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Computer Bridge: Opening-lead Inferences versus NT
(Page of 10)

(These "Computer Bridge" articles present techniques that might enable a computer-bridgeplayer to emulate certain qualities of human players, such as drawing inferences from actions taken or not taken, or imagining how a layout might be perceived by another player.)

I. Introduction

This article discusses how a defending computer-bridgeplayer (often called a "bridge robot," and which I have been calling, "Robot") might identify layouts that would be consistent with Partner's opening lead against notrump and actions taken in the auction.

II. Terminology

Some of these Computer Bridge articles use the following homegrown terms:

  • Possession—one or more consecutive tricks in which the same side leads
  • Series—one or more consecutive tricks in which the same side leads a given suit
  • Meldoid—set of two or more equals in a given suit in a given hand, or a single card in a given hand that has no equals in that suit in that hand

III. Use the Opening Lead to Identify Holdings in that Suit that Are Consistent with the Defenders' Agreements

West
Dummy
Q75
K5
AK1062
952
East
A82
42
987
KJ1073
Robot
W
N
E
S
P
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
4
5
1

According to the defenders' agreement and constrained by Robot's and Dummy's spade holdings, the opening lead could be any of the following:

  • Singleton 4
  • Doubleton 4x
  • KJ4, K104, K94, Kx4, J94, Jx4, 10x4
  • KJ94, KJx4, K10x4, J10x4, J9x4, 109x4
  • KJ94x, KJx4x, K10x4x, K9x4x, J10x4x, J9x4x, 109x4x

IV. Use the Not-visible Cards to Construct Legal Single-suit Holdings for Partner

West
Dummy
Q75
K5
AK1062
952
East
A82
42
987
KJ1073
Robot
W
N
E
S
P
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
4
5
1

In order to ultimately identify holdings in other suits that would have been more attractive to lead from than a given spade holding, Robot will first use the not-visible cards to enumerate Partner's "legal" holdings in those suits (that is, irrespective of Partner's previous actions).

Hearts:

  1. A, Q, J, 10, 9, x
  2. AQ, AJ, A10, A9, Ax, QJ, Q10, Q9, Qx, J10, J9, Jx, 109, 10x, 9x, xx
  3. AQJ, AQ10, AQ9, AQx, AJ10, AJ9, AJx, A109, A10x, A9x, Axx, QJ10, QJ9, QJx, Q109, Q10x, Q9x, Qxx, J109, J10x, J9x, Jxx, 109x, 10xx, 9xx, xxx
  4. AQJ10, AQJ9, AQJx, AQ109, AQ10x, AQ9x, AQxx, AJ109, AJ10x, AJ9x, AJxx, A109x, A10xx, A9xx, Axxx, QJ109, QJ10x, QJ9x, QJxx, Q109x, Q10xx, Q9xx, Qxxx, J109x, J10xx, J9xx, Jxxx, 109xx, 10xxx, 9xxx, xxxx
  5. AQJ109, AQJ10x, AQJ9x, AQJxx, AQ109x, AQ10xx, AQ9xx, AQxxx, AJ109x, AJ10xx, AJ9xx, AJxxx, A109xx, A10xxx, A9xxx, Axxxx, QJ109x, QJ10xx, QJ9xx, QJxxx, Q109xx, Q10xxx, Q9xxx, Qxxxx, J109xx, J10xxx, J9xxx, Jxxxx, 109xxx, 10xxxx, 9xxxx
  6. AQJ109x, AQJ10xx, AQJ9xx, AQJxxx, AQ109xx, AQ10xxx, AQ9xxx, AQxxxx, AJ109xx, AJ10xxx, AJ9xxx, AJxxxx, A109xxx, A10xxxx, A9xxxx, QJ109xx, QJ10xxx, QJ9xxx, QJxxxx, Q109xxx, Q10xxxx, Q9xxxx, J109xxx, J10xxxx, J9xxxx, 109xxxx
  7. AQJ109xx, AQJ10xxx, AQJ9xxx, AQJxxxx, AQ109xxx, AQ10xxxx, AQ9xxxx, AJ109xxx, AJ10xxxx, AJ9xxxx, A109xxxx, QJ109xxx, QJ10xxxx, QJ9xxxx, Q109xxxx, J109xxxx
  8. AQJ109xxx, AQJ10xxxx, AQJ9xxxx, AQ109xxxx, AJ109xxxx, QJ109xxxx
  9. AQJ109xxxx

 

Diamonds:

  1. Q, J, x
  2. QJ, Qx, Jx, xx
  3. QJx, Qxx, Jxx, xxx
  4. QJxx, Qxxx, Jxxx
  5. QJxxx

 

Clubs:

  1. A, Q, x
  2. AQ, Ax, Qx, xx
  3. AQx, Axx, Qxx, xxx
  4. AQxx, Axxx, Qxxx
  5. AQxxx

V. Eliminate Holdings for Partner in Non-opening-led Suits that in Isolation Would Be Inconsistent with West's Auction

West
Dummy
Q75
K5
AK1062
952
East
A82
42
987
KJ1073
Robot
W
N
E
S
P
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
4
5
1

Some hypothetical single-suit holdings for West, irrespective of the rest of the hand, would have called for a different auction by West.

Assume that over an opening 1N with a lower limit of at least 14 HCP, this partnership would intervene in direct seat at neither vul with a suit at least as good as AQJ-fifth, KQ-sixth, KJ10-sixth, A-seventh, K10-seventh, QJ-seventh, Q109-seventh, Q-eighth, J9-eighth, or J-ninth, irrespective of honors or distribution in the other three suits. (Hands where intervention would depend on holdings in more than one suit will be considered later.) These holdings are shown below in bold and will be omitted from future sections.

Hearts:

  1. A, Q, J, 10, 9, x
  2. AQ, AJ, A10, A9, Ax, QJ, Q10, Q9, Qx, J10, J9, Jx, 109, 10x, 9x, xx
  3. AQJ, AQ10, AQ9, AQx, AJ10, AJ9, AJx, A109, A10x, A9x, Axx, QJ10, QJ9, QJx, Q109, Q10x, Q9x, Qxx, J109, J10x, J9x, Jxx, 109x, 10xx, 9xx, xxx
  4. AQJ10, AQJ9, AQJx, AQ109, AQ10x, AQ9x, AQxx, AJ109, AJ10x, AJ9x, AJxx, A109x, A10xx, A9xx, Axxx, QJ109, QJ10x, QJ9x, QJxx, Q109x, Q10xx, Q9xx, Qxxx, J109x, J10xx, J9xx, Jxxx, 109xx, 10xxx, 9xxx, xxxx
  5. AQJ109, AQJ10x, AQJ9x, AQJxx, AQ109x, AQ10xx, AQ9xx, AQxxx, AJ109x, AJ10xx, AJ9xx, AJxxx, A109xx, A10xxx, A9xxx, Axxxx, QJ109x, QJ10xx, QJ9xx, QJxxx, Q109xx, Q10xxx, Q9xxx, Qxxxx, J109xx, J10xxx, J9xxx, Jxxxx, 109xxx, 10xxxx, 9xxxx
  6. AQJ109x, AQJ10xx, AQJ9xx, AQJxxx, AQ109xx, AQ10xxx, AQ9xxx, AQxxxx, AJ109xx, AJ10xxx, AJ9xxx, AJxxxx, A109xxx, A10xxxx, A9xxxx, QJ109xx, QJ10xxx, QJ9xxx, QJxxxx, Q109xxx, Q10xxxx, Q9xxxx, J109xxx, J10xxxx, J9xxxx, 109xxxx
  7. AQJ109xx, AQJ10xxx, AQJ9xxx, AQJxxxx, AQ109xxx, AQ10xxxx, AQ9xxxx, AJ109xxx, AJ10xxxx, AJ9xxxx, A109xxxx, QJ109xxx, QJ10xxxx, QJ9xxxx, Q109xxxx, J109xxxx
  8. AQJ109xxx, AQJ10xxxx, AQJ9xxxx, AQ109xxxx, AJ109xxxx, QJ109xxxx
  9. AQJ109xxxx

 

Diamonds:

  1. Q, J, x
  2. QJ, Qx, Jx, xx
  3. QJx, Qxx, Jxx, xxx
  4. QJxx, Qxxx, Jxxx
  5. QJxxx

 

Clubs:

  1. A, Q, x
  2. AQ, Ax, Qx, xx
  3. AQx, Axx, Qxx, xxx
  4. AQxx, Axxx, Qxxx
  5. AQxxx

VI. For Each Opening-lead-consistent Spade Holding for West, Eliminate Any Auction-Consistent Holdings in Other Suits that Would Have Been Led Instead

West
Dummy
Q75
K5
AK1062
952
East
A82
42
987
KJ1073
Robot
W
N
E
S
P
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
4
5
1

For each spade holding consistent with the opening lead, Robot now would use the opening-leads catalog (see "Computer Bridge: Opening Lead vs. Notrump", https://bridgewinners.com/article/view/computer-bridge-opening-lead-vs-notrump/, Dec. 20, 2018) to identify holdings outside of spades that Partner would have led instead.

