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[This 2008 study is superseded by Leading Authority, which is far more comprehensive.]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

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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 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 12 spades to be distributed in all possible combinations. 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 (spades 4-3-3-3) produces 12c3 × 9c3 × 6c3, or 369,600 combinations, while the others (spades 4-4-3-2) produce 12c3 × 9c4 × 5c3, or 277,200 combinations each, for a total of 2,032,800. In this study I have ignored the less common layouts containing a singleton or void.

1. Spades 4=3=3=3 (369600 combinations)

S ? ? ?
H A 3 2
D
C
S ? ? ? 2
H Q J
D
C
TableS ? ? ?
H K 10 9
D
C
S ? ? ?
H 6 5 4
D
C

2. Spades 4=3=2=4 (277200 combinations)

S ? ? ?
H A 5 4
D
C
S ? ? ? 2
H 3 2
D
C
TableS ? ?
H Q J 10 9
D
C
S ? ? ? ?
H K 6
D
C

3. Spades 4=4=2=3 (277200 combinations)

S ? ? ? ?
H K 6
D
C
S ? ? ? 2
H 3 2
D
C
TableS ? ?
H Q J 10 9
D
C
S ? ? ?
H A 5 4
D
C

4. Spades 4=4=3=2 (277200 combinations)

S ? ? ? ?
H 6 5
D
C
S ? ? ? 2
H A K
D
C
TableS ? ? ?
H 4 3 2
D
C
S ? ?
H Q J 10 9
D
C

5. Spades 4=2=3=4 (277200 combinations)

S ? ?
H Q J 10 9
D
C
S ? ? ? 2
H A 2
D
C
TableS ? ? ?
H K 4 3
D
C
S ? ? ? ?
H 6 5
D
C

6. Spades 4=3=4=2 (277200 combinations)

S ? ? ?
H Q J 10
D
C
S ? ? ? 2
H A K
D
C
TableS ? ? ? ?
H 3 2
D
C
S ? ?
H 9 8 7 6
D
C

7. Spades 4=2=4=3 (277200 combinations)

S ? ?
H 9 8 7 6
D
C
S ? ? ? 2
H 3 2
D
C
TableS ? ? ? ?
H A K
D
C
S ? ? ?
H Q J 10
D
C

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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 West55 holdings × 9240 combinations each = 508200
4th BestSequenceDeclarerSafe PercentRank
1. A-K-Q-23.33333.64113.650599.058422
2. A-K-J-23.25003.48543.500698.474027
3. A-K-10-23.12123.30783.365494.2424T58
4. A-K-9-23.06173.15813.270888.7338122
5. A-K-8-22.99333.06133.179088.2251126
6. A-K-7-22.96192.99713.113488.3658124
7. A-K-6-22.94522.96293.066389.6537T110
8. A-K-5-22.93812.94783.041290.660294
9. A-K-4-22.93662.94403.030091.406979
10. A-K-3-22.93662.94403.023292.088773
11. A-Q-J-23.07203.08653.400868.5714220
12. A-Q-10-22.98323.272871.1905217
13. A-Q-9-22.93783.182575.5736212
14. A-Q-8-22.87873.095378.3766204
15. A-Q-7-22.84313.021982.1212180
16. A-Q-6-22.81972.964285.5519146
17. A-Q-5-22.80892.930287.8680128
18. A-Q-4-22.80692.915789.1234T118
19. A-Q-3-22.80692.908589.8377109
20. A-J-10-22.86152.81473.153970.7576218
21. A-J-9-22.81063.084772.5866216
22. A-J-8-22.73852.991974.6645214
23. A-J-7-22.69312.920677.2511208
24. A-J-6-22.65802.864679.3398198
25. A-J-5-22.64212.828081.4069189
26. A-J-4-22.63932.812182.7165171
27. A-J-3-22.63892.806983.1926168
28. A-10-9-22.63642.58042.931770.4654219
29. A-10-8-22.58072.830675.0108213
30. A-10-7-22.53722.752978.4307203
31. A-10-6-22.51182.710780.1082197
32. A-10-5-22.50232.685781.6558185
33. A-10-4-22.50062.673782.6948T172
34. A-10-3-22.50022.669983.0303T169
35. A-9-8-22.46592.43512.631283.4740167
36. A-9-7-22.43282.581685.1190149
37. A-9-6-22.40762.550285.7359145
38. A-9-5-22.39912.533486.5693135
39. A-9-4-22.39762.524787.2944T131
40. A-9-3-22.39682.521387.5433129
41. A-8-7-22.32382.31832.421190.2706102
42. A-8-6-22.31262.404390.822593
43. A-8-5-22.30902.396091.298784
44. A-8-4-22.30832.391691.677575
45. A-8-3-22.30802.389291.883174
46. A-7-6-22.24242.24052.304893.766270
47. A-7-5-22.24052.300294.026069
48. A-7-4-22.24052.299594.101766
49. A-7-3-22.24052.298594.1991T61
50. A-6-5-22.20232.20232.256994.5346T48
51. A-6-4-22.20232.256994.5346T48
52. A-6-3-22.20232.256994.5346T48
53. A-5-4-22.18522.18522.236694.8593T42
54. A-5-3-22.18522.236694.8593T42
55. A-4-3-22.18062.231294.945941

