Virtual Reality and Opening Leads

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.

	Random	Rule	Rule B
Spades	27%	33%	39%
Hearts	27%	33%	39%
Diamonds	23%	17%	11%
Clubs	23%	17%	11%

 

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.

I	II	III	IV	V	VI
♠ 3 – 4	♠ 4 – 3	♠ 4 – 3	♠ 3 – 4	♠ 3 – 4	♠ 3 – 4
♥ 3 – 4	♥ 3 – 4	♥ 2 – 5	♥ 2 – 5	♥ 3 – 4	♥ 2 – 5
♦ 4 – 2	♦ 4 – 2	♦ 4 – 2	♦ 4 – 2	♦ 5 – 1	♦ 5 – 1
♣ 3 – 3	♣ 2 – 4	♣ 3 – 3	♣ 3 – 3	♣ 2 – 4	♣ 3 – 3
75	56	45	34	22	18

 

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:

Diamonds	4 – 2	5 – 1		Average length 4.38
	62%	38%

 

Spades	5 – 2	4 – 3	3 – 4	2 – 5	1 – 6		Spade Left	42%
(Hearts)	<1%	25%	48%	21%	4%		Spade Right	58%
Clubs	5 – 1	4 – 2	3 – 3	2 – 4	1 – 5		Club Left	45%
	0%	16%	46%	30%	8%		Club Right	55%

 

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:

Diamonds	4 – 2	5 – 1		Average length 4.55
	45%	55%

 

Spades	5 – 2	4 – 3	3 – 4	2 – 5	1 – 6		Spade Left	40%
(Hearts)	0%	12%	60%	24%	4%		Spade Right	60%
Clubs	5 – 1	4 – 2	3 – 3	2 – 4	1 – 5		Club Left	48%
	0%	21%	51%	21%	7%		Club Right	52%

 

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.

Virtual Split	Vacant Places	Probability	Vacant Place Differential
3 – 3	10 – 10	50.0 – 50.0	0
3.5 – 2.5	9.5 – 10.5	47.5 – 52.5	1
4 – 2	9 – 11	45.0 – 55.0	2
4.5 – 1.5	8.5 – 11.5	42.5 – 57.5	3
5 – 1	8 – 12	40.0 – 60.0	4

 

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.

I	II	III	IV	V	VI
♠ 4 – 4	♠ 5 – 3	♠ 4 – 4	♠ 5 – 3	♠ 6 – 2	♠ 5 – 3
♥ 3 – 3	♥ 2 – 4	♥ 4 – 2	♥ 4 – 2	♥ 2 – 4	♥ 1 – 5
♦ 3 – 3	♦ 3 – 3	♦ 3 – 3	♦ 2 – 4	♦ 2 – 4	♦ 4 – 2
♣ 3 – 3	♣ 3 – 3	♣ 2 – 4	♣ 2 – 4	♣ 3 – 3	♣ 3 – 3
100	60	56	34	22	18

 

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.

Spades	4 – 4		5 – 3		6 – 2		Average Length	4.83
	34%		49%		17%
Others	5 – 1	4 – 2	3 – 3	2 – 4	1 – 5		Club Left	45%
	<1%	18%	43%	30%	9%		Club Right	55%

 

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%).