Nine-Never and Basic Bridge Probability

July 9th, 2009  ~  Bob Mackinnon  ~  3 Comments 

There are many bad ideas floating around the bridge world concerning probability theory. Generally there is an over-reliance on the application of the a priori odds that apply to the deal of the cards before any action is taken. The oft-quoted Nine-Never Rule has this flaw. The prior odds are appropriate when one is in a state of maximum uncertainty, or if you prefer, in a state of minimum information with regard to the placement of the cards. We do know something about the probable consequences of the dealing of the cards, but the actions at the table are going to tell us a lot more about the particular hand that is being played. Naturally, as more is revealed, the odds will change. The question we address here is how to replace the a priori odds with odds more in keeping with current circumstances. Our examples are in reference to the flawed Nine-Never Rule, but the methodology is applicable to any card play situation.

Treating the problem of changing odds in a theoretical manner often leads to misunderstandings, besides which that approach requires some assumptions that get hidden in the mathematical jargon. The purpose here is to present a general framework for the calculation of odds, and to achieve that purpose we work through a simple, but realistic, example using only basic concepts which every player can understand. Along the way we shall make reference to familiar concepts, like vacant places, which may not be fully understood. Lack of basic understanding often leads to misapplication, and that in turn may lead to distrust of methods that are put forward under the guise of science. True science, first and foremost, reflects reality by complying with observation.

A Flannery Hand

For our example we assume the opening bidder (West) advertises a hand with 4 and 5♥, 10-15 HCP. We chose this bid because of the large amount of information it reveals about the distribution of the cards. If South ends up declaring the hand, that information can be used for the planning the play. This is also true: if an opponent overcalls on a weak hand and his side doesn’t end up declaring the hand, the distributional information provided can be used by the eventual declarer to the overcaller’s disadvantage. This hand provides an extreme example of that process at work. Here are the South – North hands.

South wins the ♠A and contemplates how best to draw trumps. The usual slogan is ‘Nine Never!’ Meaning that declarer should play off the ♦A –K hoping to drop the ♦Q doubleton. Not thinking beyond that, a novice may cash the ♦A and play towards the ♦K. A more knowledgeable player would play low towards the ♦K to guard against a 0 – 4 split when West is void in diamonds, then play to the ♦A when East follows twice with low diamonds. Both approaches are wrong because they are based on a blind rule, Nine Never, that has its origins in the a priori odds. We’ll examine first the effects of the bidding on the recommended play in diamonds.

Vacant Places First we register the distribution of the sides as revealed when the dummy appears. For NS the suits are numbered 5=5=9=6, so the NS side is numbered 8=8=4=7. The bidding tells us futhermore that the distribution in the majors is as follows:

The relative probability that the ♦Q being on the right (QR) rather than on the left (QL) is in the ratio of the number of vacant places available. That is, the ♦Q is more likely to be on the right in the ratio of 3:2. This assumes that neither clubs nor diamonds have been played, and that the minors suits can be lumped together as indeterminant cards. This is the odds of the deal with regard to those two suits. The missing diamonds and clubs can be reshuffled and dealt 4 to West and 6 to East without affecting the information currently available with regard to their content.

We chose to emphasize the location of the ♦Q, but the same relative probability would apply to any missing club or diamond regardless of rank, the ♣Q, for example. However, if one assumes the ♦Q is on the right, that assumption changes the number of vacant places to 4 and 5, so the odds for the ♣Q also being on the right has been reduced, although the vacant places still favor that, now in the ratio of 5:4.

So given the information possessed concerning the probable location of the ♦Q, declarer should not play for the drop, but should plan to finesse for that card in the East hand by beginning the diamond play with a low card to the ♦K in the dummy. This is not merely a safety play against a very rare void in the West, but a percentage play of greater relevance. What we want to calculate is the probability that declarer will succeed in his planned finesse.

The Magic of Vacant Places From what does the magic of vacant places derive? If we are going to use it, we should understand it. Then we see there is really no magic involved. Probability is no more than a ratio of combinations. It is assumed that the 10 missing minor suit cards can be placed at random between the West and East hands. How many card combinations exist with the ♦Q assumed to be on the left, and how many with the ♦Q on the right? The ratio of the card combinations give us the odds, assuming the cards are randomly dealt. Here there are 4 vacant places on the left and 6 on the right. There are 2 possibilities for the distribution of the 9 cards other than the ♦Q:

The ratio of the combinations is 6:4, which is also the ratio of the vacant places.

Thankfully one doesn’t need to know the individual numbers of combinations in order to obtain the ratio. One just lines up the splits as follows and chooses the last number of the more even split and the first number of the less even split. Thus,

5 – 4 vs 6 – 3 gives a ratio of 6:4.

This works for all adjacent splits. For 5-4 vs 7-2, one needs to treat 3 adjacent splits.

Thus, the ratio of combinations for a 5-4 split relative to a 7-2 split is 7:2, not 7:4.

The Distribution of Sides Once the dummy appears declarer can count separately the number of diamonds and clubs held by the opponents: 4 diamonds and 6 clubs. It is now possible to state explicitly all 5 possible combinations.

The a priori odds refer to the diamond splits. The even 2-2 split is more or less the same as the a priori value and the sum of probabilities for the 1-3 and 3-1 split is 49%, very much the same as expected initially for those splits. The difference is that the 1-3 split is much more likely than the 3-1 split due to the imbalance in vacant places.

With regard to the club splits, the 6-1 and 5-1 splits are rendered impossible by the conditions imposed by the Flannery opening bid. The 3-3 split has roughly the same probability as initially (36%) as does the sum for the 2-4 and 4-2 splits (48%), but with an imbalance in favor of the longer holding in the East hand, as expected. This illustrates that the a priori odds have some value as approximations if used in the proper sense. (If that weren’t the case, they would have gone out of use long ago.)

The Calculation of the Weights The calculations of the weights from which the probabilities are derived are merely the relative number of card combinations available on a random dealing of the cards. These weights apply before a card is played in either minor suit, so they are the probabilities of the deal only. We shall now illustrate how they come about from the numbers of card combinations available in the diamonds and clubs considered separately. It is convenient to line up the diamond splits in consecutive order and place underneath the corresponding club splits that preserve the 4-6 split in vacant places.

The numbers of combinations are entries in the Pascal Triangle, familiar to many from their schooldays. (Was that too long ago?) The greater the number of combinations, the greater is the probability that that particular configuration has been achieved. The individual percentages are merely the number of combinations divided by the total number of combinations possible (210). It is important to remember this arrangement because once cards are played in a suit, diamonds first, some combinations are ruled out of the realm of possibility. That reduces the number of combinations available in the diamond suit, while the number of possible club combinations remains unaffected. However, the changes in probability are not solely governed by the numbers of card combinations remaining. That is where some come to grief: the probability of the play is not the probability of the deal as not all cards are treated equally under the rules of bridge – some are significant, some aren’t, depending on the circumstances. That matters.

The Diamonds in Play Here explicitly are the combinations possible in the diamond suit (excluding the 4-0 and 0-4 splits) with the number of club combinations associated with each possible diamond combination. The number of club combinations serves as the weighting factor for each individual diamond combination taken 1 by 1.

We shall assume declarer begins properly with a low diamond to the ♦K in dummy and that West follows with the ♦6 and East with the ♦4. Here are the possibilities that remain. The club weights are unchanged as no club has been played.

The relative probability of the ♦Q on the left (QL) and on the right (QR) can be calculated from the weights from the associated club split.

The odds are now 5:3 that the ♦Q is on the right. What of the vacant places? They started at 4 and 6 and now they are 3 and 5, if we count the formerly unknown diamonds that have now been exposed. That’s great for simplifying the calculation of odds, but this is a special case as we shall see when we come to investigate more closely the plausible plays in the diamond suit. The special situation is this: the play of the ♦6 and ♦4 are equally probable for all remaining combinations.

Next we shall assume that declarer next leads the ♦3 from dummy and East follows with the ♦5. How does this change the probabilities? There are only 2 possibilities remaining:

The odds of the ♦Q being on the right are 4:3, in exact agreement once more with the number of vacant places remaining. Obviously declarer should take the finesse against East and not play for the drop.

Finally we assume that declarer plays the suit the wrong way around by cashing the ♦A in hand and leading towards the dummy. Assume West follows to the second round with the ♦5. The remaining possibilities are as follows:

When West plays the second low diamond, the vacant places stand at 2 and 5, which reflects the current relative probabilities, and South should play for the drop. The Nine Never Rule applies. So in one sequence declarer should play for the drop and in the other he should finesse. Here in detail are the vacant place situations described above.

Preferential Plays and Vacant Place Calculations The accuracy of the vacant place ratio in determining probabilities depends on the way in which low cards are played, in particular, the assumption is that the low cards are chosen equally at random. This reflects the conditions of the deal. From a doubleton ♦65, say, the choice of the ♦6 has a probability of 50% as does the choice of the ♦5. From ♦654, each low card has a 1/3 probability of being chosen on the first round and any permutation over 2 rounds has a 1 in 6 chance of being chosen. This assumption is equivalent to a condition of maximum uncertainty (entropy) for which the amount of information transmitted by the sequence of plays is a minimum. Any rule that gives preference of one sequence over another transmits more information. Here is an illustrative example. Suppose that defenders always play their highest spot card, in particular,

1) from a holding of ♦65 or ♦64 they always play the ♦6;

2) from a holding of ♦54 they always play the ♦5.

Applying these rules we are left with these combinations after the play of ♦6 and ♦4:

After 1 round the odds of the ♦Q on the right is 5:2, whereas the vacant place ratio is 5:3. Thus the direct correspondence between vacant place ratios and probabilities is broken. The additional information available has effected a reduction in the number of possible combinations remaining from 16 down to 2, which under the condition of maximum entropy would require, as we have seen, a third card to be revealed in order to achieve the same degree of reduction.

Permutations and Plausible Plays The order in which a defender chooses to play his cards when following in a suit represents a selection of one sequence of plays chosen from all possible permutations on the cards he holds. If all permutations are equally likely, the probability of his having chosen the observed sequence is merely the reciprocal of the total number of equal choices. This is the normal assumption, but it is subject to revision under some circumstances. The mathematics can accommodate other assumptions when appropriate. We discuss the effects later.

Assume a defender holds ♦654 and is required to follow to 2 rounds of diamonds. There are 6 possible ways to follow: 6-5, 6-4, 5-4, 5-6, 4-6, and 4-5. We term these the plausible plays. If the defender chooses to play the equivalent cards at random, the probability that a given sequence will emerge from the original holding is 1/6. On the other hand, if the defender always gives true count, the only sequence possible under that restriction is 6 followed by 5 and there exists a condition of certainty. The probability of observing 6-5 is 1. Thus, in order to calculate probabilities one must specify initially the assumptions concerning a defender’s method of choosing equivalent cards.

Suppose that a defender picks up each card as it is dealt. He ends up with ♦654. The order in which he gets to observe the arrival of the cards isn’t important. The same is true if he is dealt ♦Q64. All 6 permutations on the deal are equally likely. When he gets to play the cards the order becomes important, because the play is subject to the rules of bridge. The defender is assumed to play his cards in such a way as to optimize his chances. It would be foolish in most situations to play the unsupported honor on the first round. The plausible plays over 2 rounds are 6-4 and 4-6, and the probability of observing either is _. This is not a necessary assumption. There may be situations where the play of the ♦Q on the first round is called for. A defender may choose to cover an honor with an honor, or may attempt to create an entry to partner’s hand. Declarer needs to be aware of these situations, as does anyone wishing to calculate the odds. The mathematical formulation can accommodate any assumption in this regard.

Why is it commonly assumed a defender will play equivalent cards at random? Because that is the optimum strategy when the objective is to keep declarer in the dark to the greatest extent possible. If the objective is to inform partner to the greatest extent possible, such as by giving true count or suit preference, equivalent cards will not be played at random, but will conform to prior agreement. However, there is no guarantee the defenders will keep to such an agreement if they feel it is in their best interest to be uninformative.

Restricted Choice Consider next the combination ♦QJ6. On the first round normally a defender will follow with the ♦6. On the second round he may play either the ♦Q or ♦J with equal effect. The number of plausible plays is 2, 6-J and 6-Q, and the probability of each is _. This is a common example used to illustrate the Principle of Restricted Choice. When a declarer sees an honor emerge, naturally he takes notice. Of course, the very same principle applies to the play of low cards, as we have shown above. The selection from 2 honors is merely the last element in a sequence of equal choices.

If a defender wishes to inform partner that he holds both honors, he will not play the ♦Q on the second round, he will play the ♦J. If that takes the trick, his partner can see that declarer hasn’t got the ♦Q, otherwise he would have covered. If the defender takes the trick with the ♦Q, the possibility still exists that declarer holds the ♦J. Playing the ♦Q and ♦J at random creates a condition of maximum uncertainty in the minds of both declarer and the other defender, at least to the greatest extent that is possible given the cards they can see in their own hands.

