Bob Mackinnon

Mathematics and Feelings

Time again for New Year’s resolutions. No need to dream up new ones – I’ll just dust off the old ones which are nearly as good as new. Otherwise, it’s a time for reflection, and we note that mathematics hasn’t yet gained the prominence it deserves in the analysis of the game of bridge. Which is odd, for as a friend said to me, ‘Isn’t bridge just all about mathematics?’ Well, yes and no, for there is a lot emotion involved, but passions are wayward, and emotions require guidance and restraint. From the very first, learning to play bridge is learning to control our mental processes and channel in a useful way our God-given desire to flourish. Generally, the true aim of education is to produce a well-ordered mind open to change. The function of math is to put matters in order by using numbers. Once we understand through the simple application of statistics and probability something of how and why things work, we can broaden our horizons and increase our potential for making further progress in the New Year.

Turning our thoughts to love, isn’t love best expressed in a song? Yet a love song written down on a sheet of music is no more than a collection of mathematically related symbols. Mathematics expresses the love, but it needs a person to perform the music, be it well or poorly. Yes, one can perform brilliantly without being able to read a note, Bix Beiderbecke comes immediately to mind, just as one can play bridge well without a formal knowledge of mathematics, but is that satisfactory? What if composers couldn’t write down their music so that orchestras couldn’t play it? Music would lose its structure, and that appears to be the current trend when a receptive computer can transform anyone’s self-expressive grunts and howls into the latest, forgettable hit where the only love on display is immature self-love. Beauty is characterized by form.

Over Christmas I have re-read parts of the classic work, On Bidding, by Albert Morehead, revised by Alan Truscott and Philip Alder, Englishmen all, which gives an overview of the Standard American 5-card majors approach in the late 20th century. Page 1 introduces the 4-3-2-1 point-count method of hand evaluation, but it isn’t until page 333 that we get to the topic of mathematics per se. In between there is a practical discussion of the collective wisdom embodied in various bidding techniques that arise during a constructive auction, but not much is expressed in terms of statistics and probabilities.

At long last, on page 334, the authors take the space to note that mathematical models are crude and the required parameters are so inexact that one might as well not use the term ‘mathematics’ at all when describing the basis for decision making at the table, rather one should describe the process as ‘figuring’, implying a rough ‘guessimate’ made partly on counting, partly on experience, and partly on table feel. Even if the table estimates are inaccurate, one cannot conclude that mathematics is not applicable. Although each deal taken in isolation may appear to be a random event, there are underlying principles to which events with a random component conform. Probability is the science of uncertainty. After the game one wants to understand why one has been successful and whether one should continue in the same manner in subsequent, similar situations. Is a good result merely the consequence of a good guess, or can one hope to develop a consistent winning approach based on probability? Yes, it is possible, even though one can’t win every time.

Putting Numbers to Feelings

Often when playing a hand one has a vague feeling about what others might be doing with the same cards. This is easier to gauge in matchpoint play where one is dealing with averages over an entire field, but even so scores over a dozen tables may encompass a wide variation. During out latest game my partner and I scored 1100 in a part score deal. It can be dangerous to put too much faith in one’s feelings on the matter, yet, the feelings exist, so can one make use of it in some consistent manner? In our mathematical analysis of simple bidding decisions we used 2 probabilities: PM, the probability that a higher scoring contract would make, and PB, the probability the opponents would bid it also. Let us consider PB first. It is not sensible to assume PB equals 0 or 1, as that expresses maximum certainty that the unseen opponents will do what you expect. Our feelings might be: 1) probably they will bid it, 2) probably they won’t, and 3) I’m neutral. We might assign probabilities, PB, as follows: 1) 67%, 2) 33% and 3) 50%, representing the odds 2:1, 1:2, and 1:1. The numbers are not exact, merely representative of an average over a fair range of probabilities. 50% roughly represents a range between 2/5 and 5/9.

Judging by the predictive powers of BBO commentators who can see all 4 hands, it would be sensible in a team game to assume maximum uncertainty in this regard, that is, PB equals ½. Thus, if a player claims to ignore the possibilities concerning actions of the opponents, and makes his decisions independent of what others may be doing, he is in essence making such an assumption. Unconcern is a theoretically acceptable attitude, in the sense that the expected difference in score between bidding on and not bidding on is independent of PB, however, the result from each action varies with PB, as we showed in a recent blog. If one feels strongly, one can always act with predictable variability.

