A Four-Door Monty Hall Game

November 16th, 2009  ~  Bob Mackinnon  ~  5 Comments 

Bridge is a game of choices. A player ideally makes the choices that give him the best chance of success within the rules of the game. Decisions are based on evidence and prior knowledge, and usually there is a large degree of uncertainty involved. How to make decisions in the face of uncertainty is a topic shared by many games, some serious, some not, but the theoretically optimum approach can be expressed in terms of probabilities. It is probability theory that links many games. In order to understand the optimum decision-making process in a very complex game, one may investigate how probabilities apply in a simpler game, then seek to generalize. For this purpose the game often chosen by bridge writers is the so-called Monty Hall scenario, which is supposedly well understood.

The link between Monty Hall and bridge is the Bayes’ Equation, named after the British clergyman, Thomas Bayes (1702-1761), who first stated the ideas in the language of the time. As is common with entirely new ideas, the current language was inadequate for their expression. Intuitively Bayes knew he was right, but he just couldn’t find the phrases to convince the world at large. It was the language of 20th Century mathematics that was required to give a full and clear expression of the ideas he first conceived. Ironically, British academics were among the last to concede that their countryman Bayes had been right. Is it only a problem with the vagaries of the English language?

Bridge writers want to make matters as simple as possible for their readers, most of whom are not familiar as they should be with mathematical expressions. So they resort to common language, and avoid equations, even though they themselves may be familiar with Bayes’ Equation, as was Hugh Kelsey the great bridge author. Very often their explanations fail through being over-simplified, inexact and unconvincing on close inspection. Generalizations become hazardous. In this segment we shall introduce the exactitude of mathematical terminology for interested readers, as applied to the more complex scenario which we call ‘Four-Door Monty Hall.’

The most famous demonstration of the validity of Bayes’ Equation can be credited to Marilyn von Savant, who in her column in the magazine Parade, applied Bayes’ reasoning to the popular TV game show Let’s Make a Deal hosted by MC Monty Hall. The circumstances are described by Jeffrey S. Rosenthal in his entertaining and instructive book entitled Struck by Lightning: The Curious World of Probabilities. The setting for the game was a set of 3 doors behind one of which sat a large prize. A contestant was asked to choose one door in the hope of winning the large prize. Initially, the chance of receiving the large prize was 1 out of 3. The MC then opened one of the 2 doors remaining to show the prize did not sit behind it. The contestant was sometimes given the option of sticking with his first choice or switching to the other unopened door. Von Savant took the fun out of the game by pointing out that the odds were 2:1 in favor of switching doors. This claim was met with much opposition from her readers; even some professors, who argued it could not be true, on the grounds that Monty Hall always could open a door with no prize, so with 2 doors remaining the chances should be 50-50 regarding which one hide the prize. Wrong! A ‘free’ choice has unexpected consequences.

The matter was resolved when Marilyn had teachers play the game with schoolchildren and report the results. The statistical evidence overwhelmingly supported Bayes’ 2:1 odds. To their credit, many dissenting experts publicly apologized and it appeared the matter was settled once and for all. Civilization moved a step forward.

In the game of bridge the well-known principle of restricted choice arises from the very same Bayes’ Equation that governs the probabilities in the Monty Hall game. In order to explain its application to their readers, bridge writers discuss the solution to the Monty Hall problem in the hope that the reader is convinced and can make use of evidence that comes their way during the play of the hand when a defender follows with 1 of the 2 missing honors. Here we aim for a wider understanding and wider applications. So, we consider the Monty Hall Game where 4 doors are available. The contestant chooses one door in the hopes of winning a large prize, and the MC opens one of the 3 other doors to show that no prize sits behind it. We show how to calculate the probabilities that the prize sits behind each of the 3 remaining doors, and we express the process in mathematical terms that have a general applicability. The process can be extended to as many choices as we wish to incorporate.

The Statistical Approach to Probability

The easiest way to understand how probabilities work is to imagine an experiment where choices are presented to a great number of participants. The choices are counted and put in a table of results forming patterns from which conclusions can be drawn. This can be a convincing approach if the results are clearly in favour of one circumstance over another, as von Savant demonstrated to her readers when the odds were 2:1. We shall return to the idea of testing hypotheses from a collection of data, but first we shall assume that the contestants adhere strictly to our assumption concerning how their choices are made. This yields the expected or average numbers which one may or may not see reflected in the actual results that are subject to variation due sampling conditions. There are 2 assumptions to be studied for the purpose of exposition which are illustrated below.

The box on the left indicates the four doors, behind one of which sits the prize. The large letter denotes the door which hides the prize. Each line is given 90 samples, 360 in all, so the initial probability that the prize is hidden behind a given door equals ¼ for each door. The samples are presented to 360 school children who are asked to choose a door from B, C and D that does not hide the prize. Thus, from line 1 the 90 children have a choice of 3 doors, whereas for the other lines the other 270 children can choose from 2 doors only. First we consider the ideal experiment where the children choose exactly in accordance with a probability model. There are 2 models to consider.

Equal Choices Under this model the children have an equal chance of choosing any door that is presented to them. So with 3 choices they will line up with 30 choices each when there are 3 choices available and 45 choices each when there are 2 choices. The results are shown in the middle box. The lines top to bottom represent the samples where the prize sits behind Doors A, B, C and D, respectively. The columns left to right represent, respectively, the number of times Doors B, C, and D were chosen. The numbers in each column add to 120, but this is not the case in general. Now we may ask the question: what is the probability that the prize sits behind Door A if Door B is chosen?

Door B has been chosen 120 times out of 360, that is, for 1/3 of the samples. For 30 samples, the prize was behind Door A. Thus, the probability that the prize was behind Door A after Door B is chosen is given by the ratio of 30/120 or ¼. The same is true for opening Door C or Door D, but this result is not general, but arises from our assumption of equally likely choices. This special condition is important theoretically and is referred to as the condition of maximum entropy or maximum uncertainty.

Given that Door B has been opened, the probabilities that the prize lies behind C and D must be equal and their sum must be ¾ . This is so because the prize cannot lie behind Door B. We get the same result by adding alone the column 45 plus 45 and dividing by 120. Thus the probability that the prize lies behind Door C is 3/8, and similarly for Door D. Given that Door B has been opened, the chance that the prize lies behind Door C rather than Door A bears the favorable odds of 3:2 rather than 2:1 as in the 2-door game.

Mathematical Notation We restate these evident truths in mathematical notation which provides the means to generalize. First we state that

P (A | B) + P(C | B) + P (D } B) = 1,

where P (A | B) denotes the probability that the prize sits behind Door A given that Door B has been opened. The equation states the fact that because the prize is not behind Door B, it must be behind one of the other doors. Next,

P (A | B) ∝ P (B | A),

This line simply claims that that probability of a prize being behind Door A is proportional to the probability that Door B would be opened if the prize were behind Door A. It is common sense to say that the more likely one is to choose Door B when the prize is behind Door A, the more likely that this is indeed the situation when Door B is chosen. More generally,

P (A | B) ∝ P (B | A) P (A),

P (C | B) ∝ P (B | C) P (C),

P (D | B) ∝ P (B | D) P (D),

where P (X) is the probability that the prize was initially behind Door X. It is not necessary to assume P (X) is the same for all doors, but that is an inherent condition of our experiment where each line represents 90 samples. The relevance to bridge is that any card initially has an equal chance of being dealt to any of the four players.

Biased Choices Randomness does not require that all possibilities are equally likely. It is easy enough to introduce a bias in our experiment by assigning colours, Door B being red, the most popular colour, Door C being chartreuse, and Door D being yellow, the least favoured colour. How exactly the addition of colours would bias the results is unknown, so we shall make assumptions that are reflected in the numbers given in the right-hand box. Red is twice as popular as chartreuse, yellow is never chosen when red is available, but is an equal choice with chartreuse otherwise. This assumption could be the subject of a test on results obtained in practice.

If we now go through the process indicated above for equal choices, we find the following: P (A | B) = 2/7, P (C | B) = 3/7, and P (D | B) = 2/7. Thus, Thus, P (A | B) divided by P (C | B) is 2/3, as in the previous example, but P (A | B) is equal to P (D | B). Also, the probability that the prize sits behind Door A has changed from ¼ to 2/7, even though every child had a ‘free’ choice of a door behind which there is no prize. What we really should say with regard to the first case is that the child made an unbiased choice. Some children may prefer chartreuse to red, and they were free to make that choice based entirely on personal preference.

If Door D is chosen: P (A | D) = 0, P (B | D) = 1, and P (C | D) = 0.

The effect of our assumption with regard to selection due to colour is that Door D is chosen only when Door B is unavailable. In bridge this is equivalent to the conclusion that if a defender fourth to play takes a ten with a king, declarer may assume he was not dealt king-jack. It cannot be said that he was not dealt king-queen.

We now present the tables of conditional probabilities under each condition defined by choice of door in the 2 cases considered, equal choices and biased choices.

The initial probability, P(A), that the prize sits behind Door A is given by Bayes’ Equation:

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

where Q (X) represents the probability that Door X be chosen. We use Q (X) because we must make a distinction between the possibility that Door X might be chosen blindly and the probability that it would be chosen. This distinction is not necessary if the choice of doors is unbiased, but such an assumption doesn’t always apply. In bridge there is a difference between the probability of a card being dealt to a given defender, and the probability that he would play it under the given circumstances if he had it.

Thus, based on the number given in the right-hand box,

P (A) = (2/7) · (7/12) + (2/7) · (7/24) + 0 = 1/4. Also,

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

= (3/7) · (7/24) + (1/8) = 1/4.

Similarly, the initial probabilities for P (C) and P (D) are correctly given if we use Q (X), the probability of Door X being chosen, rather than P (X), the probability that the door might have been chosen if the choices were unbiased. In relation to cards, P (X) refers to the a priori odds of the deal where all cards are treated equally regardless of rank, and Q (X) refers to the probability that a card would be played by a defender from a selection of cards. During the play one must refer the conditional probabilities as they apply to the cards in the suit to which a defender must follow. Probabilities depend on the priori knowledge, which includes such information as an imbalance in the vacant places at the time of decision that imposes a bias on a defender’s choices.

Testing an Hypothesis Say that we enumerated 360 samples with coloured doors that were submitted by schoolchildren. We compare the matrix of selections with the box on the right and ask whether the results can be said to conform to our model. Well, we can say immediately that there will be some (probably among those who choose to sit at the back of the class) who will prefer a hideous shade of sickening yellow to a bright and cheerful crimson. So we should never model human behavior under an assumption of ‘never.’ Furthermore, even if we assumed 5% for yellow versus red, it would be hard to justify or reject that assumption on a small number of samples. To put a number to a rare occurrence requires many samples in order to achieve a high degree of confidence. In fact, the matrix of the results may fit more closely the box for equal choices (The school colours of one set of happy students was green and yellow, whereas at another school the unpopular home room teacher was wearing a red dress.) So we must always be suspicious of results from small samples where local conditions will affect the choices.

The scientific method is to gather a set of data, say, 360 samples, adjust one’s hypotheses, then test again on an entirely new set of data. As one plays hands one is gathering results to add to the collection of data stored in our memory. It is easy to be swayed by our emotions. Emotionally successes outweigh the failures, and we may continue to pursue bad practices based on the good feelings we felt when successful. It is difficult to adjust solely on the basis of one’s personal experiences. Rarely does one see a player start fresh and change his losing approach. With regard to conventions that rarely occur, one would have to play many years in order to establish on the evidence alone the efficacy of their operation. Computer simulations can be useful if done carefully.

It is fair to say that scant evidence potentially being highly variable will often appear to arise from conditions of maximum uncertainty. This leads to conclusions such as ‘system doesn’t matter’, or ‘luck plays the biggest part in determining victory’. By now we should know this isn’t so. However, when attempting to analyze the actions of a mediocre player at the table, partners included, it is often the safest approach to assume a condition of maximum uncertainty, but this may not be the optimum approach.

A Sequence of Choices Next we consider the situation in which the school children are asked to choose one door then another when there are 3 doors behind which there is no prize. It is possible, even likely that the choice of 2 doors may be biased in some way. If the doors are shown in a diagram, I would expect the middle door would be the most frequently chosen. However, let’s assume the sequences are chosen equally. There are 6 permutations of choices from 3 doors as shown below.

The requirement to choose 2 doors results in a greater resolution of the location of the prize. Given the sequence Door B followed by Door C leaves only 2 possibilities remaining, with the probability of the prize being behind the door not chosen 3 times greater than behind Door A. The same holds for any possible permutation.

Next we assume there is a bias introduced into the choice of doors, as defined by our previous assumption. Yellow door is never chosen if it can be avoided.

The prize sits behind Door A only when B and C are chosen, in which case there is an equal probability that the prize lies behind Door D, the yellow door.

An Eight-Ever Example Let’s see what is the effect of a false card on the probabilities under this condition: AT86 opposite KJ94. Declarer plays the ace followed by the 6 towards the KJ9, the defenders following with small cards all the way. The missing cards we denote as Q-u-w-y-z and the observed sequence has been card u followed by card z, then card w from the LHO on the second round, the sequence being denoted as u-z;w. What are the current relative probabilities of these 2 conditions:

If all low cards are chosen with equal probability, the probability of Condition I, the probability of Q on the left (QL), proportional to ¼ the probability of the 3-2 split, because there 4 equally probable permutations. The probability of Condition II, the probability of the queen on the right (QR) is proportional to 1/6 times the probability of the 3-2 split because there are 6 equally probable permutations. The relative probability in favour of the Q being on the left is 3:2. In mathematical terminology:

P (QL | u-z; w) = (3/2) · P (QR | u-z; w)

This is part of the justification for the eight-ever rule of always finessing for the queen. However, what if the probability of playing card z rather than card y on the first round was 1/3 rather than 1/2 due to a known tendency of the RHO to false card? Then the current probability of QL and QR would be equal after the given sequence. Furthermore, if the RHO would always false card from a combination of yz, or would always give the correct count playing card z, card y would be characterized by the same certainty of being withheld as the queen from the combination. In this case,

P (QL | u-z; w) = 3 · P (QR | u-z; w).

This merely expresses the result that Condition I is characterized by 2 possible permutations and Condition II by 6 under the assumption that the LHO needn’t consistently favour one low card over the other in a 3-card combination. Let’s now consider an example of biased selection arising from a story by David Bird, who delights in situations where the probabilities are influenced by psychological factors that produce the effect that P (X) ≠ Q (X). The tale is funnier than the equation suggests.

A Muzzy Monk’s Muddled Math

Humor often educates as it entertains; such is the case with the amusing series of monastic stories by David Bird which continue to open our eyes to a dazzling display of squeezes and endplays of which Abbot Hugo Yorke-Smith is the usual victim. It is clearly the author’s design to show in subtle ways that the sophistic Abbot gets it wrong for the wrong reasons, but when Bird’s preferred alter ego, the clever Brother Lucius, speaks, one assumes the emphasis has shifted from spoof to proof.

In the June 2009 ACBL Bulletin Lucius has again overreached to 6NT and needs to bring in a club suit consisting of ♣QJ95 in his hidden (South) hand opposite ♣A742 in the dummy. After successfully running the ♣Q on the first round, to which Brother Xavier (West) follows with the ♣3 and the Abbot (East) with a highly suspicious ♣8, declarer has to decide whether on the second round to start with the ♣5 hoping to draw the ♣K from West, or to lead the ♣J hoping to pin a now bare ♣10 in the East.

