Study 8C13   Main

Doubleton Leads

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 there’s 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 let’s 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:

Suit Pattern	Combinations	Ways	Specific	Ratio
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
Totals	27720	12	103950	608

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.

Trick Average

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 declarer’s maximum varies from 3-5 depending on the distribution.

Results are based on double-dummy play (after West’s 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 it’s generally proportional to the strength of the doubleton.

Loss Percent

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 partner’s 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.

Gain Percent

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 partner’s 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.

Low from Doubletons

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 dummy’s 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.

Result Table

The following table summarizes the results of all doubleton leads. Columns 2-5 show the trick average if led high, if led low, if partner leads, or if declarer leads. Columns 6-9 show the loss percent of high vs. partner leading, low vs. partner leading, high vs. declarer leading, and low vs. declarer leading. Columns 10-12 show the gain percent of high vs. partner leading, low vs. partner leading, and high vs. low. Tinted rows indicate touching cards, which of course have no difference in high-versus-low issues, and no trick could ever be lost by leading; 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.

Holding	High Avg	Low Avg	Partner	Declarer	H loss P	L loss P	H loss D	L loss D	H gain P	L gain P	H gain L
A-K	2.5803	2.5803	2.5803	2.5803	0	0	0	0	0	0	0
A-Q	2.1590	2.1590	2.3636	2.3636	20.4663	20.4663	20.4663	20.4663	0	0	0
A-J	1.9780	1.9780	2.1397	2.1397	16.1737	16.1737	16.1737	16.1737	0	0	0
A-10	1.8906	1.8906	2.0214	2.0486	13.0840	13.0840	15.7960	15.7960	0	0	0
A-9	1.8599	1.8599	1.9349	1.9775	7.5093	7.5093	11.7626	11.7626	0	0	0
A-8	1.8511	1.8511	1.8950	1.9424	4.3916	4.3916	9.1297	9.1297	0	0	0
A-7	1.8511	1.8511	1.8810	1.9166	2.9967	2.9967	6.5565	6.5565	0	0	0
A-6	1.8511	1.8511	1.8785	1.9070	2.7393	2.7393	5.5939	5.5939	0	0	0
A-5	1.8511	1.8511	1.8785	1.9022	2.7393	2.7393	5.1154	5.1154	0	0	0
A-4	1.8511	1.8511	1.8785	1.9022	2.7393	2.7393	5.1154	5.1154	0	0	0
A-3	1.8511	1.8511	1.8785	1.9022	2.7393	2.7393	5.1154	5.1154	0	0	0
A-2	1.8511	1.8511	1.8785	1.9022	2.7393	2.7393	5.1154	5.1154	0	0	0
K-Q	2.1590	2.1590	2.1590	2.1590	0	0	0	0	0	0	0
K-J	1.7334	1.7334	2.0194	2.0194	28.5942	28.5942	28.5942	28.5942	0	0	0
K-10	1.5829	1.5829	1.8554	1.8581	27.2558	27.2558	27.5175	27.5175	0	0	0
K-9	1.5164	1.5164	1.7536	1.7802	23.7137	23.7137	26.3752	26.3752	0	0	0
K-8	1.4922	1.4922	1.6971	1.7436	20.4929	20.4929	25.1396	25.1396	0	0	0
K-7	1.4866	1.4866	1.6712	1.7240	18.4592	18.4592	23.7381	23.7381	0	0	0
K-6	1.4866	1.4866	1.6635	1.7171	17.6897	17.6897	23.0430	23.0430	0	0	0
K-5	1.4866	1.4866	1.6621	1.7148	17.5516	17.5516	22.8175	22.8175	0	0	0
K-4	1.4866	1.4866	1.6621	1.7148	17.5516	17.5516	22.8175	22.8175	0	0	0
K-3	1.4866	1.4866	1.6621	1.7148	17.5516	17.5516	22.8175	22.8175	0	0	0
K-2	1.4866	1.4866	1.6621	1.7148	17.5516	17.5516	22.8175	22.8175	0	0	0
Q-J	1.7239	1.7239	1.6883	1.7239	0	0	0	0	3.5608	3.5608	0
Q-10	1.4718	1.4718	1.6084	1.6408	16.4047	16.4047	16.9020	16.9020	2.7468	2.7468	0
Q-9	1.3636	1.3631	1.5104	1.5539	16.0944	16.1441	19.0249	19.0746	1.4191	1.4191	0.0497
Q-8	1.3236	1.3221	1.4474	1.5182	13.4593	13.6072	19.4672	19.6151	1.0715	1.0715	0.1479
Q-7	1.3104	1.3086	1.4200	1.4994	11.9333	12.1100	18.9062	19.0829	0.9652	0.9652	0.1767
