds of play, thereby escaping from an adequate penalty. Usually I hope for a lead in an unbid suit and usually I get it.

Surprisingly, if Carl had got those 2 slams right, 2 tops instead of 2 bottoms, we would have won the game by a big margin. It doesn’t make sense that we scored so well when we bid so badly. Studying the results is like reading an Agatha Christie mystery: the characters are familiar, their actions are suspect, their motivations are questionable, and the ending is unbelievable – but it’s highly entertaining while it lasts. Maybe dear, bridge-playing Agatha was more of a social realist than critics have given her credit for.

Bridge Play and Statistics

In our previous blog we discussed a virtual experiment involving 360 school children playing a 4-door Monty Hall game. We extended of the application of the underlying principles to the selection from a club suit consisting of 3 of the 4 cards ♣Q-8-6-2.

The box on the left below indicates the possible combinations when the LHO holds 3 card and the RHO, 1. The first column represents the card held by the RHO. Each line represents 90 samples, 360 in all, so the initial probability that the ♣Q is held by the LHO is ¼ . The samples are presented to 360 bridge players who are asked to imagine themselves defending a 3NT contract and to choose a card other than the ♣Q. Thus, from line 1 the 90 players have a choice of 3 cards, whereas for the other lines the other 270 children can choose from 2 cards only. In an ideal experiment the players would choose the spot cards equally at random, so as to minimize the information transmitted to a declarer. The expected numbers of ballots are in the middle box and the resultant conditional probabilities are shown in the box on the right. All very nice and regular.

 
With regard to human behavior it is inappropriate expect perfection. Inevitably one encounters natural variability. There is always an oddball in the crowd (maybe it’s me!) To the extent that a statistical study can be thought of as being perfect, it is with regard to the conditions under which the study was conducted rather than to the results obtained. Even the conditions of an experiment may be questioned. Why 3NT? one may ask. What does the rest of the hand look like? Let’s not get sidetracked. The relevant question here is: why assume equally probable choices?

The Maximum Entropy Principle

In the 19th century applications of statistics were condemned by those who prefer to think in terms of causes and effects. Pierre-Simon Laplace (1749-1827) caused a stir when he stated publicly to Napoleon that he had no need for divine intervention in his explanation of celestial mechanics. This amused Bonaparte but subsequently angered theologians and Newtonian scientists who maintained that some unseen hand was required to keep everything eternally rotating. With regard to statistical inference, Laplace maintained that as the sun had risen regularly for 500 years, he was willing to give odds of 1,826,214 to 1 that it would do so on the following morning as well. Some have taken this jest seriously, and have continued to argue about causality and such.

More relevant to bridge (where the hidden hand is an integral part of the game) is the Laplacian concept that all possible unknown conditions are equally probable. Metaphysicians have argued that if nothing is known about conditions, they could just as easily be assumed to possess any probability distribution one might wish to assign. Modern information theory has given us this explanation. Maximum ignorance concerning a set of conditions is a state of maximum entropy in which all probabilities are equal. If some knowledge is made available concerning these conditions, their probabilities must reflect this new knowledge, and so are equal no longer.

So we come to the play of the cards at bridge. By the time the opening lead is made, much information has been conveyed through the bidding that will affect the various probabilities. A declarer should adjust the probabilities in accordance with the information he has received on this particular hand as well as with his general knowledge of how the game is played. Returning to our survey concerning cards led from ♣8-6-2, the results may indicate that the choices are equally probable, but this is true only in a statistical sense. Some players will lead the ♣2 (low from odd), others the ♣8 (top of nothing) or ♣6 (MUD). Each lead is informative as there is a deterministic rule behind it, if declarer takes the time to look at the back of the convention card. The information is degraded to the extent to which players will deviate from the stated rules. The choices would be maximally uninformative if the opening lead were chosen at random every time. In summary, one does not play against everyone at once, but against one pair at a time.

The situation is different when a defender is required to follow to a lead by a declarer. The defender may choose to play low cards at random in order to reduce the information conveyed. The statistics of these plays are relevant in the analysis of card play in a way that a survey of opening leads is not. For opening leads, the statistics of opening leads tell us how many prefer to play MUD, the least informative of the 3 possibilities. As for following to a declarer’s lead, the statistics show us how random are the choices from insignificant cards. There is an essential difference.