As part of the prior knowledge stored over several years of play, the Abbot is known for his penchant for ‘false’ cards, so he would normally play the ♣8 from 3 possible holdings, ♣108, ♣1086, or ♣1083. As 2 of those combinations are 3-card holdings, Brother Lucius erroneously concludes the odds are 2:1 that the Abbot was dealt a tripleton and Xavier a doubleton club. He must have spent considerable time in the cellar that day reclassifying the monastery’s vast reserves of Home Counties claret because his thinking was rather muzzy. Why? Answer: because one of those tripletons was ruled out when Xavier followed with the ♣3. Does that mean the odds are now 50-50? No, not even the Abbot would think so as he is an avid reader of the works of Terence Reese.

The situation as Brother Lucius ponders his second lead in the club suit has been reduced to 2 potentially favorable combinations:

If with ♣K63 Xavier would always play the ♣6 on the first lead (as Bird seems to imply), then Condition II has been eliminated entirely as a possibility, so the ♣5 should be led on the second round, regardless of which club the Abbot has chosen. The ♣K will pop up to be taken by the ♣A in dummy. If Xavier only sometimes plays the ♣6 and the Abbot always plays the ♣8, Condition I is the more probable, and the same low club play is indicated. Lucius had no problem choosing to play a low club on the second round based on the known proclivities of his opponents.

On first take the theme of the episode appears to be a condemnation of defenders who always play ‘false’ cards thus becoming their own worst enemies, however, that may be said also of players who always choose ‘true’ cards. If Brother Xavier always gives true count with the ♣3 from ♣K63 and the Abbot similarly with the ♣6 from ♣1086, the card position becomes a certainty. The defect lies in the defenders’ predictability. Their best strategy is to follow with low cards at random, making plays of equal probability out of the appearances of the ♣3 on the left and the ♣8 on the right. This transmits the minimum possible amount of information to declarer. An insightful view of the defenders’ unbiased random selection of equivalent cards is that they are merely simulating the act of dealing, thus keeping the declarer in a state of maximum uncertainty to the extent possible under the circumstances.

The Inference from the Opening Lead The a priori odds are that the 3-2 and 2-3 club splits are equally probable. Conditions I and II each have 2 plausible permutations when the defenders follow randomly with low cards to the first round. To make a decision that has better odds than a coin flip, declarer must decide whether a 3-2 split is more probable than a 2-3 split on this particular deal, and to do so he needs to consider the other suits. The only clue is that Xavier has led the •Q placing him with touching honors in that suit and perhaps length. A 4-3 diamond split gives preference to a 2-3 club split, which, perhaps fortuitously, leads to the correct (Bird’s) decision.

Let’s look at this lead of the ♦Q with 7 diamonds to the QJ10 held by the defenders. Assume the lead against an uninformative auction to 6NT is from a sequence of QJ10. The fellow defender plays his most discouraging card. There are 2 cases to consider:

On the basis that false carding is not an option it appear obvious that the diamonds are more likely to be split 4-3 than 3-4. Of course, there would be no funny story to tell if one thought that way….or maybe not. If the adventuresome Brother Cameron had been on lead and had chosen ♦Q from queen doubleton, the outcome might have been a happier one for the Abbot. Fans of the monks know that such a lead would be uncharacteristic of Brother Xavier, who, despite his perpetual criticism of the Abbot’s normal leads, wouldn’t himself dare to deviate from orthodoxy.

Against players who defend to best advantage, after the first round of clubs declarers should gather more information outside the club suit by cashing some top hearts and spades to see if the vacant places, like unreliable clarets, need to be reassessed. This may result in going down more than 1 trick in the slam, but it increases the probability that declarer will make the right decision in the club suit. By immediately playing a second round of clubs, Brother Lucius indicated he was pretty sure his opponents were acting predictably.

Rozzie and Wilma

November 9th, 2009  ~  Bob Mackinnon  ~  1 Comment 

Not long ago in a West London borough Friday evening was the time when two young couples got together for a Wives vs Husbands match which determined who in the oncoming week would be doing the wash-ups. The Husbands had worn the apron for two weeks running, when the following match took place not far from William Blake House in the home of Alfred and Meg Jones – he, a baggage expeditor at Heathrow airport, she, a brilliant story consultant for the BBC. Perhaps some will remember with fondness the popular war-time comedy series Up Periscope!? This particular match was unique in that it featured the brief participation of the Rosalind Treacher, once a well-known figure in the golden age of British bridge who had dropped out of sight three decades previous.

As our story begins, it is late in the match when the players are joined in the living room by Meg’s mother, a stranger to Peregrine Penfold, an auditor at Income Revenue. Besides having brains Meg Jones was a voluptuous blonde for whose curvaceous attractions one would be hard put to find traces on outline of the barrellous Mrs Treacher – heavy-set and grandmotherly, fully outfitted with shawl, floppy slippers, and orange tabby. A photograph of mother and daughter side-by-side at the seaside would serve admirably as a reminder to indolent dawdlers to jolly well start gathering those rosebuds while it’s still possible to bend over and touch toes.

‘Meg, Love,’ said the granddame, a cigarette dangling from her lips even as she spoke, ‘the girls is in bed, so’s maybe you can just pop up and tuck ’em in.’ Addressing the men she added politely, ‘Please excuse the interruption, Gentlemen, but you know how it is with little tykes.’

‘Quite,’ ‘Quite,’ vaguely concurred the two husbands, rising and bowing politely.

‘Peregrine, I’d like you to meet my mother, Rosalie, who’s having a short stay with us,’ said Meg pleasantly enough. ‘Mum, you remember Peg, I’m sure.’

‘Of course, Darlin’, good to see you again, and pleased to meet you, Peregrine. Call me, Rozzie,’ the pensioner wheezed, giving off the unmistakable aromas of peppermint and gin, ‘And this here is Wilma,’ she indicated, scratching behind the ear of the orange tabby she was carrying in her arms.

‘Mrw,’ greeted the tabby curtly, obviously a cat who was inclined to wait and see when it came to male visitors. Unbeknownst to the husbands their wives had been taking lessons from Rozzie on the sly when they were off to their Sunday cricket match. Unbeknownst to the wives, the previous Sunday the husbands had forgone their usual pastime to spend a few hours at the pub sharpening their bidding practices. Between them they shared a distinct aversion to soap and dry activities day after day. It had paid dividends as at half-time they had a comfortable lead under Chicago scoring.

‘Look, Mum, I did promise Joy and Myrtle their bedtime story,’ stated Meg, ‘so why don’t you take over here for me while I go up and read them Goldilocks and the Three Bears. Peregrine won’t mind, I’m sure.’

‘Delighted, do join us’, came the ready response from Penfold, unaware of the danger which lurked behind this seemingly innocent suggestion.

‘I haven’t played bridge for ever so long…but if you insist,’ said Mrs Treacher, obviously delighted to be asked to take a seat and rejoin the world at large. ‘You won’t mind will you, Puss? A bit o’ bridge, eh? You’d like that, would ya?’

‘Puurrnaow,’ replied Wilma guardedly, which in translation reads, ‘Don’t let me stop you,’ so unanimity of a begrudging sort was achieved.

‘Would you care for some tea, Mrs Treacher, we’ve just finished ours, but there may be some left in the pot?’ asked Peg. ‘Milk and sugar?’

‘Lovely,’ replied Rozzie, ‘and add a touch of sherry, would you, Dear, just to take the chill out of me feet.’

As Rozzie lit up another cigarette and blew fumes across the table, Puss jumped lightly to the floor and departed hurriedly, seeing here an opportunity answer the call of the jungle and let off some steam in the back room undisturbed. In the meantime as was his wont while others chatted inconsequentially, Peregrine allowed his mind to wander into unchartered daydreamland. Say the Jones were to put on a Christmas panto for their daughters, one could easily picture Meg as Goldilocks while their grandmother would fit admirably into the role of one of the adult bears; he himself would make a consummate Jack Frost …

‘Alfred, I don’t know if Meg told you this, but I used to play a lot of bridge in the old days before I became engaged in the fruit and vegetables trade,’ revealed the matron.

‘Meg once mentioned she had learned to play from you, but I didn’t know you were keen’ replied her son-in-law. In fact, Meg had told him very little about her mother’s past.

‘Keen as mustard, I think you could call me,’ revealed his mother-in-law as the old memories came flooding back buoyed on a stream of gin and tonic. ‘Sid, me late husband and Meg’s dad, every summer we’d go over to the Continent for a spot of holidays. There was currency restrictions in them days, so to keep us from starvation we had to win prizes or pick up some cash however we could along the way. I partnered all the great ones in those days: Cornelius, Pierre, José, Rudolf, Giorgio … all gone now.’

Sounds as if the old girl did her bit for European unity, thought Penfold, unlike Thatcher, but currency restrictions? No doubt part of the cost Britain has paid trying to save the Europeans from themselves. In response to his unspoken wish, his wife quickly reappeared with a cup of tea and a chocolate biscuit. Now they could get on with it, he thought, but Rozzie was still lost in the past within her veil of blue smoke.

‘Most of all there was dear Omar, who still sends me a postcard every time he returns to Sorrento, bless ‘im. We ‘ad a brief roundayview behind the Caffe Pinocchio, I think it was called. The sky was ever so blue, very romantic. After that we went out and won the pairs’ prize and he presented me with the whole lot, gentleman that he is. And all that time poor Sid was back at the pensione in bed with his asthma.’

At long last the cards were dealt and the reminiscences ceased temporarily as Rozzie donned her horn-rimmed spectacles. Alfred had been made somewhat uneasy by these revelations. Here’s a story that could do with some editing, he thought, nonetheless, however extensive her previous experiences, his mother-in-law was probably as rusty at the bridge table as elsewhere, so, he put it to the old girl with a flawed preemptive three no trump opening bid in third seat. Here are the four hands:

		Peg
		♠	A 9 6 4
		♥	A 10 9 7 6 3
		♦	—
		♣	10 8 5
Peregrine				Alfred
♠	Q 10 8 7 2			♠	J 3
♥	8 4			♥	5
♦	6			♦	A K J 9 8 7 3 2
♣	K J 9 4 2			♣	6 3
		Mrs Treacher
		♠	K 5
		♥	K Q J 2
		♦	Q 10 5 4
		♣	A Q 7

‘How strong is that?’ Mrs Treacher asked after rechecking her point count.

‘Ah, not strong actually ..err…Rozzie,’ explained Peregrine Penfold, somewhat warily, ‘normally a solid eight-card minor suit without an outside ace or king.’

‘What would four diamonds have meant?’ inquired Rozzie, then correcting herself quickly, she added, ‘Forget my question, it doesn’t matter, really, I’m bidding Four Hearts regardless.’

Peg Penfold, Peter Pan in Peregrine’s fantastical pantomime, was a quiet, slim woman, educated and professional, a hospital administrator with a penchant for neatness and accuracy. She looked at her hand and saw she had exceptionally fine support for her partner’s freely bid suit. Despite her partner’s inappropriate comments, she thought that no committee in the world could find fault with some show of life. Undercompensating greatly she bid a totally inadequate five hearts. Rozzie raised herself to six, Peg being such a mouse. Peregrine looked no further than the singleton in his partner’s solid suit. Would that all leads were so easy.

Meanwhile Wilma had since returned refreshed to seek the warmth of her mistress’s lap. Without the benefit of the electric fire, the coolness of the back room had been such that even a vigorous clawing of the furniture hadn’t compensated fully. With an almost insolent display of athleticism the tabby sprang lapward landing safely with a firm grip on Mrs. Treacher’s right thigh – an inch or so short of her previous record for the standing long jump.

‘Ouch! Wilma! You gave me such a start. Very nice, Peg, Dear, just what I expected,’ said Rozzie after Peg apologetically displayed the dummy.

‘Ruff with the ace, Dear, then a small trump,’ continued Mrs Treacher immediately, not allowing the sharp pain in her thigh disturb her concentration. The play went quickly as the Rozzie of old sprang into action: a heart to the king, a diamond ruff, a heart to the jack, a diamond ruff, a spade to the king and a diamond ruff with the trump ten, then a club to the ace into this position with the last trump to be played:

		Peg
		♠	A 9
		♥	—
		♦	—
		♣	10 5
Peregrine				Alfred
♠	Q 10			♠	J
♥	—			♥	—
♦	—			♦	A J
♣	K J			♣	3
		Mrs Treacher
		♠	5
		♥	2
		♦	—
		♣	Q 7

On the play of the ♥2 Peregrine could not afford a discard from either of the black suits without setting up an additional trick for declarer. After some futile computation he bared the ♣K. A spade was discarded from dummy, and Rozzie exited with a low club, soon claiming her twelfth well-earned trick in the form of the queen of clubs.

‘Ooww, that was a bit o’ luck, I must say,’ said Mrs Treacher. ‘There must be a better way to play that, as I ended up having to trust your bid, Alfred. That’s always been my weakness, being too trusting of men, but the cards have always been kind for the most part. Any more of that tea, Peg? Thanks, Love, and remember just a dash of sherry. I told you gentlemen about dear Omar sendin’ me a postcard every year. We had a song – ‘Re-torn me to Sor-rent–o’, just like Gracie used to sing it. Once on short notice I was called to her villa on Capri for a game with Ian Fleming. He was not near as charmin’ as James Bond. With writers its all in their head, you know, whereas with politicians it’s just the opposite. The one thing they ‘ave in common is you can’t believe a word.’

Adroitly reading the disapproval in her husband’s countenance, Peg had hastily produced a tepid cup of tea laced with rum and quickly returned to deal the cards. As Rozzie sorted her cards she realized that, yes, time flies: it had been ages since she last held a hand like this one – maybe all of thirty-five years, on that night her team had trounced Madame Altivolans in a qualifier so badly that the chairman of selection committee got a it’s-us-or-her midnight ultimatum from a disgruntled loser. Unfortunate, for Britain could have used the likes of Mrs Treacher in the lean ’70’s.

‘Very nice tea, Peg’ she mused while taking pleasure in the sight of so many familiar yet unwrinkled faces: ♠AQ653 ♥KQT2 ♦A ♣AKQ

After Peg had passed, Alfred cleared his throat and opened with a bid of two spades. After recent exposure to the experts on BBO he had come down with a serious case of preemptive looseness.

‘Strong this time, is it?’ inquired Mrs Treacher somewhat brusquely.

‘Nooo…. weak twos actually, 6 to 10,’ replied Peregrine, referring to high card points, not the odds against its success. ‘Something of a one-suiter.’

