Computer Bridge: Inferences from the Play to a Trick
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I. Introduction

This article presents thoughts on how a computer-bridgeplayer, often called a "bridge robot," might draw inferences strictly from how a suit is being played.

To enable this discussion to focus on inferences based on the play to a single trick, the following considerations are set aside:

• The meanings of the players' calls during the auction
• The defenders' agreements
• Which of two or more touching cards a defender would play when trying to promote the defenders' holdings or win the trick
• Broader aspects of the deal that might call for actions that otherwise wouldn't be on the player's radar

(It is acknowledged that the drawing of inferences based solely on the play to a single trick is but one of many factors that influence a player's choice of action.)

II. Dummy as Second Hand Plays the Queen from Qxx

In the partial suit layout

Dummy

Q42

Q (played)

LHO

6 (led),

LHO has opening-led the six, and Dummy has followed with the queen from Q42. A human player could infer that if Declarer held J7, for example, Dummy would have played the four or two to guarantee a stopper in the suit.

How a bridge robot might draw this inference. Our computer-bridgeplayer could be equipped with a catalog of card combinations giving LHO's lead, Dummy's holding in the suit, Declarer's holding in the suit, and Dummy's correct play(s). (These card combinations would assume that no card of the suit was previously played.) To identify layouts that are inconsistent with Dummy's playing the queen in the table situation, this catalog could be searched for layouts where the lead is the six or the equivalent, Dummy holds Q42 or the equivalent, and Dummy should play a spot.

One match in the catalog might be

Dummy (to play)

Qxx, x <= 8

LHO

x (led), x <= 10

Declarer

Jx, x <= 8.

The constraints on the x's allow setting the lead to the six and Dummy's holding to Q42; the constraint "x <= 8" on Declarer's holding is consistent with Declarer's holding an insignificant spot in the table situation; and one of Dummy's x's should be played (as indicated by the bold font).

Therefore, in the table situation, Declarer can be inferred not to hold Jx.

Another match in the catalog might be

Dummy (to play)

Qxx, x <= 8

LHO

x (led), x <= 8

Declarer

Jx, x <= 9.

Therefore, in the table situation, Declarer can be inferred not to hold J9.

(The reasoning behind the possibly mysterious constraints on the x's shown for LHO and Declarer in these two matching layouts will be supplied on request.)

III. Dummy as Second Hand Plays a Spot from Qxx

In this partial layout, LHO's lead and the holding for Dummy are as in the previous example, but this time, Dummy plays a spot:

Dummy

Q42

2 (played)

LHO

6 (led).

A human player could infer that if Declarer held 75, for example, Dummy would have played the queen because it would be the declaring side's best chance by far to win a trick in the suit.

How a bridge robot might draw this inference. Our computer-bridgeplayer's catalog that was introduced on the previous page could be searched for layouts where the lead is again the six or the equivalent, Dummy again holds Q42 or the equivalent, and Dummy should play the queen.

One match in the catalog might be

Dummy (to play)

Qxx, x <= 8

LHO

x (led), x <= 10

Declarer

xx, x <= 8.

The constraints on the x's allow setting the lead to the six and Dummy's holding to Q42; the constraint "x <= 8" on Declarer's holding is consistent with Declarer's holding insignificant spots in the table situation; and Dummy's queen should be played.

Therefore, in the table situation, Declarer can be inferred not to hold xx.

IV. Failure by Third Hand to Finesse against Dummy

In the partial suit layout

Dummy

T53

3 (played)

LHO                                        RHO

6 (led)                                     J (played),

LHO has opening-led the six, Dummy has followed with the three from T53, and RHO has followed with the jack. A human player could infer that if RHO held J92, for example, RHO would have played the nine to finesse against Dummy.

How a bridge robot might draw this inference. Our computer-bridgeplayer's catalog could be searched for layouts where LHO leads the six or the equivalent, Dummy holds T53 or the equivalent, RHO holds the jack, Dummy plays the 3, and RHO should not play the jack.

One match in the catalog might be

Dummy

Txx, x <= 8

x (played)

LHO                                        RHO (to play)

x (led), x <= 7                         J9x, x <= 7.

The constraints on the x's allow setting the lead to the six, Dummy's holding to T53, and Dummy's play to the three; RHO's holding includes the jack; and the nine should be played.

Therefore, in the table situation, RHO can be inferred not to hold J9x.

V. Defender Is Second Hand in Front of Dummy's AJT and Follows Low

In the partial suit layout

Dummy

AJT

LHO

4 (played)

Declarer

5 (led),

Declarer has led the five toward Dummy's AJT-tight, and LHO has followed with the four. A human player could infer that if LHO held KQ4, LHO would have played an honor to promote the other honor to master (or win the trick if Dummy does not play the ace).

How a bridge robot might draw this inference. Our computer-bridgeplayer's catalog could be searched for layouts where Declarer leads the five or the equivalent, LHO holds the four or the equivalent, Dummy holds AJT, and LHO should not play the four or the equivalent.

One match in the catalog might be

Dummy

AJT

LHO (to play)

KQx, x <= 9

Declarer

x (led), x <=9.

The constraints on the x's allow setting the lead to the five and LHO's holding to include the four; Dummy's holding is AJT; and one of LHO's honors should be played.

Therefore, in the table situation, LHO can be inferred not to hold KQx.

VI. Give up More Present Tricks or More Future Tricks?

In the above examples, the "inconsistent" layouts involved a player's taking an action that, based solely on the known cards in that alternative layout. would almost surely have been optimal.

This "optimality" seems unattainable to LHO in the following layout:

Dummy

AJ

LHO (to play)

KQxx

Declarer

4 (led).

If LHO ducks, then Dummy could immediately create a second winner by playing the jack, but the declaring side would take only two tricks in the suit (assuming that Declarer was dealt fewer than five pieces).

If LHO instead plays an honor, then (assuming that Dummy plays the ace) LHO would immediately establish a winner, but this might allow Declarer to ultimately take three tricks in the suit from an original holding of T9-fourth.

A play recommended in the catalog should perhaps be accompanied by whichever of the above objectives (if applicable) it is intended to satisfy. (This requires more thought.)

VII. Catalog Layouts Where Cards in a Suit Were Expended Previously

The above examples all involved table layouts where no card in the suit of interest had been played to a previous trick. But the catalog could include layouts in which fewer than 13 cards in the suit of interest are yet to be played.

If this were done for say, 12-card layouts, then their depictions in the catalog should omit the two, and in any searches against these layouts, the table layout should be altered by "promoting" any cards below the one previously played. For example, if the seven was discarded at a prior trick, then the two through six of the table layout would be promoted to the three through seven for catalog search purposes.

This could be extended to layouts involving fewer than 12 cards in the suit of interest.