It is understandable that the Principle of Restricted Choice is treated in the bridge literature solely as applying to the choice of honor cards. That is the most dramatic application as the play involves cards that have the potential of taking tricks. Also, low cards usually appear early in the play of a suit whereas honor cards emerge later at a critical stage. A defender will not usually part with an honor card unless there is a distinct advantage to doing so, whereas it is easy to part with a low card whose only function appears to be to fill in the suit. Usually there are more low cards missing than honors. For these reasons, then, the illustrations of Restricted Choice involve honors. We shall illustrate the application of Restricted Choice using a second Flannery deal.

A Second Flannery Hand

On this hand the ♦Q and ♦J are missing. Suppose that the ♦K is played to which West follows with the ♦6 and East with the ♦4, just as in the previous hand. On the second round, East follows with the ♦J. What is the probability the ♦Q will fall doubleton from the West hand? After the first round these are the possibilities remaining:

Each combination is subject to just one plausible play, so they all have the same probability of being chosen. Under the condition of equality of permutations, one can apply the vacant places to the calculation of probabilities. Adding the weights as before, we find

QL = 6 + 15 = 21 = JL; QR= 15 + 20 = 35 = JR,

and the odds of either honor being on the right is in the ratio of the remaining vacant places, namely, 5:3. The situation changes dramatically if East follows to the second round with the ♦J.

With a 2-2 split the appearance of sequence 6-4-J is a certainty, so the weight retains its full potential. However, the probability of the sequence 6-4-J given a 1-3 split is reduced by half under the assumption that it is just as likely for East to have chosen the ♦Q on the second round resulting in the sequence 6-4-Q. There are 2 equally probable sequences. The odds of dropping the ♦Q in the West has improved to 3:2. Before a diamond was played the odds favored the queen being with East 3:2. Thus, the freedom of choice of one honor over the other greatly affects the odds.

Let’s suppose that East is more likely to play the ♦J than the ♦Q, trying to keep his partner better informed. At trick 3 the situation is still unclear. If the probability of playing the jack rather than the queen were 75%, that would make the 2-2 split and the 1-3 split equally probable at this point in the play, so the odds of dropping the ♦Q in the West hand would be 50%. If East would always play the lower of 2 touching honors as a matter of general principle, alleviating the need for thought, the weight of the 1-3 split retains its full value as there is just one plausible sequence available. In that case one returns to the odds based on the current vacant places, 3:2 against the drop being successful. This is true before a diamond is played and later whenever the number of potential card sequences is the same for all remaining combinations.

What about the ♣Q? Once declarer determines the diamond split, the club split becomes a certainty and the play proceeds accordingly. Let’s suppose that against the odds according to Restricted Choice East was dealt ♦QJ4, so now there is a diamond loser after declarer plays for the drop. The slam is still assured when a heart loser can be avoided by establishing 2 tricks in clubs on which 2 losing hearts can be discarded from declarer’s hand and the losing heart in dummy can be ruffed. Declarer has been keeping count, so feels the clubs must be split 3-3. He makes his claim, showing his cards, after which the following exchange takes place.

South: I ruff the clubs good and pitch 2 hearts on the jack-ten of clubs. You can take your trump queen whenever you like, but I still make my 12 tricks.

East (showing his cards): down 1. I have four clubs to the queen.

North: Oh, Partner, you could have made it by taking a ruffing finesse in clubs!

If West holds the majors, East must hold the minors.

West: Sorry, with 6 lousy hearts it felt like a Flannery hand.

South: Grrrr….

Yes, it happens. The opening bid provided information, but it was inaccurate. Information is not exact, but one should plan according to the information available. In some situations before drawing trumps, declarer may take some seemingly pointless ruffs in a side suit in order to confirm what he thinks he knows about the distribution. In a 3NT contract declarer may duck a trick or two to good effect. These are processes of gathering information, and it may happen that this gathered information contradicts what was assumed previously. A re-evaluation of the assumptions is required.

If one is unable to safely gather more information, then it is correct to play according to what is most likely given the current information. In this deal West took an unusual action, but the probability that he did so was very low. This is generally true. It doesn’t pay to assume an unusual action and over-react in what is most likely a normal situation. More is lost by assuming aggressive opponents are trying to swindle you than is lost by assuming they are playing normally. However, it is wise to check.

Bayes’ Theorem Without stating the governing equations explicitly what we have demonstrated in these examples are simply applications of Bayes’ Theorem to card play. The mathematics involved is not difficult as it is merely a consequence of linearity, however, the notation is difficult to grasp on a casual reading. Putting the equations into words has caused confusion, beginning early in the 18th Century with Rev. Thomas Bayes himself, who was hoping to add further confirmation of the existence of God by extending the results produced by Blaise Pascal, the 17th Century co-founder of probability theory. Once philosophers got involved, matters became unnecessarily complicated. It wasn’t until the 1950’s that clarity was restored and probability was linked mathematically to information. Hopefully, all that confusion is behind us now and common sense will prevail, at least as far as the analysis of bridge play is concerned.

Warning! The application to the cases where the defenders hold an odd number of trumps is more complicated than for the examples treated above, because the number of plausible plays reaches equality at a later stage of play than for an even number of missing trumps. Expect more on the Eight-Ever Rule later.

What You Don’t Know

June 25th, 2009  ~  Bob Mackinnon  ~  No Comments 

When I was a country boy I often heard the farmers say, ‘what you don’t know won’t hurt you.’ This seemed to me to be a short-sighted attitude especially as the world had just been introduced to invisible radioactive fallout, but looking back I can see that they were saying we shouldn’t worry about things, like rainfall, over which we have no control. In a purely practical sense sometimes one is happier not knowing everything. That attitude applies well to bridge as well as to marriage. We’ll look at the consequence of not knowing in a deal played in the recent European Open Teams Championships where an English foursome faced an Irish-Welsh team.

BBO commentators, and the English ones in particular, I think it is fair to say, are often shaking their heads over optimistically bid slams, regardless of whether or not the slams make. One commentator even speculated that a pair would do better in the long run if they decided not to bid slams at all. This is going too far. If on the bidding the slam appears to be a good proposition one should bid it, especially if you have a drop of Irish blood flowing in your veins. If not, a drop of Irish whisky will do. One shouldn’t rely too strongly on the a priori odds. As an auction progresses, information is exchanged between partners. Based on this information the odds are adjusted until a player makes a decision one way or the other. The accuracy of the adjustment depends on the quality of the bidding system in operation. A player will not know everything, as do those who can see all 4 hands, but what he doesn’t know may not hurt him.

The Luck of the Irish I have been re-reading Lausanne 1979, Peter Pigot’s amusing memoir on the European Bridge Championships of thirty years ago in which Ireland won the bronze medal. I was reminded of the hard feelings that existed between the Irish and the British players at that time, and now 30 years later I was being treated to another confrontation, the intensity not in the least diminished by the presence of 2 Welsh players added to the mix. The animosity of the Welsh goes back even farther than the cruel repressions of Elizabeth I to the age of Edward I.

Here is a hand for which expert analysts with a view of all 4 hands relied too heavily on the a priori odds. Would you, like Irishman Terry Walsh, bid the vulnerable Grand Slam on the evidence available?

RKCB asks a simple question, but there are those that cannot stand giving a simple answer – they have to put their own spin on the proceedings. Here Peter Goodwin said he had the ♥Q to go along with the ♣A and ♠K, but as we can see he didn’t. The excuse is that 5 trumps to the king-jack are as good as 4 to the king-queen, and most often they are as good, but sometimes they aren’t, especially when partner may be stretching towards a Grand Slam. The auction then took on a life of its own. From Walsh’s point-of-view, partner said he had the ♥Q when obviously he didn’t, so must have had a good reason to force the auction to the 6-level. Walsh asked about kings, and was forced to bid the Grand after an inconveniently truthful response. As 13 tricks were easily taken, if one were so inclined one might imagine divine intervention at work here, a view countered by the fact that the English readily won the match despite the loss of 13 IMPs on the deal. To the grossly materialistic mind the question is: how good was the Grand given what Walsh didn’t know after Goodwin bid 5♠?

The Shifting Odds in the Spade Suit If we consider the spade suit in isolation, a simple approach of cashing the ace and king will score 5 tricks whenever the spades are split 2-2, a 40% a priori probability. This does not mean that Walsh has bid a Grand Slam with only a 40% chance of success. As commentator David Bird correctly pointed out the presence of the 10♠ gives an added chance of dropping the ♠Q or ♠J singleton in the East, in which case declarer can finesse West for the missing honor on the second round. The success rate has risen to 46%, still not high enough to justify the risk. A Grand Slam bid and made gains 13 IMPs over a small slam, but loses 17 IMPs if it goes down 1. This means one should have a winning percentage in the spade suit above 57%.

From North’s point-of-view the odds of bringing in the spade suit change with the honors held by the opener who holds 5 spades to the ♠K, leaving him with 4 vacant places in the spade suit. The defenders also hold 4 spades, so it is a question of the likely honor holdings when the spades are randomly distributed 4-4.

The overall percentage of success of bringing in 5 tricks in the spade suit is roughly 75%, or 3:1 odds in favor of bidding the Grand Slam. In fact, the odds were better than that. In this day of indiscriminate interference, when the opponents have 19 cards in the minors and fail to enter the bidding, the chances of one of them being void in spades is less than the expected 10%. As one can see from the table Walsh would be unlucky to find both the Queen and Jack held by the opposition, thus being reduced to a meager 46% chance of success before the spade suit is played. It’s time to see the full deal:

Dealer: East Vul: All		North
		♠	A 10 8 3
		♥	A Q 10 6 4
		♦	A 9 6
		♣	K
West				East
♠	Q 6			♠	J 9
♥	7			♥	5 2
♦	K J 10 5			♦	Q 8 7 3 2
♣	10 9 6 4 3 2			♣	J 8 7 5
		South
		♠	K 7 5 4 2
		♥	K J 9 8 3
		♦	4
		♣	A Q

The 2-2 spade split renders the play trivial. In other room the Hackett twins bid to 6S only, on this auction:

Once Julian discovered that the ♠Q was absent, he avoided the Grand Slam in spades. Both he and Terry made the correct final decision given what they knew at the time, but only one would score well on the board. One might say that Julian was ‘The Man Who Knew Too Much.’ Knowledge doesn’t necessarily pay off in the short term.

Yet Another Convention The RKCB auctions of both teams were not up to the task, the main reason behind this being that the asking bidder needed to know the secondary honors held in both of the suits shown by his partner. What was required was a 2-suited RKCB. Here is my version of such an asking bid, the Siamese Control Asking Bid.

When a player has shown length in 2 suits, partner can ask for the total number of controls held (Ace=2, King=1) counting all 4 aces and the 2 kings in the advertised suits. The responses are: Step1 = 0-2 controls, Step2 =3, Step3=4, etc. A subsequent ask inquiries about the number of queens held in the 2 suits with Step1 = 0, Step2 = 1, Step3=both. If there is still bidding room below slam, a bid in an off-suit can be used as a non-specific Last Train bid, seeking extras by way of critical jacks. The above deal provides an example of where 4NT can be used in this manner.

The 6D bid should be interpreted as looking for a well placed ♠J, the ♥J being redundant in a 10-card fit. If Goodwin had the ♠J the a priori odds of taking 5 spade tricks would have been 58%, just enough to justify the risk of bidding the Grand.

This treatment is useful when there is a double fit and the major concern is the quality of the long suits held. Less important is the nature of the shortages in the off-suits. If one wishes to preserve the use of one-suited RKCB which brings shortages into play, in the above example a bid of 5♣ followed by 5NT can be used as Siamese without loss.

Fresh From the 2009 World Wide Bridge Contest

June 11th, 2009  ~  Bob Mackinnon  ~  1 Comment 

Please don’t think I’m bragging when I tell you that playing in the Friday night session my partner and I score 87% on the last 6 boards and sprinted past a vast multitude of also-rans to finish in 212th place, just ahead of Julian and Justin of Center City, USA, (Sorry, Guys) and just behind Domenico and Gianna of Firenze, Italy (Saluti, Amici). Obviously our local opponents must have become weakened by their previous efforts at dodging the slings and arrows of outrageous fortune. For our part we had given away our customary overtricks here and there, but at the end had high hopes of reaching the top 200, but it was not to be. What held us back were 3 boards in which we had no role in the final outcome.

On Board 19 the opponents bid undisturbed to 3NT losing a trick in each suit.

Not an auction that would ring any alarm bells. I would bid that way myself. Declarer won the ♠Q lead in dummy (from my Qx), shunned the diamond finesse and collected 9 tricks, losing 1 trick in each suit. I felt we might have scored well as a heart lead can run to the king, after which the diamonds can be finessed successfully, putting pressure on my partner who had to guard both black suits. It looked as if 10 tricks would be taken much of the time. As is so often the case I was wrong as the frequency table on the right demonstrates. Only 363 pairs made 10 tricks or better. +400 was the most frequent result, but 478 pairs played in a NT partial and nearly half of those gathered just 8 tricks, which was good bidding as far as they were concerned.

Really, I don’t mind getting a bad score when the opponents bid reasonably and achieve their objective through competent play. I don’t like it when the opponents take a big position with a 20% chance of success that comes home, but I console myself philosophically with the thought that they gave me an 80% chance of good score without my doing anything to deserve it, and that I will benefit more times than I shall suffer from such actions. Yes, that is what I tell myself until the hurt goes away. What really pains me is when the opponents score well after missing a cold slam. Here is Board 20.