‘Three No Trumps,’ declared Rozzie firmly. “That’s right, isn’t it, Puss? You’d bid 3NT with this lot, wouldn’t ya?’ Wilma felt it would not be in the spirit of the game for a kibitzer to offer a contrary opinion at this time. Here are the four hands in full:

		Peg
		♠	—
		♥	J 8 7 5 3
		♦	Q 5 2
		♣	8 6 5 4 2
Peregrine				Alfred
♠	9 2			♠	K J 10 8 7 4
♥	A 6			♥	9 4
♦	J 10 8 7			♦	K 9 6 4 3
♣	J 10 9 7 3			♣	—
		Mrs Treacher
		♠	A Q 6 5 3
		♥	K Q 10 2
		♦	A
		♣	A K Q

After Peregrine passed, Peg faced a difficult decision. Taking into account the amount of rum she had added to the tea, or, rather, how much tea she had added to the rum, Peg realized that a jump to 3NT encompassed a wide variety of hands, yet her void in spades must be worth something extra in a suit contract and considerably less in no trump. Perhaps she should bid four diamonds as a transfer to hearts, or would four diamonds be taken as, God help us, Flint? Four clubs, natural and forcing? Better. But wait – was four clubs Stayman, or ace asking? Oh dear, her instincts told her to pass, but hadn’t Rozzie just this week insisted repeatedly that the most dangerous call of all was ‘no bid’?

‘Four clubs,’ bid Peg finally maximizing her slim chances of survival in the usual way – by making the most ambiguous call.

‘What’s our standin’ in the match, Love?’ asked Mrs Treacher.

‘Oh, well, I couldn’t really say, Mrs Treacher,’ replied Peg, blushing, ‘we were trailing at half time by a thousand or more.’

‘Call me, Rozzie, Love. I’m taking that for South African Texas – Six Hearts.’ One of Rozzie’s golden rules for living had once more come into play: never bid an unforced Grand Slam in competition.

Well, so much for the hormonal theory of bridge bidding. It was left to Peregrine Penfold to find the killing lead against six hearts or reconcile himself to being up to his elbows in soapsuds for yet another week. As a tax auditor he was trained to sense inconsistencies, and there was distinct feeling in the back of his neck that he had been presented with an improbable set of circumstances. It appeared to him that declarer had the spades well tied up leaving Alfred with undisclosed honours in the minors. Very BBO of Alfred. As Peregrine himself had both minors covered, it appeared that declarer’s best hope would be to score as many tricks as possible on a crossruff. Thinking thus, he let the ♥A and the thin dummy came down with an abject apology from a red-faced Peg.

‘Ta, Darlin’, that’s very nice, indeed. Isn’t it nice, Puss? Play low,’ commanded Rozzie smoothly, turning her thoughts to finding the best way of playing queen tripleton opposite ace tight, not a combination one finds written up in books.

Jones felt the anguish of a missed opportunity. He very much regretted his failure to gather rosebuds by doubling Peg’s nebulous four clubs, but when dummy came down with length in the suit, hope was renewed. Unfortunately his desperate signal of the ♥9 went unnoticed, as Penfold, unimaginative in real life situations, followed the ♥A stolidly with a second trump taken in declarer’s hand. Wilma, like the others around the table, was too young to remember that glorious night against Madame Altivolans, but she could appreciate the rapidity with which her companion expertly finished off the play. Rozzie cashed the top clubs then ruffed a spade to dummy, ruffed a club and ruffed a spade once more to dummy and led the last club from this 5-card position:

		Dummy
		♠	—
		♥	J
		♦	Q 5 2
		♣	8
Peregrine				Alfred
♠	—			♠	K J 10
♥	—			♥	—
♦	J 10 8 7			♦	K 9
♣	10			♣	—
		Mrs Treacher
		♠	A Q 6
		♥	K
		♦	A
		♣	—

On the play of the last club to be ruffed in declarer’s hand Jones found himself ensnared in a crisscross ruffing positional squeeze, or something or other of that sort. (Linda Lee would be able to identify it.) I don’t know if you have ever been stripped down to your bare honour, but, Dear Reader, I can assure you it is a cold and surgical feeling, the same feeling Penfold had endured on the previous hand. After much circuitous thought, Alfred bared his •K. Rozzie unfailingly cashed the diamond ace dropping the king, cashed the ♠A and ruffed a spade to dummy in order to take the last trick with the •Q. There had been a fearful symmetry in the play of the two hands.

Just in time Meg came floating into the room with a satisfied motherly smile of relief. In her husband’s eyes, she had seldom looked more beautiful fully clothed.

‘Thank you, Peg and Gentlemen,’ said Mrs Treacher, lifting herself heavily from her chair, in the process dumping an indignant Wilma onto the Persian carpet, ‘that certainly brought back old memories. We must do it again sometime soon, previous commitments permitting.’

‘Oh, absolutely,’ replied Peregrine Penfold politely without, however, attempting to affix a date to this purely hypothetical re-encounter.

‘Puurroww,’ commented Wilma impatiently, as much as to say, ‘Thank you so much, but may we not get back to some serious Telly viewing? We might just catch the tail end of The Tigers of Sumatra. Now that would be some excitement for a change!’ She led the way out with tail held high in a fine demonstration of the grand exit.

Baby Knows Best?

November 3rd, 2009  ~  Bob Mackinnon  ~  No Comments 

Last week on PBS, Charley Rose interviewed Alison Gropnick, a professor of psychology and author of the recently published book, The Philosophical Baby, about her research into the brain activities of infants. Her view is that we are all born with many innate mental facilities that are allowed to die on the vine, as it were, so growing to adulthood is not just a process of gain, but also a process of loss. For example, we know a second language is best learned earlier rather than later in life. Furthermore, according to Dr Gropnick babies have a grasp of the basic concepts of statistics, and can evaluate alternative courses of action taking probability into account, whereas, college students have difficulty absorbing the fundamentals of probability during their first academic exposure. I gather here she was speaking from a painful, personal experience. My conclusion is that children should be exposed to the study of probability at a young age, thus learning early how to cope with uncertainty in a rational manner.

There is more to this than a condemnation of current teaching methods. If one is educated to follow do-and-don’t rules rigidly rather than to think independently on the basis of evidence, natural abilities are stifled, and one becomes less adaptable to situations where uncertainty exists. Well, we can see how this affects performance at the bridge table. A bridge hand is best played with the happy expectation of a curious baby about to open a door and uncover the mysteries of a kitchen cabinet or a bedroom closet.

Don’t Worry, Be Happy In this segment in answer to a comment made by a reader of a previous blog, we consider a situation where a declarer in order to succeed in 3NT needs to develop tricks in an 8-card suit missing the AQ. Declarer can’t afford to lose the first trick in the suit to the queen on the right-hand side. There are conflicting emotions at work, the hope of succeeding and the fear of losing. Which should dominate?

The nature of the game is such that the optimum result is usually achieved by the taking of tricks rather than by the avoiding of losing them. This is true generally at matchpoint scoring, but it is also true at IMPs where there is no scope for a safety play, as in the deal to be studied, when one bids to a close game and merely hopes to make it. In these circumstances declarer’s plan should be geared towards maximizing the chance of taking tricks, rather than minimizing the chance of losing them. If the cards are as badly placed as one fears, then one shouldn’t have bid 3NT in the first place. To be consistent, declarer should assume that the contract is makeable, even if the odds are firmly against it.

Often declarers reach a decision point where the chances of success are less than 50% regardless of which choice is made. This is common in the ‘Eight-Ever’ situation when the player has 8 trumps and must decide whether or not to finesse for the queen. When the LHO follows to the second round, the finesse may be against the odds, yet we know that finessing is better than playing for the drop. So we follow through. The same principle applies to more complex problems. When 2 or more plays in a suit are required to achieve a goal, one must be patient and resist the temptation of achieving an immediate success on the first round at the expense of a reduction in the overall chances. We shall now investigate a situation where the constraints of an opening bid affect the decision.

CARDE’s Problem

The ♠2 is won by declarer with the ♠J. The bidding places North with 5 spades and the bulk of the 16 missing HCP. Declarer’s hope is to establish 4 tricks in the diamond suit in such a way that North does not win a diamond after the spades have been established. The choices are: 1) to run the ♦8 on the first round and hope not to lose to the ♦Q, or 2) to play to the ♦K and duck the second round to North’s presumed bare ♦A. In this latter case, East will take the third round of diamonds with the ♦Q but will have run out of spades. As noted above, the establishment of diamond tricks generally requires more than one round to be played. First we look at the 5 most common distributions of sides for a 7=7=5=7 division of sides with spades split 5-2.

The 2-3 split in diamonds is nearly twice as likely as the 3-2 split due to the imbalance in the vacant places with a preponderance of spades in the North hand. When a problem is complex, one first solves a similar, closely related problem in order to shed some light on the path towards a decision. Here we shall assume that North holds the ♦A, which is very likely needed to promote his hand to the status of an opening bid. There are 2 choices of play that will be successful under differing conditions:

There are 2 attractive features of the ♦K play: 1) initially a 2-3 diamond split is more likely than a 3-2 split, and 2) it would be embarrassing to go down in a makeable contract by losing to the ♦Q on the first round. It requires a certain toughness of mind to reject going up the ♦K. If one makes the correct play one needn’t feel embarrassed when it fails, as it often will. (Not many feel guilty when the wrong play succeeds, learning nothing from their mistake. It is futile to criticize a lucky play publicly, but some do.)

Placing the ♦A with North sets the vacant places North to South as 7 to 11, so before a diamond is played the odds of the ♦Q being dealt to the South hand are 11:7. Thus, a perfunctory analysis indicates playing to the ♦K, but one should look more deeply into the situation. There may be further constraints that apply because, in theory at least, the North hand must fulfill the normal requirements for an opening bid.

The Effect of the First Play in Diamonds We denote the low cards in diamonds as u,w, and y, and assume North has played card u on the lead of the ♦8 towards dummy. The successful placements are now specifically 2 in number: 1) Au opposite Qwy, and 2) AQu opposite wy. We needn’t concern ourselves about the losing combinations; we want to discover which winning condition is more probable. This depends on the associated number of card combinations in the suits that have not been played, hearts and clubs.

The Bidding Constraint North holds ♠KQxxx. Under placement #1 the ♥K is needed in the North hand whereas as under placement #2 the ♥K may be held by either defender without prejudicing the requirement for an opening bid. We shall now look at the effect of this constraint on the 5 most common distributions of sides and determine the number of combinations for each. The placement of the ♥J and ♣J are irrelevant so we treat them as x’s in the notation, denoting low cards that could be dealt to either hand without effect.

Combinations Available:

The question marks under Conditions IV and V are used to convey the fact that the ♥K is free to be placed on either side without violating the bidding constraint. Under Conditions I to III the number of allowable heart combinations is reduced by the requirement that the ♥K must sit with North. The effect of this reduction is that Conditions IV and V are now the most probable and Condition I is demoted to 3rd place.

Requiring that a specific card be placed in a particular hand greatly reduces the number of available combinations in the suit thereby affecting the probabilities in other suits. As a result, under our restrictions, there are more possible combinations for which running the ♦8 on the first round is the winning play, roughly in the ratio of 5:4. There are other, much less probable, situations where running the ♦8 is correct:

It would take a computer to calculate all the probabilities involved, but we have considered the most likely scenarios, and these situations give ample justification to the standard advice of choosing to finesse first against the lower missing honor.

Finally we note that through the bidding constraint the odds are strongly affected by the location of the ♥K. In some situations it pays to gather information on the location of a high card in one suit before making a critical decision in another, say, by leading low towards the ♥Q at trick 2. Discovery plays of this sort can be helpful especially at matchpoints, but here safety considerations preclude that operation.

Nine-Never and Restricted Choice

October 6th, 2009  ~  Bob Mackinnon  ~  107 Comments 

Mathematical models require assumptions, and it is the nature of the assumptions that limit their applicability to the real world. The Nine-Never Rule is based on a mathematically sound argument, but the assumptions behind it are not always appropriate to the situation at hand. In the recent Prince Takamatsu Cup tournament held in that beloved oasis of polite bridge in the heart of heartless Tokyo known as the Yotsuya Bridge Club, a slam hand arose which provides a rare example where theory is well-matched to reality, yet the seasoned BBO commentators did not give any clear direction to the observers on how the hand should be analyzed. In fact, the analysis was somewhat confused, the simple conclusion being that here was another example where the Nine-Never Rule failed. Why it failed was not explained, and it was not clear whether the declarer had actually made a mistake when he played for the drop rather than finessing for the queen. This leads me to conclude that it is worthwhile to go through the play of the hand in detail in order to demonstrate the mathematical assumptions that lies behind the rule, and, more importantly, the procedures that a declarer should follow when making a decision with regard to the play in a suit where 9 cards are held with the queen missing. Eventually the well-known Principle of Restricted Choice is invoked.

The Basic Suit Configuration We shall assume South is the declarer faced with the problem of finding the ♦Q in the following configuration:

(South)      ¨AJ9752   opposite ¨KT6        (North)

The ♦K is played from dummy to which East follows with the ♦4 and West with the ♦3. Next the ♦T is led to which East follows with the ♦8, and the time of decision has arrived. Should declarer finesse or play for the drop? There are 2 possible holdings remaining, namely,

On the play within the suit there is no reason to choose one configuration over the other as they share the same number of plausible plays, so the question boils down to the question of which is more likely, a 1-3 split or a 2-2 split? Declarer should play for the drop when it can be assumed that an even split in diamonds is more likely than an uneven split. This assumption is supported by the a priori odds, but it may not hold after cards have been played in another suit.

The Effect of an Uneven Split in Another Suit

In the Prince Takamatsu Cup hand hearts were trumps. The opening lead from West was a trump, and it was immediately discovered that the hearts were split 3 in the West and 0 in the East. Does this information point to an uneven split in diamonds? Yes.

The weights are an expression of the number of combinations in the black suits that are available under the 2 diamond splits. The ratio of combinations yields the odds of 11:9 in favor of the 1-3 split. Hence declarer should finesse under these circumstances, when nothing is known concerning the splits in the black suits. This is roughly the situation when trumps are drawn and diamonds are broached immediately thereafter.

The same odds are available from a vacant place analysis, following Kelsey’s Rule, which allows the use of the current vacant places when only the location of the queen remains in doubt. This is no surprise, as the assumption of maximum uncertainty with regard to the black suits is common to both methods.

The odds are 11:9 that the ♦Q sits in the East. This is the basis of the criticism the BBO commentators directed towards declarer when he played for the drop. Given the hearts had been seen to split 3-0, it seems as if the odds favor an uneven split in diamonds. However, life is seldom that simple. At the table, diamonds were not broached immediately, and several rounds of both spades and clubs were played before the critical decision was made in diamonds, so more information was available than in the simple model indicted above. To gauge the effects let’s look fully at the play of the cards.

What Really Happened

Board 3 Dealer: North Vul: E/W		Imakura
		♠	J
		♥	K J 10 8 7 3
		♦	K 10 6
		♣	9 8 6
Shimizu				Noda
♠	K 8 7 3 2			♠	10 9 6 5
♥	A 9 4			♥	—
♦	3			♦	Q 8 4
♣	A Q K 2			♣	K 10 7 5 4 3
		Ino
		♠	A Q 4
		♥	Q 6 5 2
		♦	A J 9 7 5 2
		♣	—

*Hearts Pre-empt

**Enquiry

***Good hearts

The bidding was very much to the point and showed commendable faith in partner’s non-vulnerable weak two preempts. No doubt there were hopes of a helpful lead, hopes that were not realized when Shimizu, in the face of such uncertainty, chose a passive trump lead. The ♥4 rather than the ♥A indicated he had no fear of losing his ♣A in the early rounds. Let’s look at some common distributions of sides with an established 3-0 heart split.

