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# Suit combination

A suit combination, in the partnership card game contract bridge, is the combined holding in declarer's and dummy's hands of a specific suit. The term is also used for the order in which these cards should be played to achieve a specified goal, assuming sufficient entries to both hands are available, and that multi-suit cardplay techniques, for example elimination, throw-in and squeeze play, are not used.

Suit combinations vary in complexity, but a well-defined "solution" to each is available. This may be referred to as the "correct" play of the suit. or the "optimum" play, where the word "optimum" is used in a game-theoretical sense, and denotes the play that guards against any possible line of defense, including those that might arise from defenders knowing declarer's cards.

Suit combinations occur frequently as part of the problem of how to play a given bridge hand. Therefore, players at all levels have some knowledge of suit combinations, which varies from bridge maxims ("Eight ever, nine never"), to a more detailed knowledge of probabilities and statistical considerations such as the principle of restricted choice.

Deriving optimum suit plays

The optimum play of the suit is derived by applying well-established game theoretical methods to the problem. 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 optimum plays for suit combinations were traditionally derived by hand, currently computerised methods are available.

Examples

Two tricks are required from the following combination:

The optimum play of this suit is straightforward: play small towards the queen. If it loses to the king, once you have regained the lead you play towards the ten and finesse the jack.H.Francis, A.Truscott, D.Francis: The Official Encyclopedia of Bridge, 5th Edition.] The chance you get two tricks will be close to 3/4. This is easy to see by considering the four possible ways the king and the jack can be distributed over the east-west hands. You succeed in three out of these four cases: the king and the jack in east (24% chance), the king in east (26% chance), and no honor in east (24% chance). On top of that, you also succeed in case east has singleton jack (0.5% chance). The total chance of success is therefore 74.5%.. Two tricks are also required from this combination:

The recommended optimum approachJ.M.Roudinesco: The Dictionary of Suit Combinations] M.Lawrence: How to Play Card Combinations] is to play the ace and then play small towards the jack.

If it loses, then play a small card.

The total chance of success is 94.4%.

If three tricks are required from the second combination a different line of play is recommended by Lawrence.

This is to play the ace and then duck out a round. This works not only when the suit is distributed 3-3 between the opponents or when west has doubleton honour, but also when east holds a doubleton honour. The total chance of success is 64.6%.

Exploiting defensive errors

The optimum treatment of a particular suit combination guarantees a certain minimum likelihood of success against any possible defense. However, such a treatment, whilst guarding against opponents who would exploit any error in declarer play, does not itself exploit defensive errors. In some practical cases when defensive errors are likely, it might be adviseable to deviate from the optimum play of the suit so as to benefit from the assumed defensive errors. In this example, from the Official Encyclopedia of Bridge, declarer needs two tricks from a suit in which he has three small spotcards and dummy has "K Q 10": The game-theoretical optimum approach is to lead towards the king in dummy, and subsequently - whether the king won or not - to lead to the queen.

An expert defender sitting east with the ace, but no jack, is likely to duck on the first round to protect partner's jack. Thus, if this expert defender plays the ace on the first trick, he is most likely to have either the ace singleton, or the ace and jack because with any other combination he would have ducked. In the latter case, declarer's only chance to get two tricks from this suit is to play east for ace-jack doubleton. As the chance for ace-jack doubleton (0.73%) is larger than the chance for ace singleton (0.48%), if the king loses to the ace in trick one, declarer's optimum play is to play for the drop of the jack in trick two and put up the queen.

In practice however, if in the first round the king loses to east's ace, declarer still has to make a judgement call as to whether east would indeed hold up the ace in the first round when not holding the jack. If east is judged as likely to play the ace in the first round regardless of the holding of the jack, declarer should finesse the ten in the second round. Note that an expert sitting east who deliberately makes the exploitive defense of catching the king with the ace whilst holding one or more small cards in the suit (but not the jack), is counting on the fact that declarer would judge him not to make that suboptimal play. This is a Grosvenor gambit: a psychological play that potentially gives away a trick that cannot be cashed by declarer with normal play.

