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Computer Bridge: Counting Winners versus NT
(Page of 13)

(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 count its side's possible winners against notrump and select a play accordingly.

II. Terminology

I have adapted from American football the following terms and assigned them bridge meanings:

  • 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

The following is a homegrown variant of the term "meld" in rummy and other card games:

  • 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. Gathering Inferences from the Auction, the Play, and the Partnerships' Agreements

The article "Computer Bridge: Other Players' Inferences" (https://bridgewinners.com/article/view/computer-bridge-other-players-inferences/) left Robot on lead to Trick 3 in the following deal:

West
1083
K9743
AK2
75
North
Q972
10
7543
AKQ3
East
South
W
N
E
S
P
P
1
P
1
P
1
P
1NT
P
P
P
D
1NT South
NS: 0 EW: 0
8
9
J
4
2
0
1
6
J
K
10
0
0
2
2

Robot had inferred the following:

  • Partner started with AKJx or AKJ, giving Declarer xx or xxx, respectively.
  • Partner started with 6x or 6xx, giving Declarer AQJ8x or AQJ8, respectively.
  • Partner can infer that Declarer started with AQJx(x) or QJ9x(x).

IV. Inferring What Shapes for Declarer Would Be Consistent with the Auction, the Play, and the Partnerships' Agreements

West
1083
K9743
AK2
75
North
Q972
10
7543
AKQ3
East
South
W
N
E
S
P
P
1
P
1
P
1
P
1NT
P
P
P
D
1NT South
NS: 0 EW: 0
8
9
J
4
2
0
1
6
J
K
10
0
0
2
2

The previously mentioned inferences by Robot, in conjunction with the auction and the partnerships' agreements, would suggest that Declarer has one of the following shapes:

  • 3=5=2=3
  • 3=4=3=3
  • 2=5=3=3
  • 2=5=2=4
  • 2=5=1=5
  • 2=4=3=4
  • 2=4=2=5
  • 2=4=1=6

V. Rejecting Declarer Shapes that Would Have Permitted Claiming the Contract

West
1083
K9743
AK2
75
North
Q972
10
7543
AKQ3
East
South
W
N
E
S
P
P
1
P
1
P
1
P
1NT
P
P
P
D
1NT South
NS: 0 EW: 0
8
9
J
4
2
0
1
6
J
K
10
0
0
2
2

Robot would now identify Declarer shapes in which the contract could have been claimed by playing the heart ace at Trick 2, irrespective of the distribution of the high cards that Robot has neither seen nor drawn inferences about. Being that Robot holds the master card in diamonds and that Partner is inferred to hold the master card in spades, Declarer's claim would have had to be based on the round suits only. Declarer held only one master in hearts, leaving six winners to have had to have come from clubs.

Searching the card-combination catalog (see "Computer Bridge: Card Combinations in Notrump," https://bridgewinners.com/article/view/computer-bridge-card-combinations-in-notrump/) for single-suit layouts where Dummy holds AKQx and that yield six tricks on any distribution of the defenders' cards, irrespective of who holds the unseen honors, would fetch layouts where Declarer holds any six cards of the suit. Therefore, of the Declarer shapes identified on the previous page, Robot could eliminate 2=4=1=6.

VI. Using the Available Inferences to Count the Defenders' Established Winners

West
1083
K9743
AK2
75
North
Q972
10
7543
AKQ3
East
South
W
N
E
S
P
P
1
P
1
P
1
P
1NT
P
P
P
D
1NT South
NS: 0 EW: 0
8
9
J
4
2
0
1
6
J
K
10
0
0
2
2

The defenders have taken two tricks, and Robot remains with the diamond ace and king, bringing to four the number of winners known to Robot.

If Partner started with AKJ, that would yield two future winners irrespective of who would be leading the suit, bringing the total count of winners to six.

If Partner started instead with AKJx, that would at worst yield two future winners irrespective of who led the suit. (Whether the Partner's "x" could also become a winner will be addressed later.)