Pathetically, 5♣ making 6 was the most common result, as most were tempted to play 3NT, which failed more times than it succeeded. We scored well below average because ‘the world’ couldn’t unravel the mystery of 12 easy tricks. The spades were 3-3 and if that wasn’t enough, the clubs were 2-2 as well.

Board 8, our worst result, had a similar theme. I was optimistic at the time, as there were 13 tricks to be taken off the top in 7NT, no less.

Pairs who could get to slam were good enough to bid the slam in NT, but they were unable to count to 13 tricks. The most frequent score was in a NT game making 7, which leads me to conjecture that some players may have opened 3♥ on ♠ 8 ♥ KQJT82 ♦ T85 ♣ 842. In his accompanying comments Eric Kokish gave his vote to a 2NT overcall on the grounds that it makes it easy for partner to explore alternative strains. As we all know, Eric gets criticized often for his penchant towards flexibility, but I normally agree with him. Here I disagree. In the early stages I always aim for suit contracts when holding aces, so would have bid 3♣. Obviously steering towards 3NT will be a success only if partner has some club support, and if he has it, 6♣ might prove best. So, 2NT doesn’t appear to be the right approach on this collection, or particularly flexible, for that matter.

Sharing Information

The evidence is overwhelming that players place great value on a contract of 3NT, be it at IMPs or Matchpoints. The scoring favors that approach when the hands are balanced or don’t fit that well, so most of the time it pays to conceal one’s assets and take one’s chances on a potentially unstopped suit. Interference with poor suits has become so common that often one just bids 3NT without the sign of a stopper and see if the opponents can beat it. Often they can’t. However, that playing-in-limbo approach is one of playing the score card, not with the 26 cards a partnership holds. In order to get to the correct contract on that basis, one has to exchange the details with one’s partner, like it or not. Some risk is involved, sure, but the risk is worth taking more than today’s players are willing to assume, obsessed as they are with concealment. Let’s look at our example hands again with this in mind. Here is a better auction.

Let’s suppose that after 3♣ opener evaluates his hand upwards with a view to possibly playing in 3NT, but not without a firm commitment one way or the other, being flexible, in other words. He begins with a bid of 3♥, presumably showing a stopper there while denying a full stopper in diamonds. Responder bids 3♠ to show a stopper in spades while denying a full stopper in diamonds. If opener had a partial stopper in diamonds (Jx, say) he could now show it by bidding 3NT, which responder would pass. The defenders start with diamonds, cashing 2 tricks, and declarer claims the rest. Trying to hide the situation in diamonds does no harm, and it might even help in the situation shown above where declarer is short in diamonds.

When responder bids 3♠ to deny a full stopper in diamonds, opener can be more optimistic with regard to his chances in a club slam. His 4♥ continuation is a slam invitation, and responder cooperates by showing ‘extras’ in the spade suit, a treatment made possible by the limited nature of his 3♣ raise. His clubs are certainly powerful given that partner is showing slam interest. At this point opener is a lot better off than when he jumped pessimistically to 5♣. He can still bid 5♣ if he feels that way, but 6♣ is clearly indicated.

After a good fit is found, a hand with 7 controls should be evaluated upwards to the equivalent of 23 HCP. Thus, when partner gives a limit raise, the chances of a slam should loom large. It is only a matter of placing the controls in partner’s hand. This applies to Board 8 as well, where the overcall should have been 3♣.

Advancer was willing to gamble at 4NT without the evidence of a fit, indeed, without a denial of 4 spades opposite, so how can she not make an encouraging 3♥ cue bid over a natural 3♣? True, the overcaller has a tough problem, and I would like to suggest 3♠ as a continuation, but let’s suppose that the overcaller retreats to 3NT, fearing that 3♠ would suggest an alternative trump suit. Now advancer should bid 4♣ because she knows that her partner must hold long clubs and considerable strength in the red suits. It is time to show much needed encouragement in clubs. This should not rule out playing in 4NT if the overcaller chooses to bid it. Indeed, having underbid with 3NT, the overcaller is so greatly encouraged, that she bids the 5NT Grand Slam Force asking for the 2 top club honours, and reaches the lay-down slam with more confidence than she must have had when she bid 6NT when partner might hold: ♠ KJ65 ♥ QT5 ♦ KQJ ♣ A76, where 6♣ makes and 6NT doesn’t despite the combined 33 HCP with 4 aces.

Getting Over the Hump at 3NT

I have often partnered a successful player whose attitude towards good hands is this: lacking a major suit fit, bid 3NT. To him 3NT is not just a hump in the bidding highway, it is a roadblock. Why is this true of so many players? First, because they feel that bidding a small slam is not worth the risk, and, second, because they are unable to stop at 4NT, which is always asking for aces. So there is nothing for them between 3NT and 6♣. That is why they put up the roadblock in the first place. Well, one should change that and agree to ways in which one can try for a slam and yet retreat to the relative safety of 4NT, to play.

One rule I like to use is this: once a player signs off in 3NT, direct bids of 4NT are natural and invitational. However, if the player who bid 3NT later cuebids at the 4 level, he is showing the resources necessary to cooperate in a slam exploration, always interpreted within the limitations of his previous bids. One slam try is allowable.

With regard to cuebidding, it is helpful if one can bid aces up the line, but this is not always possible when space is restricted. Players (Meckwell, in particular) have solved the problem of running out of bidding space in major suit slam auctions by introducing the Last Train convention, a bid in a suit just below the major suit game level that shows interest in slam without promising control in that suit. This is easily recognized because the trump suit has been agreed explicitly.

When 3NT is looming large, it may be that a minor suit fit has not been explicitly confirmed, and the priority may be to find stoppers, or even a suitable major suit fit. So it is difficult to maintain generally that a cue bid of 3♠ is a nebulous slam try in a minor – The Last Train to Bootyville. Here is an amusing example from the 2009 USBF Team Trials where 2 Precision pairs failed the test. I suspect it is evidence of a general malaise.

The ‘raise’ to 2♦ showed 10+HCP with a good diamond suit in the context of the Precision system for which a 1♦ opening bid doesn’t promise more than 2 diamonds. Pair A gave a perfect demonstration of how to waste one’s opportunities. Thanks to their uninformative auction, they scored 690, a minor triumph for ’conceal don’t reveal.’ Pair B got over the 3NT hump by using relay methods, but they showed us that sophistication is not a substitute for judgement, only a supplement. 2♥ showed a weak NT opening bid and 3♠ showed a singleton. It seems that all that remained was for opener, who had strictly limited his hand, to cuebid his ♣K after which responder could cuebid 4♥. They had nothing to fear but themselves. Responder might have bid 4♥ anyway. It was not their system that was at fault, but the reluctance of the players to commit to full disclosure. You won’t get to the good slams without it.

Never a Dull Deal

June 1st, 2009  ~  Bob Mackinnon  ~  1 Comment 

Probability is related to information. It changes with the play of each card. Different information sets lead to different probabilities. ‘Going with the odds’ entails ‘going with the information flow.’ Here is a simple example. Suppose South is a declarer looking for the ♥Q among 5 missing hearts and has available a two-way finesse. On the basis of the deal alone, there is a 50-50 chance that Her Majesty lies in either of the defenders’ hands The lead is the ♠2, from which declarer deduces that the spades are split 4 on the left and 3 on the right. This information leads to the conclusion that the ♥Q is more likely to have been dealt on the right in the ratio of the vacant places 10:9, so it is correct to finesse through East. Next suppose that at the other table North has become declarer in the same contract through a transfer sequence. The lead is the ♦2. By the same reasoning, North should finesse West for the ♥Q. He also is correct, but only one declarer is going to be successful. The information about one suit affects the probabilities in the other suits, but because the information is different, the deduction is also different.

Both the a priori odds and the vacant place calculation are based on the assumption that unknown cards can be placed randomly between the two defenders. That is an assumption of a condition of maximum uncertainty with regard to the placement of the unknown cards. It loses validity if there are indications one way or the other. Let’s next suppose that the South declarer doesn’t immediately take the heart finesse which he thinks is favored in the ratio of 10:9, rather he plays on diamonds and finds they are split 4-3 the other way. Now the vacant places are even at 6 and 6 between West and East. Due to this additional information the odds on the heart finesse have returned to their a priori value of 50-50. One feels that the more information one obtains the better will be the estimates probability and the better the chances of making the right decision.

If the contract were 6NT, declarer might postpone the heart finesse until some information is obtained from the club suit. So declarer safely ducks a club just to see what happens. If he can deduce the club split, he will be able to deduce the heart split and can play accordingly. The answer will not be a 50-50 decision anymore as one player will be assumed on the basis of the information available to hold 3 hearts. However, there are no guarantees as even a 60-40 finesse will lose 40% of the time. One has been taught to assure the contract against bad breaks at team play and to forget about overtricks. As a consequence one is discouraged from making an exploratory move if that might lead to the defeat of the contract on a bad split. That in turn results is an over-reliance on the a priori odds which, as we noted, are based on a condition of maximum uncertainty.

At matchpoints overtricks risking the contract for the sake of an overtrick can be the correct play. Exploratory moves for the purpose of gathering information make good sense. Let’s look at my misplay in a recent local Regional. It was a typical inexpert matchpoint game in which success often depends on the opponents not defending to best advantage. The deal was a routine 3NT across the whole field, so the play was all about overtricks. Some find such contracts dull, but I find even they contain elements of great interest. Besides which, our mistakes are stepping stones to improvement. Sure they are. First and foremost we must prepare ourselves to adapt to changing circumstances.

As West I covered the lead of the ♠3 with the ♠9 which won the trick as South, using standard carding, followed with the ♠2. How would you play this hand playing Teams where the primary objective is to make the contract? After you answer that, the next question is: how would you play the hand at matchpoints where the primary objective is to win as many overtricks as possible?

The simplest way to assure 9 tricks is to play the ♦Q at trick 2, establishing 2 tricks in the suit. A spade continuation will set up the ninth trick, so the defenders’ best return is a passive diamond to the now bare ace. No problem, as a 3rd heart trick is assured by riding the ♥J from dummy. At most declarer loses 2 spades and 2 tricks in the red suits. Giving up on the red suit finesses in this manner is playing the hand with extreme short-sighted pessimism as declarer has a double stop in every suit. There must be a reasonable limit to our fear of loss. However, if one plays that way, 3NT does indeed become a very dull deal. As with our lives, there are no dull deals but a lack of initiative makes them so.

At matchpoints there is a chance of 2 overtricks in a common contract, so declarer must go all out to obtain them if available. We want to find the ♦K located in the North and the ♥Q in the South. The opening spade lead gives us the immediate odds of 10:9 for the second condition. With this in mind I led the ♥J from dummy, hoping for a cover, passing it, and losing to the Queen. Unlucky? Or should I have followed Zia’s advice that if they don’t cover they don’t have it? More on this later. Next North cashed his 2 spade winners and led a low diamond. Should I finesse or go up with ♦A and rely on bringing in 4 club tricks?

The a priori odds that clubs will split evenly is just 36%, so it would appear at first glance that the diamond finesse is better. However, the location of the ♣10 gives an a priori chance for 4 club tricks of about 60%. To judge the probabilities at this late stage requires more information on the expectation of the splits in the other suits. If North is expected to hold 4 clubs, then the play of clubs for 4 tricks would lose its luster.

My thinking at the time turned away from card combinations to psychology and motivation. I viewed North’s strategy with suspicion. Why cash his spade winners and why not exit passively with a heart or even a spade? It would appear that I am committed to the diamond finesse regardless. It didn’t occur to me at the time that he had no devious plan in mind and was merely happy to be ahead in the game by taking a trick in hearts to which he was not entitled. In a confused state of mind I decided to take the diamond finesse on the grounds that, after all, every declarer in the field would be taking that finesse. Wrong! Not only on the play itself, but, more importantly, with the thinking behind it. We should always keep in mind that a defender is working with different information and can’t see the cards we hold, and vice versa. Here is the full deal.

Dealer: West Vul: All		North
		♠	A Q 5 3
		♥	Q 8 4
		♦	10 5 2
		♣	J 5 3
West				East
♠	K 7			♠	J 10 9 4
♥	A 7 2			♥	K J 10 3
♦	J 8 7 3			♦	A Q
♣	A K 10 8			♣	Q 4 3
		South
		♠	8 6 2
		♥	9 8 5
		♦	K 9 6 4
		♣	6 5 2

The Objective This hand provides a fine example of ‘do as I say, not as I did’. I indulged in the kind of fragmented thinking that goes along with the acceptance of arguments based on the a priori odds in each suit taken without reference to the deal as a whole. The first step in the process after the opening lead is to examine dummy and set a realistic target for the number of tricks one hopes to take. The lead gives us 2 tricks in spades, there are 3 tricks in clubs, and 2 finesses to be taken in the red suits. So a realistic target is one of 10 tricks. Of course, as this is matchpoints, declarer should strive for one trick more. The question arises as how one might negotiate to the desired ending without unduly risking what appears to be the normal result of making one overtrick.

