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Computer Bridge: Other Players' Inferences
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(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 presents thoughts on how a computer-bridgeplayer (often called a "bridge robot," and which I have been calling, "Robot") could identify inferences that other players draw from cards played or not played to a given trick.

II. Gather Inferences from the Auction and the Partnerships' Agreements

West (Robot)
1083
K9743
AK2
75
W
N
E
S
P
P
1
P
1
P
1
P
1NT
P
P
P
 

Assume that the partnerships' agreements allow Robot to infer from the auction, that Partner holds fewer than five spades and that Declarer has

  1. Four or five hearts,
  2. Fewer than four diamonds, and
  3. Either two spades, or three spades and a non-minimum 1 response.

III. Draw Inferences from the Play at Trick 1

Robot
1083
K9743
AK2
75
Dummy
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
1
 

Before attempting to identify inferences that other players might draw from the play to this trick, Robot would try to place the unseen spades. To identify Declarer's holdings that would be consistent with Dummy's playing the nine and Declarer's playing an x under Partner's jack, Robot would search the card-combination catalog (see "Computer Bridge: Card Combinations in Notrump", https://bridgewinners.com/article/view/computer-bridge-card-combinations-in-notrump/) for positions matching the following (where bold indicates the cards played to this trick):

                    Q 9 7 x

8 (led)                              J

                        x (from two or three cards, without the 10)

Assuming that this search would yield no results where Declarer holds the ace or king, Robot can infer that Partner holds those two cards.

IV. Infer Partner's Inferences from the Play at Trick 1

Robot
1083
K9743
AK2
75
Dummy
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
1
 

To construct how the position might appear to Partner, Robot would then emulate how Partner might search the card-combination catalog in order to infer the suit's layout. Given that Partner is inferred to hold the ace and king, the presumed objective of Partner's search would be to place the ten. To identify Declarer's holdings that from Partner's perspective would be consistent with Dummy's playing the nine and Declarer's playing an x under Partner's jack, Robot would search the card-combination catalog for positions matching the following:

                         Q 9 7 x

8 (led)                                  J (from 3–4 cards, with AK but not 10)

                            x (from two or three cards)

Assuming that this search would yield no results where Declarer holds the ten yet played the nine from Dummy, Robot would infer that Partner can place the ten with Robot.

V. Infer Declarer's Inferences from the Play at Trick 1

Robot
1083
K9743
AK2
75
Dummy
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
1
 

To construct how the position might appear to Declarer, Robot would then, based on Robot's inference that Declarer is missing the ace, king, and ten, emulate Declarer's hypothetical search of the card-combination catalog for positions where RHO would cover the nine with the jack. Such positions would match the following:

                     Q 9 7 x

8 (led)                                J

                         (without ace, king, or ten)

Assuming that the above search would fail to yield any results where Partner holds the ten, Robot can infer that Declarer will place the ten with Robot. (Choosing the ten in third seat after having decided to play an equal from JT could instead be an explicit agreement.)

VI. Draw Inferences from the Play at Trick 2

Robot
1083
K9743
AK2
75
Dummy
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

Before attempting to identify inferences that other players might draw from the play to this trick, Robot would try to place the unseen hearts.

From the players' actions thus far, the visible cards, and the defenders' agreement, Robot would infer that Partner's 6 might be top of a tripleton, top of a doubleton, or lowest from H86. Robot would now test each possibility.

If Partner led from A86, that would give Declarer QJ52. Searching the card-combination catalog for the equivalent single-dummy layout would yield

                         T (Dummy)

                                                     x (led)

                      QJxx, x < 9

which would reveal that Second Hand would play an x rather than the jack (to prevent being finessed against if LHO's hearts were headed by AK9). This indicates that Partner did not lead from Axx.

 

If Partner led from Q86, that would give Declarer AJ52. Searching the card-combination catalog for the equivalent single-dummy layout would yield

                        T (Dummy)

                                                         x (led)

                      AJxx, x < 9

which would again reveal that Second Hand would play an x rather than the jack (to prevent the defenders from establishing a second heart trick by force). This indicates that Partner did not lead from Qxx.

 

If Partner led from 652, that would give Declarer AQJ8. Searching the card-combination catalog for the equivalent single-dummy layout would yield

                         T (Dummy)

                                                             x (led)

                       AQJx, x < 9 (where the cards in bold are alternatives)

indicating that Partner's leading from xxx would be consistent with Declarer's playing the jack. (This entry in the card-combination catalog means that whether Second Hand should play the jack or the x would depend on the broader layout.)

If Partner led from 6x, that would give Declarer AQJ8x. Searching the card-combination catalog for the equivalent single-dummy layout would yield

                         T (Dummy)

                                               x (led)

                      AQJxx, x < 9

indicating that Partner's leading from xx would again be consistent with Declarer's playing the jack.

The above searches would lead Robot to infer that Declarer started with AQJ8 or AQJ8x.

VII. Infer Partner's Inferences from the Play at Trick 2

Robot
1083
K9743
AK2
75
Dummy
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 then consider how Partner, after seeing Trick 2, would search the card-combination catalog to try to work out the heart position. If Partner holds xx, then from Partner's perspective, the relevant holdings for Declarer would be four or five cards that include the jack and exclude the king and ten. To identify the holdings that Partner would deem consistent with Robot's having played the king, Robot would emulate Partner's searching the card-combination for single-dummy layouts that match

                       T (Dummy)

K                                            x (led)

                        J (from four or five pieces)

Assume that this search would yield layouts in which Robot started with AKxxx(x) or Kxxxx(x), x < 10 in either case. Robot infers that this would enable Partner to place the queen with Declarer. (Choosing the queen in third seat after having decided to play an equal from KQ could instead be an explicit agreement.)

Robot would then emulate Partner's searching the card-combination catalog for positions where Second Hand started with the queen but not the king and that match the following:

                        T (Dummy)

                                               x (led)

                        J

Assume that this search would yield layouts in which South's remaining cards would be headed by (1) ace-queen (at least two winners despite having expended the jack), or (2) queen-nine (still being able to create a winner by forcing out the ace). Robot would conclude that if the 6 was from a doubleton, Partner would place Robot with hearts headed by AKx, x < 9 or by Kx, x < 10.

Robot would repeat this exercise for where Partner's lead would be from 6xx. For brevity, it is not shown, but it also would prompt Robot to infer that Partner would place Robot with hearts headed by AKx, x < 9 or by Kx, x < 10.

VIII. Infer Declarer's Inferences from the Play at Trick 2

Robot
1083
K9743
AK2
75
Dummy
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

Given that Robot inferred Declarer as having been been dealt AQJ8 or AQJ8x, Robot can emulate Declarer's using the defenders' actions thus far and their agreement, to infer that RHO's 6 lead was from 6xx or 6x and that LHO was dealt K97x or K97xx.

IX. Incorporating Other Players' Inferences When Planning the Defense

Future articles will explore how Robot might incorporate other players' inferences when planning the defense.

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