Suit combinations
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In contract bridge the term suit combination refers to the combined holding in declarer's and dummy's hand of one specific suit, but is also used to denote the order in which dummy's and declarer's cards in that isolated suit need to be played in order to optimise a specified goal (e.g. highest likelihood of making a specified minimum number of tricks in the suit). In defining such optimal single suit plays, it is commonly assumed that the remaining three suits provide enough entries to both hands to allow full freedom of choice on the card to be played at each turn. Also, optimal defense is assumed. Note that this assumption implies the likelihood of falsecarding on certain hands.
Obviously, multi-suit playing techniques like elimination, throw-in and squeeze play do not feature in suit combinations. In practice, a full hand is therefore seldomly played solely by invoking the optimum treatment of a suit combination. In other words: in declarer play the four suits 'interact' and can not be treated in isolation.
Yet, suit combinations do occur frequently as part of the problem how to play a given bridgehand. Therefore, bridge players at all levels have some knowledge of suit combinations. For beginning players this may take the form of bridge maxims ("Eight ever, nine never"), whilst for expert players this often takes the form of a profound insight into probabilities and statistical considerations such as the principle of restricted choice.
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[edit] Examples
You are tasked to maximise the chance to get two tricks out of the following lay-out:
| A 10 4 |
| Q 3 2 |
The correct approach 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. The chance you get two tricks will exceed 75%. 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, the king in east, and no honor in east. On top of that, you also succeed in case east has singleton jack (0.5% chance). The total chance of success is therefore 75.5%.
Now consider the following lay-out:
| J 10 5 4 3 |
| A 2 |
How do you maximise your chance to secure two tricks from this suit? The correct approach is again straightforward: play the ace and then play small towards the jack. If it loses to the king or queen, once you have regained the lead you play a small card. This secures two tricks with a 94.4% likelihood.
More interesting is the problem how to play for three tricks from this suit. The correct approach is to play the ace and then to 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 likelihood that you get three tricks amounts to 64.6%.
[edit] Optimum versus correct play
The optimal treatment of particular suit combinations by declarer, can depend on the playing strength of the opponents. One usually assumes that the opponents defend optimally, but this assumption can be flawed in practice.
An example to clarify this point:
You are tasked to maximise the chance of getting two tricks out of a suit in which the dummy has K Q 10 opposite three small spotcards in your hand:
| K Q 10 |
| 4 3 2 |
The correct approach against expert players is to lead towards the king in dummy, and subsequently - whether the king won or not - to lead to the queen.
Any rational declarer would indeed play the queen at trick two if the king made the trick. An expert defender sitting east with the ace (but no jack!) would therefore duck in the first round. Hence, if this expert defender takes the king with the ace, she is bound to have A) the ace singleton, or B) the jack as well. In the latter case, your only chance to get two tricks from this suit is to play east for ace-jack doubleton. In the former case, you should finesse the 10. As the chance for ace-jack doubleton ( 0.73%) is larger than the chance for ace singleton (0.48%), you should play for the drop of the jack in trick two and put up the queen.
Note that an expert on east who would catch the king with the ace whilst holding one or more small cards in the suit (but not the jack) clearly counts on the fact that the declarer is an expert as well. In that case the defender on the east seat made a Grosvenor gambit: a play that gives away a trick that, however, can not be cashed by declarer when he plays rationally.
[edit] Complex suit combinations
Even without psychological factors playing a role, the analysis of suit combinations is often far from straightforward. Suit combinations have appeared in the Official Encyclopedia of Bridge that were later demonstrated to be erroneous.[1]
An example is the following:
| A 10 4 2 |
| 9 5 3 |
Your task is to get two tricks out of this suit.
The correct play is to lead small to the nine. For the continuation, two relevant cases need to be distinguised:
A) If the nine is captured by west with a high card, once you have regained the lead you play the ace. If the ace captures a honor from west, you duck a trick. If the ace captures a honor from east, you next finesse towards the ten.
B) If east plays high, once you have regained the lead you play the nine and let it run. Later you finesse towards the ten.
[edit] Goal setting
Another complication to suit combinations is that the specification of the goal depends on the type of scoring. In team matches with IMP scoring, the objective of maximising your team's 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 rather is to ensure that your line of play beats alternative approaches in term of scoring more tricks on as many lay-outs as possible. In game theoretical terms this corresponds to determining the Nash equilibrium. An example illustrates the point:
| K 10 8 4 |
| Q 3 2 |
What is the best matchpoint play? 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. 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 matchpoints. 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, in matchpoints the play that leads to 2.09 tricks on average is the optimal play.
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.
[edit] Mixed strategies
Further complications can arise as in some cases no single deterministic strategy leads to an optimal result. A small change in the lay-out of the last example illustrates this:
| K 10 8 7 |
| Q 3 2 |
What is the best matchpoint play? 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 matchpoints, 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 be 'matchpoint optimal' 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 it's left neighbor. As a result, the optimal approach in matchpoints corresponds to a so-called mixed strategy and is probabilistic in nature: the declarer has to stochastically select one of the eight lines of play.
[edit] Tables of suit combinations
The correct treatment of the various suit combinations obtained with the computer program SuitPlay[2] 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.
| Suit combinations in contract bridge |  |
|---|---|
| Missing honours:
J10 | Q10 | QJ | K10 | KJ | KQ | A10 |AJ | AQ | AK QJ10 | KJ10 | KQ10 | KQJ | AJ10 | AQ10 | AQJ | AK10 | AKJ | AKQ KQJ10 | AQJ10 | AKJ10 | AKQ10 | AKQJ |
|
[edit] See also
[edit] 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.