The Testing of a Hypothesis

When one wishes to devise a statistical test, one must first have in mind what one is attempting to discover. The conditions of a test should be tailored with a particular question in mind. Assume we have devised a scenario discussed above where the LHO holds 3 of the 4 club cards and it doesn’t matter in any practical sense which insignificant card he chooses. Does the probability that the ♣Q is held by the RHO remain unchanged regardless of which low card appears on the first round from the LHO? In other words, in this situation do players choose from their low cards equally at random (our null hypothesis) or is there a bias? We collect the results from the 360 bridge players in the manner indicated previously and perform a test of the results to see to what degree the null hypothesis can be said to be confirmed. Here is a set of results we might obtain.

 
There are methods one can use to discover to what degree one may say the results are obtainable when taken from a uniform distribution of choices. In total we have 360 choices represented of which 107 were of the ♣8, 126 of the ♣6, and 127 of the ♣2. The expected number was 120 for each. The null hypothesis that each card is equally likely to be played can be accepted at the 25% level, meaning that such variation from the norm would be generated by a random sampling of 360 trials more than 25% of the time.

Of course, the numbers represent not just one experiment but 4. The results in the first line give rise to suspicions that the ♣8 is less likely to be played from a combination of ♣8-6-2 than either the ♣6 or the ♣2. The variation evident in this mix would occur from a random sampling of equal distributions less than 10% of the time. One might conclude that more experiments are required for this combination in particular.

Alternatively, one might change the assumptions for this line. The hypothesis one is testing should not be formulated from the data themselves, for in that case one would always get a good fit, but must be proposed before the experiment is performed. Let’s assume the expected numbers are 15, 30, and 45, respectively, our guess expressed in the previous blog. The goodness-of-fit for this hypothesis is very good as variations greater than this would occur in more than 50% of samples of the same size. The biased choice model is more acceptable than the unbiased choice model.

Our Bridge Experiences Our encounters at the bridge table, successful or otherwise, represent but a very small sample of experiences from the great experiment which is Life. If it is difficult to draw conclusions from a controlled experiment, how much harder it is to do so from the chaotic conditions we encounter at the local bridge club. Our results, good or bad, are subject to a natural, random variability. Some impatiently attempt to ‘time the market’ by taking huge risks, thus increasing the variability, while others, akin to bond holders, ride out the storms with stable, but uninspired, adherence to standard textbook advice. Most tend to ‘go with the field’, which involves guessing the actions of the majority of the surrounding players. This acts to widen the statistical base, and has the advantage of minimizing variability at the cost of not attempting to maximize gain. Rarely does an expert play for averages; the late Paul Soloway said he never did so. Ideally one should prefer to employ methods based on the sound principles of probability theory, tempered by experience, without the egotistical expectation of always being right.

Four-Door Monty – A Recapitulative Continuation

In our previous blog we maintained that the easiest way to understand how probabilities work at bridge is to imagine a simpler game of chance where choices are presented to a great number of participants. In this segment we return to our scenario where schoolchildren are used as contestants in a 4-four Monty Hall game. We assume they adhere strictly to our probabilistic model concerning how choices are made, although in practice we can never expect such perfection. The aim is to demonstrate simply the concepts behind Bayes’ Theorem and indicate how they apply to card play.

The box on the left below indicates the four doors behind one of which sits the prize. The bold letter denotes the door which hides the prize. Each line represents 90 samples, 360 in all, so the initial probability that the prize is hidden behind a given door equals ¼ for each door. The samples are presented to 360 school children who are asked to choose a door from B, C and D that does not hide the prize. Thus, from line 1 the 90 children have a choice of 3 doors, whereas for the other lines the other 270 children can choose from 2 doors only. In an ideal experiment the children choose exactly in accordance with a rule which makes one choice no more likely than another.

 
The symbol P(X | Y) represents the probability that the prize lies behind Door X after Door Y has been opened to reveal the prize is not behind that door. The probability of the prize being behind Door A is ¼ regardless of which door is chosen by the participants. Presumably is what commentators have in mind when they say, ‘the probabilities don’t change’. The message is that P(A), the a priori odds of the prize being behind Door A, equals P(A | Y) regardless of which of the 3 is Door Y. The validity of this statement depends on the way the choice of doors is made. If we introduce a bias in the choice, the probabilities P(A | Y) vary as indicated in the example from our previous blog where the doors are painted in different colors to induce various degrees of preference from unsuspecting juveniles, as is shown below.

 
Under the stated rules of preference, P(A | B) and P(A | C) are equal even though they represent vastly different numbers of choices, and they are not equal to P(A).