Q-6	1.3088	1.3069	1.4090	1.4934	10.9663	11.1587	18.4550	18.6474	0.9522	0.9522	0.1924
Q-5	1.3088	1.3069	1.4066	1.4917	10.7320	10.9244	18.2912	18.4836	0.9522	0.9522	0.1924
Q-4	1.3088	1.3069	1.4066	1.4917	10.7320	10.9244	18.2912	18.4836	0.9522	0.9522	0.1924
Q-3	1.3088	1.3069	1.4066	1.4917	10.7320	10.9244	18.2912	18.4836	0.9522	0.9522	0.1924
Q-2	1.3088	1.3069	1.4066	1.4917	10.7320	10.9244	18.2912	18.4836	0.9522	0.9522	0.1924
J-10	1.4480	1.4480	1.3492	1.4480	0	0	0	0	9.8761	9.8761	0
J-9	1.2817	1.2812	1.3106	1.4167	9.6519	9.7016	13.4983	13.5480	6.7613	6.7613	0.0497
J-8	1.2217	1.2196	1.2635	1.3813	8.6132	8.8220	15.9626	16.1714	4.4292	4.4292	0.2087
J-7	1.2002	1.1968	1.2349	1.3656	7.3862	7.7324	16.5358	16.8820	3.9161	3.9161	0.3462
J-6	1.1960	1.1922	1.2245	1.3600	6.6286	7.0140	16.3924	16.7778	3.7848	3.7848	0.3854
J-5	1.1956	1.1916	1.2219	1.3584	6.4205	6.8138	16.2878	16.6811	3.7848	3.7848	0.3933
J-4	1.1956	1.1916	1.2219	1.3584	6.4205	6.8138	16.2878	16.6811	3.7848	3.7848	0.3933
J-3	1.1956	1.1916	1.2219	1.3584	6.4205	6.8138	16.2878	16.6811	3.7848	3.7848	0.3933
J-2	1.1956	1.1916	1.2219	1.3584	6.4205	6.8138	16.2878	16.6811	3.7848	3.7848	0.3933
10-9	1.2599	1.2599	1.1169	1.2599	0	0	0	0	14.2997	14.2997	0
10-8	1.1738	1.1722	1.1047	1.2462	3.3995	3.5637	7.2338	7.3980	10.3110	10.3110	0.1642
10-7	1.1432	1.1399	1.0877	1.2307	2.9723	3.3053	8.7467	9.0798	8.5265	8.5265	0.3331
10-6	1.1356	1.1314	1.0788	1.2258	2.3688	2.7883	9.0145	9.4340	8.0499	8.0499	0.4194
10-5	1.1345	1.1301	1.0763	1.2244	2.2025	2.6376	8.9941	9.4293	8.0207	8.0207	0.4352
10-4	1.1345	1.1301	1.0763	1.2244	2.2025	2.6376	8.9941	9.4293	8.0207	8.0207	0.4352
10-3	1.1345	1.1301	1.0763	1.2244	2.2025	2.6376	8.9941	9.4293	8.0207	8.0207	0.4352
10-2	1.1345	1.1301	1.0763	1.2244	2.2025	2.6376	8.9941	9.4293	8.0207	8.0207	0.4352
9-8	1.1597	1.1597	1.0070	1.1597	0	0	0	0	15.2669	15.2669	0
9-7	1.1169	1.1154	1.0035	1.1560	1.0995	1.2447	3.9134	4.0586	12.4346	12.4346	0.1453
9-6	1.1056	1.1032	0.9989	1.1516	0.6919	0.9353	4.6039	4.8473	11.3648	11.3648	0.2434
9-5	1.1035	1.1009	0.9966	1.1506	0.5609	0.8200	4.7168	4.9759	11.2435	11.2435	0.2591
9-4	1.1035	1.1009	0.9966	1.1506	0.5609	0.8200	4.7168	4.9759	11.2435	11.2435	0.2591
9-3	1.1035	1.1009	0.9966	1.1506	0.5609	0.8200	4.7168	4.9759	11.2435	11.2435	0.2591
9-2	1.1035	1.1009	0.9966	1.1506	0.5609	0.8200	4.7168	4.9759	11.2435	11.2435	0.2591
8-7	1.1093	1.1093	0.9600	1.1093	0	0	0	0	14.9260	14.9260	0
8-6	1.0934	1.0927	0.9600	1.1085	0	0.0707	1.5117	1.5823	13.3397	13.3397	0.0707
8-5	1.0901	1.0893	0.9600	1.1075	0	0.0825	1.7344	1.8169	13.0111	13.0111	0.0825
8-4	1.0901	1.0893	0.9600	1.1075	0	0.0825	1.7344	1.8169	13.0111	13.0111	0.0825
8-3	1.0901	1.0893	0.9600	1.1075	0	0.0825	1.7344	1.8169	13.0111	13.0111	0.0825
8-2	1.0901	1.0893	0.9600	1.1075	0	0.0825	1.7344	1.8169	13.0111	13.0111	0.0825
7-6	1.0912	1.0912	0.9495	1.0912	0	0	0	0	14.1715	14.1715	0
7-5	1.0864	1.0864	0.9495	1.0912	0	0	0.4785	0.4785	13.6930	13.6930	0
7-4	1.0864	1.0864	0.9495	1.0912	0	0	0.4785	0.4785	13.6930	13.6930	0
7-3	1.0864	1.0864	0.9495	1.0912	0	0	0.4785	0.4785	13.6930	13.6930	0
7-2	1.0864	1.0864	0.9495	1.0912	0	0	0.4785	0.4785	13.6930	13.6930	0
6-5	1.0864	1.0864	0.9486	1.0864	0	0	0	0	13.7836	13.7836	0
6-4	1.0864	1.0864	0.9486	1.0864	0	0	0	0	13.7836	13.7836	0
6-3	1.0864	1.0864	0.9486	1.0864	0	0	0	0	13.7836	13.7836	0
6-2	1.0864	1.0864	0.9486	1.0864	0	0	0	0	13.7836	13.7836	0
5-4	1.0864	1.0864	0.9486	1.0864	0	0	0	0	13.7836	13.7836	0
5-3	1.0864	1.0864	0.9486	1.0864	0	0	0	0	13.7836	13.7836	0
5-2	1.0864	1.0864	0.9486	1.0864	0	0	0	0	13.7836	13.7836	0
4-3	1.0864	1.0864	0.9486	1.0864	0	0	0	0	13.7836	13.7836	0
4-2	1.0864	1.0864	0.9486	1.0864	0	0	0	0	13.7836	13.7836	0
3-2	1.0864	1.0864	0.9486	1.0864	0	0	0	0	13.7836	13.7836	0

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!

Study 8C13   Main

Top   Doubleton Leads

© 2018 Richard Pavlicek