Even though the hearts are split 3-0, the even 2-2 diamond split is more likely than the uneven 1-3 split over this small selection. Why? Because the imbalance in the vacant places is more readily compensated for by uneven splits in the longer black suits. These 5 cases represent a snapshot rather than a panoramic view, however, as cards are played, it is to be expected that the play will ‘zoom in’ on the most likely possibilities shown above. Of course, the discovery of uneven splits in the black suits may alter our view.

The play in slam contracts is often a long journey of discovery. Declarer may postpone the critical play in the diamond suit until he has learned more about the splits in the black suits. Let’s follow the perilous path as it occurred at the table.

Both defenders have played 4 spades and only the ♠K is now missing. It could sit on either side. East has played 5 clubs, ♣10-7-5-3-2, and the ♣K-Q-J are still missing. From the play in the club suit we may assume the clubs were not dealt 5-5, for that would mean West began with ♣AKQJ2, which is contrary to the evidence of the lack of bidding, the opening lead and the subsequent play. Thus it is reasonable to assume clubs were split 4-6, and it is only the split in spades that remains in question at trick 11. At this point the split in diamonds depends solely on the split in spades. Here are the 2 live possibilities:

Under Condition I

		Dummy
		♠	—
		♦	10 6
		♣	8
West				East
♠	—			♠	K
♦	Q			♦	8
♣	Q J			♣	K
		Declarer
		♠	—
		♦	A J 9
		♣	—

Under Condition III

		Dummy
		♠	—
		♦	10 6
		♣	8
West				East
♠	K			♠	—
♦	—			♦	Q 8
♣	Q J			♣	K
		Declarer
		♠	—
		♦	A J 9
		♣	—

Of the 5 most likely distribution of sides listed above, only Conditions I and III remain as possibilities at the critical point in Trick #11. The problem has boiled down to this: does West or East hold the ♠K? The answer will determine who holds the ♦Q.

Restricted Choice

So far the numbers indicate that declarer should play for a 2-2 diamond split as on the deal Condition I is the most likely distribution of sides in the proportion of 3:2. However, as Shakespeare once wrote, ‘the play’s the thing’; there is the evidence of the play to be considered. The problem in probability is this: given the defender’s sequence of plays, is it more likely East holds the ♠K and, if so, by how much? This card play problem is related to the Principle of Restricted Choice.

At Tricks #4 and #5 EW followed suit with low spades. At Trick #7, when declarer ruffed the ♠Q the defenders’ ♠K109 were equivalent cards. If East had been dealt ♠K10965 he could have followed with any of the 3 equals, but if he had held only ♠10965 his choice would be restricted to 2 cards. In retrospect, Conditions I and III had become equally probable when East played the ♠9. At Trick #9, both defenders had to find a discard on a trump lead. The ♠8-♠10 play was 1 of 8 possible plays under Condition III and 1 in 9 under Condition I. Thus, at Trick #10 Condition III becomes the more likely in the proportion of 9:8. The BBO commentators, along with everyone else who could see all 4 hands, were correct in maintaining declarer should have finessed at Trick #11.

At the other table North opened with a weak 2♥ bid and ended up in slam. West first competed in spades at the 3-level on a poor 5-card suit, then in the end, egged on by East, doubled 6♥. With this wealth of information made freely available to him North was able to play the diamonds correctly and claim his doubled contract for a gain of 16 IMPs. Lest this be construed as a condemnation of hyper-aggressive bidding practices, we hastily add that the aggressors’ team won their semi-final match handily, only to lose by a narrow margin in the final.

We next present a mathematical description of the play process.

Bayes’ Theorem at Work

Bayes’ Theorem is a simple idea easily expressed in the form of a mathematical equation. The difficulty experienced by many in understanding the equation lies primarily in a lack of previous exposure to the mathematical notation. It is worthwhile to go through the formulation for the above problem, as Bayes’ Theorem is fundamental to our understanding of how probabilities change due to the exposure of cards during the play.

We begin with the distributions of sides of which there are many, but with hindsight we may limit our attention to Conditions I and III. Condition II is a remote possibility which we shall put aside for the sake of simplification. The probability of Condition I relates directly to the probability that spades are split 4-5 [denoted as P(4-5)] and the probability of Condition III relates to the probability that the spades are split 5-4 [denoted as P(5-4)]. The beauty of probabilities is that we need concern ourselves only with ratios which represent relative probabilities. Thus, we can state that:

P(4-5) / P(5-4) equals 105/70 or 3/2.

In their ratio, one may take P(4-5) and P(5-4) as representing the probabilities of a particular combination of spades rather than all possible combinations. Once cards are played a new type of probability arises which requires a new notation, as follows. Let S represent a sequence of plays in which the cards are revealed. The probabilities that a particular sequence would appear under Condition I or Condition III are denoted as P(S │4-5) and P(S │5-4), respectively. What one seeks is the relative probability that the spades are split 4-5 after the sequence S has been observed, and this probability is denoted as P(4-5 │S) . Bayes’ Theorem is a statement of this linear proportionality:

P(4-5 │S) is proportional to P(4-5) times P(S │4-5) ;

P(5-4 │S) is proportional to P(5-4) times P(S │5-4).

This means that the probability of Condition I after the sequence S is observed is proportional to the initial probability of Condition I before a spade is played times the probability that the sequence S would be chosen if indeed Condition I were to apply. Similarly for the probability of Condition III after the sequence S has been observed.

As noted above, probabilities vary as the sequence grows. Before a diamond is led at Trick #10, the last action of the sequence involves a restricted choice of discards in the black suit at which point:

P(S │4-5) divided by P(S │5-4) equals 16/27. Thus, we can conclude that

P(4-5 │S) divided by P(5-4 │S) equals (3/2) times (16/27) which equals 8/9.

In plain words Condition III has become more likely than Condition I once the discards have been observed. At this juncture each condition has been reduced to just one possible spade combination. It is usually the case after a long sequence of plays that restrictions in the play weigh more heavily than the restriction of the deal.

Comments on the Bayes Approach

Truth will sooner come out of error than from confusion.

– Sir Francis Bacon (1561 -1626)

Seldom can it be said that a model fits reality to perfection; models are perfect, but reality is messy. The concept of ‘best chance’ is an abstraction in itself that requires the incorporation of all the information that is available at the time of decision. Some simplification is required, how much partly depends on a declarer’s powers of observation. Although one may not be able to reach the correct conclusion through common sense alone, in retrospect a mathematically derived conclusion should be seen to reflect common sense. In simple language Bayes’ Theorem tells us the following:

(1) If a player hasn’t played a particular card, it is more likely he hasn’t got it than that he has it but chose to play an equivalent card instead.

(2) The more likely a player is to have played a particular card if he had it, the more likely it is that he doesn’t have it if he hasn’t played it.

The Assumption of Perfect Play At the table it is a player’s task to induce errors, and suppress the fear that an opponent can see through the cards, yet the mathematical model assumes the play is perfect. It is all too easy to assume a defender will not err, but then, how can one expect to win against a faultless opponent? In the above example it is in the defenders’ best interest not to give away information with regard to the distribution of the cards, which means that they should play equivalent cards at random rather than attempt to give partner the count, which would have made declarer’s task much easier.

Memory Problems Not many declarers could remember at Trick #11 the sequence of cards played by the defenders, yet a complete analysis requires one make use of this information. The task is made easier if a declarer is alerted to the main features of interest. That is why when the dummy comes down, declarer should focus the mind on the most likely distributions of sides, which are the models for what is most likely to transpire. In the above example, it should be immediately clear that a major objective will be to determine the diamond split, so declarer should make plans to achieve that objective. He goes about his business looking for clues, planning his sequence of plays accordingly.

With regard to Ino’s journey of discovery, might he not have adopted a different route to better induce a revealing play in spades by leading the ♠J from dummy at Trick #2 to see whether East would cover? Of course, East does not cover, and declarer goes about his business. Suppose that subsequently East shows up with the ♠T9. Now looking back on Trick #2 declarer may draw the correct conclusion based on the observation that if East held ♠KT9 he most certainly would have covered the ♠J at Trick #2. On this partial basis he might choose correctly to finesse for the ♦Q. This is a matter of anticipation and recall for which an understanding of the mathematical treatment prepares the mind.

Quick Japanese Yotsuya means‘Four Valleys’. Takamatsu means ‘Loftly Pine’. Bush is ‘Yabu’. Obama is a seaport in Western Japan.

Probability of HCP Distributions

September 23rd, 2009  ~  Bob Mackinnon  ~  No Comments 

Conditions and Constraints

Imagine you are playing in a 2♠ contract without the opponents’ interference. Dummy come down and you count up 20 HCP for each side. Well bid! The defenders lead clubs and sooner of later you ruff and draw trumps. Now you are faced with playing on diamonds with KJxx in dummy and xxx in hand. So far the RHO has shown up with 8 HCP and the LHO with none. Who is the more likely to hold the ♦A? On the hundreds of hands you have played 2♠ against silent defenders holding 20 HCP your memory tells you that most of the time the HCP are pretty evenly divided between them. Does that mean the LHO is more likely to hold the ♦A, because so far he has shown up with fewer HCP? No, even though the a priori odds tell you the HCP are likely to be split evenly.

This is similar to a classic problem that arose with coin tosses. It wasn’t until the 17th Century that thinkers got it right. Consider a sequence of tosses with all heads, say, HHHH. How often does it arise? Not very often – the a priori odds are definitely against such an occurrence. So, if someone has thrown HHH would you think that tails (T) is now more likely than heads? Of course not. The probability of a tails is 50%, just as it was at the beginning of the sequence and all the way through. At this point the relevant odds are the a posteriori odds, and HHHH and HHHT are equally likely. This is still very difficult for some to grasp. They think the more heads that appear in sequence, the greater the chances of a tails on the next toss. They give odds against the string continuing. Wrong. One has to abandon the a priori odds, and concentrate on the current odds.

Consider 4 players sitting around the table as a dealer gives them cards one-by-one. The expectation is that by the end of the deal each will receive 4 court cards (A,K,Q,J). As the cards are dealt one-by-one you find that your first 4 cards are court cards. Does that mean you are likely subsequently to receive fewer court cards than a player who has received none as yet. No. The chance of receiving a court card in the next 9 rounds is the same for all players regardless of their current holding.

Of the 20 HCP missing initially East has shown up with 8 HCP so far, thus there are 12 HCP remaining. These are likely to be split in accordance with the number of vacant places. If there is a balance in vacant places, the odds favor 6 HCP on each side for an overall split of 6 HCP in the West and 14 in the East. However, suppose that East has passed in first seat. This constrains East to at most 12 HCP. He cannot have 14 HCP and pass. This constraint acts to eliminate the possibility of some card combinations. To calculate the resulting probabilities one merely discards the now impossible combinations, and adds up the numbers of combinations still within the bounds of the constraint. This is the way of conditional probability, and it is the proper guide to the play the hands.

The best process with regard to estimating probabilities is to get (safely) as much information as possible concerning the distribution of cards before making a decision. Ideally to find the ♦A, say, one would like to know how the diamond suit is split. If one can’t accomplish that, then one may have to work with 2 suits, treating them as equals during the deal. This is the situation we shall treat to show how the calculation proceeds in several simple examples. One may be able to use the number of vacant places to calculate the current probabilities, but sometimes constraints apply and the vacant places won’t yield a good approximation.

NS Bid Game, EW Have 15 HCP

In Case 1 West leads the ♣ J (Jack Denies, so the top of a sequence). South takes his ace, ruffs a club and draws trumps in 2 rounds. With some luck in diamonds declarer just might get rid of a heart loser. There are 9 HCP remaining in the red suits. West has shown up with 1 HCP (♣J) and East with 5 (♣KQ). Does that mean West is more likely to hold the ♦A than the ♦Q? No.

The key to the locations of the ♦A and ♦Q lies in the number of vacant places remaining before declarer breaches the diamond suit. Let’s assume the following splits to this point based on the evidence of the opening lead and East’s plays in the club suit.

One cannot be certain of the club split as the defenders may be false-carding, but when there exists a high degree of uncertainty, it is best to assume the most even split consistent with the play to this point, there being nothing to suggest otherwise. In the absence of interference this is the most probable condition. Under that assumption, the probability of the ♦A being in the West is 50%. The same applies to the ♦Q, or any other red card. On the basis of a random deal of the missing red cards, we can calculate the probabilities of the number of HCP held in the red suits by West.

The symmetry is evident. The median number of HCP held is 4.5 for each side. The chance of finding the ♦A or the ♦Q in the West is 50%. This is because the vacant places are equal at 7 -7. It doesn’t matter that East has shown up with just 1 HCP to West’s 5. The same figures would apply if West had led the ♣ K from ♣KQJ. A random deal is blind with regard to the ranks of the cards. The evidence of the bidding slightly affects the odds. With 14 HCP and KQJxx, some West players might overcall, but East would be less inclined to take action over 4♣ with a similar holding.

The equality of vacant places is a very significant feature. The a priori odds are also a consequence of symmetry with 13 vacant places to a side. The reduction from 13 a side to 7 a side changes the probabilities, but the conclusions are not unexpected. The 9 HCP that are missing are most likely to be evenly divided 4-5 or 5-4. Before play in the suit starts, there is nothing to choose between ♦A on the right or ♦A on the left. Next we look at a case where the bidding tells us there is an imbalance in vacant places.

Case 2: East has 7 clubs and doubles for the lead

The ♣ J is led, declarer ruffs a club and draws trumps that prove to be split 2-2. The evidence of the bidding and play point to an imbalance of vacant places for the red suits.

Here are the probabilities for red suit HCP held by West:

Obviously symmetry is destroyed and West is likely to hold more HCP than East. The median value is 6.5 for West, hence 2.5 for East. The chance of finding the ♦A or the ♦Q in the West is 2:1, in accordance with the ratio of the vacant places (8 against 4). East has not limited his HCP by his double, so there is no restriction in that regard. Next we shall look at the situation where East has limited his HCP range.

Case 3: East preempts and a constraint applies

Now the bidding tells us something about the distribution of the HCP. Let’s bravely assume that East cannot hold the ♦A along with 7 clubs to the KQ. Here are the probabilities for red suit HCP held by West.

Based on our working assumption the probability of the ♦A in the West is 100%. The probability of the ♦Q being with West is 63.6% based on a compilation of the possible card combinations without an ace in the East. This result agrees exactly with a vacant place calculation. Here we have placed the ♦A in the West, so the vacant places are now 7-4. The probability of any red card other than the red ace being with West is 7/11.

Case 4: A conflicting constraint

The above cases are simple illustrations of how probabilities are calculated using vacant places when there is no conflicting constraint. The mathematical procedure conforms to what can be classified as a common sense approach by a competent declarer. The distribution of the cards is the foremost consideration. In the case where the bidding has gone (P) 1♠ (P) 2♠ All Pass one may assume initially the hands are balanced in shape and HCP content, but maybe not. The play of the cards may reveal that an abnormal constraint applies. We’ll now examine a case where the distribution of sides is the most common 8-7-6-5 configuration.