The sum of the columns must equal 1. That is represents the fact that after Door Y is opened the prize must sit behind one of the other 3 doors. The sum of the rows is not equal to 1. We note that the sum of the choices along a row must equal 90 as a consequence of our choice of presenting 90 prizes for each door. (We could have biased that as well.) In order to recover the a priori probability of the prize being behind Door A, one must factor in the probabilities of each door being chosen. Thus,

P(A) = P(A | B) · Q(B) + P(A | C) · Q(C) + P(A | D) · Q(D) ,

where Q(X) represents the probability overall that Door X would be chosen. For an unbiased choice of doors, Q(X) is the same 1/3 for all doors. For a biased choice, the Q(X) vary in such a way that the a priori odds, P(A) through P(D), are recovered.

The above arguments are intended merely to present a reasonable scenario as to how the various probabilities apply in a Platonic experiment. Mathematicians would prefer a more rigorous (and obscure) argument. A statistical approach will be considered later, but for now, we shall pass quietly on to the consideration of how these ideas apply to bridge.

Application to Card Play

We can translate the door matrix into a configuration in which a suit played by declarer has 4 missing cards, ♣Q-8-6-2. It is assumed the LHO holds 3 of the 4 cards. For this trial the participants play the role of the LHO and are asked to choose any card they wish excepting the queen. We aim to estimate the probability that the RHO holds the ♣Q. First, we assume the spot cards are chosen at random. We isolate the ballots that have chosen the ♣8.

 
Given that the LHO cannot choose the ♣Q, we arrive at the result that the ♣Q sits on the right in 30 cases, and on the left in 90 cases, so the odds of the ♣Q being on the left is unchanged from the probability associated with the initial distribution (the deal). Of course, the chance of the ♣8 being on the right is now zero. If we use the same ballots to calculate the odds that the ♣2 is on the left, we find the probability has changed from 3:1 to 5:2. As a bridge player one is more interested in the location of the ♣Q than that of a spot card.

Suppose we next impose upon the participants the severe restriction of always playing the highest spot card (giving false count.) The numbers of ballots on which the ♣8 has been chosen are as follows:

 
The odds of the ♣Q (or the ♣2) being on the left are now 2:1, so the odds on the location of the ♣Q have been altered by the rules which have greatly restricted the number of plausible plays. At the other extreme in the scale of permissibility, each card can be chosen without restriction as with the dealing of the cards, which rule leads to the following configuration.

 
The number of plausible plays is the same for all remaining conditions. This is reflected in the number of vacant places. The process began with 3 vacant places on the left and 1 on the right. After a card is played by the LHO, the vacant places are reduced to 2 on the left and 1 on the right. Thus, when the plausible plays have reached a status of equality between the remaining conditions, the vacant place ratio under the assumption of a given split equals the probability of the location of a particular missing card. Note that this same property is possessed as well by the results obtained under the previous rule, because, there too, the plausible plays attained equality.

Distinguishable Spot Cards

Spots cards cannot be said to be indistinguishable. Fans of Right Through the Pack have happily recognized this for a long time. Given a choice of cards from ♣8-6-2, most bridge players initially would play the ♣2 without giving the matter much thought. Some seeking to deceive would try the ♣8. Those with an innate love of obscurity are drawn inexorably to the ♣6. Taken together this unequal treatment of the spot cards alters the probabilities. Let’s suppose the ♣8 appears with a frequency of 1 in 6, the ♣6, 1 in 3 and the ♣2, 1 in 2 times. Here is the matrix for the play of the ♣8.

 
Under the assumption that the spot cards are indistinguishable in the 3 combinations of Q-x-x, when the ♣8 appears the odds become 6:1 that the ♣Q is on the left. When the ♣6 appears the odds are unchanged from the a priori odds of 3:1. When the ♣2 appears, the odds are 2:1. So, when the ♣2 appears we are less confident than initially that the ♣Q is on the left, whereas when the ♣8 appears we are more confident in that regard. When the ♣6 appears we are left to puzzle its significance. Here are full results in matrix form with the first column representing the cards held by the RHO.

