Study 8C01   Main

Lead Safety Analysis

by Richard Pavlicek

[This 2008 study is superseded by Leading Authority which is far more comprehensive.]

As a member of the Consumer Protection Agency, I have discovered that unleaded gasoline is not what it claims to be. Spectroscopic analysis of samples from 48 states reveals an average lead content of 1.3 mg/liter, which is 88 times the acceptable level. Further, I’m convinced this is a political conspiracy, as the condition did not exist before Obama. A full investigation is under way, from the major oil companies all the way to Obama’s mama. No, wait! Never mind.

This study is about bridge, or more specifically an analysis of the safety of various leads. Should you lead from a queen, or from a king? With a two-card sequence should you lead an honor, or low? Is there any merit in advice such as “Never lead from a jack”? These and other questions have been debated for years, but I’ve never seen any hard theoretical evidence. This study will attempt to produce some, based on an exhaustive analysis of card combinations.

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The Experiment

Assume West has a four-card spade suit, and the suit splits in common fashion around the table; i.e., 4-3-3-3 (one way) or 4-4-3-2 (six ways). This comprises over 6 million combinations of the 13 spades. To facilitate the study and reduce the playing field, I decided to assign West the 2 and distribute only 12 spades, which produces 2+ million combinations — still a formidable number but potentially within grasp.

Each of the 2+ million combinations must be analyzed three ways: (1) West leads fourth-best, (2) West leads top of a sequence if he has one, and (3) declarer leads the suit first. The last scenario is complex, because declarer must be able to start spades from either hand, and lead whichever card is to his advantage. Further, for a true comparison, declarer must be required to lead spades; else he could pursue other means when appropriate. For example, it wouldn’t benefit this study if declarer could strip the hand and endplay a defender.

Double-Dummy Issues

The analysis of 2+ million endings, each in multiple ways, is a task only a computer could do — at least it didn’t appeal to me — which entails double-dummy play, so nobody misguesses. While this tends to balance out, it favors declarer in this arena, so some of West’s leads would fare better in practice than the “safe percent” indicates. For example, a lead from 10-x-x-x will usually survive when dummy has J-9-x and South A-K-x (declarer misguessing) but not here.

This four-eyed aspect also affects the trick averages when leading from a sequence instead of fourth-best. For example, from Q-J-x-x it makes no difference which card is led when dummy has K-10-8 and South A-x-x at double-dummy, but in practice leading low gains a trick. Therefore, the averages shown here tend to favor a sequence lead; so in close cases, leading low is probably better.

When declarer is the one to lead spades first, knowing how the missing cards lie will trigger extraordinary techniques that are illogical in actual play. To level the playing field somewhat, I imposed a restriction on declarer’s manipulation: On the first round of spades, a high or middle card can be led only if his side has a touching card.*

*This prevents a grossly anti-percentage backward finesse; e.g., with K-x-x opp. A-J-9 and the queen offside, declarer could lead the jack to force a cover, then finesse the nine on the way back. This is more like stealing than guessing, so it’s barred, because neither hand has a card touching the jack.

Ending Templates

To frame the experiment I created seven templates, one for each spade pattern, as shown below. Spades shown as ‘x’ are to be distributed in all combinations (note 2 always West). Each ending requires a second suit (I chose hearts) not only to balance the diagram but to allow both sides to achieve their positionally due tricks in the spade suit. Essentially this required a careful arrangement of heart entries to retain flexibility yet foil endplays, which wasn’t easy. Indeed, I’m still not sure they are fair to every possible combination.

Note that the first template yields 12c3 × 9c3 × 6c3, or 369,600 combinations, while the others (spades 4-4-3-2) yield 12c3 × 9c4 × 5c3, or 277,200 combinations each, for a total of 2,032,800. In this study I have ignored layouts containing a singleton or void.

