Suit combination

Specific subset of the cards of one suit

In the card game contract bridge, a suit combination is a specific subset of the cards of one suit held respectively in declarer's and dummy's hands at the onset of play. While the ranks of the remaining cards held by the defenders can be deduced precisely, their location is unknown. Optimum suit combination play allows for all possible lies of the cards held by the defenders. The term is also used for the sequence of plays from the declarer and dummy hands, conditional on intervening plays by the opponents; in other words, declarer's plan or strategy of play given his holdings and his goal for the number of tricks to be taken. In addition to understanding the possible initial combinations and probabilities for the location of the opponents' cards in a suit, declarer can further inform himself from the bidding, the opening lead and from the prior play of cards in establishing the probable location of remaining cards.

WestEast  K 10 8 —  —  K 10 8  K 10 8  K 8  10  10 8  K  8 K 10  10  K 8  K  10 8 The diagram at left shows a heart suit combination with six cards in dummy (North, at top) and four in declarer (South, at bottom). Declarer can deduce that the two opposing hands hold only three hearts the king, the ten and the eight but their exact location are unknown. The table at right shows the eight possible lies of those three cards; the suit combination and its diagram implicitly include all eight possibilities. As the number of cards in a particular suit held by declarer and dummy decreases, the number held by the opposing side must increase since there are always thirteen cards in each suit. Expressed mathematically, the number of possible combinations of n cards held by the opponents is 2n. In the above example, three cards are held 23 or 8 ways (2x2x2 = 8). In this example, the opponents hold four cards in 24 or 16 ways (2x2x2x2 = 16). In this example, the opponents hold five cards in 25 or 32 ways (2x2x2x2x2 = 32).

Typically in standard bridge exposition, not all small cards are explicitly identified and the representation of the hand is made more generic by replacing certain cards with an 'x' where the 'x' represents the 2 or any other card low enough to be equivalent to the 2. The 'x' represents a card below any other that is specified and has no trick taking capability or potential. The following progression of alternatives allows for higher and higher spot cards to be deemed insignificant to the analysis.

One other convention is to put the greater number of cards in dummy, North, if the suit combination comprises two unequal holdings. Given the simplified setting, that makes no difference except for occasional psychological considerations, Crowhurst says. At the table, against two defenders who do see the open hand and don't see the closed hand, the difference may be very important.

Crowhurst generally covers two alternative objective functions, (maximum) expected number of tricks won, or tricks expectation, and (maximum) probability of winning a salient specific number of tricks such as three for a combination with four cards in each hand. That set of two objectives is limited in some ways that are practically important, so they may have a big impact on the application of any findings to "real deals". It turns out that the findings are not simply applicable to trump contracts or to notrump contracts; nor generally applicable to a trump suit or a side suit in a trump contract. The crux of the matter is that the number of winning tricks in a suit is too simple. The number of losing tricks is not redundant and the sequence of winning and losing tricks may be significant. First, consider the given suit combination in a heart contract. If the suit splits 0=5, or – at left and K10876 at right, then the defense has a fifth round winner in hearts, which cannot be avoided. (The fifth trick in a suit may never be played, but the fifth card in trumps is a winner if played on a side suit trick.) In a four card suit combination such as this one, "three winners" usually means "one loser" but that is not redundant, and the distinction between three with one loser and three with two losers may be vital to the objectives of the two sides on a real deal. Second, consider the given suit combination in a spade contract. Three winners on the first three hearts and a loser on the fourth trick — say, T876 opposite singleton king, and dummy leads the queen — leave open the possibility of losing no heart tricks, if the fourth one can be discarded or trumped. Three winners on the first, third, and fourth heart tricks — say, 87 opposite KT6, and declarer leads the ace — imply a loser on the second trick which cannot be avoided (or only rarely). The number of winning tricks for the declaring side, out of four cards in the suit, only approximately matches the objectives of the two sides on a real deal.

Within the simplified setting, declarer's optimal play of a suit combination may be derived using well established game theory, namely the theory of two person zero sum games. Crowhurst generally covers two alternative objective functions for every suit combination in the catalog. One is the (maximum) expected number of tricks won, or tricks expectation. Another is the (maximum) probability of winning a salient specific number of tricks such as three for a combination with four cards in each hand. This means that an objective function to be maximised is specified. For suit play purposes, this objective function (or goal) is usually taken to be the likelihood of making a specified minimum number of tricks. Given this objective, all lines of play are checked against all possible defenses for each distribution of opponent's cards, and the objective function is determined for each of these cases. Each line of play combined with each distribution of opponent's cards can then be assigned a minimum value of the objective function resulting from the best defense for that layout. The optimum line of play is selected as the line that maximises the minimum value of the objective function averaged over all possible layouts. As a result, the optimum solution to the suit combination takes into account all lines of defense (including all forms of falsecarding), and guards against the best lines of defense, but is not necessarily optimal in terms of exploiting errors made by the defense.

Although there can be little debate on what is the game theoretically optimum play of a suit given the suit lay out and the objective function to be maximised, the choice of what constitutes the right objective function for a given practical situation can be subject of debate. Generally, the specification of the objective function depends on the type of scoring. In team matches with IMP scoring, the objective of maximising the imp score usually corresponds to the goal of maximising the likelihood of obtaining a specified number of tricks from the suit under consideration (see above examples). In matchpoint scoring, one usually assumes that the objective of maximising your matchpoint score corresponds to the goal of maximising the expected number of tricks from the suit under consideration. This assumption is not always correct. The goal for declarer in matchpoint scoring rather is to ensure that his line of play beats alternative approaches in term of scoring more tricks on as many lay outs as possible. When applying this 'matchpoint objective' to the line of play for a single suit, optimum lines of play originate that may differ from the non exploitative line of play that optimises the expected number of tricks from the suit. A well known result in game theory states that in such cases an optimal mixed strategy must exist. A small change in the lay out of the last example illustrates this:

What is the best matchpoint play for this suit? The line of play that maximises the expected number of tricks is to finesse by playing to the ten. If the ten loses to the jack, you next play towards the king. If the ten loses to the ace, you next play the queen. Again, this play is not optimal in terms of matchpoint objective, as it gets beaten by the following line of play: take a deep finesse by playing to the eight. If the eight loses to the nine, next play the ten and finesse the jack. If the eight loses to the jack, next let the ten run. If the eight loses to the ace, let the queen run and then finesse over the jack. A similar analysis as in the previous example shows that the line of play that starts with a deep finesse in 31.43% of the cases leads to more tricks than the line of play starting with a finesse. The reverse result holds only in 23.18% of the cases. The above line of play starting with the deep finesse also fails to optimise the matchpoint objective as it gets beaten by another line of play. In turns out that there are a total of eight lines of play that are non transitive: the eight lines of play can be thought to be placed on a circle such that each line of play beats its left neighbor. As a result, the optimal approach in the context of the matchpoint objective corresponds to a so called mixed strategy and is probabilistic in nature: the declarer has to select randomly one of the eight lines of play.