South ducks the opening lead and West, after some thought, helpfully continues with the ♣T leaving declarer in peace to work things out for himself. We don’t mind that. South wins the second club, goes to dummy in trumps, ruffs a club, and draws trumps, West having been dealt ♠Jxx. East has dropped the ♣KQx so we determine the vacant places to be 5 with West and 8 with East. Because he passed initially, we assume East cannot hold 2 red aces to go alone with his club honors. Under these restrictions the distributions for the 13 remaining HCP in the East hand are:

East has more red cards than West, but cannot hold more than 9 red HCP (AQQJ), so the constraint on HCP in the East acts contrary to the tendency of the vacant place imbalance to favor East as the holder of the ♦A. It is very important that declarer note this conflict which arises when a defender has revealed a long, topless suit. The HCP distribution appears truncated at the top end of the scale due, of course, to the constraint placed on the total number of HCP. East holds 7-9 red HCP in 58% of the cases, but did not open the bidding with12+HCP. This points to a flat shape: ♠xx ♥QJxx ♦Axxx ♣KQx ? It was not very sporting of him to pass, reasonably conservative perhaps, albeit a rare and unexpected occurrence. Immediately a long topless suit is revealed, declarer should become aware that something unusual has occurred.

Double dummy ‘errors’ occur most commonly when a player holds a topless long suit. It is more probable that a long suit contains its fair share of court cards. When a long suit is deficient in HCP the distribution of HCP will not conform to expectations. Bidding systems are designed for normal conditions. If a player preempts in a long, topless suit with honors in his short suits, the other players will reasonably assume otherwise. Swing results are likely to occur. Similar considerations apply during the play of the hand. A declarer who finds evidence of a long, topless suit, must adjust his prior expectations.

The probability of the ♦Q in the East is 65%, of the ♦A in the East, 40%. How to play the diamond suit? If one simply plays to the ♦K and that loses, the ♥A is sure to be offside. But if the ♦K holds, there is a very good chance the ♥A is also onside. So the optimist plays to the ♦K and then to the ♥K and claims 8 tricks! This approach has a 48% chance of success.

Finally we note that East made a discard on the 3rd spade. Was it a low diamond? That points to a hand like this: ♠ xx ♥ AJx ♦ Qxxxx ♣ KQx. Usually one ignores such slim indications, however, an attempt should be made to speculate without being unduly swayed. Here a diamond discard supports the direct approach, so merely reinforces a decision rather than being an essential element in its construction.

Case 5: a normal situation

In the previous example West led from ♣J109xx opposite ♣KQx, an abnormal configuration. This led to constraints on East that resulted in an abnormal distribution of HCP far different from the prior expectations. Now we consider the more normal situation where West leads the ♣10 (coded 9’s and 10’s) from ♣KJ109x opposite ♣Qxx. This is a somewhat venturesome attacking lead, but we assume West didn’t have a better lead. The upshot of this is that the constraints on the total HCP held in the West or East are not onerous. In the previous Case 4, one had to count up the excluded combinations, but in this case West could have opening points and not wish to overcall or balance with a club suit. Consequently, due to the lack of constraint, uncertainty is at a high level and declarer may use the vacant places as the primary tool in estimating probabilities.

The probability that the ♦A sits in the West and the ♥A sits in the East is 26%, so the optimistic approach of drawing trumps and playing of low to the ♦K has lost its glow.

[The odds of success are (8/13) times (5/12), or 26%] On the other hand, the chance that West holds at least one of the diamond honors is 64%. [The odds are 1 minus (8/13) times (7/12).] So rather that adopting a Kamikaze attacking approach as before, it makes better sense for declarer to play a diamond to the jack initially and hope West holds a doubleton honor. Try to get the cooperation of the defenders with regard to the play in the red suits. This is a hand where declarer wishes he had better spot cards.

It pays to envision the distribution of the sides based on the evidence. What is the single most probable distribution?

Condition I looks to be much less likely on the bidding and play than indicated by the weights which are based solely on the number of card combinations on the deal. Rule it out. It is better to play West for a doubleton diamond rather than a doubleton heart. If the diamonds are split 2-4, the chance of West holding at least one top honor is 60%. So choosing to play for the single most likely distribution (Condition II) makes sense as it is in accordance with the vacant place analysis. Both methods are based on the assumption of an unconstrained random deal in the red suits. So, declarer should finesse the ♦J, expecting to lose, duck the heart return, hoping to hold the losses to 2 hearts and 2 diamonds. The ♥10 may prove crucial.

Concluding Remarks Probabilities should be no more or less than a numerical expression of common sense. If an expert, like Bob Hamman, say, plays the hands well consistently over a decade or more, one may assume that he has followed sound mathematical principles even though he may or may not express his procedures in mathematical terms. Artificial intelligence computer programs that adapt to the expert’s procedures may grow to simulate his performance with greater and greater efficiency without providing less gifted players with insight into how to improve their game. It is the task of the mathematician to provide an explicit model that functions as a general guide for good decision making. Simplifying assumptions are necessary to make the model useful at the table, or, at the very least, to provide insight into an expert’s approach.

The application of vacant place estimations of the probability of the location of a card of interest is a consequence of such a model. Vacant places yield a good approximation under the right conditions, namely, when there is maximum uncertainty with regard to the distributions of suits which are yet to be played. The degree to which a mathematical model fits reality is the true test of its worth. Those who insist in applying the a priori odds inappropriately are doing a disservice, because their simplistic model is inflexible with regard to changing circumstances, especially with regard to imbalances in the vacant places.

Hooray for the Chinese Ladies Team

September 17th, 2009  ~  Bob Mackinnon  ~  3 Comments 

Bridge has limited appeal for spectators, but all agree that for colour, excitement, and general value for money, there is nothing to touch the women’s international trials.

– M. Harrison-Gray from When Ladies Meet, in Country Life, May 9, 1957

After many years of diligent effort the Chinese ladies have finally triumphed in the Venice Cup competition recently held in Brazil. A Precision enthusiast myself, I feel I have learned much from watching them perform on BBO. Recently my scores have improved temporarily. The question I want to address is, what can we learn from watching the ladies compete at the highest level?

Women’s events have always attracted interest from sympathetic males who, like Harrison-Gray, take a tolerant and largely humorous view of the human condition. Ever since becoming civilized, Man has encountered difficulty answering the question, ‘what do women really want?’ Males unable to fathom the female psyche (and I agree there is still much research to be done in that area) in self-defence make jokes about inadequacies that all humans share to a greater or lesser degree. Linda Lee has in a recent blog described the scene at the VuGraph auditorium when China faced USA1 in the Venice Cup finals. Glaring errors are made. The male commentator gasps. The predominantly male audience, seeing all 4 hands, bursts into laughter. Hot tears of female resentment well up in Linda’s eyes, she bites her lip, and then finds that she, too, is laughing along with the rest. That’s the spirit, Linda.

Charlie Chaplin was a great comedian, but when you come to think of it rationally, what is funny about being hit on the head, being chased by a policeman, or slipping on a banana peel? Our laughter signifies that ‘there but for the grace of God fly I’. Laughing at the disasters of others can be healthy, but only when you can see yourself erring in the same way. One must tolerate the errors of others and thereby come to tolerate one’s own mistakes. To attempt to play a faultless game is counter-productive, as it promotes anxiety, generates inner tensions, stifles initiative, and inevitably leads to disappointment.

‘Learn from your mistakes’ is good advice, but given a choice, naturally one would prefer to learn from the mistakes of others. Do mistakes have a gender? No. To prove that, I give you a quiz on some howlers I witnessed in the recent championships in Brazil. I challenge the reader to distinguish between which of the following laughable blunders were made by a man, and which by a woman.

Howler #1 On lead against 3NT West held: ♠ J93 ♥ 654 ♦ KJ3 ♣ 8653

The auction had gone as follows: 1NT(14-16) – 2♣; 2♥ – 2NT; 3NT All Pass.

Who made the ‘thoughtful’ lead of the ♦J into declarer’s ♦Q105 thus allowing the contract to make for a loss of 14 IMPs where a simple club lead in the other room set the contract 4 vulnerable tricks?

Was it: (a) Beth Palmer, (b) Glenn Groetheim, or (c) Gabriel Chagas?

Howler #2 The South player on lead held ♠ T632 ♥ A6432 ♦ Q4 ♣ 72.

The Precision auction went as follows: 1♣ (1♦) 1NT; 3NT – 4NT; 5NT – 6NT (That sure looks like women’s bidding, doesn’t it?)

Instead of leading partner’s advertised diamond suit and defeating 6NT, who chose to lead the ♥3 and let the contract through?

Was it: (a) Peggy Sutherlin, (b) Sjoert Brink, or (c) Kit Woolsey?

Howler #3 The contract was an adventuresome 4♠. Dummy held ♠A984.

With ♠J6532 South was in a hopeless situation as 2 trump losers are certain.

In desperation declarer made the psychological play of leading the ♠J.

Which West defender holding ♠K107 killed partner’s singleton trump queen by following the beginner’s rule of always covering an honor with an honor?

Was it: (a) Wang Hongli, (b) Jeff Meckstroth, or (c) Zia Mahmood?

Howler #4 South opened 1♥ showing at least 5 hearts and North had a decent but not spectacular hand in support: ♠A654 ♥ Q72 ♦ Q9 ♣ AQ73.

In the other room the natural bidding proceeded in a perfunctory manner: 1♥ – 1♠; 2♥ – 4♥, All Pass, making 6, so there was a chance of a much needed pickup. North–South had at their disposal the scientific methods that enabled an exploration of even the remotest chances of a slam. Their luck was in, until after 6 rounds of revealing bids, North lost concentration, forgot the trump suit was hearts, and bid 6♠!

Was it: (a) Irina Levitina, (b) Joey Silver, or (c) Lorenzo Lauria?

Howler #5 Non-vulnerable versus vulnerable this North player made a second seat preemptive 2♥ bid on: ♠ 864 ♥ QJ632 ♦A73 ♣ J8.

After a 2NT Ogust enquiry, North bid game on a 7-card fit, and the other circumstances were not favorable resulting -800 in 4♥* against a failing 4♠ in the other room – a 12 IMP loss on the last board of an otherwise fairly even session.

Was the North player: (a) Sabine Auken, (b) Karen McCallum, or (c) Pam Wittes?

Well, I suppose you got the last one right. Yes, indeed, it was a woman who bid outrageously in the manner indicated. As for the other hands, it was not a woman who made those laughable blunders. (The answers are given at the end of this article.) Again I emphasize the laughable part. We all blunder at times, so it is just a matter of reducing such errors to a tolerable level. Modern research into how humans make decisions has shown conclusively that we are not nearly as rational as one would hope. Reason tempered by emotion is the way to proceed, as without emotion we humans are not going to get it right. But we must limit the emotions to a secondary role. When emotions rule, mistakes are made. The trick is to get our emotions working for us, not against us. It is all a matter of paddling downstream, not upstream.

Playing for Increased Uncertainty

In the land of the blind, the one-eyed man is king.

– Erasmus (1465-1536)

There are those who would say the last example does not qualify as a howler, or even as a mistake. They would argue that this preempt turned out to be costly, but that there is a price that one is willing to pay occasionally for the many triumphs that have gained in this manner. The strategy is to create uncertainty in the hope that the uncertainty costs the opponents more than it costs your side. The worse the hand, the better the chances the opponents can make something! Unlucky if partner holds a good hand, for then you may be in trouble. One improvement I have noticed in the Chinese Ladies’ performance is that they are now much less likely to be pushed around by light actions. They are willing at times to be talked out of a part score. Being disciplined, they are less likely to get rattled. Let’s look at the deal in full to illustrate the bad effects of increasing uncertainty.

At the other table:

After Sun passed as North, the auction took a normal course according to today’s loose standards, namely, overbidding to a hopeless game that went down 2 undoubled. This is the normal action if one thinks the rival pair will do the same. McCallum’s opening preempt in hearts has little to recommend it apart from the favorable vulnerability. Baker had a 5-loser hand, unfortunately without the quality of heart support that McCallum required. when she raised herself to game,‘Kate’ obviously was hoping for better hearts from Baker. Wang may not have doubled 3♥, but 4♥ proved to be too much of a temptation even though there was no guarantee she could set the contract. At it turned out, the silent Liu could provide considerable help.

Planning for Uncertainty

It is one thing to take an occasional flyer in the hopes of a lucky result that may shake the opponent’s confidence, it is another to design your system around bids intended to confuse. When Howard Schenken first introduced the weak 2, it was as a constructive bid that allowed a partnership to reach major suit games on hands that were lacking in HCP but strong on distribution. There was a requirement that the long major contained 2 top honors, the normal complement for a 6-card suit. Later experts adopted the weak 2 as a preemptive bid and did away with the honor requirement that was normally fulfilled anyway. The resulting increase in uncertainty could be offset by the introduction of an asking bid, Ogust 2NT. It appears today that Ogust is insufficient for the task at hand. At one time the type of hand on which McCallum preempted was acceptable only in third seat, but now the contagion has spread. Does increased uncertainty constitute progress?

To consider this further let’s look at 4 hands in the quarterfinal match between the Netherlands and USA2 in which 2 youthful Dutch players employed a consistent strategy of unsound actions against Meckwell, who play a disciplined pressure game. On three consecutive boards (25 thru 27) Meckwell gained 39 IMPs, the gains being attributable directly or indirectly to the brash, uninformative style of their opponents. The reader should take particular note of the indirect effects.

Driver opened with a (self-) destructive preempt in hearts. Meckstroth had an easy overcall, Rodwell an easy raise. Brink couldn’t without loss lead the suit partner had bid and he had supported. He chose to lead from his worthless doubleton, very often a dangerous choice. Good lead, thought the viewers, at least on this hand. The play was simple enough. Meckstrorth took the second round finesse in trumps and made his vulnerable game. What could be easier? But in the other room there was no NS bidding. Declarer went down playing normally for the drop in trumps for a loss of 12 IMPs.

A rationalist would say that the opening bid was foolhardily bad, but wait! Aren’t we told that type of action often pays off? Defenders of this style can point to a board in the Round Robin where Meckwell missed a vulnerable 6NT after Bertens of the Netherlands opened a Multi 2♦ on ♠ 75 ♥ JT752 ♦ A4 ♣ T874. (Resembles the McCallum hand, doesn’t it?) As Buffy the Vampire Slayer commented on BBO, ‘chalk one up to that aggressive Multi opening.’So one might conclude on the basis these 2 boards taken in isolation that the Netherlands’ strategy had put them slightly ahead of the game.

Well, everything works some of the time. On Howler #2 described above, it was Brink who on Board 27 of the quarterfinals was reluctant to lead his partner’s suit, and consequently lost 14 IMPs. Good lead on Board 25, bad lead on Board 27, no matter; one might say the losses due to uncertainty were mounting up in the all-important quarterfinals. Board 26 was to provide evidence that consistently inconsistent behavior on previous hands can indirectly cause a loss in an otherwise normal situation.

This was a deal where one expects a highly competitive auction with a whole lot of guessin’ going on – intelligent guessin’ in the Bermuda Bowl. Although the South and East hand patterns are unusual, and an 11-card fit is rare, the situation is normal under the circumstances.