Assume that for this auction, the opening-leads catalog would reflect the priorities recommended by Richard Pavlicek (at http://www.rpbridge.net/4g00.htm), of which the top five are as follows:

  1. Five-card or longer suit
  2. Safe honor sequence (which, based on Richard's examples, I understand to mean a holding headed by at least three equals, or a doubleton consisting of two equals, and I'll assume that either must contain at least one card higher than the nine and that AK-tight would be an exception)
  3. Four cards with neither ace nor king
  4. Worthless three cards
  5. Worthless doubleton

 

Within a category, Richard suggests choosing a major suit over a minor suit.

Suppose that Robot considers 43 first. Robot will dismiss any non-spade holdings that match any of the first four items in this list, on the grounds that Partner would have led from that holding instead of from a worthless doubleton. (This would not bar those non-spade holdings from being considered as accompanying other spade holdings that would be consistent with the opening lead and Partner's auction.)

Hearts:

A, Q, J, 10, 9, x

AQ, AJ, A10, A9, Ax, QJ, Q10, Q9, Qx, J10, J9, Jx, 109, 10x, 9x, xx

AQJ, AQ10, AQ9, AQx, AJ10, AJ9, AJx, A109, A10x, A9x, Axx, QJ10, QJ9, QJx, Q109, Q10x, Q9x, Qxx, J109, J10x, J9x, Jxx, 109x, 10xx, 9xx, xxx

AQJ10, AQJ9, AQJx, AQ109, AQ10x, AQ9x, AQxx, AJ109, AJ10x, AJ9x, AJxx, A109x, A10xx, A9xx, Axxx, QJ109, QJ10x, QJ9x, QJxx, Q109x, Q10xx, Q9xx, Qxxx, J109x, J10xx, J9xx, Jxxx, 109xx, 10xxx, 9xxx, xxxx

AQ109x, AQ10xx, AQ9xx, AQxxx, AJ109x, AJ10xx, AJ9xx, AJxxx, A109xx, A10xxx, A9xxx, Axxxx, QJ109x, QJ10xx, QJ9xx, QJxxx, Q109xx, Q10xxx, Q9xxx, Qxxxx, J109xx, J10xxx, J9xxx, Jxxxx, 109xxx, 10xxxx, 9xxxx

AJ9xxx, AJxxxx, A109xxx, A10xxxx, A9xxxx, QJ109xx, QJ10xxx, QJ9xxx, QJxxxx, Q109xxx, Q10xxxx, Q9xxxx, J109xxx, J10xxxx, J9xxxx, 109xxxx

J109xxxx

Diamonds:

Q, J, x

QJ, Qx, Jx, xx

QJx, Qxx, Jxx, xxx

QJxx, Qxxx, Jxxx

QJxxx

Clubs:

A, Q, x

AQ, Ax, Qx, xx

AQx, Axx, Qxx, xxx

AQxx, Axxx, Qxxx

AQxxx

VII. Use a Given Spade Holding and the Auction- and Lead-consistent Holdings in the Other Suits to Construct Candidate Hands for West

West
Dummy
Q75
K5
AK1062
952
East
A82
42
987
KJ1073
Robot
W
N
E
S
P
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
4
5
1

Having eliminated non-spade holdings that would have caused West to have chosen a different auction or opening lead, Robot would now use the remaining holdings to construct plausible hands for West. Assume that Robot considers 43 first, for which the following single-suit non-spade holdings remain:

Hearts:

  1. A, Q, J, 10, 9, x
  2. AQ, AJ, A10, A9, Ax, Q10, Q9, Qx, J9, Jx, 10x, 9x, xx
  3. AQJ, AQ10, AQ9, AQx, AJ10, AJ9, AJx, A109, A10x, A9x, Axx, QJ9, QJx, Q109, Q10x, Q9x, Qxx, J10x, J9x, Jxx, 109x, 10xx, 9xx
  4. AQJ10, AQJ9, AQJx, AQ109, AQ10x, AQ9x, AQxx, AJ109, AJ10x, AJ9x, AJxx, A109x, A10xx, A9xx, Axxx

 

Diamonds:

  1. Q, J, x
  2. Qx, Jx, xx
  3. QJx, Qxx, Jxx

 

Clubs:

  1. A, Q, x
  2. AQ, Ax, Qx, xx
  3. AQx, Axx, Qxx
  4. AQxx, Axxx

 

