Main Study 8C13 by Richard Pavlicek
Doubletons are popular leads for a variety of reasons. With an honor sequence they offer great attacking ability without risk, and from weak holdings they offer a passive opportunity, rather than risk leading from an honor in another suit. At suit contracts theres also the fair chance of a third-round ruff. Out of curiosity, I checked my major-event database, and the opening lead was a doubleton 11,105 times not exactly overwhelming out of 79,000+ contracts, but certainly worth a study. So lets begin.
There are 78 possible doubletons, from A-K through 3-2, and each can occur in 176,685 specific layouts. For this study I will exclude layouts with a singleton, void or 6+ card suit, not only because they are rare but because the bidding would usually divulge the extreme distribution, giving the leader special knowledge, so the situation would not fit the general scenario sought to be emulated here. This reduces the field to 103,950 common layouts per doubleton, or 8,108,100 in all. The following table shows the breakdown by distribution:
|2 (4-4-3)||11c4 × 7c4 = 11550||3||34650||275|
|2 (5-3-3)||11c5 × 6c3 = 9240||3||27720||198|
|2 (5-4-2)||11c5 × 6c4 = 6930||6||41580||135|
The Ratio column shows the relative probability of a specific layout in that distribution group. This is used as a weight factor to obtain true averages and percents, i.e., 4-4-3 distributions (in any order) are the most common, so their results get the greatest weight.
The basic calculation was to determine the number of tricks won by the defense on each of the 8,108,100 layouts in four cases: (1) West leading high, (2) West leading low, (3) Partner leading first to best advantage, and (4) Declarer leading entirely. In Cases 1-3, declarer was required to lead after the first trick, and in all cases declarer could lead from either hand to best advantage. The 32+ million calculations are not something I would attempt (or live to complete) by hand, but an easy task by computer with the right programming.
For each layout, tricks won by the defense are the difference between the maximum declarer could win and the actual result. For example, if declarer has four cards in dummy and three in hand, the most he could win is four; so if declarer wins three, the defense gets one. Note that declarers maximum varies from 3-5 depending on the distribution.
Results are based on double-dummy play (after Wests forced lead in Cases 1 and 2) which of course is not necessarily what would occur in practice. Nonetheless, other studies have shown a close correlation, with differences tending to even out, so the evidence is generally valid. Correlation should be extremely close after a doubleton lead, because the layout is often an open book to both sides.
For each doubleton, the average tricks won is then calculated in each of the four cases. Weight factors (275, 198, 135) are applied for each distribution group to make this a true average based on frequency of occurrence. Trick average is most relevant in comparing the lead cases within each doubleton, rather than one doubleton to another, since its generally proportional to the strength of the doubleton.
Doubleton leads are notorious for losing tricks, especially when containing an honor not in sequence. For each doubleton, a running count is kept of the times a trick is lost in four scenarios: (1) High lead versus partner leading first, (2) Low lead versus partner leading first, (3) High lead versus declarer leading entirely, and (4) Low lead versus declarer leading entirely.
It is interesting to note that the count of times a trick is lost is identical to the number of tricks lost. That is, assuming declarer has the needed communication (a premise for this study) it is impossible to lose more than one trick with a single lead. Over the years I have often questioned this fact from my own or partners aberrations, but an exhaustive search has proved it to be true.
The counts for each distribution group are then weighted (same as trick averages) to obtain the true percent of the time that a trick is lost in leading the doubleton.
Leading a suit can never gain versus declarer leading entirely. In other words, any lead by declarer can only help the defense, so there is no way to gain a trick by leading it yourself. This is true regardless of length (not just for doubletons) from the positional nature of all card play. The side leading a suit can only break even or lose compared to the result of the other side leading.
A doubleton lead can gain a trick in three scenarios: (1) High lead versus partner leading first, (2) Low lead versus partner leading first, and (3) High lead versus low lead. The first two should be obvious, as it is often necessary to lead partners suit, rather than him starting it first.
The counts for each distribution group are weighted (as previously) to obtain the true percent of the time that a trick is gained in leading.
A trend (abomination?) of modern times is the second and fourth philosophy, which dictates leading low from doubletons. (Honor-doubletons are excluded by most advocates.) This practice is inferior. For example:
|Any contract||A Q 8 4|
|9 5||K 10 6 3 2|
|West leads||J 7|
If West leads the standard nine, declarer can win only two tricks (East ducks to preserve his tenace); but if West leads the five, declarer is gifted a third trick simply by playing low from dummy. In fact, leading the five is the only way to blow a trick. Even East can start the suit by leading into dummys A-Q, and declarer can win only two tricks.
This study revealed a remarkable statistic: Not once in all the suit layouts did leading high lose a trick versus leading low, while about 13,000 tricks were lost the opposite way. This would not hold true if North or South could hold a singleton (excluded in this study), e.g., leading high from K-x could lose to a blank ace; but even without restrictions, leading high is still a big favorite.
The following table shows the complete results of all doubleton leads. Tinted rows indicate touching cards, which have no relevance in the high-versus-low issues, and no trick could be lost leading either card; hence all zeros in the Loss columns.
The table clearly shows the worst doubleton leads to be from the king, with K-J topping the list. (Many would have guessed A-Q to be worst, but all ace leads are safer than kings.) As the cards become lower, the risk is reduced, with the notable exception that nearly touching cards are safer than lesser holdings within the same high-card group; e.g., J-9 is safer than any other J-x. Eventually the risk vanishes to complete safety in low spot cards.
|Trick Average||Loss Percent||Gain Percent|
|High||Low||Partner||Declarer||High vs P||Low vs P||High vs D||Low vs D||High vs P||Low vs P||High vs L|
The table provokes some curiosities: Why is leading from Q-8 more risky than Q-9 or Q-7? Or how can you lose a trick by leading from 7-5? Only The Shadow knows!
© 2018 Richard Pavlicek