Templates

No.	West	North	East	South	Combinations
1	xxx2 QJ	xxx A32	xxx KT9	xxx 654	369600
2	xxx2 32	xxx A54	xx QJT9	xxxx K6	277200
3	xxx2 32	xxxx K6	xx QJT9	xxx A54	277200
4	xxx2 AK	xxxx 65	xxx 432	xx QJT9	277200
5	xxx2 A2	xx QJT9	xxx K43	xxxx 65	277200
6	xxx2 AK	xxx QJT	xxxx 32	xx 9876	277200
7	xxx2 32	xx 9876	xxxx AK	xxx QJT	277200
Total combinations					2032800

Study 8C01   Main

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The Results

Tables show the average number of spade tricks won by the defense for each of the 220 West holdings, according to whether West leads fourth-best, from a sequence (if he has one), or if declarer leads spades first. Column 5 shows the percent of the time that West’s lead* does not lose a trick; i.e., declarer does just as well on his own. The last column shows the rank of each holding by its “safe percent” (T=tie).

*If West has a sequence, he is presumed to have led consistently the card that performed better overall. In other words, it is not based on switching back and forth according to what works better in each instance.

Ace West

West	4th Best	Sequence	Declarer	Safe Percent	Rank
A-K-Q-2	3.3333	3.6411	3.6505	99.0584	22
A-K-J-2	3.2500	3.4854	3.5006	98.4740	27
A-K-10-2	3.1212	3.3078	3.3654	94.2424	T58
A-K-9-2	3.0617	3.1581	3.2708	88.7338	122
A-K-8-2	2.9933	3.0613	3.1790	88.2251	126
A-K-7-2	2.9619	2.9971	3.1134	88.3658	124
A-K-6-2	2.9452	2.9629	3.0663	89.6537	T110
A-K-5-2	2.9381	2.9478	3.0412	90.6602	94
A-K-4-2	2.9366	2.9440	3.0300	91.4069	79
A-K-3-2	2.9366	2.9440	3.0232	92.0887	73
A-Q-J-2	3.0720	3.0865	3.4008	68.5714	220
A-Q-10-2	2.9832		3.2728	71.1905	217
A-Q-9-2	2.9378		3.1825	75.5736	212
A-Q-8-2	2.8787		3.0953	78.3766	204
A-Q-7-2	2.8431		3.0219	82.1212	180
A-Q-6-2	2.8197		2.9642	85.5519	146
A-Q-5-2	2.8089		2.9302	87.8680	128
A-Q-4-2	2.8069		2.9157	89.1234	T118
A-Q-3-2	2.8069		2.9085	89.8377	109
A-J-10-2	2.8615	2.8147	3.1539	70.7576	218
A-J-9-2	2.8106		3.0847	72.5866	216
A-J-8-2	2.7385		2.9919	74.6645	214
A-J-7-2	2.6931		2.9206	77.2511	208
A-J-6-2	2.6580		2.8646	79.3398	198
A-J-5-2	2.6421		2.8280	81.4069	189
A-J-4-2	2.6393		2.8121	82.7165	171
A-J-3-2	2.6389		2.8069	83.1926	168
A-10-9-2	2.6364	2.5804	2.9317	70.4654	219
A-10-8-2	2.5807		2.8306	75.0108	213
A-10-7-2	2.5372		2.7529	78.4307	203
A-10-6-2	2.5118		2.7107	80.1082	197
A-10-5-2	2.5023		2.6857	81.6558	185
A-10-4-2	2.5006		2.6737	82.6948	T172
A-10-3-2	2.5002		2.6699	83.0303	T169
A-9-8-2	2.4659	2.4351	2.6312	83.4740	167
A-9-7-2	2.4328		2.5816	85.1190	149
A-9-6-2	2.4076		2.5502	85.7359	145
A-9-5-2	2.3991		2.5334	86.5693	135
A-9-4-2	2.3976		2.5247	87.2944	T131
A-9-3-2	2.3968		2.5213	87.5433	129
A-8-7-2	2.3238	2.3183	2.4211	90.2706	102
A-8-6-2	2.3126		2.4043	90.8225	93
A-8-5-2	2.3090		2.3960	91.2987	84
A-8-4-2	2.3083		2.3916	91.6775	75
A-8-3-2	2.3080		2.3892	91.8831	74
A-7-6-2	2.2424	2.2405	2.3048	93.7662	70
A-7-5-2	2.2405		2.3002	94.0260	69
A-7-4-2	2.2405		2.2995	94.1017	66
A-7-3-2	2.2405		2.2985	94.1991	T61
A-6-5-2	2.2023	2.2023	2.2569	94.5346	T48
A-6-4-2	2.2023		2.2569	94.5346	T48
A-6-3-2	2.2023		2.2569	94.5346	T48
A-5-4-2	2.1852	2.1852	2.2366	94.8593	T42
A-5-3-2	2.1852		2.2366	94.8593	T42
A-4-3-2	2.1806		2.2312	94.9459	41