King West45 holdings × 9240 combinations each = 415800
4th BestSequenceDeclarerSafe PercentRank
1. K-Q-J-22.90913.08653.154493.203571
2. K-Q-10-22.85062.95713.053090.411398
3. K-Q-9-22.75762.80282.979082.3810176
4. K-Q-8-22.68592.66962.914577.1429209
5. K-Q-7-22.63902.57782.852778.6255201
6. K-Q-6-22.61212.52732.807180.4978T194
7. K-Q-5-22.59642.50312.781681.4827188
8. K-Q-4-22.59262.49292.773481.9264184
9. K-Q-3-22.59262.49112.772482.0238182
10. K-J-10-22.71592.66502.950677.0130210
11. K-J-9-22.64882.874277.7814207
12. K-J-8-22.58152.801778.2468205
13. K-J-7-22.52752.737779.2424199
14. K-J-6-22.49452.688480.8658190
15. K-J-5-22.47512.658381.9372183
16. K-J-4-22.47062.647582.5649175
17. K-J-3-22.47062.646282.6948T172
18. K-10-9-22.47942.42852.747873.8095215
19. K-10-8-22.43252.674876.3095211
20. K-10-7-22.37822.603877.9654206
21. K-10-6-22.35102.554780.1515196
22. K-10-5-22.33542.525281.5368187
23. K-10-4-22.33102.514882.1320179
24. K-10-3-22.33102.512482.3701177
25. K-9-8-22.30452.25622.519378.9610200
26. K-9-7-22.26762.455681.6342186
27. K-9-6-22.24312.411083.6364164
28. K-9-5-22.22912.388484.5022156
29. K-9-4-22.22502.381084.8377152
30. K-9-3-22.22502.379085.0325151
31. K-8-7-22.16362.14122.295087.2944T131
32. K-8-6-22.14892.270988.2359125
33. K-8-5-22.13822.255888.6688123
34. K-8-4-22.13452.249988.8961120
35. K-8-3-22.13452.247689.1234T118
36. K-7-6-22.07352.06752.166391.147289
37. K-7-5-22.06832.160491.2229T85
38. K-7-4-22.06672.158891.2229T85
39. K-7-3-22.06672.157791.331283
40. K-6-5-22.02982.02922.119891.428678
41. K-6-4-22.02922.119891.3745T80
42. K-6-3-22.02922.119891.3745T80
43. K-5-4-22.01542.01542.103191.6558T76
44. K-5-3-22.01542.103191.6558T76
45. K-4-3-22.01402.096492.186172

Queen West36 holdings × 9240 combinations each = 332640
4th BestSequenceDeclarerSafe PercentRank
1. Q-J-10-22.41672.57912.676290.2922101
2. Q-J-9-22.37772.45092.612184.5238155
3. Q-J-8-22.32642.33552.548580.4978T194
4. Q-J-7-22.29352.23692.493882.6623174
5. Q-J-6-22.27612.17512.462084.1017159
6. Q-J-5-22.26692.14362.441685.2273148
7. Q-J-4-22.26472.13162.433585.8117144
8. Q-J-3-22.26472.13062.431985.9740141
9. Q-10-9-22.27922.27402.489680.7468191
10. Q-10-8-22.23812.449580.5952193
11. Q-10-7-22.19702.391982.2294178
12. Q-10-6-22.17622.358083.5390165
13. Q-10-5-22.16622.337884.5671154
14. Q-10-4-22.16362.330485.0433150
15. Q-10-3-22.16362.328185.2706147
16. Q-9-8-22.09842.08202.328478.5281202
17. Q-9-7-22.06832.276880.6710192
18. Q-9-6-22.04792.242982.0346181
19. Q-9-5-22.03802.222983.0303T169
20. Q-9-4-22.03552.215883.4957166
21. Q-9-3-22.03552.214083.6797163
22. Q-8-7-21.95671.94062.125184.7727153
23. Q-8-6-21.94522.102485.8983143
24. Q-8-5-21.93712.085786.7532134
25. Q-8-4-21.93432.079587.0887133
26. Q-8-3-21.93432.077487.3052130
27. Q-7-6-21.87081.86701.998588.8420121
28. Q-7-5-21.86731.992089.1450117
29. Q-7-4-21.86611.989989.2316116
30. Q-7-3-21.86611.989289.3074115
31. Q-6-5-21.82981.82971.946689.9242107
32. Q-6-4-21.82971.946289.9567T105
33. Q-6-3-21.82971.946289.9567T105
34. Q-5-4-21.81531.81531.929490.1948T103
35. Q-5-3-21.81531.929490.1948T103
36. Q-4-3-21.81301.924190.497896