In the other room the Netherlands team had bid to 6♠ and had gone down 1. Meckwell had stopped in 5♠, but all was not lost for the Netherlands as declarer was booked to lose the ♦A, a diamond ruff, and an eventual ♣Q. Tie board? No! According to custom Brink did not lead a suit both he and his partner had bid, but this time he had a very good reason for it. Yes, he had found the killing lead of the ♦2, but under the ♦A Meckstroth discarded the ♦K. Could this pitiful attempt at deception really work? Not really. Driver wouldn’t be fooled that easily by an opponent, but every hand has a history. It would be just like Brink to preempt vulnerable to the 3-level on just 6 hearts. To the astonishment of all who witnessed it, Driver tried to cash a heart trick. Meckstroth ruffed and was able to make his contract for a gain of 13 IMPs.

The Importance of Reliable Information

A good system is designed around what is normal, hence most probable. Seeking an abnormal result is what one does when hopelessly behind in a match. It seldom works and the usual effect is to increase the losses. It’s a reasonable approach under those circumstances, but why should one adopt this strategy from the start? From the howlers noted above, one needn’t assume that world’s champions will not make mistakes.

On Board 30 the Driver and Brink got to a solid 7NT after 7 rounds of relays on this combination of cards:

One won’t reach the optimum contract without exchanging information with one’s partner in a clear, efficient manner. Try it. The Dutch gained 14 IMPs when in the other room the USA pair, now world champions, on natural bidding and without interference, stopped in 4♥ with 15 tricks on view. After the previous disasters the great Dutch auction turned out to be so much wasted effort. Now if a pair can spend the time necessary to perfect their relay bidding system, isn’t it somewhat ironic that they take the view unsound preempts are worthwhile? Hopefully in future they will decide to follow the example of the Chinese Ladies and adopt a reliable competitive bidding style along with good constructive methods. Youth isn’t everything, it’s only a beginning.

Information is the key to success. At the bridge table reality is represented by the composition of the deal. One needs reliable information in order to have one’s action conform to reality, information which comes from the other players at the table. If partner’s bids are nebulous and admittedly designed to cause confusion, then one is left with having to trust the opponents whose best strategy is to bid boldly to games and slams and see if you can defeat them with your side’s disadvantage of self-imposed uncertainty. Such conditions are highly dangerous for defenders. My last piece of futile advice is this: stop making these futile preempts in hearts.

Answers to the Howler quiz: #1 (b), #2 (b), #3 (c), #4 (c), #5 (b).

The Chinese Way

August 31st, 2009  ~  Bob Mackinnon  ~  1 Comment 

Many observers have stated that ‘the future of bridge belongs to China.’ As we approach the 2009 world’s championships in Brazil, I am wondering whether China will fulfill its promise and become a dominant force in international competition. The women’s teams have done well enough in the past, but never have reached the summit. The Open team has still to establish itself as a front-runner. The US may fight hard to keep its place at the top, but many players are senior citizens, and youth tends to prevail.

I have been reading Bobby Wolff’s sad and bitter memoir, The Lone Wolf, a self-portrait of an insider who still considers himself an outsider. One issue foremost in his mind is the sponsorship of national teams, and here there is a major difference between the US and China that is of particular interest. Bridge in the US has had a ‘grass-roots’ development. Ely Culbertson was the genius who changed the game into a business, while at the same time improving the method of play, but there came a time when the gulf between the aspiring world champion and the merely good player became too wide to be encompassed within one philosophy. I don’

t think there are many local players who know (or care) the extent to which the international scene has become largely dominated by pairs who employ highly artificial systems.

The leading American teams have been formed by sponsors willing to pay for the chance to become known as a world champion. Ideally the sponsor is the worst player on his team. Bobby Wolff begins his autobiography by accounting how he had to tell Ira Corn, sponsor of the legendary Aces, that he was not good enough to play as a member of the team. Corn was a business tycoon who felt that with the application of his natural ability and drive he could become an expert bridge player in a short period of time and personally lead his troops to a world’s championship. It is part of the grass-roots hype that anyone can win given hard work, desire, and a little luck. Corn knew how to organize, but, alas, he was not good enough as a player, and Bobby had to tell him so. (The unpleasant task of telling it like it is has fallen often upon Wolff’

s broad shoulders.) To his credit Corn stepped aside, but continued to foot the bills as the Aces went on to victory in 1970 and 1971 using the training methods imposed upon them from above. Wolff and Hamman developed their own version of the Neapolitan Club and became the unofficial leaders of many championship teams that followed over the next 25 years.

The development of bridge in China has been quite different from that in the West. The government was the first sponsor, a task now being assumed by large companies for the sake of prestige. Bobby Wolff relates how as president of the ACBL he went to China in 1993 to talk to government officials about the advantages of developing bridge as an international sport. He suggested one way to develop the game was to introduce it to children in primary school as a curriculum subject worth learning for its own sake. Somewhat to his surprise, the Chinese officials immediately accepted his off-hand proposal and asked him to see to the production of suitable textbooks in Chinese. To his regret he was unable to carry this through due to bridge politics back home. One hang-up was which system to foist upon innocent young minds. I suspect another was an aversion to the Chinese leadership in those days before China had become an economic necessity.

Kathy Wei had been at work in China before Wolff’s visit and had formed a working agreement with Deng Xiaoping, premier of China and a keen player. As a result of her work and the prestige associated with a head of state, Precision had become popular, a system imposed from above rather than grown from below. It was an elitist approach. Bridge was considered not only a worthwhile intellectual pursuit but also a discipline from which real-life lessons could be drawn. Let’s list some of these: patience, tolerance, politeness, honesty, humility in victory, graciousness in defeat, perseverance, anticipation, cooperation –

ironically, characteristics that may still be appreciated in the East, but which are falling out of favor in the West.

With ambitious sponsors winning is everything. With committees winning is paid lip-service, but the manner of winning takes precedence. That entails following established practices. Within past British selection committees the choice of Martin Hoffman was frowned upon, not because he was not a proven winner, but because his methods were considered too individualistic and somewhat unorthodox. The committee hoped to win with lesser talents. It is hard to say they were wrong, because that is how committees think. Will the Chinese officials do a better job?

To me there is a Platonic ideal –

a few mature, dedicated players get together voluntarily with an all-for-one-one-for-all attitude. They submit to a regimen of healthy food and vigorous exercise. They practice constantly together primarily to correct their psychological flaws. They must learn to stop blaming others for their own mistakes. They work as a team over a limited period of time with a limited objective in mind. We have a prime example to emulate: Iceland 1991. There are others. It is not a matter of size or money, but of devotion to a common cause. (Cue the theme from The Chariots of Fire.)

To want to win while remaining free to ‘do your own thing’, to me is misguided. I may have to eat those words after China faces USA in the Women’s Final. The Chinese ladies play a sedate version of Precision, but have been practicing against men’s teams in order to ‘toughen up’. They were manhandled by the redoubtable veteran pairs of the English team that defeated them narrowly in 2008. I have found Liu rather shaky, and I wonder how she will cope with the trauma of facing the highly individualistic and psychologically based style that Linda Lee (not me) has characterized as ‘women’s bridge at its best.’

China vs USA should be a most amusing clash of cultures.

The question that remains to be answered at this time is whether China will develop its own style. Coming late to the game, they have the benefit of all that has gone before, so they can pick and choose what to them appear to be the best methods. Being elitists, they don’

t have to start with SAYC and work their painful way up. Their leading pair, Zhong and Fu, are not even burdened with standard Precision c.1979. It may be too early to say there is a Chinese Way, but one may speculate while watching the top Chinese pairs perform in the recent Inter-Cities Championships. One distinctly Chinese tendency that has been noted is that of opening 1NT with a 4-4-4-1 shape. I can see responding a strictly limited NT with this shape, and I do so after a Big Club opening bid, but to open a strong NT with this shape may well cause insurmountable problems, and not just on minor suit slams. I was happy to find the following example of such damage.

Safly, safly, catch monkey

– from Scottish Proverbs (1832)

A strong NT structure has long been a primary feature of the Standard American approach, albeit one that is schizophrenically at odds with the rest of the system, including as it does a limited opener, asking bids, transfers, relays, and a space-saving version of Blackwood. The aim is to pass information efficiently and accurately. Despite their Precision heritage, the Chinese are no sticklers for accuracy as evidenced by their off-shape takeout doubles and their inclusion of 4-4-4-1 shapes to strong NT hands. Here is a hand from the finals where the winning team picked up 5 IMPs against such a bid.

BBO failed at a critical point so we are not certain of Cheng’s bid, but EW ended up in 3♠ making +140, when NS are cold for 4♥. What is most remarkable is Dai’s pass to partner’s 2♠ balancing bid. It makes sense, however, as it appears to be a part-score deal. Softly, softly: East can always bid one more spade if required to do so. At the other table the auction followed a course more in line with Western sensibilities.

The trick in bidding 4-4-4-1 hands is to uncover the degree of fit in 2 suits. Fu’s fit-showing bid confirmed for Gan a double fit in the red suits. Fu and Gan followed up with the correct bids based on the information made available during the auction. That’s good.

Virtual Reality and Opening Leads

August 17th, 2009  ~  Bob Mackinnon  ~  1 Comment 

In this article we look at the connection between information and probability in the case of an unusual opening lead for the not uncommon 7=7=6=6 division of sides. There has been confusion on the effect of the opening lead on probabilities, which we hope to set straight by studying an example which is of interest for its own sake. The probabilities on opening lead are transitory, but it is important for declarer to start play in the right frame of mind and to proceed logically from that point. At the end we look at virtual vacant places, an interesting concept that furnishes insight and may have a wider range of applicability.

The Improbable Can Happen

The improbable grabs our attention. ‘Man bites dog’ is newsworthy, but not when the situation is vice versa. One shouldn’t fear the improbable, but one shouldn’t ignore it, either. It is a matter of degree. ‘Don’t undress in front of your dog’, is sound advice as such action poses an unnecessary risk with little to gain, but it is not a rule over which one becomes obsessive. I suppose it depends to some extent on the height of the dog. On the other hand, if your dog growls in the other room, you should pay attention as there must be a reason, even if the dog can’t reason. We digress. At the bridge table, a predictable opening lead doesn’t greatly affect normal expectations, whereas an unusual opening lead does. It is newsworthy, and there is more than the ordinary amount of information to be gleaned from it. One shouldn’t be afraid of taking a risk if the circumstances merit it, but one should evaluate the unusual situation realistically. This is where probability comes in.

The 7-7-6-6 Division of Sides

It is said that Dame Fortune favors the bold, which, in my opinion, is fair and just. Why should Dame Fortune, or any other dame for that matter, favor a do-nothing who is afraid to get involved? The suggestion that the Meek will eventually inherit the earth to me sounds like a shameless appeal to the latent greed that lies dormant within the indolent breast. A one-sixth portion is a more reasonable expectation, and we are not talking sea-view property. In the here and now the Meek get rewarded far beyond a limit commensurate with the risks they are willing to take. At the local club, and it is bridge we’re mainly thinking about, one may score well enough by sitting quietly and taking profit from the opponents’ errors, but it is sheer greed to expect to win without ever stepping outside one’s comfort zone. In addition there is a small domain where the rewards for a lack of initiative are immediately forthcoming due to a low expectation of total tricks.

The center of the domain to which I refer is defined by a 7-7-6-6 division of sides where the number of Total Trumps is a measly 14. It constitutes 10.5% of the deals, so is entered about 3 times per session. This is approximately what the Meek deserve, besides which, it is a reasonable proportion for keeping a lid on excesses. Within its bounds those who don’t open on a perfectly good 12 HCP because they don’t like the look of their cards get rewarded for their timidity and those who don’t balance get praise from their partners rather than the customary scorn. Here players who bid to normal contracts that run aground in the shallows of the distribution are forced to listen respectfully as the Meek excitedly explain how unfounded suspicions led them to underbid profitably.

Bold ones mustn’t resent yielding up this small portion of infrequent victories. The best we can do as declarers is modify our procedures and take a different tack, one modified to take into account that we are in a strange land where aggressive play may not be superior to a wait-and-see approach that risks less. When the dummy comes down and we note the 7-7-6-6 division of sides, that is the time to reconsider our strategy with a fresh mind. What does the opening lead tell us? In what follows we consider the blind minor suit lead against an uninformative 1NT -3NT auction when the defence holds 7=7=6=6.

Rule A and Rule B

In order to calculate probabilities after a blind lead is made, one relates the opening lead strategy to the lengths of the suits held by the opening leader. The rule adopted is generally statistical in nature, and may be modified by knowledge based on the proclivities of the player involved. This facet of Bayesian probability is beneficial because it is sensible. We consider two rules formulated as follows:

Rule A The longest suit is led. Suits of equal length have an equal chance of being led.

Rule B A major suit is led whenever it is equal in length to or longer than either minor suit. Suits of equal length have an equal chance of being led.

These rules represent extremes. Rule A amounts to ‘4th highest from the longest and strongest’ regardless of suit rank. Rule B gives full preference to the major suits. Some habitually play that way. There are intermediate variations.

The frequency of opening leads in 4 suits are given below (on the assumption neither hand features a void.) Random refers to a random choice from a pack of 7-7-6-6 cards.

The probabilities of Rule B may be closest to our everyday experiences. This can be tested using statistics from actual bridge deals. The question here is ‘how unusual is a minor suit lead?’ It depends on which rule is the more likely to be employed by a particular player. Rule B players are strongly affected by the bidding and will not lead a minor suit unless there is no reasonable alternative. They are sometimes victims of those who will open 1NT with a 5-card major. I do that in third position, and am surprised at how often the lead is in my long major. Rule A players operate on a hand-by-hand basis, putting less trust in the bidding and more on the quality of their suits. They tend to be more passive than Rule B players. A major suit lead is normal, so it’s a matter of degree.

A Low Diamond is Led

The lead is likely to be from either a 4-card or a 5-card suit. It is best to start with a look at some of the more probable the distribution of sides and their initial weights.

On the deal alone, the first 4 conditions for which the diamonds split 4-2 are more likely than the conditions for which the diamonds split 5-1. In addition there are companions to Conditions II-IV for which the hearts are longer than the spades. Under Rule B Conditions II and III don’t get counted as possibilities as a major suit would have been led instead of a diamond. Under Rule A they are included but their weights are reduced by half. The weights for the 5-1 diamond split are not affected by the difference in the rules, so their relative contributions increase under Rule B more than under Rule A. Condition I represents the most likely distribution regardless of which rule is applied.

The most likely distribution on an a priori basis has a weight of 100. It is missing because it consists of the following splits: ♠ 4-3, ♥ 3-4, ♦ 3-3 and ♣ 3-3. A spade lead is indicated. A companion shape has 4 hearts instead of 4 spades.

In order to calculate probabilities one merely amasses the distributions that apply under the rules, adjusts the weights accordingly, and adds up the total weights under the various conditions of interest. Once one has defined the process, it is easy enough to have a computer program do the work (and do it better) for all possible division of sides and all possible reduction factors, but for the present case the calculations were done by hand and distributions with voids were excluded.

Under Rule A we find:

Under Rule A the modal distribution is ♠ 3-4, ♥ 3-4, ♦ 4-2 and ♣ 3-3 which corresponds to the maximum likelihood estimate (Condition I). The distributions of the major suits are centered about a 3-4 split. It is perhaps the club suit that is of most interest to a declarer as usually he will aim to develop tricks in that suit. The probability of any particular club, the ♣Q, say, being on the right is much greater than it being on the left, but the 3-3 split far outweighs the 2-4 split. There are no surprises here.