The Deep Finesse Effect The program Deep Finesse is a wonderful tool as it provides us with the optimal results obtainable on the given lie of the cards. It knows all the cards so does not deal with probabilities. To obtain a perfect result usually declarer has to take any finesse that works. The wide-spread availability of this analysis acts as an encouragement to take every finesse in sight and hope it succeeds. As with references to the a priori odds, this treatment fragments one’s thinking into a suit-by-suit approach. It freezes the approach to be taken. In the above deal one focuses attention on the obvious finesses in the red suits hoping first and foremost to take the heart finesse in the right direction.

If a declarer makes the optimal number of tricks as determined by Deep Finesse, he is entitled to feel he has done his part well enough, but to insure a good score he may have to exceed the optimum. This usually comes about because the defence has erred, so a secondary aim of a declarer is to play the hand in such a way that the defenders won’t find the optimum defence. On this hand declarer aims to avoid the diamond finesse if possible. Why? Well, let’s not forget the club suit. As noted above the club suit is expected to provide 4 tricks about 60% of the time, which is a better percentage than the diamond finesse taken in isolation, so, although the doubleton ♦A Q stands out in the dummy like a sore thumb, declarer wants to blind North to its importance when he doesn’t hold the ♦K until it is too late. With a bit of luck in the club suit, the ♦Q may be discarded on the 4th club, and the finesse avoided. So timing becomes important. If the diamond finesse can’t be avoided, then reluctantly one has to take one’s chances along with everyone else.

If declarer were to play a spade at trick 2 to establish a second spade trick, North should become aware of the danger and lead a diamond before taking his second and last spade trick. More simply, North might be one of those stubborn folk who see that if declarer wants to play on spades, it must be correct on general grounds to switch suits. Taking the heart finesse immediately at trick 2 and hoping for a defensive error is better tactics. The question arises as to what is the best direction for that move. Preliminary evidence suggests that finessing through South has the greater chance of success (10:9 odds), but it appears ‘more natural’ to finesse through North. (My partner certainly thought so.) If South wins the ♥Q, she will undoubtedly return a spade, her partner’s suit. If North then cashes a second spade before returning a diamond, good timing will have been achieved. If he doesn’t take a second spade, declarer resorts to the diamond finesse as a matter of necessity, losing nothing in the process. The question arises as to how much is risked by taking the heart finesse ‘the wrong way’? Is it a good investment? To answer that we should look more closely at the possible distributions involved.

The Distribution of Sides

If one accepts that the deduction that the spades are split 4-3, the most common distribution of the NS cards are as follows:

The relative weights are based on the number of card combinations that result from a random deal of the other 3 suits. From the weights we can be a rough estimation of the probability of each condition having been dealt. The assumption is the same as that behind the vacant place calculation, but here we display only the most common combinations of splits in hearts, diamonds, and clubs. This focuses the mind on what is most probable given the information available so far from the bidding and the opening lead. It is of interest to note that the single most likely distribution (Condition I) is the actual distribution encountered at the table, the flattest of flat 4-3-3-3 on both sides of the table. If declarer had to choose just one condition to play for, that would be it.

If we were to add more possibilities, we would be adding more uneven splits, but the bidding has not indicated there are extreme distributions of concern. Also, if there North were to hold two 4-card suits, one can imagine that the opening lead might have been at least part of the time a passive lead in the other 4-card suit, rather in a suit headed by the ♠A Q which might give away a trick immediately. Even so, it appears that the ♥Q is more likely have been dealt to South than to North, the odds being much the same as the vacant place odds.

However, holding the ♥Q, South may cover the ♥J as often as 1 time in 10, balancing the odds as to the location of that card. Playing the ♥J from dummy and overtaking with the ♥A when it is not covered appears to be a dangerous and unreliable way to collect information, but if North doesn’t follow to the second heart, declarer wins the ♥K in dummy and leads a spade. The odds for the diamond finesse have improved, as North’s most likely shape is now 4=1=4=4, and the ♥10 3 still stand guard if the diamond finesse loses. There is another consideration at matchpoints, which is this: how will the majority of declarers play the hand? I imagine that most will take the ‘natural’ heart finesse, playing to the ♥A in hand and finessing through North, in part to protect against an early diamond switch. It risks little to go this route even if the odds are against it. The strategy is to play for an average and hope the defenders make a mistake along the way.

My error was not so much in trying for a cover of the ♥J, but in not following through and playing the ♥A when South played low. When North won the ♥Q and cashed his spade winners, my luck changed and I was presented with the very mistake for which I should have been hoping all along. As the reader can see, when North cashes both his spade winners, he set up my ninth trick (2 spades, 3 hearts, 1 diamond and 3 clubs), so it was now a matter of the overtrick. Fearing that North might hold 4 clubs was misguided because if North held the ♦K and 4 clubs he would be squeezed when I ran off the hearts and the ♠J in the dummy from this position:

With the diamond finesse still at the forefront of my thinking, I didn’t follow through by rejecting the diamond finesse and playing for the better odds of 4 tricks in clubs. I should have been focused from the very beginning on the most probable conditions with regard to both minors taken simultaneously. (Just envisioning Condition I would have been a good start.) Taking the ♦A and playing off the hearts would determine the split in the heart suit to have been 3-3, so the remaining possibilities are Conditions I, II, and VI, and one may observe some discomfort on the play of the last spade under Condition II when South holds the ♦K and 4 clubs to the CJ. As at worst only one trick remains to be lost, there is an average score to gain and nothing to lose by refusing the diamond finesse. With all hands revealed, failure to count becomes the most glaring of errors.

The bidding and the play are important factors in determining probabilities. Generally speaking the a priori odds represent a reasonably accurate guess of one’s chances during the play as long as even splits are encountered. That may change significantly if some splits are known to be extreme. There was no evidence that such was the case on this deal as bid and played. If one calculates from Conditions I, II and VI the odds for obtaining 4 club tricks by playing ♣A – ♣Q – ♣K, one finds it to be 70%, much better than the a priori odds, because of the squeeze possibilities that have evolved.

NEXT!

Old Meanings for an Old Bid

May 19th, 2009  ~  Bob Mackinnon  ~  2 Comments 

In the good old days a double said what it was meant to say, ‘increase the penalty’. Well, that is the view of the purist with failing memories, but life was, and is, richer and more varied than the idealists lead us to believe. Of course, the intent was to inflict punishment on errant opponents, but the call itself took on various meanings according to the circumstances under which it was invoked. Here are a few of the interpretations that I still encounter at my local bridge club and even observe on BBO from time to time.

The Undouble My Double Double This rare type was introduced to me through a 1973 book by Robert Ewen entitled Doubles for Takeout, Penalties and Profit. I’ve never used it myself, but recently it almost came up in a major tournament on BBO. The circumstances are that one has made a double during the auction indicating a particular suit, but now that the opponents have bid to slam regardless, you wish to cancel your previous message and have partner lead a different suit. Say, you doubled a cue bid of diamonds to suggest sacrificing in that suit because you held 6 to the queen-jack, but now wish to have partner lead another suit, because the defensive values in diamonds are negligible against the freely bid slam. This idea has been generalized as follows.

The Cancel My Previous Bids Double This is a very useful double in our age of weak suit interference. One sees frequently that bidding bad suits leads to bad defence. One may bid ‘em up early with bad suits that could be used profitably in an offensive mode, but once the priority shifts to defence a different set of criteria applies. The double conveys the meaning that the opponents may have been pushed profitably in the wrong direction by your deceptive maneuvers. Classified as an expert maneuver.

The Lead Your Suit Double A variation of the above against 3NT contracts that conveys the message ‘hope your suit is good enough to beat this contract, because mine isn’t.’

From my experiences at the table I can list several other interpretations which can be put to penalty doubles. Although they may not be as theoretical sound as the Cancel My Bids Double, they are common enough.

The Point Count Double After the opponents bid to a cast-iron contract the doubler announces he has 16+HCP. This is the most common and least successful double.

The Picket Fence Double A defender doubles a suit contract partly on the strength of a gappy 5-card holding in their trump suit, only to find he has helped declarer make it.

The Get Your Wish Double A variation of the previous type in which one doubles a transfer bid to advertise the suit only to end up defending unsuccessfully in that strain with nowhere to escape. A prime example is: 1NT (Pass ) 2♥ (Dbl) All Pass. -670.

The Trump Void Double A player doubles with a void in trumps in an attempt to protect partner’s presumed QTxx or such. Altruism at its finest.

The Suicide Double A short-sighted double of a doomed contract that pushes the opponents to a more profitable contract they wouldn’t have bid. An example: 4♣* moved to 4♥ on a Moysian fit, making due to the marked ruffing finesse in clubs. Fairly common. If partner doubles 4♥, it becomes the rare Double Suicide Double.

The 3NT Chicken Takeout Double The RHO preempts in a minor and the LHO bids a cold 3NT, but a confident balancing double by a previously silent defender induces the LHO to change his mind and takes out to 4 of the minor, because he trusts his opponent more that he trusts his preempting partner. This works even at the highest levels.

The 6NT Option Double A player makes a Lightner Double of a slam contract, giving the opponents the chance to escape to a makable 6NT. Only Zia would employ the anti-Lightner variation in which he wants to be on lead against a shaky 6NT.

The I’m on Lead Lead-Directing Double Technically, an anti-positional double, but how the opponents react to this otherwise meaningless action may help determine the best lead. Another Zia specialty employed during an extended slam auction.

The Honeymoon Double ‘Be my dream come true,’ and become the only one to find the obscure defence that beats this contract. Employed exclusively by new partnerships.

The Confusion Double ‘Too many bids, partner.’ A double made by a mediocre player who hasn’t been able to follow the lengthy auction in its entirety. Often works!

The Suspicion Double ‘These guys have a history.’ Generally not a good reaction.

The Pessimistic Double ‘A bottom is a bottom, partner.’ A double with a low probability of success that is employed exclusively at matchpoints.

The Admiral’s Prerogative ‘I assume you can beat this.’ Some players are not content with being captain of the constructive auctions, they also want to make all the decisions in a competitive auction as well. That constitutes a promotion from the captain of a sunken ship to the admiral of a doomed fleet. Otherwise unfathomable.

The Clever Monkey Double A fatuous double of two of a minor made in the hope that the opponents will take the inadequate reward of two doubled overtricks rather than bid game. Here is the situation: 1♣ (1♥) 2♦ (Dbl) All Pass. The clever monkey holds a poor hand with 4 worthless hearts, so rather than raise hearts preemptively, he doubles, and no one at the table can be sure of what that means. If challenged with a redouble he will raise hearts which may get doubled in turn and yet be a good save, which is what he was aiming for all along. (It has been suggested that this cousin of the Striped Tail Ape Double is Chinese in origin and based on the legend of the Monkey King and his partner, Libidinous Hog, traveling to the West, but I very much doubt that.)

The Sayonara Double A blatantly hopeless double made at the end of a game as a polite, unspoken suggestion that this end the partnership once and for all.

Total Trumps and 5-4-3-1

May 14th, 2009  ~  Bob Mackinnon  ~  1 Comment 

In this segment we consider the implications with regard to card distributions of doubling an opponents’ preemptive raise at the 3-level that implies a 9-card fit. In particular, we wish to determine the probabilities of the number of total trumps based on an assumption of a random deal of the unknown cards, a flawed assumption to be sure, but one that enables a reasonably accurate estimate as a first approximation. As in the case of a 4-4-4-1 hand, only the most likely shapes opposite are considered as extreme distributions are rare, and in such cases partner will take appropriate action independently.

¶

Here are the 5 most frequent conditions when the opponents have a 9-card heart fit and one holds a 5=1=4=3 hand.

It may be surprising to those who rely on a priori odds that a 5-3-3-2 shape opposite is more likely than a 4-4-3-2 shape. The above figures are estimates of the a posteriori odds under the given restrictions. There are 6 possible divisions of sides with a 5-3-3-2 shape and only 3 with a 4-4-3-2 shape. More conveniently, one should deal with the total number of trumps. The percentages are as follows: 18 trumps (55%), 19 trumps (40%), and 20 trumps (5%). In 11 out of 20 deals, a player holding 5-4-3-1 shape will be in a situation where the total trumps number 18.

The most important suit is spades in which it is assumed the player holds 5. One wishes to estimate what are the percentages for holdings of 7, 8, or 9 cards in the suit when consideration is limited to these 5 most common conditions.

¶

One has maximum uncertainty as to whether there are 6-7 spades, 8 spades, or 9-11 spades opposite, each with a probability of 1/3. Guessing 8 opposite is smack in the middle of the expectations, in which case the total number of trumps is 17. It would appear then that it is OK to bid 3♠ on the expectation of partner’s shape, or to stretch to 4♠ with appropriate controls.

There is a 53% chance that partner will hold 3 or fewer diamonds. What if one holds just 4 spades? One may convert the diamonds to spades and apply the percentages given above for diamonds, but if we assume the opponents would not preempt to 3♥ when holding as many as 7 spades, we can eliminate the lower two holdings when considering a 4-card spade suit. It would be better to double to show 4 spades and let partner decide whether to support spades or not. The odds are roughly 60-40 that you will find a good fit there, so a double is not aggressive since it is significantly on the right side of the odds.