Complex suit combinations

Even without psychological factors, the analysis of complex suit combinations is not straightforward. Human analysis can lead to oversight of certain possibilities, and supposedly optimum approaches to suit combinations were published in the Official Encyclopedia of Bridge, 5th edition, which automated analysis later demonstrated to be incorrect. [http://home.planet.nl/~narcis45/BE3.htm Warmerdam: Improvements on the Suit Combination section of The Official Encyclopedia of Bridge, 5th edition] ]

This suit combination is an example.

Two tricks are required.

The line of play claimed by The Official Encyclopedia of Bridge to guarantee 51% success, is: "Lead low to the nine. If this loses to West, finesse the ten next. If an honor appears from East on the first round, lead low to the nine again; if East shows out or plays another honor, finesse the ten next; otherwise play the ace."

However, using computerised exhaustive searches of his own design, Warmerdam found a play that he claims leads to at least 58% success against any possible defense: "Lead small to the nine. If this loses to West, cash the ace. If an honor appears from East on the first round, run the 9 and if it loses finesse the ten."

The 6th edition of The Official Encyclopedia of Bridge recommends the same line of play as Warmerdam but states that the chance of success is 51%.

Goal setting

Although there can be little debate on what is the game-theorically optimum play of a suit given the suit lay-out and the objective function to be maximised, the choice of what consititutes 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-exploitive line of play that optimises the expected number of tricks from the suit. [ [http://home.planet.nl/~narcis45/SuitPlay/SuitPlayHelp.txt Warmerdam: SuitPlay helpfile] ] An example illustrates the point: What is the best matchpoint play? The line of play that maximises the expected number of tricks from this suit 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. This approach results in three tricks in 28.7% of the cases, two tricks in 54.4% of the cases, and one trick in 16.9% of the cases. The expectation value for the number of tricks is therefore 2.12 tricks. However, this play is "not" optimal in the sense of optimising the above described matchpoint objective. Consider the line of play that starts by taking a deep finesse by playing to the eight. If the eight loses to the nine, next play to the king. 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. This play results in 2.09 expected tricks, a results slightly less than the above 2.12 tricks obtained by playing to the ten. Yet, the play that leads to 2.09 tricks on average, beats the play leading to an average of 2.12 tricks in terms of matchpoint objective.

This can be seen by considering the lay-outs on which the line of play that starts with a deep finesse takes more tricks than the line of play starting with a finesse and vice-versa: it follows that the deep finesse beats the finesse in 22.95% of the cases, while the finesse beats the deep finesse only in 18.33% of the cases. In the remainder of the cases (58.72%) both lines of play lead to the same number of tricks.

Mixed strategies

Further complications can arise as in some cases no single deterministic strategy leads to an optimal result.Jeroen Warmerdam, "Speelfiguren in paren", Bridge Magazine IMP, December 1998 (in Dutch)] Jeroen Warmerdam, "Maniements de couleur en tournoi par paires", Le Bridgeur, no 781, Fevrier 2005 (in French)] . 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.

Interestingly, 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.

Tables of suit combinations

The correct treatment of the various suit combinations obtained with the computer program SuitPlay [http://www.suitplay.com SuitPlay by Jeroen Warmerdam] ] is listed in separate tables. These are classified according to the honour cards (ace, king, queen, jack and ten) missing from the two hands. The tables can be reached using the links provided below.

ee also

*Finesse
*Principle of restricted choice (bridge)
*Safety play
*Bridge probabilities

Literature

* J.M.Roudinesco: The Dictionary of Suit Combinations
* H.Francis, A.Truscott, D.Francis: The Official Encyclopedia of Bridge, 5th Edition.
* M.Lawrence: How to Play Card Combinations
* A.Truscott: Standard Play of card Combinations.

References

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