55 holdings × 9240 combinations = 508200 tests

Study 8C01   Main

Top   Lead Safety Analysis

King West

West	4th Best	Sequence	Declarer	Safe Percent	Rank
K-Q-J-2	2.9091	3.0865	3.1544	93.2035	71
K-Q-10-2	2.8506	2.9571	3.0530	90.4113	98
K-Q-9-2	2.7576	2.8028	2.9790	82.3810	176
K-Q-8-2	2.6859	2.6696	2.9145	77.1429	209
K-Q-7-2	2.6390	2.5778	2.8527	78.6255	201
K-Q-6-2	2.6121	2.5273	2.8071	80.4978	T194
K-Q-5-2	2.5964	2.5031	2.7816	81.4827	188
K-Q-4-2	2.5926	2.4929	2.7734	81.9264	184
K-Q-3-2	2.5926	2.4911	2.7724	82.0238	182
K-J-10-2	2.7159	2.6650	2.9506	77.0130	210
K-J-9-2	2.6488		2.8742	77.7814	207
K-J-8-2	2.5815		2.8017	78.2468	205
K-J-7-2	2.5275		2.7377	79.2424	199
K-J-6-2	2.4945		2.6884	80.8658	190
K-J-5-2	2.4751		2.6583	81.9372	183
K-J-4-2	2.4706		2.6475	82.5649	175
K-J-3-2	2.4706		2.6462	82.6948	T172
K-10-9-2	2.4794	2.4285	2.7478	73.8095	215
K-10-8-2	2.4325		2.6748	76.3095	211
K-10-7-2	2.3782		2.6038	77.9654	206
K-10-6-2	2.3510		2.5547	80.1515	196
K-10-5-2	2.3354		2.5252	81.5368	187
K-10-4-2	2.3310		2.5148	82.1320	179
K-10-3-2	2.3310		2.5124	82.3701	177
K-9-8-2	2.3045	2.2562	2.5193	78.9610	200
K-9-7-2	2.2676		2.4556	81.6342	186
K-9-6-2	2.2431		2.4110	83.6364	164
K-9-5-2	2.2291		2.3884	84.5022	156
K-9-4-2	2.2250		2.3810	84.8377	152
K-9-3-2	2.2250		2.3790	85.0325	151
K-8-7-2	2.1636	2.1412	2.2950	87.2944	T131
K-8-6-2	2.1489		2.2709	88.2359	125
K-8-5-2	2.1382		2.2558	88.6688	123
K-8-4-2	2.1345		2.2499	88.8961	120
K-8-3-2	2.1345		2.2476	89.1234	T118
K-7-6-2	2.0735	2.0675	2.1663	91.1472	89
K-7-5-2	2.0683		2.1604	91.2229	T85
K-7-4-2	2.0667		2.1588	91.2229	T85
K-7-3-2	2.0667		2.1577	91.3312	83
K-6-5-2	2.0298	2.0292	2.1198	91.4286	78
K-6-4-2	2.0292		2.1198	91.3745	T80
K-6-3-2	2.0292		2.1198	91.3745	T80
K-5-4-2	2.0154	2.0154	2.1031	91.6558	T76
K-5-3-2	2.0154		2.1031	91.6558	T76
K-4-3-2	2.0140		2.0964	92.1861	72