 
Plausible Plays In situations where the card choices are of equal probability, the number of plausible plays equal to the number of choices. Therein lies the origin of the terminology. In the combination of ♣Q-8-2, it is assumed the ♣8 and the ♣2 would be chosen equally. There are 2 real and immediate alternatives, and the probability of either being chosen is the reciprocal of the actual number of apparent choices. However, the probability of appearance of the ♣8 from ♣8-6-2 is 1 in 6. In this case there are not 6 immediate alternatives, there are only 3, and the various choices have differing probabilities of being chosen. However, in a virtual experiment choices are made repeatedly as many times as we specify. The expected number of ballots expressing the choice of the ♣8 is one-sixth of the total number cast, in the above case 15 out of 90. The expected number of ballots expressing a particular choice is the total number cast divided by the number of ‘plausible plays’, where fractional plays are allowed. If the ♣2 was expected over a long sequence of choices to be chosen at a frequency of 2 out of 3 times, say, the effective number of plausible plays would be 1.5, which is the reciprocal of 2/3. The ♣2 is chosen on average once out of every 1.5 opportunities.

The Enlightenment of Scrooge

“Bah! Humbug!”
- Ebenezer Scrooge (1843)

Charles Dickens, were he alive today, could teach us something about becoming imbued with the proper spirit in which to approach bridge. Recall how Ebenezer Scrooge became enlightened on a Christmas Eve long ago. The Old Scrooge would have been in line for the Nobel Prize for Economics, if it has existed in his time, before he disqualified himself on Boxing Day by raising Bob Cratchit’s minimum wage unnecessarily without the prerequisite upward pressure from the labor market. Until then some economies had thrived without the need for Christmas; any time was a good time to buy a girlfriend a diamond bracelet. In our more enlightened times, it is acknowledged far and wide the New Scrooge was the better man, and the better economist to boot, a pre-Keynesian, no less. As with Scrooge and Christmas so with those who say Humbug! to Bridge Probability. Yes, one may struggle on miserably without it, but how much richer our lives become when we can enjoy its benefits to the full.

Humbuggers deserve a night visitation from the three spirits of Probability: Probability Past, Probability Present, and Probability Future. The Ghost of Probability Past will speak of what cannot be changed and is forever fixed, the a priori odds with which every hand begins. One awaits a new hand with great expectations, but, alas, some times as with poor Fan, one is dealt a Yarborough. Furthermore, the Ghost will recall the numerous hands when Scrooge played against the odds and went down in a contract that was destined from the beginning to succeed. This will be his final warning, ‘don’t be a Fezziwig who couldn’t adapt to changing circumstances, use conditional probabilities.’

Probability Present will show the partners Scrooge threw away by a stubborn refusal to change his ways. At this year’s Christmas party he will see his ex-partners having a good time without him. Some are obviously happy with their new partners, others, like dear, sweet Alice, not. Her bridge seems to have improved remarkably since their breakup. The Spirit will tell him, ‘the past is ever lost, the future is beyond reach, so all you have left, Scrooge, is the present. Learn to live in it. Take heed and fare thee well.’

Lastly, Probability Future will speak of the uncertain times to come when one must speculate as to what is likely to happen, good or bad but mostly bad, if one keeps going along the same track. He will pose questions like, ‘if I play ace-king and a low club, what are the chances I can catch an unwary Fred in a ruff-and-sluff endplay?’ He will let Scrooge in on a future post mortem where the discussion of his abilities becomes pretty frank when he’s not there to defend himself. He may say to the Spirit, ‘despite what Mrs Dilber says, I’m sure I would always do what was proper at the time,’ but that just doesn’t cut it in the face of the harsh comments of the accusatory team-mates.

Hopefully, the Ebenezers of the bridge world will have a change of heart, learn the basics of probability, endeavor to be sympathetic to their partners, and live happily ever after in their new-found state of self-awareness. As Tiny Tim might say in conclusion, ‘God bless us every one, even those unfortunates who don’t play bridge.’

Experts Prefer Endplays

In the December 2009 issue of the ACBL Bulletin Mark Horton, editor of Bridge Magazine, gives this advice to the readers: ‘As declarer, eliminate a suit if you can even if you don’t know why you are doing it. Good things can happen.’ There is no warning label; Horton leaves it to the reader not to get carried away and to use some common sense in its application. I like that. Another good piece of advice would be, ‘always make the play that has the greatest probability of success’, but here the difficulty is that a declarer may not know which choice fits the description. That is where Jeff Rubens comes in. In his recent book Expert Bridge Simplified, Rubens presents the reader with methods that allow a player to sift out the plays that stand the best theoretical chance of being successful, always assuming that the player is capable of the necessary rapid mental arithmetic. This book is on my Christmas list.