The most important criterion with regard to bidding on or not is the probability, PM, that the higher scoring contract will make. Here we may pride ourselves in being able to distinguish between several possibilities. After an informative auction but before seeing dummy most would be able to judge on a 5-level basis: sure (75%) confident (67%), doubtful (50%), wishful (40%), and apprehensive (25%). The 50% level is the level at which it pays to take a chance and bid on, hoping for some extra help from the defenders, that is to say, we think on the basis of the bidding the odds are very close, but the conditions of the game dictate we bid games that are no worse than a finesse away from making. When the odds are close to 50-50 it doesn’t pay to worry too much over what to do, as the results are likely to be largely random. At matchpoints one may decide for safety’s sake to do what most of the field is expected to do, thus assuring oneself of ‘company’ if the decision is wrong. That is, one estimates PB to be much greater than ½ and decides accordingly. That is why we often bid 1NT – 3NT, without much concern.

The Long-term and the Short-term

Having assigned numbers to feelings, one is in a position to apply the methodology of mathematics in order to draw logical conclusions expressed in terms of probabilities. A principal result is the average gain and loss that results from a particular decision assuming values for PM and PB. Some argue that they are not concerned with the long-term gains, only with the probable outcome with the hand before them under the current conditions. This expresses the short-term view. Some successful players are known for their ‘table feel’, an ability to play against the odds and come out a winner, based their intuition or skill in reading the psychological tea leaves. This ability should not be dismissed, but then it should not be the total basis for decision making. At the very least the mathematical results are there to be used as guidelines. There may be hidden factors which are not well reflected in the usual models from which the mathematical results are derived, but the task of the analyst is to search out these factors, put numbers to them, and improve the numerical models by including them in future.

One should not be reliant psychologically on short-term results, as there is a variability to random events that may point to trends which are not directly attributable to cause-and-effect. Sometimes one gets a bad result or two for no good reason, and such occurrences are natural to the process. It is better if one takes the long-term view. One may sit back and hope to benefit from an opponent’s mistake, but the mistakes are more likely to come if the opponents feel under pressure. If one gets an early gift, expect that sooner or later it will be balanced with a fix that is beyond one’s control. If partner makes a costly play, don’t despair – assume that later he will make a brilliant one.

Part of the long-term strategy at teams is to continue to play your game consistently while expecting random gains and losses, and to continue to do so even towards the end of the match with just a few boards remaining. Last minute desperate measures are just that – desperate. Playing conservatively to preserve a lead can be fatal. It is better to adopt a comfortable strategy and follow it through, treating the last board with the same approach as the first one. Don’t assume you know what an unseen opponent is doing. This is consistent with playing for the expected result on each board to the limit of one’s ability to judge the prevailing conditions and to react to them.

Following the Rules

Bidding rules are based on the 4-3-2-1 point-count, and players are accustomed to following the rules in the hope that the final contract will produce the desired result. Of course, the rules have a basis in probability of which most are unaware. Let’s take the simple example of the auction 1NT – 3NT. Partner opens 1NT (15-17 HCP) and you hold Q3 983 KQ74 AQT6. Probably you bid 3NT without much thought and are not greatly disturbed if 6 would have been a better contract. The actions appear to be automatic, and an entire field may be in 3NT.

On the other hand there may be hands on which one bids 3NT on 28 HCP and it goes down on normal defence because there is just not enough combined power in a particular suit. No one is overly disturbed by getting to the obviously wrong contract. Following the rules becomes the primary aim of the average player. However, the rules themselves must be such as to reflect a wide range of probable outcomes. Most of the time 3NT will be the right contract, but sometimes it will fail and sometimes slam is makeable.

The conclusion in On Bidding is, roughly speaking, that one should bid higher scoring contracts that have a 50% chance of success. That translates into bidding 3NT if it depends at most on a finesse. A better way of thinking is that one should bid game if the probability of its making is no worse that a coin-toss, thus removing the focus on finesses. Even better, one bids games for which there is maximum uncertainty as to whether it makes or not. It follows that a bidding system that tells a player to bid game when holding such-and-such number of points must include within its boundary of definition games that are maximally uncertain to produce the best result. That is to say, if 3NT always makes, the rule are too constrictive; sometimes it must fail in order to justify the requirements of the game. At the other end of the scale, 6 makes with this combination:


KJ102 Q3 1NT 3NT 1 1
A2 983 Pass 1 2 * (FSF)
AJ2 KQ74 3 4
KJ53 AQ106 4 4NT
5 6


One might argue that the auction that arrives at 6 is based on better hand evaluation that the blind auction, but there are risks involved in a revealing auction that stops short of slam, apart from the increased probability of a bidding error because two players are making decisions instead of one. Declarers in 3NT will score well without a heart lead (12 tricks) and may score poorly with it (9 tricks). Many players are reluctant to make revealing bids that may guide the opening leader, so they deliberately bid blindly relying on the probability of outcomes to support their decision. The probability in an average filed that someone will bid 6 is low. This is how bad bidding is self-propagated and bad systems are justified.