There is only a 23% chance the partnership holds 8 or more clubs. What if one holds just 3 spades? One may convert the clubs to spades and apply the percentages. Normally it is not best to suggest playing in spades when holding 5-4 in the minors. If one assumes the opponents don’t hold as many as 7 spades for their preemptive action, the odds swing to roughly 60-40 that you will find a good fit in spades. It is acceptable to double with only 3 spades if the opponents are assumed short in spades, dangerous otherwise. After 2♥ -3♥, one may double hopefully, but after 1♥ – 3♥, one has to be more cautious.

The Mode and the Most Likely Distribution of Sides The mode is the most likely number of trumps for each suit. One sees the modes for spades, diamonds, and clubs are 8, 7, and 7, which is agreement with the single most likely side, 8=4=7=7. It is to be noted that given the 3-1 heart split, there are 22 cards remaining to be divided between 3 suits. The most even division is the most probable, that being an 8-7-7 split. In order of probability the component sides are 8=4=7=7 (13%), 7=4=8=7 (9%), and 7=4=7=8 (8%). One may wait a long time for a specific distribution to turn up at the table, but in total the most likely division of sides (8-7-7-4) occurs roughly 30% of the time.

Competing over 3♠ After the opponents have bid preemptively to 3♥, the odds favor taking action when holding 5-4-3-1 shape with a singleton heart, even with only 3 spades. The same percentages can be used to gain a rough estimate of the number of hearts held when the opponents preempt to 3♠. The difference is that one needs more to bid a level higher, so a 9-card fit is the norm for bidding at the 4 level, only a 33% chance with a 1-5-4-3 shape. At IMPs scoring, vulnerable against not, bidding 4H is not unreasonable when holding 4 hearts when the high card content otherwise justifies the risk.

Larry Cohen Defies the Law

The tradition in America is that a double of one major promises at least 4 cards in the other, but there are cases where an off-shape double is made on general strength. This can lead to problems later. The idea is that the doubler can get around to describing the nature of his call on the next round, but this method is fraught with danger as it is susceptible to preemptive action. The following example involves Larry Cohen and Dave Berkowitz playing in the finals of the 2004 Vanderbilt Cup against a famously active Italian pair. On Board 60, the opponents preempted, Cohen held a 4-4-4-1 hand, but it was Berkowitz who doubled first to show strength. It appeared they held at most 8 hearts between them, nonetheless, Cohen bid to 4♥, vulnerable vs not.

Dealer: North Versace Vul: E/W		Versace
		♠	10 8 7 6 3
		♥	7 2
		♦	5 4
		♣	J 10 6 2
Berkowitz				Cohen
♠	A K			♠	2
♥	K Q 4			♥	A J 8 5
♦	Q 3 2			♦	J 10 9 8
♣	A Q 9 8 5			♣	K 7 4 3
		Lauria
		♠	Q J 9 5 4
		♥	10 9 6 3
		♦	A K 7 6
		♣	—

Michael Rosenberg commented dourly on this deal in the Nov 2004 issue of The Bridge World. 4♥ is off 1 and 4♠ is off 2, so the total (major suit) tricks are 17 and the total trumps are 17, a perfect match. However, the best fit EW is in clubs, making 10 tricks, but the method of scoring dictates that EW play in the inferior fit. There is one other consideration: EW can make 3NT. The only alternative Cohen had in order to maintain the possibility of reaching the right contract was a space-saving double. That would make sense if the proper interpretation could be placed on that bid. It can’t be a penalty double, can it?

Cohen’s decision to bid 4♥ is wrong in theory, but the Italian pair could not afford to gamble that EW might make their game. 4♠ was ‘good insurance’. How could Lauria tell that the defense can take the first 4 tricks off the top? There is enough uncertainty that South makes his decision on general grounds. Cohen’s bold move has paid off.

This illustrates how the game is played. South opens light in third seat, West doubles on an off-shape hand too strong for an overcall, North preempts with a poor hand driving the opponents to an unmakeable contract, after which partner saves them by pulling to another unmakeable contract. Everyone feels they have routinely made the right decision. Only those who can see all 4 hands feel some improvement can be made.

Let’s consider Cohen’s decision to bid 4♥. As he holds a singleton spade, Cohen can place Berkowitz with at least 2 spades, possibly 3. Berkowitz may have a hand too strong for a NT overcall. Quite possibly the number of total trumps is 17 and the best fit is in a minor suit. One sees that the best game is 3NT. Of course, Cohen can’t know for sure the best available alternative. It would be convenient if 4♥ proved the best spot, but there is no reason for thinking it is, except on general grounds. This is a good situation for a doubtful double that leaves the decision to partner, who, after all, stands to have the best hand at the table. Double cannot be unilaterally for penalty, but all alternatives remain in place.

3 NT is To Play

The best answer to frivolous preempts is to bid an uninformative 3NT directly and see if the opponents can find the defence to beat it. After all, the opponents are reducing the information content of their bids, so why should the stronger side help them out by expressing doubt and giving away information unnecessarily? Bidding what you think you can make is the other side of the application of the doubtful double. A problem may occur if a thinking partner has other ideas, as in this example from the finals for the 2009 Vanderbilt Cup.

Dealer: North Vul: None		Elahmady
		♠	K J 10 8 7 2
		♥	K 9 8 7 3
		♦	2
		♣	3
Moss				Gitelman
♠	Q 6 5 4			♠	A 8
♥	5			♥	A Q 10 6 4
♦	K 10 9 7			♦	A Q 4
♣	A J 8 7			♣	K 5 2
		Sadek
		♠	3
		♥	J 2
		♦	J 8 6 5 3
		♣	Q 10 9 6 4

**Looking for a 4-4 fit?

The match was tied at the half-way point when Elahmady used a preemptive 2♥ opening bid that showed hearts and another unspecified suit. Obviously he was not trying to describe his hand accurately, but rather to disrupt the opponents. Gitelman bid what he thought he could make. He would welcome a heart lead, so his shut-out bid put the pressure on Sadek to find a better lead. It was akin to a ju-jitsu move where the aim is to use the opponent’s energy to work against them – a good strategy.

On such jumps to 3NT the bidder has a right to expect a few useful pieces to appear in the dummy, but some partners are very protective of their right to be heard from. Thinking, perhaps, that one doesn’t preempt after a preempt, Moss felt an ace and a king were enough to make an attempt to improve the contract, perhaps to 4♠. Elahmady’s uninformative 2♥ bid was working its magic. Gitelman signed off, but Moss was still not satisfied that a power slam wasn’t possible. 5NT was not forcing and makeable, but Gitleman could see extras for his previous 4NT sign-off, and went on to the hopeless slam, losing 11 IMPs, a prime example of two heads being worse than one.

One might imagine an auction where Gitelman doubles, Moss shows interest in a spade contract, and Gitleman signs off in 3NT. In this sequence Moss is given the opportunity to have his say, which can then be over-ruled graciously. If Moss persists with a 4♣ bid, 4NT reinforces the opinion that 3NT was best, and all’s well that ends well. This is contrary to the concept of a doubtful double and bidding what you think you can make when you have no interest in slams, but it massages a partner’s ego who no longer feels neglected. ‘Well, I tried’, he may say with a sigh of resignation.

If one has to bid psychologically to keep a partner’s ego satisfied, science takes a back seat. Second guessing becomes part of the system when neither player can assume the captaincy of the auction. In my opinion, the disaster on Board 33 led to further loss of 13 IMPs three boards later on this combination of hands.

In the other room on a similar auction Levin cue bid his ♥A and Weinstein proceeded to slam in the hope his diamonds would prove in the end a source of tricks. They did. Gitelman proved to be a bit shy despite his fine support for partner’s presumed lengthy diamond suit. The East hand is much better than might be expected from the initial free bid. It would cost to make a slam try only if an aggressive Moss would over-react to any modest move in that direction. The way I see it, the partnership psychology was wrong.

Total Trumps and 5-4-2-2

In the above example Gitelman held a 5-4-2-2 shape, and partner had advertised length in his short suit, albeit one with KJ doubleton. The 5-4-2-2 shape has gained a bad reputation over the years even though the a priori probability of finding an 8-card fit in one of the long suits is the same as for the 5-4-3-1 shape (74%). The total trumps are one less for the former shape, and the latter shape has the advantage of a built-in shortage.

One may analyze the distribution of sides for 5-4-2-2 in the same way as described above for the 5-4-3-1 shape when the opponents have advertised a 9-card heart fit. We spare the reader the details, only to show below the most likely distributions which illustrate the potential problems encountered with the 5-4-2-2 shape.

The estimated total trumps are what partner would calculate if he assumed the standard 4=1=4=4 shape opposite. This would be a very bad estimate when he held 5 clubs, the suit in which his partner holds only 2. The mathematical reason why Cases II and IV have such a high probability of occurrence derives from the fact that the most even split between 11 unknown clubs is 6-5, in the ratio of 7:5 with regard to a 7-4 split. This argues against doubling for takeout with a 5-4-2-2 shape. The temptation is to try to hit the best fit in the long suits, while the danger is that partner’s best suit is the one in which one holds a doubleton.

If one simply bids one’s 5-card suit, there is a 20% chance that the 4-card suit provides the better fit. Of course, if the 5-card suit is spades, there is no concern about that when competing for a part score or a game. If the 5-card suit is diamonds and the 4-card suit is spades, it is a different situation. One may double and correct to diamonds if partner bids clubs. (Remind yourself that partner is not trying to be difficult, he is merely bidding his longest suit. Of course if he jumps a level to do so, one may question his judgment.) If the 5-card suit is clubs, one needs must correct 4♦ to 4♠, and partner must interpret this as suit correction and not an indication of slam invitational values.

Finally, one may ask whether over a preempt to the 3-level in spades one should bid a 4-card heart suit to the 4-level as Larry Cohen did in the above example. Even if you are wrong, the opponents may ‘save’. Otherwise, provided that the opponents don’t hold more than 6 hearts, there is roughly a 2 out of 3 chance that one will find an 8-card or longer heart fit in partner’s hand. As noted above it pays to have a good holding in the doubleton suit when that is probably partner’s longest suit. Here is a case where a doubleton honors do not constitute ‘wasted’ values, but, in fact, are an essential component in the hands taken as a complementary whole. (In terms of the losing trick count, one doesn’t like to gamble with 2 losers in the short suit.)

Total Trumps with a 4-4-4-1 Shape

May 7th, 2009  ~  Bob Mackinnon  ~  No Comments 

In To Bid or Not To Bid, his book on the Law of Total Tricks, Larry Cohen often estimates the number of total trumps on the assumption that partner holds a singleton and a 4-4-4-1 shape. On the surface this is a rather a strange assumption, because a 5-4-3-1 shape is more likely on an a priori basis. One might think vaguely that 4-4-4-1 is reasonable as the average of 3, 4, and 5 is 4, but that is not a fruitful way of thinking. In this study we consider the exact consequences of Cohen’s assumption with regard to probability considerations to be taken into account during the auction.

Here are 6 such consequences of assuming partner holds a 4-4-4-1 shape:

1) your side’s best fit is in the longest suit in your own hand;

2) the total number of trumps is 16 + the difference between your longest suit and the opponents’ trump suit;

3) the estimate of total trumps thus obtained is optimistic on average;

4) given that partner has a singleton in the opponents’ trump suit, the resultant distribution of sides is the single most likely one on a random deal;

5) the 4-4-4-1 shape represents a condition of maximum uncertainty with regard to the number of cards held by partner in the potential 3 trump suits.

6)  The assumption conforms to Jaynes’ Principle, which states that in a condition of partial knowledge one should assume the distribution that is most probable, that is, the one associated with the greatest number of card combinations on the basis of a random deal and whatever else is otherwise known.

We shall now illustrate how the consequences apply to common competitive bidding situations in which the Law of Total Tricks plays a central role.

The Total Trump Calculation

When the dummy appears a declarer can count the total trumps by subtracting number of cards in the longest combined suit from the number of cards in the shortest and adding the result to 13. The total of trumps is an unambiguous characteristic of the division of sides.

The number of total trumps = 13+ (longest combined suit – shortest combined suit)

Of course, the defenders’ hands produce that same number of total trumps albeit generally with different independent hand distributions. Their division of sides may be different from that of the declarer. For example, if declarer has an 8-7-6-5 division, so will the defending side, but in the order 5-6-7-8. The total number of trumps is 16. If declarer has a 9-6-6-5 division, the defenders will have a 4-7-7-8 division, and the total number of trumps is 17, no matter which division of sides one uses to calculate it. A difference in the division of sides is a characteristic of deals that produce an odd number of total trumps.

The problem faced by a player in the middle of an auction is to estimate the total number of trumps knowing only for sure his own hand shape. Naturally, one would attempt to use the estimate that is most probable given what is assumed at the time from the auction. Suppose the opponents have preempted and partner has doubled for takeout. The calculation of total trumps is easiest for a 4-4-4-1 shape opposite. That shape contributes a difference of 3 between the long suit and the short suit regardless of which of 3 suits provides the best fit. This produces a minimum of 16 total trumps. To obtain an estimate of the total trumps with both hands taken into account, partner needs to add to 16 the difference between his longest suit and the presumed trump suit of the opponents. Thus, with a 5-3-3-2 shape the number of total trumps will be either 18 or 19 depending on whether the player holds 3 or 2 cards in the opponents’ trump suit.