For Rule B, we find:

The 5-1 split in diamonds is the most likely and the average number of diamonds lies closer to 5 than to 4. There is no correspondence between the modes of all 4 suits to any one distribution of sides. There is a strong tendency for the majors to be split 3-4 and the clubs to be split 3-3, which tends to place the diamonds at 4-2. There is an inconsistency due to the great reduction in the number of 4-card leads from a diamond suit. If we look at the weights of the distributions, the most likely distribution stands out like a giant among the pygmies, but the latter outweigh it on accumulation.

Virtual Vacant Places

A fistful of numbers may not come out and hit you between the eyes. How can one discern some order in the above probabilities? It would be convenient if one could translate the numbers in terms of vacant places, but life isn’t always so convenient. Let’s give it a try nonetheless. The concept of vacant places relating to probabilities is based on the following argument. There are 6 diamonds held by the defenders. Suppose these are split 5-1, leaving 8 vacant places on the left and 12 on the right. The differential in vacant places is 4. If the remaining spades, hearts and clubs are shuffled and dealt, the chance of any card ending up on the right is 12 out of 20, or 60%, regardless of the rank of that card. If the diamonds are split 4-2, the probability of a card being dealt to the right becomes 55%. The differential in vacant places is 2.

In practice, the diamond lead is either from a 4-card suit or a 5-card suit and there are just 2 vacant place differentials possible, but on average the number of cards in the suit could lie anywhere between 4 and 5. This idea gives rise to the concept of a virtual vacant place differential that represents an average that can’t exist in reality. Let’s see how it works with 6 cards in a suit and 20 other cards divided between the two defenders.

The interpretation of the probabilities calculated under Rules A and B in terms of virtual vacant places fits the results rather well. Differentials of 1, 2 3, and 4 are represented. An interesting facet is that clubs have lesser differentials than the majors, so the indication is that the vacant places that need to be filled varies with the available length of the suits. In other words, the probability of a club being on the right is different from the probability of a heart or a spade being on the left, which is contrary to the classical approach to vacant places with regard to probabilities, where each card is treated as independent contribution.

The most remarkable result is under Rule B where the virtual split in diamonds with regard to clubs is 3.5 – 2.5, remarkable in the fact that the diamond suit must be at least 4 cards in length. But it makes sense. If the diamonds are 4-2, there are 2 vacant places to be filled on the right. The 3-4 splits in the majors fill those vacant places quite nicely, leaving the clubs to maintain a balance at 3-3. This is the characteristic of the most likely distribution, Condition I given above. If the diamonds are split 5-1, the majority situation under Rule B, then there are 4 vacant places to be filled, most readily accomplished by 3-4 splits in the majors and a 2-4 split in clubs. That is the characteristic of Condition IV which is the most likely distribution when diamonds are known to split 5-1. On average, then, the clubs tend to fill 1 vacant place on the right.

Under Rule A a large majority of distributions feature a 4-2 diamond split. The major suit distributions correspond to a virtual diamond split of 4.5 – 1.5, a difference of 3 virtual vacant places, the additional vacant place arising from the possible 5-1 split. Removing 6 diamonds from the defenders’ hands with a 4-2 split is not the same as noting that the opening lead was from a 4-card diamond suit. Why? Simply because the 6 cards removed by the opening lead are akin to Shylock’s pound of flesh, there is spillover involved. The connecting tissues between suits means the combinations with longer spades, hearts, and clubs are no longer possible. Under Rule B many weighty distributions featuring a 4-card major are removed entirely from consideration because the preferred lead is a major rather than an equal-length diamond. This accounts for the differences in probabilities under the application of different rules for the opening leads.

Conclusion It appears in our pursuit of mathematical beauty we have wandered far from our starting point which was how to take advantage of the unusual minor suit lead when the division of sides is 7=7=6=6. Not so. As long as one keeps to a reasonable path one won’t get lost in the undergrowth. Good mathematics will support reasonable decisions. The advice is always the same: be aware of the circumstances which have been thrust upon you. Focus on the most likely distribution of sides. This direct approach puts the emphasis on counting out the hand with all suits involved. Each suit should be treated not individually, but in conjunction with the other suits. When the lead is from a suit of indeterminate length more than one distribution of sides needs to be kept in mind, but the most even splits are the most likely and these are closely related. The first goal is to safely resolve uncertainty, so ducking a diamond lead will often combine safety with resolution of the diamond split. Losing a tempo is not likely to be costly.

Be valiant, but not too venturous – John Lyle (1554-1606)

Virtual Vacant Places with 8=6=6=6

The concept of virtual vacant places should work well when the opening lead is very often from the most plentiful suit where the restrictions on the distributions are not severe. The most obvious candidate is the 8=6=6=6 division of sides for which a low spade lead is not at all surprising. The Total Tricks equal 15, so we find ourselves in the borderlands of the Meek. Here are some of the more likely distributions of sides.

Condition I is unique, whereas Condition II has 3 variations, any suit being capable of splitting 2-4. This tells us that the spade lead will often be from a 5-card suit. Condition III has 6 variations 2 of which contain a 4-card heart suit, and so are subject to a reduction by half on the grounds a heart could have been led as well as a spade. A lead from a 6-card suit will not be uncommon as Condition V has 3 variations.

Here are the overall results under Rule A which doesn’t distinguish between the ranks of the suits. They support the application of virtual vacant place approximations.

The most likely distribution of sides is Condition I, but because it is unique, it gets outweighed by the variations on Condition II. The modal distribution features 3-3 splits in 3 suits, but this is not matched by a 4-4 split in spades, which is impossible. In practice then, declarer anticipates a 3-3 split in any one of the sparse suits, but is aware that spades are probably not split 4-4. If spades are 5-3 look for 2-4 in one of the other suits.

With regard to virtual vacant places, it so happens that the probabilities of a non-spade card being on the left or right is given by the virtual vacant places for the average split in spades, namely, 4.83 – 3.17. The virtual vacant places are 8.17 and 9.83, so the probability of a card being on the right is approximated by 9.83 divided by 18 (54.6%).

If one applies Rule B which distinguishes between hearts and the minors with regard to the choice of opening lead, differences arise, because the distributions with 4 of a minor are given full weight. A minor on the right has a probability of 53%, whereas a heart on the right has a probability of 55%. The average spade length is 4.72 (54%).

The Most Likely Distribution

August 4th, 2009  ~  Bob Mackinnon  ~  No Comments 

Jaynes’ Principle of Uncertainty

In making inferences on the basis of partial information we must use the probability distribution which has maximum uncertainty subject to whatever is known

– after E.T. Jaynes (1957)

The central idea behind Jaynes’ Principle is that one accepts whatever information is available then interprets it in the widest sense consistent with the evidence at hand. The consequences to the play of a bridge hand are easily stated. Whether we think of maximum uncertainty or ratios of card combinations, it comes to the same thing: assume the most even splits possible under the current set of circumstances. Of course, calls and plays provide additional amounts of partial information, so the most likely distribution of the sides may change. One starts with a candidate distribution consisting of even splits, but one must not be over-reliant of just one distribution. That having been said, one must start somewhere, and the most likely distribution of sides is the preferred choice. This focuses the mind in the correct manner, at least as far as probability is concerned.

We illustrate below how this works in the simple situation where the bidding has gone 1NT – 3NT and a spade is led. This is the classical ‘blind lead’ situation where the 4th highest lead in a major suit is the common choice. We have discussed previously how an opening lead places restrictions on distributions, the more informative the lead, the more severe the restrictions and the fewer the distributions that remain to be considered. Now we consider how the most likely distribution of sides relates to declarer’s inferences.

The 8=7=6=5 Division of Sides

The opening lead is chosen the dummy is tabled, and declarer attempts to form a plan based on the information that has been made available to him. With regard to the division of sides, declarer knows immediately how many cards the opponents hold in each suit. In the absence of interference bidding he may obtain from the opening lead a good idea of how the cards in each suit are distributed between the defenders. The distribution of the cards encompasses the greatest number of card combinations is the most probable single distribution. We call this distribution the maximum likelihood estimate. As an example we shall consider the sides of 8=7=6=5. The most even splits are: Spades: 4-4, Hearts 4-3 or 3-4, Diamonds 3-3, and clubs 3-2 or 2-3. These even splits have to be assembled into combinations that make up 13 cards to a side, so some restrictions apply. Here are some of the more likely splits:

The number below a distribution is the relative number on a scale of 100 of the card combinations encompassed by that distribution. Condition I is the most likely single distribution after a spade is led from length. Although Condition II encompasses the same number of card combinations, the hearts are of equal length so there is a good chance that a heart might be led instead of a spade. In the absence of clues from the bidding, it is reasonable to assume that a heart would be chosen for half of the combinations encompassed, so the weight after a spade is led should be reduced by half to 50, making it less likely that Conditions III – V.

The splits in the minors are of interest as well. The 5 conditions shown encompass the most even split in diamonds, 3-3, but also the 2-4 and 4-2 splits. The fact that a spade was led from length alters the odds in favor of the even split. The club splits divided between 3-2 and 2-3 with the latter more favored on the limited selection shown.

The Spot Card Effect

For the purposes of our illustration, the assumption is that a spade will be led whenever spades is the longest suit, the spades are equal in length to a minor, and half the time spades are of equal length with hearts. When one reads in an analysis of a deal that ‘a low spade was led’, one has been poorly informed. Which spade makes a difference to the odds, because sometimes declarer can make a pretty good guess as to whether or not the lead was from a 4-card or a 5-card suit. First we assume the lead is the ♠ 2 which looks very much like a lead from a 4-card suit. Given this is a long-suit lead, the possible distributions are reduced in number to 10, 4 of which contain a 4-card heart suit. For these we reduce the number of combinations by half. We can calculate the number of combinations for the relevant splits in hearts, diamonds, and clubs and express them as percentages of the total available for a given suit, as follows.

It is quite according to expectations that the hearts are most likely to split 3-4, as some combinations of 4-3 are partially eliminated by our assumption of parity. It is also expected that the diamonds would split evenly at 3-3, but there is an unexpected bias to the left in favor of the 4-2 split. Clubs also exhibit a left-hand bias with the 3-2 split half again as likely as a 2-3 split. The conclusion is that, although the spade lead appears at first glance to create a vacant place on the left, it cannot be concluded that a missing honor in a minor is more likely to be on the right. In fact, the contrary tendency applies.

How do we make sense of this conclusion? Easily, if one considers the maximum likelihood estimate expressed under Condition I shown above. Putting together the most frequent splits (modes) in each suit as shown above, and one obtains ♠ 4-4, ♥ 3-4, ♦ 3-3, and ♣ 3-2, which constitutes Condition I. It is good practice to consider the maximum likelihood distribution as a starting point as one is made conscious of the modes of the distributions of the suits taken individually. Remember this: the suit combinations can vary independently only to the degree that the sum of the cards in the suits must come to 13 in the end, and the degree of variation depends on the number of cards held in the suit.

Of interest is the probability that a given card in a suit, say a queen, will be dealt to the opening leader on the left or to his partner on the right. The ♥ Q is likely to be on the right roughly in the proportion of a 3-4 split. The ♦ Q is just slightly more likely to be on the left. The ♣ Q is actually more likely to be with the opening leader, roughly in the ratio of a 3-2 split. Thus Condition I provides a first approximation of the suit-dependent odds of finding a queen on the left or right.

The Lead from a Sparse Suit

The restrictions are more severe after a diamond is led, under the assumption that it is the longest suit or in a tie with clubs.

The maximum likelihood estimate of the distribution given a diamond is a long-suit lead is ♠ 3-5, ♥ 3-4, ♦ 4-2, ♣ 3-2 (weight =60) which is also the combination of the most likely splits taken individually in each suit. The information in the diamond lead indicates a major honor card is strongly favored to be on the right, that is, not in the hand of the opening leader. The most likely splits give a fairly good approximation of the probability of catching a major suit queen on the right (3:2).

The uncertainty is again a maximum for the club suit, as it is nearly 50-50 whether the ♣ Q would be on the right. The average number of clubs on the right or left is 2.5, an impossible number of cards to be dealt, which reflects the uncertainty. It goes against intuition, perhaps, that a 3-2 club split is more likely than a 2-3 split, that is, the longer club is more frequently with the opening leader. This apparent inconsistency can be resolved by considering the maximum likelihood estimate which is a result of taking into account all suits and how they interact.

The Effects of Uncertain Length

The effect of uncertainty is to increase the number of possible combinations that must be taken into consideration. If the spot card lead is unreadable, one must allow more suit combinations to enter the mix. We shall assume that spades can be 4, 5 or 6 cards in length. Adding these possibilities we find the following frequency of splits.

The modes for the splits are: ♥ 3-4, ♦ 3-3, and ♣ 2-3 as before, so Condition III becomes the modal estimate. The 3-4 split in hearts is favored greatly, and overall any particular heart, say, the ♥ Q, is more likely to sit on the right with odds approximately those of a 3-4 split. The odds of particular diamond (♦ Q) or a particular club (♣ Q) sitting on the right is close to 50%, but with a slight bias towards the right. The margin is roughly that provided by a vacant place split of 12 on the left and 13 on the right in keeping with the a priori odds adjusted to the exposure of one card on the left.

Why has Condition I lost the status of the modal distribution? The reason is obvious: the inclusion of the possibilities of 5-card and 6-card spade suits, but not of 3-card suits, results in the average length of spades being 4.6, even though spades will be only 4-cards in length nearly half of the time (48%). The pressure of ‘virtual’ vacant places on the right is overcome by a flip from a 3-2 club split to a 2-3 club split, as one finds in Condition III. The heart and diamond splits are the same for both conditions, so the flip in the shortest odd-numbered suit, clubs, by itself accommodates the additional spade length. Common sense tells us that we shouldn’t put all our money on Condition I, which requires a lead from a 4-card suit, but keep in mind Condition III just in case the lead was from a 5-card suit. (Frivolous overcalls help.)

Opening Leads and Honors

Books have been written on how to choose an opening lead. The defender must take into account the opposition bidding and the location of his high cards. It is considered dangerous to choose a suit with gaps in the honors, because even a lead of Q from QJ98 can come to grief. Declarer knows the division of sides and which high cards are missing, but he doesn’t know on which side of the table they sit unless the opening lead is from an honor sequence. Generally in the absence of bidding the honors are divided evenly between the defenders, and consideration of the most likely distribution of sides remains a valid approach. If the lead is an honor card, that occurrence constitutes additional information that isn’t dictated by length alone, but it is more likely to have been made from a sequence in a longer suit rather than a shorter one. The most difficult situation to read when a long-suit lead is avoided because of gaps in the honors held.

Symmetry and the A Priori Argument

Some bridge analysts are reluctant to use the evidence of the opening lead as justification for an adjustment of the odds on the location of a particular card of interest, say the ♥ Q. A characteristic of the a priori conditions is symmetry, which is destroyed on the opening lead, but this feature remains central in the minds of some. To see how the opening lead has affected the odds, we can look at the most likely candidates involving the 4-3 and 3-4 splits in hearts which initially are equally probable.