45 holdings × 9240 combinations = 415800 tests

Study 8C01   Main

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Queen West

West	4th Best	Sequence	Declarer	Safe Percent	Rank
Q-J-10-2	2.4167	2.5791	2.6762	90.2922	101
Q-J-9-2	2.3777	2.4509	2.6121	84.5238	155
Q-J-8-2	2.3264	2.3355	2.5485	80.4978	T194
Q-J-7-2	2.2935	2.2369	2.4938	82.6623	174
Q-J-6-2	2.2761	2.1751	2.4620	84.1017	159
Q-J-5-2	2.2669	2.1436	2.4416	85.2273	148
Q-J-4-2	2.2647	2.1316	2.4335	85.8117	144
Q-J-3-2	2.2647	2.1306	2.4319	85.9740	141
Q-10-9-2	2.2792	2.2740	2.4896	80.7468	191
Q-10-8-2	2.2381		2.4495	80.5952	193
Q-10-7-2	2.1970		2.3919	82.2294	178
Q-10-6-2	2.1762		2.3580	83.5390	165
Q-10-5-2	2.1662		2.3378	84.5671	154
Q-10-4-2	2.1636		2.3304	85.0433	150
Q-10-3-2	2.1636		2.3281	85.2706	147
Q-9-8-2	2.0984	2.0820	2.3284	78.5281	202
Q-9-7-2	2.0683		2.2768	80.6710	192
Q-9-6-2	2.0479		2.2429	82.0346	181
Q-9-5-2	2.0380		2.2229	83.0303	T169
Q-9-4-2	2.0355		2.2158	83.4957	166
Q-9-3-2	2.0355		2.2140	83.6797	163
Q-8-7-2	1.9567	1.9406	2.1251	84.7727	153
Q-8-6-2	1.9452		2.1024	85.8983	143
Q-8-5-2	1.9371		2.0857	86.7532	134
Q-8-4-2	1.9343		2.0795	87.0887	133
Q-8-3-2	1.9343		2.0774	87.3052	130
Q-7-6-2	1.8708	1.8670	1.9985	88.8420	121
Q-7-5-2	1.8673		1.9920	89.1450	117
Q-7-4-2	1.8661		1.9899	89.2316	116
Q-7-3-2	1.8661		1.9892	89.3074	115
Q-6-5-2	1.8298	1.8297	1.9466	89.9242	107
Q-6-4-2	1.8297		1.9462	89.9567	T105
Q-6-3-2	1.8297		1.9462	89.9567	T105
Q-5-4-2	1.8153	1.8153	1.9294	90.1948	T103
Q-5-3-2	1.8153		1.9294	90.1948	T103
Q-4-3-2	1.8130		1.9241	90.4978	96

36 holdings × 9240 combinations = 332640 tests

Study 8C01   Main

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Jack West

West	4th Best	Sequence	Declarer	Safe Percent	Rank
J-10-9-2	1.9913	2.1289	2.1601	96.8831	39
J-10-8-2	1.9777	2.0152	2.1346	88.0519	127
J-10-7-2	1.9509	1.9226	2.1062	84.4697	157
J-10-6-2	1.9380	1.8574	2.0957	84.2316	158
J-10-5-2	1.9281	1.8187	2.0890	83.9177	T160
J-10-4-2	1.9246	1.8017	2.0865	83.8095	162
J-10-3-2	1.9246	1.7988	2.0854	83.9177	T160
J-9-8-2	1.8829	1.8853	2.0251	86.0173	140
J-9-7-2	1.8640		2.0003	86.5368	136
J-9-6-2	1.8477		1.9853	86.4177	137
J-9-5-2	1.8367		1.9779	86.0498	139
J-9-4-2	1.8333		1.9754	85.9632	142
J-9-3-2	1.8333		1.9739	86.1147	138
J-8-7-2	1.7773	1.7601	1.8807	89.9134	108
J-8-6-2	1.7623		1.8690	89.5887	112
J-8-5-2	1.7512		1.8597	89.4048	114
J-8-4-2	1.7476		1.8555	89.4697	113
J-8-3-2	1.7476		1.8537	89.6537	T110
J-7-6-2	1.7010	1.6950	1.7976	90.5952	95
J-7-5-2	1.6953		1.7944	90.3571	100
J-7-4-2	1.6938		1.7926	90.3788	99
J-7-3-2	1.6938		1.7919	90.4545	97
J-6-5-2	1.6646	1.6644	1.7588	90.8442	92
J-6-4-2	1.6644		1.7583	90.8658	T90
J-6-3-2	1.6644		1.7583	90.8658	T90
J-5-4-2	1.6523	1.6523	1.7433	91.1580	T87
J-5-3-2	1.6523		1.7433	91.1580	T87
J-4-3-2	1.6515		1.7405	91.3636	82