The Most Probable Conditions

To illustrate how to estimate relative probabilities, we consider the following simple example from Cohen’s book where a player holds this hand: ♠ A43 ♥ QJT54 ♦ 963 ♣ 82, a 3=5=3=2 shape. The LHO opens the bidding with a preemptive 5♣. Partner doubles and the RHO passes. Cohen assumes the LHO holds 8 clubs for his bid and partner holds a singleton in that suit for his double. What is the number of total trumps one should use as a basis for the decision of whether or not to pass the double for penalty? Cohen assumes partner holds a 4-4-4-1 shape, so the division of sides should be 7=9=7=3. The total trumps add up to 19 (13+9-3). The division of sides for the opponents is 6=4=6=10. Cohen concludes that one should pass the double with such a low number of trumps available.

Let’s look at the relative probabilities of a 4-4-4-1 or a 5-4-3-1 shape opposite. Here are 6 possible distributions.

I                 II                III              IV               V             VI

♠ 3 – 4        ♠ 3 – 4        ♠ 3 – 5       ♠ 3 – 5      ♠ 3 – 3      ♠ 3 – 4

♥ 5 – 4       ♥ 5 – 3        ♥ 5 – 3       ♥ 5 – 4       ♥ 5 – 4     ♥ 5 – 5

♦ 3 – 4        ♦ 3 – 5        ♦ 3 – 4       ♦ 3 – 3       ♦ 3 – 5      ♦ 3 – 3

♣ 2 – 1        ♣ 2 – 1       ♣ 2 – 1      ♣ 2 – 1       ♣ 2 – 1     ♣ 2 – 1

Sides    7=9=7=3    7=8=8=3    8=8=7=3    8=9=6=3    6=9=8=3   7=10=6=3

Trumps     19                  18            18                  19                 19           20

Weights    100                 96            96                  69                 69          46

The probability weights are a reflection of the number of card combinations available for each pairing. Once a player sees his hand the distribution on the left is fixed (at 3=5=3=2) and it becomes a question of how many card combinations are available to his partner on the right-hand side. The more combinations available on a random deal basis, the greater the probability that the given condition exists. Condition I encompassing a 4-4-4-1 shape is the single most likely distribution, however, it is much more likely overall that partner holds a 5-4-3-1 shape as there are 6 such possibilities(only 5 are shown).

Calculation of Weights

There are available these many cards with which to fill out the hand: 10 spades, 8 hearts, and 10 diamonds. Conditions I and II has these many card combinations chosen at random from the pool of unknown cards:

Condition I    (10! 8! 10!) divided by (4! 6!)(4! 4!)(4! 6!)

Condition II    (10! 8! 10!) divided by (4! 6!)(3! 5!)(5! 5!)

Ratio II to I    24 divided by 25 which is the same as 96 to 100.

Similarly for Conditions III through VI. These calculations don’t take into account high-card content, but they provide reasonably accurate guidelines in this free-wheeling age where consideration of shape plays the dominant role in competitive bidding.

Note that the most even distribution in the 3 suits, 4-4-4, produces the maximum number of card combinations. Condition I is the condition of maximum likelihood, that is, it is the single most likely configuration given the player has counted his own hand. The fit in the heart suit is the key factor. There is a 9-card heart fit for Conditions I, IV, and V, hence 19 total trumps, and a total weight of 238 (46% of all cases). There is an 8-card heart fit for Conditions II and III, hence 18 total trumps and a sum of weights of 192(37%). It is most likely that the total number of trumps is 19. Least likely is that the total trumps are 20, 2 conditions with a total weight of 92(18%). There is a low probability that the longest suit in one hand matches the longest suit in the hand opposite. The average number of total trumps is 18.5, so 19 represents a slightly optimistic estimate on average.

Subjectivity These probabilities are based on the dealing of the cards, a situation of maximum uncertainty with regard to the placement of the cards. The auction may provide clues that constitute information that gives greater weight to one condition over another.  There is another factor to be taken into account, which is: with which shape is partner most likely to have doubled? The most advantageous situation for takeout is represented by ConditionVI for which the doubler holds 9 cards in the majors with longer hearts that spades. If partner takes out to 5♥, all is well, as from his point-of-view that should represent the best fit. Not surprisingly this is also the condition that represents the greatest number of total tricks. Thus, it is also the condition under which it is most dangerous to pass the double for penalty.

The degree to which one might wish to adjust the probabilities depends on the known behavior of one’s partner. If his doubles are generally penalty orientated and send the message, ‘they can’t push us around’, then one might favor leaving the double in as penalty. If his message is more likely to be, ‘we should play this hand’, then Condition VI becomes more likely and one would be more inclined to take out to 5♥.

Theoretically there is nothing wrong with adjusting probabilities according to what you expect from the known inclinations of one’s partner and/or one’s opponents. To be realistic, probabilities must reflect the current state of partial knowledge. That is ‘un-mathematical’ unless we can assign some numerical percentages to our subjective bias. So, in the above example, one should make an estimate of how likely it is that the doubler is operating under Condition VI rather than Condition I. Probabilities are adjusted accordingly. If one is maximally uncertain about partner’s action, one accepts the weights as shown which represent the probabilities of the deal, the a priori condition of maximum uncertainty. Partners being partners, that may be the best policy overall. However, Larry Cohen has given us many examples where experts bid one more than the rest of us, not always best.

A Mathematical Way to Choose a Partner

April 27th, 2009  ~  Bob Mackinnon  ~  No Comments 

There is a special game coming up at my club and I would like to win it. I have 2 very good non-expert partners whom I could ask to play. One is very cautious (Player A) and the other is very aggressive (Player B). Which one is more likely to produce the 3 tops over average which are needed to stand a chance of coming away with the trophy?

I decided to do a simple mathematical analysis of our chances based solely on the expected distribution of tops and bottoms. Player A is a very good card player who never takes big risks. He seldom generates a bottom and not very often generates a top through overt action. On most of the boards we are at the mercy of the opposition who provide us with a scattering of average pluses and minuses that average out over the session. My task is to be a reliable partner who occasionally adds a dash of initiative.

Player B is like a stock speculator who gambles on emerging markets with unsubstantiated optimism. He gets in early and often, forcing the action with aggressive bidding based largely on distribution rather than HCP. Many of the opposing pairs who would have made a mistake if given the chance are forced to silent but effective defence against a hopeless contract. Player B has a tendency to aim always for the best possible result – too narrow a target.  Sometimes it works, and partners must get out of the way and accept some bottoms along with the more frequent tops.

One might think that a swinging player is more likely to produce a big score than a cautious player, on the grounds that swingers are masters of their own fate and may pile up many more tops when the conditions prove favorable.  Assume the upcoming session is 26 boards, top of 12. On similar sessions at the club Player A and I produce on average an estimated 3 tops and 1 bottom through our own overt actions, that bottom may be due to my over-reaction to his quiet approach. Let’s suppose that Player B produces 5 tops and 3 bottoms, my role being reduced largely to that of scorekeeper. Both methods produce 2 tops over average, a decent 58% score. The question is this: with which partner am I more likely to score more than 3 tops over average (61.5% or more)?

To obtain a rough estimate of our future chances we assume the tops are randomly distributed in time so as to conform to a Poisson distribution with a given average. The average is also the variance of the distribution, so 5 tops can produce a large number of tops at times. That’s encouraging to those who habitually play for tops, but we can’t overlook the bottoms. Psychologically we may think of a top as a result of good practice and a bottom as a result of bad luck, but the two belong to the same family for which not every member turns out to be a smashing success.

We assume the number of bottoms also conforms to a Poisson distribution, one independent of the distribution of tops. Thus the joint probability density function of tops and bottoms taken together is merely the product of their individual probabilities. This is not true in general as early results affect later actions, however, we assume that such is not the case, and that each board is played on its own merits in a consistent manner.

Poisson Probability Distributions

Player B                Player A

Events Tops (5)    Bottoms(3)        Tops(3)    Bottoms(1)

0        0.0067        0.0498            0.0498        0.3679

1        0.0337        0.1494            0.1494        0.3679

2        0.0842        0.2240            0.2240        0.1839

3        0.1404        0.2240            0.2240        0.0613

4        0.1755        0.1670            0.1670        0.0153

5        0.1755        0.1008            0.1008        0.0031

6        0.1462        0.0504            0.0504        0.0005

7        0.1004        0.0216            0.0216           —

8        0.0653        0.0081            0.0081           —

9        0.0363           —

10        0.0181           —

For those who love numbers as I do, the columns show how the number of tops and bottoms (events) are likely to be distributed over many sessions. The numbers in each column sum to 1, so their products also sum to 1, as all cases are covered. We wish to extract those cases for which the tops exceed the bottoms by 3 or more. The combinations are 3-0, 4-0, 4-1, 5-0, 5-1, 5-2, and so on.  The sum of the products in these cases gives the proportion of sessions in which the condition for a good score holds.

For Player B I estimate the proportion of good scores is just above 30%, whereas for Player A it is about 40%. Thus, swinging for tops lessens the chance of a good score when the associated number of bottoms is also high. Surprisingly, just playing a large number of hands for boring averages is a good policy if one can combine that with the ability to score well when the few opportunities arise. Failure to take advantage of erring opponents by avoiding penalty doubles is carrying caution to an unacceptable extreme. One needs to generate some tops in order to expect to win, but careful Player A appears to be the better choice.

And yet… the winners seem to score many tops, so it can’t be an entirely bad strategy to force the action during the auction. The secret is to avoid the bottoms while striving for tops. If we assume that the average number of tops created by overt action is 5, what is the average number of bottoms that is required to match the performance of Player A? It turns out that average has to be reduced to 2.2 per session.  Two bottoms are an acceptable number of disasters provided that one generates 5 tops through hyperactivity. Player B is a little too error prone and doesn’t always come up to his full potential.

Memo to Myself

When playing with cautious Player A, remind him not to always play down to the field. Encourage him to take advantage of clearly advantageous positions, such as bidding to a cold minor suit slam rather than stopping much too short in 3NT.

Remember: all raises are invitational and the minor suits are mere stepping stones.

Confession

I must now confess that the results given above are restricted by practical considerations based on my experiences with these players. I have not included cases where we might generate 6 or more boards above average (73%). Quite probably it is as much as, if not more than, my fault than theirs, but there it is. One shouldn’t be a slave to mathematical theory when practical experience over-rides the idealistic assumptions. However, suppose that Player B and I were in the expert class so that extremely good scores should not be ruled out, then we can add 10% to the total of Player B, giving him about a 40% chance of scoring over 61.5%. Those great sessions, although rare, add up. If Player A were promoted to expert class as well, he would have a 4% chance of scoring over 73%, thus keeping ahead of Player B in the ‘good score’ category, while falling behind in the ‘excellent score’ category. Thus, if the game were a world-wide pairs contest, I should choose Player B, as a big score is needed to place amongst all those unknowns who get over 70% playing in small clubs in remote locations.

These are the results for a one-session matchpoint event. One difficulty for swingers is that there may not be enough boards on which to swing tops to make up for the inevitable self-generated bottoms.  Often it is merely a matter of running out of boards. Eight tops and five bottoms constitute half the boards in play. There is a logical reason for playing a tight game in a short contest and the mathematics should reflect that.

In a 2-session (or longer) event there is more scope for recovery from early bottoms. The Poisson distributions for larger averages more closely resemble a normal (Gaussian) distribution symmetrical about the mean. The cautious player loses some of the advantage due to the skew of the Poisson distribution concentrated between zero and two bottoms. (Refer to the right-hand extreme of the table above for evidence of this effect.) As a result of the symmetry the key characteristic becomes the difference between the average number of tops and bottoms.

Swingers must combine patience with their aggression. A convenient way of achieving this is to play a system that differs from that of the majority and then to keep faithfully to that system. This removes the emotional element. One waits for the opportunities for tops that will inevitably arise because one is operating under different bidding conventions. At my club the 2/1 system is the almost exclusive choice. I prefer Precision, but suspect that Polish Club is best. The measure of success should be how many more tops than bottoms are produced systemically. Moving to Poland, I might switch to ACOL, but I very much doubt that would be a move in the right direction.

If one’s main concern is avoiding bottoms, one should adopt the communal bidding system, perhaps, for the sake of ego, adding a few trendy conventions that seldom arise. The worse the communal system the better it is for the experienced players who can make intelligent adjustments, (sometimes incorrectly referred to as ‘lies’), and who can take comfort in the belief that, come what may, they will never do worse than the dumbest pair in the me-too crowd.

Why lie?

March 30th, 2009  ~  Bob Mackinnon  ~  1 Comment 

He who recognizes that all men are born liars best knows all the Truth that needs concern him, because he will then know himself and have an advantage over others.

–  from The Handbook of Lies (1554) by Bartholomaeus Ingannevole

The questions we wish to address with regard to bridge are these: what constitutes lies and do they pay off in the long run? With regard to liars, St Augustine cast a very wide net indeed. Subsequent saints accused him of over-fishing, recognizing that some lies are like marriage, sometimes a necessary means of reaching a desirable end. Today the general public, attuned to moderation in all things, agrees that some lies are forgivable, yet the attitude of Bartholomaeus is still condemned on the basis of insufficient proof. For the purpose of this article, let’s consider it a working hypothesis.

