Main   Study 8C01 by Richard Pavlicek  

Lead Safety Analysis

[This 2008 study is superseded by Leading Authority which is far more comprehensive.]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

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 S 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 ending templates, one for each spade pattern, as shown below. Question marks (?) represent the spades to be distributed in all combinations (except S 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 circumvent 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.

1. S ? ? ?
H A 3 2
D
C
Spades 4=3=3=3
369600 combinations
S ? ? ? ?
H Q J
D
C
Table S ? ? ?
H K 10 9
D
C
S ? ?
H 6 5 4
D
C

2. S ? ? ?
H A 5 4
D
C
Spades 4=3=2=4
277200 combinations
S ? ? ? ?
H 3 2
D
C
Table S ? ?
H Q J 10 9
D
C
S ? ? ?
H K 6
D
C

3. S ? ? ? ?
H K 6
D
C
Spades 4=4=2=3
277200 combinations
S ? ? ? ?
H 3 2
D
C
Table S ? ?
H Q J 10 9
D
C
S ? ?
H A 5 4
D
C

4. S ? ? ? ?
H 6 5
D
C
Spades 4=4=3=2
277200 combinations
S ? ? ? ?
H A K
D
C
Table S ? ? ?
H 4 3 2
D
C
S ?
H Q J 10 9
D
C

5. S ? ?
H Q J 10 9
D
C
Spades 4=2=3=4
277200 combinations
S ? ? ? ?
H A 2
D
C
Table S ? ? ?
H K 4 3
D
C
S ? ? ?
H 6 5
D
C

6. S ? ? ?
H Q J 10
D
C
Spades 4=3=4=2
277200 combinations
S ? ? ? ?
H A K
D
C
Table S ? ? ? ?
H 3 2
D
C
S ?
H 9 8 7 6
D
C

7. S ? ?
H 9 8 7 6
D
C
Spades 4=2=4=3
277200 combinations
S ? ? ? ?
H 3 2
D
C
Table S ? ? ? ?
H A K
D
C
S ? ?
H Q J 10
D
C
TopMain

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
55 holdings × 9240 combinations = 508200
West4th BestSequenceDeclarerSafe PercentRank
A-K-Q-23.33333.64113.650599.058422
A-K-J-23.25003.48543.500698.474027
A-K-10-23.12123.30783.365494.2424T58
A-K-9-23.06173.15813.270888.7338122
A-K-8-22.99333.06133.179088.2251126
A-K-7-22.96192.99713.113488.3658124
A-K-6-22.94522.96293.066389.6537T110
A-K-5-22.93812.94783.041290.660294
A-K-4-22.93662.94403.030091.406979
A-K-3-22.93662.94403.023292.088773
A-Q-J-23.07203.08653.400868.5714220
A-Q-10-22.98323.272871.1905217
A-Q-9-22.93783.182575.5736212
A-Q-8-22.87873.095378.3766204
A-Q-7-22.84313.021982.1212180
A-Q-6-22.81972.964285.5519146
A-Q-5-22.80892.930287.8680128
A-Q-4-22.80692.915789.1234T118
A-Q-3-22.80692.908589.8377109
A-J-10-22.86152.81473.153970.7576218
A-J-9-22.81063.084772.5866216
A-J-8-22.73852.991974.6645214
A-J-7-22.69312.920677.2511208
A-J-6-22.65802.864679.3398198
A-J-5-22.64212.828081.4069189
A-J-4-22.63932.812182.7165171
A-J-3-22.63892.806983.1926168
A-10-9-22.63642.58042.931770.4654219
A-10-8-22.58072.830675.0108213
A-10-7-22.53722.752978.4307203
A-10-6-22.51182.710780.1082197
A-10-5-22.50232.685781.6558185
A-10-4-22.50062.673782.6948T172
A-10-3-22.50022.669983.0303T169
A-9-8-22.46592.43512.631283.4740167
A-9-7-22.43282.581685.1190149
A-9-6-22.40762.550285.7359145
A-9-5-22.39912.533486.5693135
A-9-4-22.39762.524787.2944T131
A-9-3-22.39682.521387.5433129
A-8-7-22.32382.31832.421190.2706102
A-8-6-22.31262.404390.822593
A-8-5-22.30902.396091.298784
A-8-4-22.30832.391691.677575
A-8-3-22.30802.389291.883174
A-7-6-22.24242.24052.304893.766270
A-7-5-22.24052.300294.026069
A-7-4-22.24052.299594.101766
A-7-3-22.24052.298594.1991T61
A-6-5-22.20232.20232.256994.5346T48
A-6-4-22.20232.256994.5346T48
A-6-3-22.20232.256994.5346T48
A-5-4-22.18522.18522.236694.8593T42
A-5-3-22.18522.236694.8593T42
A-4-3-22.18062.231294.945941

