Probability, Information, and the Opening Lead

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.

	♠	♥	♦	♣	♣
	♠4 – 4	♠3 – 5	♠3 – 5	♠3 – 5	♠3 – 5
	♥3 – 4	♥4 – 3	♥3 – 4	♥3 – 4	♥3 – 4
	♦3 – 3	♦3 – 3	♦4 – 2	♦3 – 3	♦2 – 4
	♣3 – 2	♣3 – 2	♣3 – 2	♣4 – 1	♣5 – 0
Weights	100	80	60	40	6

 

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:

Suit Led	Random	Rule A	Rule B	My Guess
Spades	32%	49%	43%	50%
Hearts	27%	37%	43%	35%
Diamonds	23%	11%	11%	10%
Clubs	19%	3%	3%	5%
Entropy	0.595	0.463	0.466	0.475

 

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.

Suit Led	Random	Rule A	Rule B
Spades	31%	56%	51%
Hearts	23%	21%	25%
Diamonds	23%	12%	12%
Clubs	23%	12%	12%
Entropy	0.598	0.505	0.521

 

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.