Some cynics claim that language is the stuff that lies are made of. A bidding system constitutes a language, so of its very nature must engender a pack of lies.  However, to paraphrase Sir Francis Bacon, bids are the images of matter, and to fall in love with a bid is to fall in love with a picture. So we mustn’t put too much credence in what a bid purports. It’s our own fault if we do. Perhaps Bacon’s imaginary was inspired by a famous historical lie, Holbein’s flattering portrait of Anne of Cleves that was brought to Henry VIII then considering re-marriage. In the context of bidding, a lie is a call that grossly misrepresents reality and is intended as a push in the wrong direction.

Misrepresentation is relative to a predefined structure limited by the HCP content and the distribution of cards. The more vague the definitions the less information the bids contain. Various hands can be included within the definition of a call, and the greater the variety the greater the uncertainty and the less the information content. The rare exception does little to change the information content, as the players involved will assume initially that the most likely hand is the one being represented, a view in which they, like Henry VIII, may be sadly mistaken. A lie occurs as an outlier far beyond the expected limits of variation. Is there a possibility of defining lies in terms of statistical variation? No. Even a miracle can be judged to be within finite mathematical bounds of probability. What about intent? Hard to judge. Most would conclude that although we can’t give an exact definition of a lie, we can recognize one when we see one.

Here is a simple example from my experience to illustrate the difficulty. A weak 2♠ bid is defined as showing a 6-card spade suit within the range of 6-10 HCP. Would you say this hand falls within the definition: ♠AJ9765 ♥4 ♦JT986 ♣9? Many would agree 2♠ is a legitimate bid in third seat but would condemn it in second position on the grounds that partner may hold a good hand and that 6♦ may be the best contract. This argument is based on the principle of self-interest. Next suppose that the RHO has spent some time considering her hand before passing in first seat. Well, that is as good as an opening bid in my book, and the possibility of game or slam in diamonds has diminished to the vanishing point. So I make a lead-directing preempt of 2♠, and it works. The information content of my 2♠ bid has been little affected because it is still much more likely that my shape is 6-3-2-2 or 6-3-3-1 than it is 6-5-1-1, however, my partner and the opponents are much more likely to assume the former cases than the latter one. I do not consider this bid to be a lie under the inadequate definition, but the risks were judged to be greater for the opposition than for my partner, and my intent was to take advantage of the uncertainty created.

Bridge World Views Quite often problems arise where there is no bid that quite conveys what one wishes to tell partner. The Master Solvers Club thrives on situations where there are at least 3 viable alternatives to a bidding problem. This tells us immediately that we are close to a condition of maximum uncertainty where 3 bids can be chosen with equal probability. It is a consequence of the inadequacy of the communication system. There is no real solution to such problems, even long-time partners often disagree, so the MSC may continue on for years to come without resolution.

For Problem B of the March 2009 issue the top score was awarded to an overcall of 1NT with a hand that contained a singleton. BBO commentators have informed us that bidding 1NT with a singleton is common practice in China. Does that mean most Chinese players are liars? No. Lies through custom lose the name of falsehood. The experts’ answers to Problem B tell us that the definition of a 1NT overcall is being altered in America, because, presumably, it works. For a while non-experts will be bamboozled by professionals promoting the interests of their clients until the dupes catch on and try the same thing at the local club. Eventually it may become a part of ‘Standard American.’

Here is Problem E from the same issue.  Matchpoints, None Vulnerable:

1♦   (1♥)    ???    You hold:  ♠A86 ♥942 ♦T93 ♣AQT2.     Your bid?

Here we are faced with a situation where we might bid 1NT without a stopper in the overcaller’s suit. Playing Precision where 1♦ is nebulous and limited to at most 15 HCP, I just might try that and term it a ‘pre-balance NT’. Today so many overcalls are made on worthless suits that it is not guaranteed that I’ll get a heart lead. Also, as we saw in the recent Vanderbilt Cup, ♥942 can act as a stopper opposite ♥QJ tight, nevertheless I would classify 1NT without a stopper as a lie, but not a lie that experts yet condone.

Only 2 experts could bring themselves to pass, because the modern game demands immediate action. This meant the other Master Solvers had to ‘lie’, as no bid was pre-defined in such a way as allow this hand to be included. Of course, because they bid, their definitions changed through usage. In this case the biggest change was to the meaning of double, which previously promised 4 spades. The lesser fib was a bid of 2♣ which would now no longer promises 5. Because the definitions were changed by their actions, their bids can no longer be classified as lies, but as established agreements. The hand certainly looks like one best suited for 1NT, stopper or no stopper.  As noted by Anders Wirgren, a Swede, there is a classical solution that solves Problem E in a pristine manner: a double denies as many as 4 spades. To avoid lying should we redefine our doubles?

Players continue to play systems in which they are forced to go against prior agreements. Most bidding systems, like Bridge World Standard are ‘kludges’, consensual patchworks that have evolved to combat changes in practices as they occur. It is dishonest to continue to maintain we play the old definitions designed in gentler times when we don’t. It is hypocritical to interpret one’s own agreements freely, then to demand that the opposition have unassailable agreements in rare situations and to appeal an unfortunate result on the grounds of being misinformed. Fans who watched the Vanderbilt Cup on BBO know to whom I refer. It is permissible to be inexact when there is no specific agreement, as inexactitude is the normal condition of the game. An admission of uncertainty does not constitute a lie, in fact, it may be the most truthful description in many situations. We should never classify reasonable departures from expectations as ‘lies’, but as a normal part of the game. The convention card presents only a vague outline with questionable limits, but the idealistic proposition that an opponent should know everything a long-term partner knows is unsound. Suppose my partner were to inform an opponent, ‘Bob sometimes bids 2♠ with 6-5-1-1 shape.’ That would be a true but misleading statement because it is most unlikely to happen again.

Is Honesty the Best Policy in Bridge?

All that one gains from falsehood is not to be believed when speaking the truth

– Aristotle (384BC – 322BC)

Aristotle put his finger on the pulse of poker where bluffing is an essential element of the winning strategy. Of course, lying is pleasurable in and of itself, otherwise, I assume, it would not have merited the status of a sin and poker would not be as popular as it has become. Poker is essentially a psychological game of percentages where the percentages include not just the probability of the card distributions but also the probability that an opponent is bluffing or can be bluffed.

Poker analysis is reduced to card combinations when an opponent invariably bets on his good hands and folds on his bad hands. The information gathered from a predictable opponent influences the probability of the card distributions, and so makes it easy for an expert to prevail on that basis alone. Players who go strictly by the book are easy victims of those who can translate their actions correctly. When only experts are involved, masters of the odds and good bluffers to boot, psychology comes to the fore, and it becomes a clash of wills. One has to create an atmosphere in which yours is the dominant personality. You set the agenda by bluffing. Not only that, but it is to your advantage to be caught bluffing early to show you are not afraid. To fold with a good hand is equally deceptive, but it is very bad psychology. So you lie when you hold nothing of value.

The strategy of bluffing applies to bridge insofar as bridge can be considered in the context of psychology. It appears that some leading players have adopted the idea that lying is beneficial in the long run. That has always been a minor element in bridge, usually restricted to bids in third seat on the grounds that passing partner is less likely to act on an optimistic misrepresentation of one’s assets. The Drury Convention testifies to that practice. The difference today is that experts will bluff in first and second seat before partner has had a chance of express his own values.

Does bluffing (lying) pay off at bridge? It is a hard question to answer. The bluff and its resulting loss are sure to be revealed, so that is not of concern to the practitioner. He expects to make up for the loss in many following hands where he tells the truth but the opponents don’t believe him. For example, this may affect an opening lead against a close game. So, the current loss is obvious, but the future gains are hidden.

The trouble with applying poker strategy to bridge is that it is not expensive for an opponent to call the bluff – he simply bids 3NT. That is a good bet as only seldom will he miss a slam, and the cost of going down may not be great. The upside of 3NT is that it may be slated for defeat but actually makes on poor defence based on the misinformation provided by the bluff bid. It may even be doubled and making when partner takes the bluffer seriously or when declarer is able to read the distribution. We encounter this phenomenon frequently on BBO. Here is an example from the 2009 Vanderbilt semi-finals.

Dealer: South Vul: None		North – Moss
		♠	AKJT32
		♥	K7
		♦	T6
		♣	762
West – Zia				East – Hamman
♠	Q4			♠	875
♥	T2			♥	AQ9
♦	742			♦	K983
♣	QJT543			♣	AK9
		South – Gitelman
		♠	96
		♥	J86543
		♦	AQJ5
		♣	8

This was Board 11 of a 64-board match, early enough to try a bluff and reap the rewards immediately, always a hope, or make up for the loss on subsequent sessions. A lawyer might argue that Gitelman’s 2♥ bid is within the letter of the law, but to me it is a gross misrepresentation because the HCP are concentrated in diamonds and the hearts are weak. I conclude that Gitleman may have been influenced by co-residents of Las Vegas, the poker capital of the world, and that deception was the obvious intent.

Zia had nothing to contribute to the auction and Moss gave a typical unsound raise, leaving Bob Hamman with a tough decision. He followed the rule named after himself, namely, when in doubt bid 3NT, thus reaching a poor contract that could be set by the taking of at least 8 tricks off the top. But something happened on the way to the bank. Gitelman led an attitude ♥8, but it was a bit late to show a dislike of hearts. Now it was Hamman who had 8 tricks off the top, and in the time honored manner he ran the long suit in an attempt to pressure the opponents into giving him a 9th on a misreading of the cards. It worked! On the last club Gitelman discarded his life line to Moss’ spades and was endplayed into giving up a trick in either diamonds or hearts. In the confusion signaling again proved to be a weakness in the expert’s game.

What would be the effect of a pass in the South seat? We know the answer, because in the other room Nickell passed and Freeman bid 2♠, making 3, for +140 NS. So Gitleman’s brilliant bid was slated to pick up 3 IMPs; instead it lost 11 IMPs. That represents good odds in favor of Hamman’s Rule. Furthermore, it reduced the deficit from 21 IMPs to 11 IMPs giving new life to the Nickell team – to no avail as it happened.

Oh What a Tangled Web!

The Nickell team is justly famous for its tremendous late-stage comebacks. (Perhaps one should consider more carefully how they get so far behind in the first place.) Meckstroth was in a desperate situation by the time Board 31 was played in the Vanderbilt semi-finals with his team down by 47 IMPs. The best lies are those that have some element of truth. In bridge that translates to bidding good suits especially on bad hands. The main advantage is that when one defends it won’t cost to lead the suit from either side of the table. When one opens light on a garbage suit, the consequential lowering of expectations on the number of total tricks due to poor trump quality makes it all the safer for the opponents to stop in 3NT …. and then to make it, because, unlike poker, there is always a partner there to mess it up.

Dealer: Vul:		North – Meckstroth
		♠	6
		♥	K86532
		♦	9432
		♣	AJ
West – Greco				East – Hampson
♠	42			♠	AKQ95
♥	QJ74			♥	AT
♦	KQJT8			♦	5
♣	63			♣	Q9752
		South – Rodwell
		♠	JT873
		♥	9
		♦	A76
		♣	KT84

Meckstroth’s 1♥ opening bid was deceptive except with regard to distribution. Once Hampson was able to show a strong hand, Greco applied Hamman’s Rule, and it was up to Meckwell to defeat a 3NT contract that wouldn’t be bid in the other room. It was a bit late to get the necessary close co-operation that was not part of the opening bid strategy.

Meckstroth got off to a good spade lead. Greco set about establishing 4 diamond tricks in the hidden hand without having an entry to cash them all. On winning the ♦A Rodwell returned the top spade from ♠JT8 towards dummy’s ♠Q95. Greco cleverly ducked and Rodwell had to find the right switch. Those seeing all 4 hands saw the winning defence was to lead a club to Meckstroth’s ace, win the club return with the ♣K and lock declarer in dummy.  The setting trick would come either from a black suit winner in the South hand or from Meckstroth’s ♥K which would at long last prove its worth by snatching victory from the jaws of defeat. Unfortunately, Justice was served when Rodwell returned a heart giving Greco 5 undeserved tricks in the red suits. This nail-in-the-coffin cost 12 IMPs as the Nickell teammates, EW at the other table, played undisturbed in 2♦ going down 1.

Once again defensive signaling proved inadequate. Meckstroth had the opportunity of making 2 discards with which to make the Ultimate Signal, which is, ‘forget what I told you before’, much needed in current state of bridge bidding.  His discards, upside-down attitude, count, and suit preference, were the ♥5 and ♥2. The clear message of ‘clubs, not hearts’ was not emphatic enough to avoid the losing play. Obviously, if Meckstroth’s hearts had been as good as ♥KJ8652, there would have been no bad play for Rodwell.

Finally we look at Board 4 in the Vanderbilt semifinals, a deal that illustrates a real gamble at bridge, the redoubling of a contract which may not make. Like a raise in poker, the potential for a gain is substantially increased, as real pressure is placed on the opposition to make a costly move. Is it surprising that Zia was the perpetrator?