King West
45 holdings × 9240 combinations = 415800
West4th BestSequenceDeclarerSafe PercentRank
K-Q-J-22.90913.08653.154493.203571
K-Q-10-22.85062.95713.053090.411398
K-Q-9-22.75762.80282.979082.3810176
K-Q-8-22.68592.66962.914577.1429209
K-Q-7-22.63902.57782.852778.6255201
K-Q-6-22.61212.52732.807180.4978T194
K-Q-5-22.59642.50312.781681.4827188
K-Q-4-22.59262.49292.773481.9264184
K-Q-3-22.59262.49112.772482.0238182
K-J-10-22.71592.66502.950677.0130210
K-J-9-22.64882.874277.7814207
K-J-8-22.58152.801778.2468205
K-J-7-22.52752.737779.2424199
K-J-6-22.49452.688480.8658190
K-J-5-22.47512.658381.9372183
K-J-4-22.47062.647582.5649175
K-J-3-22.47062.646282.6948T172
K-10-9-22.47942.42852.747873.8095215
K-10-8-22.43252.674876.3095211
K-10-7-22.37822.603877.9654206
K-10-6-22.35102.554780.1515196
K-10-5-22.33542.525281.5368187
K-10-4-22.33102.514882.1320179
K-10-3-22.33102.512482.3701177
K-9-8-22.30452.25622.519378.9610200
K-9-7-22.26762.455681.6342186
K-9-6-22.24312.411083.6364164
K-9-5-22.22912.388484.5022156
K-9-4-22.22502.381084.8377152
K-9-3-22.22502.379085.0325151
K-8-7-22.16362.14122.295087.2944T131
K-8-6-22.14892.270988.2359125
K-8-5-22.13822.255888.6688123
K-8-4-22.13452.249988.8961120
K-8-3-22.13452.247689.1234T118
K-7-6-22.07352.06752.166391.147289
K-7-5-22.06832.160491.2229T85
K-7-4-22.06672.158891.2229T85
K-7-3-22.06672.157791.331283
K-6-5-22.02982.02922.119891.428678
K-6-4-22.02922.119891.3745T80
K-6-3-22.02922.119891.3745T80
K-5-4-22.01542.01542.103191.6558T76
K-5-3-22.01542.103191.6558T76
K-4-3-22.01402.096492.186172

Queen West
36 holdings × 9240 combinations = 332640
West4th BestSequenceDeclarerSafe PercentRank
Q-J-10-22.41672.57912.676290.2922101
Q-J-9-22.37772.45092.612184.5238155
Q-J-8-22.32642.33552.548580.4978T194
Q-J-7-22.29352.23692.493882.6623174
Q-J-6-22.27612.17512.462084.1017159
Q-J-5-22.26692.14362.441685.2273148
Q-J-4-22.26472.13162.433585.8117144
Q-J-3-22.26472.13062.431985.9740141
Q-10-9-22.27922.27402.489680.7468191
Q-10-8-22.23812.449580.5952193
Q-10-7-22.19702.391982.2294178
Q-10-6-22.17622.358083.5390165
Q-10-5-22.16622.337884.5671154
Q-10-4-22.16362.330485.0433150
Q-10-3-22.16362.328185.2706147
Q-9-8-22.09842.08202.328478.5281202
Q-9-7-22.06832.276880.6710192
Q-9-6-22.04792.242982.0346181
Q-9-5-22.03802.222983.0303T169
Q-9-4-22.03552.215883.4957166
Q-9-3-22.03552.214083.6797163
Q-8-7-21.95671.94062.125184.7727153
Q-8-6-21.94522.102485.8983143
Q-8-5-21.93712.085786.7532134
Q-8-4-21.93432.079587.0887133
Q-8-3-21.93432.077487.3052130
Q-7-6-21.87081.86701.998588.8420121
Q-7-5-21.86731.992089.1450117
Q-7-4-21.86611.989989.2316116
Q-7-3-21.86611.989289.3074115
Q-6-5-21.82981.82971.946689.9242107
Q-6-4-21.82971.946289.9567T105
Q-6-3-21.82971.946289.9567T105
Q-5-4-21.81531.81531.929490.1948T103
Q-5-3-21.81531.929490.1948T103
Q-4-3-21.81301.924190.497896