The following deal as presented by the ACBL reviewer, Paul Linxwiler, combines these two pieces of advice as it involves the possibility of a triple elimination with the estimation of the probability of success. Will Horton’s advice prove its worth on careful examination?

 
West reaches slam in spades and receives the opening lead of the ♥10. Going to dummy in diamonds to take a winning spade finesse would assure the contract, but that play has a probability of success slightly less than 50%. The alternative is to play the ♠A, hoping but not expecting to drop the ♠K, then cashing all the top side winners before end playing a defender who started with doubletons in both black suits. The central question is how likely is it that a player holding the ♠K doubleton also was dealt a doubleton club. Rubens points out that the latter play should be chosen as the success rate is about 54%.

Linxwiler agrees, but is at pains to point out that for most readers it may be too close to call. The best approach for many may be to save energy, take the finesse, and move quickly to the next hand. I object to that. A margin of 4% is not too low to matter, and good players should attempt to make the distinction at least part of the time.

The simplest advice one can imagine with regard to going with the odds is Bob’s Blind Rule: always play for the most likely splits possible given what you know at the time of decision. How much easier can it get?

Declarer controls a division of sides 10=4=4=8 (all even), so the defenders hold a sides of 3=9=9=5 (all odd). There has been no interference and the opening lead is a nondescript ♥10, so no alarm bells are ringing, and there is no reason to assume that the suit aren’t splitting normally. Let’s look at pairs of hands composed of the most even splits in each suit, which are also the most probable combinations on the evidence so far.

 
Condition IV is less likely that Condition I by a factor of 1/3, so we ignore the possibility. For Conditions I through III the odds are 2 to 1 that the doubleton spade is in the same hand as the doubleton club. The obvious reason is that there are more possibilities when it is not required that both red suits split 5 in one hand and 4 in the other. Therefore, under Bob’s Blind Rule, declarer should adopt Rubens’ suggestion: play to the ♠A and if the ♠K doesn’t drop, play off the side suit aces and kings (as Horton suggests) then exit with a second spade hoping the winner has no more clubs. Note that it doesn’t matter which defender has the double black suit doubletons. Of course, one cannot say the odds are anything as good as 2 to1 in favor, but it appears likely that this sequence gives better odds than an initial trump finesse.

Let’s look at the suit combinations that lie at the bottom of this simple method. The first critical move is the play of a low spade towards declarer’s hand to which South follows with a low spade. What are the probabilities at this critical moment of first decision? By ignoring the implications, if any, of the red cards played to the first two tricks, one may use the initial vacant places as a rough guide to the number of combinations. These are as follows:

 
The weights are the relative number of card combinations that accompany the given splits in spades. Next we shall consider the play of low trump to which South follows with a low card. If the missing spades are denoted as K,u, and w, and the spade played is card u, this is the situation at the time of decision:

 
To obtain the current odds the combinatorial weights must be adjusted through division by the number of plausible plays available in the play of card u. If South held both cards u and w, he might have played card w instead, so the probability that he would play card u is halved. Now one can calculate the probability of the finesse winning by comparing the number of combinations for which the finesse wins (FW) to the number for the finesse losing (FL) or the drop winning (DW).

FW = 26 + 11 = 37; FL = 26 + 13 = 39; and DW = 13.

The finesse has a winning percentage of 49%, the drop, 17%. Thus, if it were just a matter of finesse versus drop, declarer would finesse, however, as the saying goes, 2 chances are better than 1. The probability of success for the subsequent endplay is the product of the probability of king now being singleton (52/76) multiplied by the probability that the doubleton club sits with the singleton ♠K. The following configuration shows the weights to be attached to the relevant conditions.

 
The chance of the doubleton club having been dealt to the hand containing the ♠K doubleton is 10/19, slightly more than 1/2. So we find the probability, PE, that the endplay will be successful after card u appears:

PE = 13/76 + (52/76) times (10/19) = 0.53 > 0.49

The conclusion is that the endplay is the preferred strategy by about 4%. This is in close agreement with Rubens. The calculation confirms the more general advice of Horton and Bob’s Blind Rule. One might conclude that happiness in bridge as well as life depends largely on one’s ability to recognize possibilities as they arise.

Here is an obvious variation on the Rubens hand that involves an elimination play when the defenders hold 7 cards in a suit. It is more readily recognized as there will be a loser in clubs should the spade finesse fail. How does Bob’s Blind Rule fare?