VII. Searching for Diamond Layouts that Would Set When Partner Started with AKJ

If Partner started with AKJ, the only suit that would offer a chance to build and take a seventh winner on the current possession would be diamonds. Robot would search the card-combination catalog for layouts equivalent to the following and that would yield at least three winners (that is, at least one winner in addition to Robot's two masters) without losing possession.

               7 5 4 3 (Dummy)

A K 2                       (3 to 5 diamonds)

               (1 to 3 diamonds)

Assume that this would yield the following layouts:

  • Declarer's RHO holds any five diamonds, with either defender on lead
  • Declarer's RHO holds any four diamonds, with either defender on lead
  • Declarer's RHO holds Qxx (x < Q), with either defender on lead
  • Declarer's RHO holds JTx (x < T), with RHO's leading the first or second round
  • Declarer's RHO holds J98, with RHO's leading the first two rounds
  • Declarer's RHO holds T98, with RHO's leading the first two rounds

VIII. Constructing Successful Multi-suit Plans Where Partner Started with AKJ

To meet the most stringent lead constraint identified in the previous section, Partner would need two entries for leading diamonds. If Partner remains with the spade ace and king, they would provide the necessary entries. Robot could determine this by constructing multi-suit plans akin to what was discussed in "Computer Bridge: Single-suit Plans in Notrump II" (https://bridgewinners.com/article/view/computer-bridge-single-suit-plans-in-notrump-ii/). This would call for Robot to lead a spade to Trick 3.

IX. Emulating Partner's Thinking When Dealt AKJ

Robot would emulate Partner's thinking when dealt AKJ, as follows:

"Robot could on this possession take at most one more trick in hearts (with the ace), and doing so would not depend on which of us led the suit. Adding the already taken heart trick, the second possible heart winner, and my three spade winners would leave a shortfall of at least two winners. Being that Dummy holds the master club, it is clear for me to shift to diamonds if we're to set the contract on this possession."

Therefore, Robot could infer that Partner would shift to diamonds at the first opportunity.

X. Searching for Spade Layouts that Would Set When Partner Started with AKJx

To determine how many tricks might be available from spades if Partner started with AKJx, Robot would search the card-combination catalog based on the spade cards remaining after the first round of the suit. Being that Partner's holding the spade six or the spade five would be consistent with the play thus far, Robot would need to search for both.

                Q 7 2 (Dummy)

T 3                                 A K 5

                     6

 

                Q 7 2 (Dummy)

T 3                                 A K 6

                     5

After promotions, these layouts would effectively become

                 Q T 6 (Dummy)

J 7                                  A K 8

                     9

 

                 Q T 6 (Dummy)

J 7                                  A K 9

                      8

The catalog would indicate that in either layout, a total of four tricks could be won in the suit only if Declarer's LHO led the ten on the second round and the three on the third round.

XI. Constructing Successful Multi-suit Plans Where Partner Started with AKJx

To take four spade tricks altogether, Robot would need to lead the suit twice more. Robot is currently on lead and has the AK as entries (based on the inference that Partner holds at least three diamonds), leaving one "surplus" entry. Therefore, if Partner started with AKJx, Robot could lead the spade ten now or first cash a diamond.

XII. Emulating Partner's Thinking When Dealt AKJx

Robot would emulate Partner's thinking when dealt AKJx, as follows:

"Robot could on this possession take at most one more trick in hearts (with the ace), and doing so would not depend on which of us led the suit. Adding the already taken heart trick, the second possible heart winner, and my four spade winners would leave a shortfall of at least one winner. Being that Dummy holds the master club, it is clear to me to shift to diamonds if we're to set the contract on this possession."

Therefore, Robot could infer that Partner would shift to diamonds at the first opportunity.

XIII. Selecting the Card to Lead to Trick 3

Which of Robot's possible leads to Trick 3 would enable setting the contract in the current possession would depend on the diamond and spade positions.

If Partner was dealt AKJx and diamonds cannot yield a third winner, then the spade ten lead is the only one that would suffice.

Given that the spade ten would also suffice when diamonds can yield a third winner, it's clear for Robot to lead that card to Trick 3.

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