28 holdings × 9240 combinations = 258720 tests

Study 8C01   Main

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No HCP West

West	4th Best	Sequence	Declarer	Safe Percent	Rank
10-9-8-2	1.6675	1.7137	1.7262	98.7554	23
10-9-7-2	1.6555	1.6524	1.7158	94.4048	55
10-9-6-2	1.6463	1.6122	1.7089	94.1775	63
10-9-5-2	1.6422	1.5855	1.7060	94.0584	68
10-9-4-2	1.6409	1.5739	1.7036	94.1667	64
10-9-3-2	1.6409	1.5712	1.7027	94.2532	57
10-8-7-2	1.5995	1.6035	1.6560	95.1840	40
10-8-6-2	1.5891		1.6514	94.1991	T61
10-8-5-2	1.5841		1.6475	94.0909	67
10-8-4-2	1.5826		1.6458	94.1126	65
10-8-3-2	1.5826		1.6447	94.2208	60
10-7-6-2	1.5436	1.5411	1.6036	94.4372	53
10-7-5-2	1.5399		1.6018	94.2424	T58
10-7-4-2	1.5393		1.6003	94.3290	56
10-7-3-2	1.5393		1.5995	94.4156	54
10-6-5-2	1.5149	1.5149	1.5746	94.4697	52
10-6-4-2	1.5149		1.5740	94.5238	51
10-6-3-2	1.5149		1.5737	94.5563	47
10-5-4-2	1.5048	1.5048	1.5621	94.6970	T45
10-5-3-2	1.5048		1.5621	94.6970	T45
10-4-3-2	1.5038		1.5603	94.7835	44
9-8-7-2	1.4412	1.4576	1.4594	99.8160	18
9-8-6-2	1.4378	1.4378	1.4578	97.9978	36
9-8-5-2	1.4360	1.4239	1.4565	97.9545	38
9-8-4-2	1.4360	1.4170	1.4563	97.9762	37
9-8-3-2	1.4360	1.4155	1.4558	98.0195	35
9-7-6-2	1.4227	1.4239	1.4407	98.3225	31
9-7-5-2	1.4210		1.4398	98.1169	34
9-7-4-2	1.4210		1.4393	98.1710	33
9-7-3-2	1.4210		1.4389	98.2143	32
9-6-5-2	1.4097	1.4097	1.4259	98.3874	30
9-6-4-2	1.4097		1.4255	98.4199	29
9-6-3-2	1.4097		1.4254	98.4307	28
9-5-4-2	1.4034	1.4034	1.4185	98.4848	T25
9-5-3-2	1.4034		1.4185	98.4848	T25
9-4-3-2	1.4029		1.4174	98.5498	24
8-7-6-2	1.3410	1.3436	1.3449	99.8701	T16
8-7-5-2	1.3410	1.3410	1.3440	99.6970	21
8-7-4-2	1.3410	1.3391	1.3436	99.7403	T19
8-7-3-2	1.3410	1.3384	1.3436	99.7403	T19
8-6-5-2	1.3406	1.3406	1.3419	99.8701	T16
8-6-4-2	1.3406		1.3415	99.9134	T12
8-6-3-2	1.3406		1.3415	99.9134	T12
8-5-4-2	1.3386	1.3386	1.3395	99.9134	T12
8-5-3-2	1.3386		1.3395	99.9134	T12
8-4-3-2	1.3384		1.3389	99.9567	11
7-6-5-2	1.3053	1.3053	1.3053	100.0000	T1
7-6-4-2	1.3053	1.3053	1.3053	100.0000	T1
7-6-3-2	1.3053	1.3053	1.3053	100.0000	T1
7-5-4-2	1.3053	1.3053	1.3053	100.0000	T1
7-5-3-2	1.3053		1.3053	100.0000	T1
7-4-3-2	1.3053		1.3053	100.0000	T1
6-5-4-2	1.2906	1.2906	1.2906	100.0000	T1
6-5-3-2	1.2906	1.2906	1.2906	100.0000	T1
6-4-3-2	1.2906		1.2906	100.0000	T1
5-4-3-2	1.2837		1.2837	100.0000	T1

56 holdings × 9240 combinations = 517440 tests

Totals: 220 holdings × 9240 combinations = 2032800 tests

Study 8C01   Main

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© 2008 Richard Pavlicek