The distributions form symmetric pairs with equal probabilities before a lead is made. The opening lead destroys the symmetry in probabilities. Under Condition II a heart is as likely to have been led as a spade, hence the number of combinations represented must be reduced by half. Under Conditions IV and VI a spade would not be led. So for these 6 conditions a spade lead adjusts the probabilities as follows:

Of course, if the spade lead were to come from the right instead of the left, the odds would be reversed with the 4-3 split the more probable. As the opening lead can be made from either side with equal probability, it is correct to say that the 4-3 and 3-4 splits are equally probable before a lead is made. But such is not the case after a spade is led from the left. Hypothetically, the lead could have been a spade from the right, not the left, but there is no evidence to support that assumption on this particular deal.

We note that on average the probability of the location of the ♥ Q corresponds to 12 vacant places on the left and 13 on the right. This average involves all remaining possible heart splits, but because the number of hearts is an odd number, there is no single split that closely reflects those odds, as there is with a 3-3 split in diamonds. The hearts can be split 4-3 or 3-4, but not both at the same time. The odds on the location of the ♥ Q with the hearts taken in isolation will be either 4:3 or 3:4. Rather than look at averages over several possible splits, one should consider the mode of the splits, and initially a 3-4 split clearly represents the greatest frequency of occurrence. As play progresses the general trend is to retain the more even splits. If a split in another suit gets established, that information will affect the odds of the heart splits. It may turn out eventually that a stage is reached where the 4-3 and 3-4 splits again become equally likely, but we can’t assume that will happen. An example: with ♠ 5-3 and ♦ 2-4, the distribution ♠ 5-3; ♥ 4-3; ♦ 2-4; ♣ 2-3 has the same weight as the non-symmetric ♠ 5-3; ♥ 3-4; ♦ 2-4; ♣ 3-2. This stage will be reached rarely, as it is normal for declarer to play on clubs before diamonds.

Probability, Information, and the Opening Lead

July 21st, 2009  ~  Bob Mackinnon  ~  No Comments 

Summary

In the next few pages we demonstrate how the fundamental principle of information theory that links information to probability applies to the opening lead. That principle is expressed by the equation Information = -log (Probability). The demonstration is given in terms of possible card combinations after a low card lead against a NT contract.

Here are the basic conclusions:

1) the less probable an opening lead, the more information it provides, and

2) the greater the amount of information in the opening lead, the fewer the number of card combinations that remain to be considered.

This conclusion is simple enough and amount to commonsense when one thinks about it, provided always that one has thought about it. Nonetheless, a demonstration is desirable to bolster the statement without extensive computer simulation (the results of which may not convince the skeptical mind), so one must make some simplifying assumptions. The major assumption we make is that the lead is a low card from the defender’s longest suit. Not always true, of course, but true most of the time I venture to say. For the great majority of cases where it is true, the results follow like clockwork.

Here are the bridge rules that arise from this analysis:

1) if the opening lead is in the shortest suit jointly held by declarer’s side, the information content is relatively low, and many distributions of sides remain as possibilities;

2) If the opening lead is in the longest suit jointly held by declarer’s side, the information content is relatively high, and relatively few possible distributions of sides remain.

3) The difference between the information in one suit relative to that in another increases with the difference in the number of cards held in the suits.

As the difference between the longest suit and the shortest suit determines the number of total trumps of the deal, there is a connection between information and total trumps yet to be explored. The first part of the text is an introductory discussion of how probability applies to situations where a non-random choice is being made. It shouldn’t be necessary to establish this, but there it is. We recall the words of Louis Pasteur, ‘Analogy is not proof’, but maybe it helps in getting started.

The City Council’s Secret Ballot

Put yourself in the shoes of a local reporter keen to predict the outcome of a critical vote of your city council on a controversial development that divides 2 major factions; the Red Party has come out for it on the grounds that it will provide an immediate and much needed boost to the economy, the Blue party against it on the grounds that more stringent regulation is required in the long-term interests of the public. There are 13 councilors, 6 Reds and 6 Blues, and 1 Yellow independent, who has previously said she hadn’t yet made up her mind on which way to vote. As they arrive for the closed door meeting you have the opportunity to ask just one of the councilors how he is going to vote. It is tricky to predict the outcome of 13 votes from a sample of 1, but if you could choose just one, which one would it be? As the Yellow member represents the swing vote, it is pretty obvious that you would choose to ask her. There is much more relevant information to be had from the singular case than from the common case. Sure, in a secret session some Reds might vote No and some Blues might vote Yes, but individual preferences will be rare and may tend to balance themselves out. So the Yellow vote is the best single predictor. So it is on opening lead: a lead from a sparse suit contains more information than a lead from a plentiful suit. We shall show later how this works.

Next suppose that the interviewee is selected for you at random. One weighs the evidence of a single vote on the basis of his or her relationship with the other council members. The public statement of a compliant party member doesn’t add to what is already known and it is only rarely you will solicit an answer from the one person who represents the swing vote. Similarly, low card lead selected at random from a large number of available low cards is not going to tell declarer much about the hand as a whole. That is the principal argument for choosing a passive lead. Something that occurs against the odds, such as receiving a highly unusual opening lead from a short suit, is very informative, but by its very nature, it doesn’t occur often. If a reporter happened to ask the leader of the Reds his view, and he said on his daughter’s urging he was changing his vote to No, that would represent real news. Another way of putting it is this: the greater the surprise, the greater the information. Think of an opening lead as a message, and when you receive an unusual lead, think hard about what it is telling you.

One hang-up we have encountered in the bridge literature is the idea that an opening lead is not a random choice but a conscious choice, so not subject to the laws of probability which deal with random events. There have been caucuses for both parties and the arguments for and against have been gone over thoroughly. A reporter is not privee to the deliberations, yet is called upon to make a prediction without detailed knowledge of the pros and cons. Even with such knowledge, the reporter would be presumptuous to let his assessment sway his prediction as different priorities undoubtedly apply all around. Randomness comes from the fact that in a secret ballot some may cross party lines and that one person is uncommitted up to the last minute. It would add to the uncertainty if a councilor when asked in public didn’t answer truthfully. One must judge on scant evidence, past experience, and interrelationships.

To Assume Nothing is to Assume Something

How can nothing be something? This question for centuries has plagued philosophers between breakfast and lunch. We recall that the concept of zero was slow in gaining general acceptance and that the symbol for zero didn’t appear in the Western World until the year 967 (AD, that is). More recently philosophers have argued whether or not zero probability can exist in an infinite universe, an important point, I gather. They have no such problems after supper when they go out to play bridge and enter the finite and interdependent world of cards. With just 52 cards to deal with they can see that more of this means less of that, and vice versa. A zero score on a deal is not only possible, but probable. Nonetheless, the idea that nothing can be something may appear strange at first glance. Here is the context.

An opening lead is made, play continues, cards are revealed, yet many argue that the a priori odds still apply. By staying with the probabilities of the deal, the only action that is guaranteed to be random, they feel they have made no assumptions, yet by ignoring the evidence before them, they are assuming no information has been provided as any information obtained would necessarily affect the probabilities. Their assumption is one of total mistrust. To give such arguments some leeway, we might say the assumption is that the information provided is so unreliable as not to be trusted until a defender shows out of a suit, at which time vacant places can be adjusted on the basis of an incontrovertible fact. If one excludes the opening lead from consideration, one prefers to make a decision based on the basis of maximum uncertainty rather than be swayed by meager evidence. Variability swamps the mean. This attitude is akin to the advice that in a storm at sea one should stay with a foundering ship rather than trust a leaky lifeboat. A more fruitful way of thinking of this is that there must be information in the opening lead because the reasonable choices are few. The fact that the lead is not random is what provides the information!

I often have wondered why some have a predisposition to ignoring the evidence. Then this week I saw on television that some war veterans who have lost limbs still feel pain in their missing members. They can experience itchiness and cramps where there is neither flesh nor bone. Their brains do not accept what is apparent even to themselves. Reasoning doesn’t help, because logically they already know the facts. Recently doctors have learned that mercifully the brain can be fooled by mirrors. By showing the patient’s other hand or arm in a mirror, when a corrective action is applied to the remaining limb the brain is fooled into thinking the action has been applied to the missing one, and the symptoms are relieved. To translate this into bridge terms, irrational distrust may not go away even if one knows logically we are not being continuously deceived. The cure is to look at yourself in the mirror and ask how often you make opening leads with deceit uppermost in your mind. The next question is, how often has it paid off? Just as the best spies are those least suspected, the best deceptions are rare and disguised as normal. If one continually suspects a normal lead, on the basis of frequency of occurrence it can be said that one has found another way to lose by defying the odds.

When the dummy appears it present declarer with an incontrovertible fact in the form of the division of sides. This by itself shifts the odds, a fact hardly worth a mention in the bridge literature, but it doesn’t change the principle characteristic, which is, that the missing cards are most likely to be divided according to the most even distributions still possible. The opening lead shifts the vacant places temporarily, but a complete round to which both defenders follow with low cards in the suit maintains a balance of possibilities between the defenders’ hands. Nonetheless possibilities have been eliminated and fewer remain. That is due to an acquisition of knowledge. Now we look in detail at a common situation from what to most will be a new perspective.

The Lead from Length

The bidding has gone 1NT – 3NT and the opening lead is a low spade, and as declarer we think automatically of ‘4th highest from the longest and strongest’. Before drawing inferences let’s look at the cards in dummy and form the distribution of sides to see if, indeed, the spade lead is what we should expect. Form the dummy we calculate that the defenders hold 8 spades, 7 hearts, 6 diamonds, and 6 clubs, thus a sides of 8=7=6=5 and, indeed, spades is their most plentiful suit, so no surprise a spade was led.

There will be times when shorter suits are led. One will have formed, consciously or unconsciously, a set of prior probabilities for leads in the various suits. The reader is asked at this point to make an estimate based on experience as to how often a lead is made in each suit. Express this estimate in terms of a probability, P(suit). I assume on a tentative basis the following set for the sake of illustration: P (♠) = 0.50, P(♥) = 0.35, P(♦) = 0.10, and P(♣) = 0.05, but maybe you have a better estimate.

When a low card is led, I estimate it will be a spade half the time. A club lead would be unusual, occurring just once in 20 occasions. A heart is expected once in 3 occasions, and a diamond once in 10. The numbers could be the subject of a test using computer simulations, but let’s assume there are close enough for now. Next we consider the most likely distributions in each suit. The weights are the relative number of combinations.

As the number of cards available to the defenders decreases from spades to clubs, the probability of that suit being led decreases (the manner of this being open to question), and the number of combinations for the most likely distribution decreases. Overall there will be fewer combinations available that are attributable to the variations that lie behind these most frequent distributions. The possibilities are reduced as the number of available cards in the suit led is reduced. The fewer the possibilities, the more severe the restrictions on the properties of hidden hand, and the greater the information conveyed by a lead in that suit.

The last 2 columns are the 2 most frequent cases where clubs is the longest suit. Of low probability the last column will be quickly eliminated when the RHO follows suit. So we may conclude that a club lead on the first round establishes the full distribution of the defenders’ cards. There was an impressive amount of information in that lead.

Mathematical Models for Opening Leads

Mathematical models are based on assumptions. Simple is best, but if a model doesn’t predict accurately enough, one must question first the underlying assumptions. There is a relationship between the choice of an opening lead and the lengths of the suits held, but such a relationship observed from the outside must be of a statistical nature. In this section we study the consequences of three rules that relate suit lengths to the frequency of choices of the opening lead.

Random Rule

From a pack of 8 spades, 7 hearts, 6 diamonds and 5 clubs, draw a card, then lead a low card in that suit. The chances of a spade being led are 8 out of 26.

Rule A

a) Lead a spade if it is the longest suit, when it is equal in length to a minor, and half the time when it is of equal length with hearts;

b) lead a heart when it is the longest suit, when it is equal in length to a minor, and half the time when it is of equal length with spades;

c) lead a minor only when it is the longest suit or equal to the other minor.

A popular idea with some logic behind it is, ‘when in doubt, lead hearts’. We shall look at the consequences to information when the following rule is applied:

Rule B

a) Lead a spade if it is the longest suit, or equal in length to a minor;

b) lead a heart when it is equal in length to another suit or longer;

c) lead a minor only when it is the longest suit or equal to the other minor.

Given those rules it is an easy but tedious task to count up the number of combinations that will produce a lead in a given suit. I did so, excluding hands that contain a void, cases of low probability made lower under the circumstances of no interference. Here are the probabilities that result:

Entropy is a measure of the degree of uncertainty given that the only information transmitted is the suit denomination. The mathematical expression for entropy is the sum over the 4 suits of – P(suit) times Log [P(suit)]. The uncertainty is a maximum when all 4 suits have an equal probability of being led (0.25), so the maximum entropy possible is 0.600. A random choice from cards whose composition restricted to 8=7=6=5 is just slightly less than the theoretical maximum. If a major suit lead is much more likely than a major suit lead, uncertainty is reduced, and there is not much difference in that regard between Rules A and B. Where Rule B gains an advantage is that the probabilities of a heart lead and a spade lead are equals and provide the same amount of information, as there is (near) equality in the number of combinations associated with each play. Thus, there is an information-theoretic reason for adopting that strategy on these most common choices (86%), the same reason that governs Restricted Choice and tells defenders to follow to declarer’s suit plays with low cards chosen at random.

What Should Declarers Assume?

It is obvious that the selection of an opening lead is a more complex operation that the application of Rule B. It is also obvious that the opening lead is not chosen at random. Our experience amounts to a statistical survey of leads made against us and the evidence points to probabilities not far different from Rules A and B. The analysis above may serve to direct our attention to implications of which we were only vaguely aware. If so, our models may have served a purpose, and may lead to further refinements. At best, Rule B should be adopted as a working assumption until further information becomes available. The leading candidates for the distribution of sides are the most likely distributions listed above for each suit.

One can observe the effect without discerning the cause, so why not adopt an ad hoc model for the sake of convenience? If one assumes nothing, but notes the division of sides, the amount of information one can gather from an opening lead is near minimum. Yet our experience tells us that Rule B better reflects our observations overall. One card will not tell everything, but it tells something, and that is the best attitude with which to start. Hope to gather more information before a critical decision is required.

Another Division of Sides

The 8-7-6-5 division is the most common, so it is easier to gauge than the rarer types. The same principles apply throughout, so one can calculate the information available under the different divisions and there should be no surprises. A computer program is desirable for going through the process, but here is one more set of results got by hand calculation, this time for a sides of 8=6=6=6. Declarer’s division is 5=7=7=7.

Although the same number of spades is held as in the previous example, there is less of an inclination to lead a major suit, more inclination to lead a minor suit, so the overall effect is to increase uncertainty, as can be seen by a comparison of the entropies involved.

The Message

Think of the opening lead as a message that relates to the operation under which it has been selected. If the lead is to be considered a random choice from a pack of cards jointly held by the defenders, it has the status of the first low card dealt to the LHO. Knowing the first card tells us nothing about the relationships between the cards that will be dealt subsequently, and very little that a declarer doesn’t know already. On the other hand, there are many practical advantages to choosing a lead from the longest suit held, so it is reasonable to assume the message most often relates, in some mysterious way, to the relative lengths of the suits held by the opening leader. Start there.