Analysis 7A47 Main

Echelon Play at IMPs

by Richard Pavlicek

The following play problem was recently presented by Leonard Helfgott, New Jersey, though the actual hands occurred about 25 years ago. After a simple auction, South is declarer in 3 NT with eight top tricks. The diamond suit will always provide at least two more but might require giving up the lead twice, which declarer cannot afford after the attacking spade lead. The club suit might also provide a ninth trick, so the goal is to find the best way to combine chances in these two suits, sometimes called an “echelon play,” to achieve the maximal success rate.

	A 5			West	North	East	South
Both vul	K 8 4						1 NT
	10 9 5 3			Pass	3 NT	Pass	Pass
	A J 9 4			Pass
Lead: 3
	K 6
	A Q 7
	A J 8 4 2
3 NT South	K 6 3

A preliminary analysis reveals two viable plans: (1) Win the A and lead another diamond, then fall back on the club finesse if diamonds do not behave, or (2) Win the A, then if no honor drops, win the A, K and lead a club toward the J-9. Either plan is strong and will succeed a great majority of the time. But which is better?

Regardless of the plan chosen, declarer should win the K and cash two top hearts (ace and king) in case there is a revealing show-out, then lead the 10 from dummy to the ace. Assuming everyone follows suit and no diamond honor drops, you are now at the crossroads: Continue diamonds? Or switch to clubs?

Problems like this are difficult, because the probabilities of the division of any particular suit are dependent on the distribution of other suits, a virtual merry-go-round. The best way to get an accurate assessment is to consider hand patterns that fit the scenario. First, let’s assume spades are 5-4 (either way), not guaranteed because West could have led from three, but practical for analysis. From winners cashed, hearts must be 4-3 or 5-2 (either way) and diamonds 2-2 or 3-1 (either way). Further, West’s choice of a spade lead should preclude a longer suit. Based on these conditions, my Hand Pattern Analyzer lists 18 possible distributions with the relative percent chance of each.

Hand Pattern Analysis for West-East 9=7=4=6
Known: Ws4-5 Wh2-5 Wd1-3 Ws)Wh Ws)Wc
Contracts compared Vulnerable: 3N 3N
18 Distributions Sorted by Frequency

Weight		Tricks Won	West	East	Factors	Percent	Make Pct		Trick Avg		IMP Avg		MP Avg
1	100	B9	5323 2	4423 1	126 35 6 20	13.33	100	100	11.00	9.00	2.00	0	2.00	0
2	100	CC5 B94 B86	5422 3	4324 1	126 35 6 15	10.00	100	60.00	11.33	9.60	5.73	0	1.67	0.33
3	100	CC B92	4324 1	5422 3	126 35 6 15	10.00	100	100	11.33	10.00	1.33	0	1.67	0.33
4	100	BB2 A9 79	5413 4	4333 0	126 35 4 20	8.88	75.00	100	9.75	10.00	0.25	3.25	1.00	1.00
5	100	BB2 A9 79	4333 0	5413 4	126 35 4 20	8.88	75.00	100	9.75	10.00	0.25	3.25	1.00	1.00
6	100	CC5 BB10 AA 994 794 786	5332 2	4414 3	126 35 4 15	6.66	66.67	80.00	9.53	10.00	0	2.33	0.67	1.33
7	100	CC5 BB10 996 A94 7A5	5314 4	4432 1	126 35 4 15	6.66	83.33	100	9.97	10.33	0.13	2.17	0.97	1.03
8	50	B9	4423 1	5323 2	126 35 6 20	6.66	100	100	11.00	9.00	2.00	0	2.00	0
9	100	CC B92	5224 3	4522 3	126 21 6 15	6.00	100	100	11.33	10.00	1.33	0	1.67	0.33
10	100	BB2 A9 79	5233 2	4513 4	126 21 4 20	5.33	75.00	100	9.75	10.00	0.25	3.25	1.00	1.00
11	100	CC5 BB10 996 A94 7A5	4234 1	5512 6	126 21 4 15	4.00	83.33	100	9.97	10.33	0.13	2.17	0.97	1.03
12	50	CC5 BB10 995 794 786	4432 1	5314 4	126 35 4 15	3.33	66.67	80.00	9.50	9.97	0	2.33	0.67	1.33
13	50	CC5 BB10 996 A94 7A5	4414 3	5332 2	126 35 4 15	3.33	83.33	100	9.97	10.33	0.13	2.17	0.97	1.03
14	100	CC BB5 99 89 884	5431 4	4315 4	126 35 4 6	2.67	58.33	66.67	9.67	9.75	0	1.00	0.92	1.08
15	40	CC5 BB10 AA 994 794 786	5512 6	4234 1	126 21 4 15	1.60	66.67	80.00	9.53	10.00	0	2.33	0.67	1.33
16	100	CC BB5 995 AA	5215 6	4531 4	126 21 4 6	1.60	100	100	10.17	10.17	0	0	1.00	1.00
17	40	CC B9 BB4	5521 6	4225 3	126 21 6 6	0.96	100	100	11.17	10.83	0.33	0	1.17	0.83
18	40	BB 88	5530 7	4216 6	126 21 4 1	0.11	50.00	50.00	9.50	9.50	0	0	1.00	1.00
Averages or totals			2.31	2.34	3,970,764	100.00	86.86	92.74	10.43	9.81	1.27	1.35	1.33	0.67