Dealer: Vul:		North – Moss
		♠	Q92
		♥	Q97542
		♦	—
		♣	QJ95
West – Zia				East – Hamman
♠	KJT5			♠	A63
♥	83			♥	A
♦	K876432			♦	Q95
♣	—			♣	KT8732
		South – Gitelman
		♠	874
		♥	KJT6
		♦	AJT
		♣	A64

Gitelman followed normal practice by not overcalling 1♥ over Hamman’s natural 1♣. On the next round he gave a gentle raise of partner’s weak jump overcall despite his prime support: 2 outside aces and 3 heart honors.  Some might admire his restraint at this vulnerability. Imagine his surprise when given room Hamman raised Zia to game at the 5-level. This seemed too good to be true, so he doubled with good prospects for 3 tricks outside the heart suit. So far the auction had taken a normal course. In the other semifinal match both teams played in 5♦* (Versace and Katz being the successful declarers), but here Zia added to the excitement by redoubling. If Greco and Hampson at the other table had also reached 5♦*, Zia’s redouble would gain 6 IMPs if the contract made, and lose 5 IMPs if it went down 1 at both tables. If NS had escaped to 5♥, the upside for Zia would be considerable, but Gitelman had no inclination to take what would be an expensive phantom sacrifice. He probably felt his plan had worked. It didn’t.

The action at the other table could be taken as an illustration of where previous bluffs pay off. The fear engendered without evidence is that one is being talked out of something. It is difficult to measure the effectiveness of previous deceptions, because even the victim may not be aware of the full extent of the influence on his decision.

The auction began poorly for the Precision pair in the EW seats. 2♣ was natural with 6+clubs, 11-15 HCP. Precision players may open with a long club suit and not much else, as Sontag did to his regret in an earlier round, and Meckstrorth did later in this match. On this hand it looks wrong to balance in the North seat. One shouldn’t suspect skullduggery at this vulnerability. South holds an opening bid, but he took no action even though his clubs must be weak. West might have given a weak raise to 3♣ on 3 poor clubs, so it doesn’t appear the opposition has a fit. No controls and QJ95 in the opposition’s only guaranteed long suit are bad omens. Who has the diamonds? Ah, there’s the clue, as West may have them and couldn’t bid because the Precision system doesn’t provide a natural nonforcing diamond bid. These considerations scream ‘PASS’, but what if one suspects a bluff? What are the chances of missing game?

Many feel that in today’s game one has to get into the bidding on what previously would have been considered inappropriate values. We have discussed 2 examples from the MSC. The most convenient change involves the double, and some would suggest that Nickell should have doubled on his hand in the Italian style. It would have been counter-productive here, as it gives West a chance to enter the auction freely. Whether one is sympathetic or not, bidding 2♥ represented an expensive failure to take advantage of a weakness in the opponents’ bidding system. All was not lost as Greco and Hampson still couldn’t find their diamond fit, but ended up playing in a 4-3 fit in 4♠, a contract that could be defeated. So maybe bidding 2♥ was not so bad after all. Nickell led a heart to Hampson’s ♥A. The ♦9 followed and it was up to Nickell to cover with a low honor. Instead he went up with the ♦A and gave Freeman a ruff, removing his nuisance trump from ♠Qxx. Understandable, but the real damage had been done before, perhaps long before.

Afterthought:  Beauty and Bridge

Accuracy is essential to beauty. – Ralph Waldo Emerson (1803-1882)

I do not know what the philosopher was thinking of when he wrote the statement above – perhaps of the fancy clock on his mantelpiece that couldn’t keep time, or of a coloratura’s faulty rendition of the Queen of the Night aria still ringing in his ears from the night before, or maybe of a shot curving past a goalkeeper’s outstretched arm only to deflect off the left-hand goal post.  Whatever the case, a bidding sequence can be judged beautiful if and only if it accurately reaches the optimum contract. It is most annoying when a beautiful sequence of bids results in failure due to an ‘unfair’ distribution of the cards. Accuracy is a higher form of neatness, a characteristic we all admire in others, but it represents a cool kind of perfection that appeals to the part of the brain capable of reasoning, a part that has a depressingly small role to play in our everyday decision making.

The realm of top-flight bridge is a small one with the same few players competing against each other repeatedly for the highest prizes after the vast majority of lesser talents have fallen by the wayside. Superior cognitive powers are enough to get one to the territorial boundary, but once admitted to the tribe, psychology comes to play an increasingly important role as each player strives to achieve or maintain a dominant position. Applying pressure through bluffs becomes common practice against other experts. Logically, this shouldn’t work.

Some idealists, and many commentators fall within the category, wish the game would provide a refuge within which what might be termed ‘the higher functions’ can freely roam. They treat a deal primarily as a puzzle capable of being solved by an application of logic. They write books embodying the concept that there is a right contract and a right way of playing the hand. They would be happy if the bidding process was strictly regulated so that the information provided would provide a sound and familiar basis for reasoning from the known to the unknown. To those idealists I say, control your rancor, embrace uncertainty. The closer the world of bridge reflects the real world, the more amusing it becomes.

(I highly recommend a new book, How We Decide by Jonah Lehrer (2009), a terrific read. He has a section on poker, but none on bridge. Don’t wait for the paperback edition.)

Nothing is beautiful viewed from every aspect. – Horace (65BC – 8BC)

The 3 D’s of Bidding – Describe, Deceive, and Declare

March 11th, 2009  ~  Bob Mackinnon  ~  6 Comments 

In a previous blog I put forth my opinion that the success of Meckwell lies primarily in their keeping to their agreements through thick and thin. I gave an example where Jeff Meckstroth made a bad 1♥ overcall of 1♦ on ♠ K843 ♥AJ98 ♦Q63 ♣86. There are hands with which one might overcall profitably on a 4-card suit, a favorite tactic of Marshall Miles, but this wasn’t one of them. I commended Eric Rodwell for keeping to their agreements by raising to 2♥ on ♠ QJT6 ♥ T62 ♦A82 ♣742, even though the result was a loss of 12 IMPs when the other room their rivals played in 2♠ making 140 on their 4-4 fit. Now we have been told that this overcall was not an aberration on the part of Meckwell, but more or less standard expert spin with an upside. (See the expert comments to the previous blog, for which I am grateful.)

Well, because a cold is common doesn’t mean you want to catch it. Let’s uncover recent evidence of the cost of violating partnership agreements by following the adventures of the USA Hampson Team, Precision players all, who finished a disappointing 4th in the competition for the Yeh Bros Cup won by the Netherlands’ team. I am a great admirer of the Precision System and feel that in the hands of great players it provides a distinct advantage over the opponents without the need for misdirection. Here a master-mind at work against an underdog team of very capable Indian players.

Dealer: South Vul:  None		North: Gupta
		♠	A3
		♥	Q753
		♦	Q43
		♣	KJT5
West: Cheek				East: Grue
♠	Q762			♠	J984
♥	A964			♥	KT8
♦	AKJ8			♦	72
♣	2			♣	9873
		South: Satya
		♠	KT5
		♥	J2
		♦	T965
		♣	AQ64

The result was down 3 for a loss of 7 IMPs on the board when the EW opposition played in 2♠ making 3. West might have a flat 11 HCP hand, but East is required in today’s game to reply on a bare 4 HCP. If normal light action results in disaster, partner will be sympathetic. Competing on slim values is going to be advantageous when EW have a spade fit, otherwise one is courting disaster. Declaring in 1NT without an 8-card fit may actually keep the opponents out of a 3NT that fails because of a poor 7-7-6-6 division of sides. As one can see, Cheek held an excellent hand and would have raised 1♠ to 2♠ whereupon a normal result is achieved. Unlucky? No, just another deserved and unnecessary loss due to masterminding on insufficient evidence.

Quite often in competition one should merely describe as best one can the nature of one’s hand and let the IMPs fall where they may. Here is another horrible result from the USA-India match brought on by reluctance to bid descriptively.

Dealer: South Vul: Both		North: Gupta
		♠	Q872
		♥	9
		♦	QJT654
		♣	52
West: Cheek				East: Grue
♠	5			♠	9
♥	A843			♥	K52
♦	AK82			♦	92
♣	T743			♣	AKQJ986
		South: Satya
		♠	AKJT643
		♥	QJT76
		♦	7
		♣	—

First we analyze the auction involving Meckwell. Over a limited 1♠ doubled Meckstroth, North, bid a straightforward 4♠ that forced East to guess at the 5-level, which normally belongs to the opponents. Rodwell bid one more. 5♠ makes, but Chokshi made the good decision to pull to 6♣ which was fated for down 1. Can we blame Rodwell for taking out insurance at the 6-level based on his hand? No.

Contrast that with the actions at the other table where Gupta took his time and ended up stealing the pot. Over 1♠* he felt that immediate action was not required – after all, unlike Rodwell, Satya had not limited his hand significantly. If Grue had merely described his hand by a jump to 5♣, there may have been a different story to tell, but he stalled with a non-descriptive cue-bid, eventually backing in with the 5♣ bid he was always going to make. Unfortunately Gupta proved to be the better general when he was able to pressure Cheek into an ill-advised double based on what appeared to be 3 quick tricks with more to come in clubs. The resultant loss was 13 IMPs. Could it have been worse if initially East had bid a straightforward and informative 5♣?

Here is a comic example of being too clever for one’s own good, by which we mean, by making a deceptive bid that serves to steer the opposition to a winning contract that they would not have reached if left to their own devices. We catch the world’s best pair indulging in ineffective deceptive bidding on the first board against a supposedly lesser Norwegian team that knocked them out of the finals. Board 1 – experts like to get in early.

Dealer: North Vul: None		North: Meckstroth
		♠	8
		♥	432
		♦	98654
		♣	QJ83
West: Brekka				East: Salensminde
♠	AQJT742			♠	53
♥	K75			♥	AQJ9
♦	A			♦	J2
♣	AK			♣	T9765
		South: Rodwell
		♠	K96
		♥	T86
		♦	KQT73
		♣	42

Rodwell’s 1♦ promised 11-15 HCP with as few as 2 diamonds. After 2 passes one might want to throw sand in the opponents’ faces, but the question always to resolve is, ‘which way is the wind blowing?’ A descriptive weak 2♦ might tempt me, and North would make a precipitous jump raise to 5♦, slated for a loss of -1400, but in Precision 2♦ is not a weak 2. The best descriptive call undoubtedly is pass. ‘Saved by the system’ works for me.

Brekka had his double, and Salensminde made a straightforward jump to show where his values lay. Isn’t it nice when partner bids a suit in which he holds honors? Brekka began an asking sequence. Rodwell took the opportunity to show he really had diamonds, which only served to convince Brekka that the spade finesse was working and that there were no wasted values in diamonds. So he bid the wrong grand slam! And it worked! The Norwegian’s bad bidding aided by Rodwell’s indiscretion gained 11 IMPs when in the other room Cheek-Grue stopped in a sensible 6♠. So we ask once more, would the result have been worse if Rodwell had bid systematically and passed throughout? What does ‘being clever’ entail beyond keeping to one’s system, drawing the right conclusions from partner’s bids, and going with the odds at the time of decision?

My confession in the spirit of St. Augustine. At the local club on the day I am writing this, against the top pair, I opened an abhorrent 2NT on ♠ AQ2 ♥ KQ965 ♦KQ76 ♣A. I would classify this as a mastermind bid related to the Standard American system we were playing.   Had I been playing Precision, I would have been worried about missing a slam in a 4-4 fit in diamonds, and after a strong 1ß opening bid I would have had the methods available for reaching it, but under the circumstances the chances of reaching 6♦ were slim. 6♦ would have gone down on an unsuccessful spade finesse, but no one here bids minor suit slams, even ones that make. Garozzo smiles. Yes, we had an 8-card heart fit, but happily partner raised to 3NT with help in clubs. On a club lead we scored a top at 660 on a logical misdefence. With everyone in 4♥, we would have scored a bottom if the ♠K were onside, but a 50% plus chance for a top represented good odds against this pair who normally defend well. Playing a bad system gives one excuses for taking matters into one’s own hands. There is a need in some to take charge. I think that is why so many masterminds like to play loose systems for which even a weak hand has the built-in opportunity of busily screwing up the auction and forcing a bad decision.

In the days of whist during Victorian times it was considered akin to cheating to act out of the ordinary in order to confuse an opponent. The term ‘false card’ has come down to us, but not the ethical implications. Although it still causes considerable annoyance to its victims, misinformation no longer a question of morality but one of practicality in the face of the uncertainty with which the game is imbued. In our machiavellian world of rampant individualism, there is no bad bid, only bids that turn out badly. There are 3 aspects to bidding common to most public communications: deficient description, deplorable deception, and dauntless declaration. As Hilary Clinton said with regard to her ‘smart’ diplomacy, it is primarily a matter of maximizing the chances of success.  This resemblance of our game to Realpolitik makes it worthy of serious study.

Bidding what you think you can make has its attractions to those who like to make the decisions for the partnership, but as every farmer who wants to make cheese knows, you have to have a co-operative cow, and there’s the rub. Information nourishes the brain. There is a close mathematical connection between probability and information, so uninformative bids, like false cards, result in an inaccurate estimation of the probabilities involved. The information content of bids is where dispassionate analysis should be focused. It is time to devise effective countermeasures to misinformation. God knows, we have enough evidence by now.