According to Morehead et al the basic tenet of bidding theory is that ‘each partner must be permitted to speak for himself’, which indicates to me that each player must be permitted to make decisions either directly or through judgmental selection of the bids they choose. Such a concept is popular in North America where individual rights are often the excuse for egotistical misadventures. This is contrary to the principle of captaincy, to say the least. My feeling is that asking bids, like 4NT Blackwood, are a very useful device in which one partner asks directly and the other answers truthfully thus ensuring an accurate one-way exchange of information without the usual spin. The time for judgment comes later. To put it another way, the right of the individual to decide is put aside temporarily in the cause of the common good with the assurance that one’s chance to shine will be forthcoming later in a prescribed manner.

A bidding scheme in which either partner can decide the final contract must of necessity provide a sufficient flow of information in both directions. This exchange may prove beneficial to the defenders, hence there is a reluctance among ‘natural’ bidders to bid in a truly informative manner. I was guilty of this when partner opened 1 and rebid 2 after my response of 1 . What is your bid holding my 20 HCPs: KT9 KQJ8 AKQ65 Q ? Some players bid 4NT (RKCB), found partner with 3 key cards, then bid 7NT, doubled, going down 1 on a club lead. Partner held: AJ A3 743 K76542. I bid 6NT directly and made 7 by playing a club towards the hidden hand, naturally ducked by my RHO. This made up for the previous hand where I lazily jump bid 4NT RKCB on 18 HCPs and stopped in 5 with 6NT making. So, just another average round at the club!


Bobby WolffJanuary 9th, 2011 at 1:14 pm

Hi Bob,

Bingo! Recently, I have had presented to me the combination of a good friend, Blair Fedder, gifting me with a book, titled: Innumeracy, mathematical illiteracy and its consequences, by Mr. John Allen Paulos, published simultaneously in 1988, by both Canada (Collins Publishers, Toronto) and printed in the USA. It is, into its 5th printing and gives as its pedigree and acknowledgment the following description, written by Douglas Hofstadter, the author of Godel, Escher, Bach:

Innumeracy–the mathematical counterpart of illiteracy–is a disease that has ravaged our technological society. To combat this terrible disease, John Allen Paulos has concocted the perfect vaccine: this book, which is in many ways better than an entire high school math education! Our society would be unimaginably different if the average person truly understood the ideas in this important little book. It is probably hopelessly optimistic to dream this way, but I hope that Innumeracy migh help launch a revolution in math education that would do for Innumeracy what Sabin and Salk did for polio.

While I (and my wonderful wife, Judy) have our hands full promoting bridge playing on a world wide stage, I am not advocating, nor attempting to crusade, for everyone to stop what they are doing and become numerate, but rather it may be helpful and certainly bridge topical to suddenly have others realize what they are missing.

In short, this book has helped me understand, after many unknowing years, why some take to and become immediately addicted to the joys of our game, while most others, though at the very least, just as bright, resourceful, and industrious, fail to get out of the batter’s box in their real understanding of how mathematics (sometimes better described as numbers) plays such an integral part in one’s rise to be extraordinary as a bridge player.

That book combined with your enthusiastic approach to discussing arithmetical application in the game itself is probably all that is necessary to unlock a mystery which has always confused me, and no doubt, others, as to why some of the brightest people in the world have so much trouble at the bridge table. “Know the truth and the truth shall make ye free”.

Likely, the interest I have always had in searching out the answer to that mysterious question is not shared by everyone, but, at least to me, finally I think I understand why and by doing so I will be a much happier and relaxed person.

Thank you, Bob (and Blair), for being part of the experiment which, for whatever reason, has finally given me, after so many years, an incredible inner peace.

Bob MJanuary 9th, 2011 at 10:06 pm

Excuse my blushes. I feel that geniuses show us how to play, and if we look closely at their actions we can usually find the underlying principles that guide them. The mathematical theory has to follow reality closely, so if the math comes out wrong, it is probably the underlying assumptions that need to be re-examined so as to conform to excelllence, such as you have demonstrated over the years.

One critic found some of my examples of probability at work to be rather uninspiring as they just followed normal, good practice. Thus I think he missed the point that probability and good practice should arrive at the same decisions.

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