If West led from 43, the only distribution for West that can be constructed from the above holdings is 2=4=3=4. This would entail combining the following suit holdings:

Hearts:

  • AQJ10, AQJ9, AQJx, AQ109, AQ10x, AQ9x, AQxx, AJ109, AJ10x, AJ9x, AJxx, A109x, A10xx, A9xx, Axxx

Diamonds:

  • QJx, Qxx, Jxx

Clubs:

  • AQxx, Axxx

The resulting constructed hands would be as follows:

  • 43 AQJ10 QJx AQxx
  • 43 AQJ10 QJx Axxx
  • 43 AQJ10 Qxx AQxx
  • 43 AQJ10 Qxx Axxx
  • 43 AQJ10 Jxx AQxx
  • 43 AQJ10 Jxx Axxx
  • 43 AQJ9 QJx AQxx
  • 43 AQJ9 QJx Axxx
  • 43 AQJ9 Qxx AQxx
  • 43 AQJ9 Qxx Axxx
  • 43 AQJ9 Jxx AQxx
  • 43 AQJ9 Jxx Axxx
  • 43 AQJx QJx AQxx
  • 43 AQJx QJx Axxx
  • 43 AQJx Qxx AQxx
  • 43 AQJx Qxx Axxx
  • 43 AQJx Jxx AQxx
  • 43 AQJx Jxx Axxx
  • 43 AQ109 QJx AQxx
  • 43 AQ109 QJx Axxx
  • 43 AQ109 Qxx AQxx
  • 43 AQ109 Qxx Axxx
  • 43 AQ109 Jxx AQxx
  • 43 AQ109 Jxx Axxx
  • 43 AQ10x QJx AQxx
  • 43 AQ10x QJx Axxx
  • 43 AQ10x Qxx AQxx
  • 43 AQ10x Qxx Axxx
  • 43 AQ10x Jxx AQxx
  • 43 AQ10x Jxx Axxx
  • 43 AQ9x QJx AQxx
  • 43 AQ9x QJx Axxx
  • 43 AQ9x Qxx AQxx
  • 43 AQ9x Qxx Axxx
  • 43 AQ9x Jxx AQxx
  • 43 AQ9x Jxx Axxx
  • 43 AQxx QJx AQxx
  • 43 AQxx QJx Axxx
  • 43 AQxx Qxx AQxx
  • 43 AQxx Qxx Axxx
  • 43 AQxx Jxx AQxx
  • 43 AQxx Jxx Axxx
  • 43 AJ109 QJx AQxx
  • 43 AJ109 QJx Axxx
  • 43 AJ109 Qxx AQxx
  • 43 AJ109 Qxx Axxx
  • 43 AJ109 Jxx AQxx
  • 43 AJ109 Jxx Axxx
  • 43 AJ10x QJx AQxx
  • 43 AJ10x QJx Axxx
  • 43 AJ10x Qxx AQxx
  • 43 AJ10x Qxx Axxx
  • 43 AJ10x Jxx AQxx
  • 43 AJ10x Jxx Axxx
  • 43 AJ9x QJx AQxx
  • 43 AJ9x QJx Axxx
  • 43 AJ9x Qxx AQxx
  • 43 AJ9x Qxx Axxx
  • 43 AJ9x Jxx AQxx
  • 43 AJ9x Jxx Axxx
  • 43 AJxx QJx AQxx
  • 43 AJxx QJx Axxx
  • 43 AJxx Qxx AQxx
  • 43 AJxx Qxx Axxx
  • 43 AJxx Jxx AQxx
  • 43 AJxx Jxx Axxx
  • 43 A109x QJx AQxx
  • 43 A109x QJx Axxx
  • 43 A109x Qxx AQxx
  • 43 A109x Qxx Axxx
  • 43 A109x Jxx AQxx
  • 43 A109x Jxx Axxx
  • 43 A10xx QJx AQxx
  • 43 A10xx QJx Axxx
  • 43 A10xx Qxx AQxx
  • 43 A10xx Qxx Axxx
  • 43 A10xx Jxx AQxx
  • 43 A10xx Jxx Axxx
  • 43 A9xx QJx AQxx
  • 43 A9xx QJx Axxx
  • 43 A9xx Qxx AQxx
  • 43 A9xx Qxx Axxx
  • 43 A9xx Jxx AQxx
  • 43 A9xx Jxx Axxx
  • 43 Axxx QJx AQxx
  • 43 Axxx QJx Axxx
  • 43 Axxx Qxx AQxx
  • 43 Axxx Qxx Axxx
  • 43 Axxx Jxx AQxx
  • 43 Axxx Jxx Axxx

VIII. Eliminate Hands with Which West Would Have Had a Different Auction

West
Dummy
Q75
K5
AK1062
952
East
A82
42
987
KJ1073
Robot
W
N
E
S
P
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
4
5
1

Robot would now eliminate candidate hands with which West's auction would have been different. Assume that Robot first considers hands in which the opening lead was from 43.