Jack West28 holdings × 9240 combinations each = 258720
4th BestSequenceDeclarerSafe PercentRank
1. J-10-9-21.99132.12892.160196.883139
2. J-10-8-21.97772.01522.134688.0519127
3. J-10-7-21.95091.92262.106284.4697157
4. J-10-6-21.93801.85742.095784.2316158
5. J-10-5-21.92811.81872.089083.9177T160
6. J-10-4-21.92461.80172.086583.8095162
7. J-10-3-21.92461.79882.085483.9177T160
8. J-9-8-21.88291.88532.025186.0173140
9. J-9-7-21.86402.000386.5368136
10. J-9-6-21.84771.985386.4177137
11. J-9-5-21.83671.977986.0498139
12. J-9-4-21.83331.975485.9632142
13. J-9-3-21.83331.973986.1147138
14. J-8-7-21.77731.76011.880789.9134108
15. J-8-6-21.76231.869089.5887112
16. J-8-5-21.75121.859789.4048114
17. J-8-4-21.74761.855589.4697113
18. J-8-3-21.74761.853789.6537T110
19. J-7-6-21.70101.69501.797690.595295
20. J-7-5-21.69531.794490.3571100
21. J-7-4-21.69381.792690.378899
22. J-7-3-21.69381.791990.454597
23. J-6-5-21.66461.66441.758890.844292
24. J-6-4-21.66441.758390.8658T90
25. J-6-3-21.66441.758390.8658T90
26. J-5-4-21.65231.65231.743391.1580T87
27. J-5-3-21.65231.743391.1580T87
28. J-4-3-21.65151.740591.363682

No HCP West56 holdings × 9240 combinations each = 517440
4th BestSequenceDeclarerSafe PercentRank
1. 10-9-8-21.66751.71371.726298.755423
2. 10-9-7-21.65551.65241.715894.404855
3. 10-9-6-21.64631.61221.708994.177563
4. 10-9-5-21.64221.58551.706094.058468
5. 10-9-4-21.64091.57391.703694.166764
6. 10-9-3-21.64091.57121.702794.253257
7. 10-8-7-21.59951.60351.656095.184040
8. 10-8-6-21.58911.651494.1991T61
9. 10-8-5-21.58411.647594.090967
10. 10-8-4-21.58261.645894.112665
11. 10-8-3-21.58261.644794.220860
12. 10-7-6-21.54361.54111.603694.437253
13. 10-7-5-21.53991.601894.2424T58
14. 10-7-4-21.53931.600394.329056
15. 10-7-3-21.53931.599594.415654
16. 10-6-5-21.51491.51491.574694.469752
17. 10-6-4-21.51491.574094.523851
18. 10-6-3-21.51491.573794.556347
19. 10-5-4-21.50481.50481.562194.6970T45
20. 10-5-3-21.50481.562194.6970T45
21. 10-4-3-21.50381.560394.783544
22. 9-8-7-21.44121.45761.459499.816018
23. 9-8-6-21.43781.43781.457897.997836
24. 9-8-5-21.43601.42391.456597.954538
25. 9-8-4-21.43601.41701.456397.976237
26. 9-8-3-21.43601.41551.455898.019535
27. 9-7-6-21.42271.42391.440798.322531
28. 9-7-5-21.42101.439898.116934
29. 9-7-4-21.42101.439398.171033
30. 9-7-3-21.42101.438998.214332
31. 9-6-5-21.40971.40971.425998.387430
32. 9-6-4-21.40971.425598.419929
33. 9-6-3-21.40971.425498.430728
34. 9-5-4-21.40341.40341.418598.4848T25
35. 9-5-3-21.40341.418598.4848T25
36. 9-4-3-21.40291.417498.549824
37. 8-7-6-21.34101.34361.344999.8701T16
38. 8-7-5-21.34101.34101.344099.697021
39. 8-7-4-21.34101.33911.343699.7403T19
40. 8-7-3-21.34101.33841.343699.7403T19
41. 8-6-5-21.34061.34061.341999.8701T16
42. 8-6-4-21.34061.341599.9134T12
43. 8-6-3-21.34061.341599.9134T12
44. 8-5-4-21.33861.33861.339599.9134T12
45. 8-5-3-21.33861.339599.9134T12
46. 8-4-3-21.33841.338999.956711
47. 7-6-5-21.30531.30531.3053100.0000T1
48. 7-6-4-21.30531.30531.3053100.0000T1
49. 7-6-3-21.30531.30531.3053100.0000T1
50. 7-5-4-21.30531.30531.3053100.0000T1
51. 7-5-3-21.30531.3053100.0000T1
52. 7-4-3-21.30531.3053100.0000T1
53. 6-5-4-21.29061.29061.2906100.0000T1
54. 6-5-3-21.29061.29061.2906100.0000T1
55. 6-4-3-21.29061.2906100.0000T1
56. 5-4-3-21.28371.2837100.0000T1

Totals: 220 holdings × 9240 combinations each = 2032800

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© 2008 Richard Pavlicek