Under Make Pct and Avg columns, first result is Plan 1 (gold) and second is Plan 2 (green).

The Factors cell in each row, when multiplied out, shows the number of specific layouts for that case. For example, in Case 1 spades can be distributed 126 ways (9c5), hearts 35 ways (7c3), diamonds 6 ways (4c2) and clubs 20 ways (6c3) to produce 529,200 specific layouts.

An adjustment should be made, analogous to restricted choice, in cases where West has equal length (4-4 or 5-5) in the majors. On average he would probably select a spade lead only half the time, so the weight of these cases is halved (50 instead of 100). Also, when West is 5-5, I reduced the weight even further to 40, because he might have bid on some of those hands — subjective of course, but the adjusted amount is surely more accurate than no adjustment at all. Weight adjustments are reflected in the Percent column and all totals.

To compare two plans, it is necessary to determine the tricks won by each plan in each case. Cases 1 and 8 are simple: Plan 1 always wins 11 tricks (diamonds 2-2) and Plan 2 always wins 9 (fourth club sets up). Tricks won by each plan are given respectively in hexadecimal (to ensure a single digit for both) followed by the portion to which it applies. A portion of 1 is presumed if no number is given.

Case 2 is not as easy. When West has Q-10 or Q-x (5/15 of the club layouts) both plans win 12 tricks, because Plan 2 can safely switch to diamonds when the Q drops, so the first entry is CC5. When West has 10-x (4/15 of the club layouts) Plan 1 wins 11 tricks, but Plan 2 wins only 9, so the second entry is B94. Finally, when West has x-x (6/15 of the club layouts) Plan 1 still wins 11 tricks, but Plan 2 fails (8 tricks) so the last entry is B86. Note that the sum of portions (5+4+6) equals the number of club layouts.

Case 6 is more difficult, because with diamonds 3-1 there are two equally likely occurrences: an honor drops, or it doesn’t; so the 15 club portions must be considered twice, effectively 30 portions in all. For the 15 club layouts with a stiff diamond honor, both plans succeed, winning 12 tricks if the Q drops (5 layouts) or 11 tricks if it doesn’t (10 layouts), so the first two entries are CC5 BB10. When no diamond honor drops, both plans win 10 tricks when West has Q-10 (1 layout) but only 9 tricks when West has Q-x (4 layouts) so enter AA 994. Now for the bad news: When West has 10-x (4 layouts) Plan 1 not only fails but is down two ( A is lost as E-W scarf up the rest), while Plan 2 survives (794); and when West has x-x (6 layouts) both plans fail but Plan 2 by only one (786). Check the sum of portions (5+10+1+4+4+6) to verify it equals 30.

Results clearly show Plan 2 to be better. The totals of the Make Pct column show that Plan 1 makes 3 NT 86.86 percent of the time, which is certainly good, but not as good as Plan 2 with 92.74 percent, almost 6 percent higher. The most meaningful column, however, is the IMP Avg which indicates the average number of IMPs won per board if every possible layout were played once. Note that these are IMPs won (not net IMPs) so the crux is the difference in the two averages, which shows that Plan 2 gains 0.08 IMPs per board.

Vulnerability matters

What I found most interesting about this problem is that the proper play depends on the vulnerability. As given with North-South vulnerable, Plan 2 is clearly superior as shown. When nonvulnerable, however, the IMP results change considerably, because game swings are lessened, and overtricks remain the same. By my account, with North-South nonvulnerable Plan 1 averages 1.19 IMPs per board, and Plan 2 only 1.03, which means Plan 1 now gains 0.16 IMPs per board, or twice the amount lost when vulnerable. Quite a turnaround.

Analysis 7A47 Main

Top Echelon Play at IMPs

Matchpoints

While the problem occurred at IMPs, the preceding table also shows a matchpoint comparison of the two plans. No contest! Plan 1 is a clear winner, because it wins more tricks on average, which is the name of the game at matchpoints. Making a contract is only secondary — which brings to mind the old question of whether matchpoints is really bridge, but I refuse to answer on the grounds it may be incriminating.