 
One of the interesting features of the above hand is that the numbers of missing cards in a suit are a combination of evens and odds. This profoundly affects the card combination probabilities as shown in the following list of the 5 most likely distributions.

 
Given that the spades split 2-1, creating an imbalance in vacant places of 1, the other odd numbered suit, hearts, can split 3-4 and allowing the even numbered suits to attain an even-split status (Condition I). If the longer card in hearts and spades lie to the same side, it creates a vacant place imbalance of 2, so one of the even numbered suits must split unevenly to fill the vacant places, thus reducing the numbers of possible card combinations (Conditions II thru IV). It is much more probable that the longer minor (diamonds) is split unevenly (conditions II and V). Taken all together, one sees that on these 5 common conditions, the clubs are most likely to split 3-3.

Usually playing in a matchpoint contest, simple is best. If the contract is 4♠, taking the spade finesse is the recommended line, as the field may make 12 tricks on this line of play. If the contract is 6♠, other factors require consideration. What proportion of the pairs will reach this slam? If fewer than half, one will score above average just by making the contract, whereas going down 1 will be a disaster. The situation is such that assuring the contract is the paramount consideration, just as it is at IMPs. Rather than maximizing the number of tricks won, one strives to minimize the number of tricks lost.

If we return to the situation where a spade is led towards the trump tenace and South follows with card u, the probability of the drop succeeding is 17%, so under those fortunate circumstances 12 tricks are assured. When the drop fails to produce the desired result, declarer must revert to the endplay hoping not to lose a trick in clubs. It would be extremely unethical at this point to mutter ‘Oh Damn!’ or your preferred equivalent. Hearts are eliminated and a spade ducked to the ♠K. A club is returned and declarer plays generally on the hope that the ♣QJ are split between North and South. Are the odds good enough to give an overall probability of success greater than 49%?

Suppose first declarer always plays for split club honors. The probability, PE, that the endplay will be successful on those grounds alone:

PE = (13/76) + (52/76) times the probability that the ♣Q-J are split.

One may estimate the probability of the ♣Q-J being split on an a priori basis as follows:

 
In the case of a 6-0 split, the endplay will be successful at least half of the time. So we arrive at an estimated overall rate of success, PE, of 53% before a club is played. This is about the same chance as with the first problem considered. The next question is whether anything can be gained or lost from the enforced return in the club suit.

Suppose South wins the ♠K and returns the ♣Q. Should declarer continue to play for split honors, or does that lead make it more likely South also holds the ♣J? If so, declarer can win in the dummy and finesse against the jack presumed to be in the South hand. Here is the situation with a 3-3 split. The ratios are a measure of the relative conditional probabilities once the queen has been placed on the table.

 
With QJx South must lead an honor, but he could choose either, so there are 2 plausible plays with 4 possible combinations. With Qxx South must find North with the jack, so he can lead with equal effect any of 3 cards with 6 possible combinations. On the assumption of perfect knowledge the lead of the queen is equally probable from Qxx or QJx. This illustrates the general principle that maximum uncertainty is achieved by defenders who choose equally between equivalent cards.

Well, that’s the way one might program a computer to defend, but reality is different. The computer strategy may represent the distillation of results for defenders of wide ranging abilities, but in practice one plays against one pair at a time. By those who see only their own cards, an honor would be led only when obviously necessary, that is, when holding both queen and jack. It is (almost) certain that the lead would have been low from Qxx, in which case declarer should let the lead ride, win the jack with the king, and finesse for the queen on the way back.

An imaginative player can see that declarer must hold the ♣10 in order to spurn the spade finesse, so that leading low from Qxx is a losing proposition. Such a player will most often lead the queen from Qxx hoping declarer will play him for both honors. Against such a player declarer should assume the lead is from Qxx, win the ace and finesses North for the jack. The reason is that the queen will be played from more combinations of Qxx than of QJx. There may be a nasty surprise in store, but one must expect occasionally to pay off to an unusual play. Advice to declarers: play for split honors unless a poor player leads an honor.

The above argument is based on a 3-3 split which is the most likely situation even when South has shown out on the third round of hearts (Condition V above), the reason being that the longer diamonds will split unevenly more readily than the shorter clubs. Here is the rare situation where South leads the queen from Qx:

 
The odds for leading the queen from Qx greatly outweigh the odds for QJ, the reason being that a QJ doubleton is a rare occurrence. On any lead, declarer should play for split honors.