Jack West
28 holdings × 9240 combinations = 258720
West4th BestSequenceDeclarerSafe PercentRank
J-10-9-21.99132.12892.160196.883139
J-10-8-21.97772.01522.134688.0519127
J-10-7-21.95091.92262.106284.4697157
J-10-6-21.93801.85742.095784.2316158
J-10-5-21.92811.81872.089083.9177T160
J-10-4-21.92461.80172.086583.8095162
J-10-3-21.92461.79882.085483.9177T160
J-9-8-21.88291.88532.025186.0173140
J-9-7-21.86402.000386.5368136
J-9-6-21.84771.985386.4177137
J-9-5-21.83671.977986.0498139
J-9-4-21.83331.975485.9632142
J-9-3-21.83331.973986.1147138
J-8-7-21.77731.76011.880789.9134108
J-8-6-21.76231.869089.5887112
J-8-5-21.75121.859789.4048114
J-8-4-21.74761.855589.4697113
J-8-3-21.74761.853789.6537T110
J-7-6-21.70101.69501.797690.595295
J-7-5-21.69531.794490.3571100
J-7-4-21.69381.792690.378899
J-7-3-21.69381.791990.454597
J-6-5-21.66461.66441.758890.844292
J-6-4-21.66441.758390.8658T90
J-6-3-21.66441.758390.8658T90
J-5-4-21.65231.65231.743391.1580T87
J-5-3-21.65231.743391.1580T87
J-4-3-21.65151.740591.363682

No HCP West
56 holdings × 9240 combinations = 517440
West4th BestSequenceDeclarerSafe PercentRank
10-9-8-21.66751.71371.726298.755423
10-9-7-21.65551.65241.715894.404855
10-9-6-21.64631.61221.708994.177563
10-9-5-21.64221.58551.706094.058468
10-9-4-21.64091.57391.703694.166764
10-9-3-21.64091.57121.702794.253257
10-8-7-21.59951.60351.656095.184040
10-8-6-21.58911.651494.1991T61
10-8-5-21.58411.647594.090967
10-8-4-21.58261.645894.112665
10-8-3-21.58261.644794.220860
10-7-6-21.54361.54111.603694.437253
10-7-5-21.53991.601894.2424T58
10-7-4-21.53931.600394.329056
10-7-3-21.53931.599594.415654
10-6-5-21.51491.51491.574694.469752
10-6-4-21.51491.574094.523851
10-6-3-21.51491.573794.556347
10-5-4-21.50481.50481.562194.6970T45
10-5-3-21.50481.562194.6970T45
10-4-3-21.50381.560394.783544
9-8-7-21.44121.45761.459499.816018
9-8-6-21.43781.43781.457897.997836
9-8-5-21.43601.42391.456597.954538
9-8-4-21.43601.41701.456397.976237
9-8-3-21.43601.41551.455898.019535
9-7-6-21.42271.42391.440798.322531
9-7-5-21.42101.439898.116934
9-7-4-21.42101.439398.171033
9-7-3-21.42101.438998.214332
9-6-5-21.40971.40971.425998.387430
9-6-4-21.40971.425598.419929
9-6-3-21.40971.425498.430728
9-5-4-21.40341.40341.418598.4848T25
9-5-3-21.40341.418598.4848T25
9-4-3-21.40291.417498.549824
8-7-6-21.34101.34361.344999.8701T16
8-7-5-21.34101.34101.344099.697021
8-7-4-21.34101.33911.343699.7403T19
8-7-3-21.34101.33841.343699.7403T19
8-6-5-21.34061.34061.341999.8701T16
8-6-4-21.34061.341599.9134T12
8-6-3-21.34061.341599.9134T12
8-5-4-21.33861.33861.339599.9134T12
8-5-3-21.33861.339599.9134T12
8-4-3-21.33841.338999.956711
7-6-5-21.30531.30531.3053100.0000T1
7-6-4-21.30531.30531.3053100.0000T1
7-6-3-21.30531.30531.3053100.0000T1
7-5-4-21.30531.30531.3053100.0000T1
7-5-3-21.30531.3053100.0000T1
7-4-3-21.30531.3053100.0000T1
6-5-4-21.29061.29061.2906100.0000T1
6-5-3-21.29061.29061.2906100.0000T1
6-4-3-21.29061.2906100.0000T1
5-4-3-21.28371.2837100.0000T1

Totals: 220 holdings × 9240 combinations = 2032800 TopMain

© 2008 Richard Pavlicek