I was recently asked by Antonio Cocco, France, if it would be better at matchpoints to take two diamond finesses. That is, to maximize the chance to win four diamond tricks but fail miserably (no second chance in clubs) when West has K-Q doubleton. Good question! I couldn’t provide a definite answer, so I decided to run another comparison, this time pitting the previous Plan 1 against Plan 3: Win the K and cash two hearts (as with each plan) then run the 10; if it loses, finesse again (assuming East follows low). Stand back for another table!

Hand Pattern Analysis for West-East 9=7=4=6
Known: Ws4-5 Wh2-5 Wd1-3 Ws)Wh Ws)Wc
Contracts compared Vulnerable: 3N 3N
18 Distributions Sorted by Frequency

Weight		Tricks Won	West	East	Factors	Percent	Make Pct		Trick Avg		IMP Avg		MP Avg
1	100	BB5 B8	5323 2	4423 1	126 35 6 20	13.33	100	83.33	11.00	10.50	2.17	0	1.17	0.83
2	100	CC5 BB10 C8 B82	5422 3	4324 1	126 35 6 15	10.00	100	83.33	11.33	10.78	2.17	0	1.17	0.83
3	100	CC5 BB10 C8 B82	4324 1	5422 3	126 35 6 15	10.00	100	83.33	11.33	10.78	2.17	0	1.17	0.83
4	100	BB2 AB 7B	5413 4	4333 0	126 35 4 20	8.88	75.00	100	9.75	11.00	0	3.50	0.50	1.50
5	100	BB2 AA 77	4333 0	5413 4	126 35 4 20	8.88	75.00	75.00	9.75	9.75	0	0	1.00	1.00
6	100	CC5 BB10 AA 994 774 786	5332 2	4414 3	126 35 4 15	6.66	66.67	66.67	9.53	9.73	0	0.60	0.80	1.20
7	100	CC5 BB10 9B6 AB4 7C5	5314 4	4432 1	126 35 4 15	6.66	83.33	100	9.97	11.33	0	2.70	0.50	1.50
8	50	BB5 B8	4423 1	5323 2	126 35 6 20	6.66	100	83.33	11.00	10.50	2.17	0	1.17	0.83
9	100	CC5 BB10 C8 B82	5224 3	4522 3	126 21 6 15	6.00	100	83.33	11.33	10.78	2.17	0	1.17	0.83
10	100	BB2 AA 77	5233 2	4513 4	126 21 4 20	5.33	75.00	75.00	9.75	9.75	0	0	1.00	1.00
11	100	CC5 BB10 AA4 996 775	4234 1	5512 6	126 21 4 15	4.00	83.33	83.33	9.97	9.97	0	0	1.00	1.00
12	50	CC BB2 99 772	4432 1	5314 4	126 35 4 15	3.33	66.67	66.67	9.50	9.50	0	0	1.00	1.00
13	50	CC5 BB10 9B6 AB4 7C5	4414 3	5332 2	126 35 4 15	3.33	83.33	100	9.97	11.33	0	2.70	0.50	1.50
14	100	CC BB5 99 885	5431 4	4315 4	126 35 4 6	2.67	58.33	58.33	9.67	9.67	0	0	1.00	1.00
15	40	CC5 BB10 AC 9C4 7B10	5512 6	4234 1	126 21 4 15	1.60	66.67	100	9.53	11.33	0	4.67	0.50	1.50
16	100	CC BB5 9B5 AC	5215 6	4531 4	126 21 4 6	1.60	100	100	10.17	11.17	0	1.00	0.50	1.50
17	40	CC5 BB25 C8 B85	5521 6	4225 3	126 21 6 6	0.96	100	83.33	11.17	10.64	2.17	0	1.17	0.83
18	40	BB 88	5530 7	4216 6	126 21 4 1	0.11	50.00	50.00	9.50	9.50	0	0	1.00	1.00
Averages or totals			2.31	2.34	3,970,764	100.00	86.86	83.46	10.43	10.49	1.02	0.71	0.95	1.05

Under Make Pct and Avg columns, first result is Plan 1 (gold) and second is Plan 3 (red).

Yes, indeed. Plan 3 (two diamond finesses) outscores Plan 1 at matchpoints. The comparison is done a la board-a-match, scoring 2 for a win and 1 for a tie (no fractions). Hence each deal decides 2 matchpoints, of which Plan 1 averaged 0.95 and Plan 3 got the better with 1.05.

Note, however, that Plan 1 beats Plan 3 by a wide margin at IMPs.

Analysis 7A47 Main

Top Echelon Play at IMPs

© 2016 Richard Pavlicek