Assuming that this partnership would not have intervened with any of those 2=4=3=4 hands over a 1N opener having a lower bound of at least 14 HCP, they all make it to the next stage.

IX. Eliminate Layouts that Would Give Declarer a Hand that's Inconsistent with that Side's Agreements

West
Dummy
Q75
K5
AK1062
952
East
A82
42
987
KJ1073
Robot
W
N
E
S
P
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
4
5
1

For each opening-lead-consistent hand for West, Robot would exclude any that would give Declarer a hand that doesn't match the declaring side's agreement. We'll assume that Robot starts with the hands where West led from 43.

Suppose that the opponents open 1N only with 4-3-3-3, 4-4-3-2, 5-3-3-2, or 5-4-2-2 distribution (where these represent combinations rather than permutations).

Each of the previously identified candidate hands for Partner would give Declarer 5=5=2=1, which would fail to match any of the above distributions. If Robot was set up to extend the permissible distributions to include 5-4-3-1 or 6-3-2-2, 5=5=2=1 still would not qualify.

Suppose also that the opponents open 1N only with 14 to 16 HCP.

The candidate hands for Partner range from 16 to 9 HCP, giving Declarer 4 to 11 HCP. Even the upper end of this range would leave Declarer 3 HCP under the 14 HCP lower bound for a 1N opener. If Robot were to allow for a one HCP deviation from this range, that would still leave Declarer at least 2 HCP below the more permissive 13 HCP lower bound.

Therefore, Robot would reject 43 as a possible spade holding for West.

X. Eliminate Opening-led Suit Layouts that Would Be Inconsistent with Dummy's First Play

West
Dummy
Q75
K5
AK1062
952
East
A82
42
987
KJ1073
Robot
W
N
E
S
P
1NT
P
3NT
P
P
P
D
3NT South
NS: 0 EW: 0
4
5
1

For each opening-led suit holding for Partner that hasn't been eliminated, Robot would consider whether Dummy's Trick 1 play would be consistent with the declaring side's cards in that suit and with the inferences available to Declarer from the defenders' auction and lead agreements.

Being that West's having led from 43 was eliminated in the previous section, it will not be considered here. We'll have Robot examine KJ1043 instead. This holding would imply the following single-suit layout (where bold indicates a played card to this trick):

                     Q 7 5 (Dummy)

K J 10 4 3                         A 8 2 (Robot)

                        9 6

From Declarer's perspective, this would yield

                    Q 7 5 (Dummy)

4 (LHO)

                      9 6

To look this up in the card-combination catalog (see "Computer Bridge: Card Combinations in Notrump", https://bridgewinners.com/article/view/computer-bridge-card-combinations-in-notrump/, Jan. 15, 2019), the insignificant spots would be replaced by x's:

                   Q x x (Dummy)

x (LHO)

                     9 x

To visualize Declarer's hypothetical situation, Robot would search the card-combination catalog for single-suit layouts consistent with the above and where Dummy's playing low would be depicted as valid. One such layout would be

                    Q x x (Dummy)

K 10 x x x x (LHO)             A J (RHO)

                     9 x (Declarer)

For this and any other single-suit layouts found, Robot would determine whether the opening leader's hypothetical holding would be consistent with the defending side's auction and lead agreements. The holding K10xxxx would be consistent with Partner's auction, in that the suit would not be categorically stout enough to intervene with directly over 1N with neither side vulnerable. (Robot assumes that Declarer is aware of the defenders' thresholds for suit-quality that were presented earlier in this article.)

Robot would also determine that the card led in this card-combination catalog layout would be consistent with the defenders' agreements. Therefore, Robot would find that Dummy's playing low at Trick 1 is consistent with Declarer's holding 9x, because that play would be necessary in at least one consistent layout. (This does imply that it's necessarily Dummy's optimal play, but Robot is satisfied that it's a non-nullo play.)

Robot would therefore not reject the hands from earlier stages where Partner holds KJ1043. Robot would also consider the other opening-lead suit layouts that reached this stage.

How Robot might proceed after having identified Partner's hands that would be consistent with prior inferences is yet to be addressed.

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