Suit combinations
Encyclopedia
In the partnership card game
contract bridge
, a suit combination is the holdings of one suit
in declarer's and dummy's hands. The holdings in two opposing hands are unknown; one suit combination covers all possible lies of the remaining cards in those two closed hands.
A bridge deal diagram usually shows dummy at the top, North, and declarer at the bottom, South. The given diagram shows a suit combination with seven hearts in dummy and four in declarer, or a "7–4 fit". The two opposing hands hold only two hearts, the king and ten. There are four possible lies of those two cards; the suit combination and its diagram implicitly include all four.
The term suit combination is also used for the sequence of plays from declarer and dummy, conditional on intervening plays by the opponents; in other words, the plan or strategy of play with a given two-hand holding. It may be convenient to attribute a plan or strategy of play to the opponents also.
For example dummy and declarer, North and South, may both hold four of the thirteen hearts as depicted. That pair of four-card holdings is a suit combination. So is a conditional plan such as "Lead the queen and play low unless the king covers the queen (thence play the ace). If the queen wins ... If the queen loses to the king ... If the queen is covered by king and ace ...".
The first alternative differs from the original in replacing each of the 5, 4, 3, and 2 by 'x', which conventionally represents the 2 or any other card low enough to be equivalent to the 2 for the purpose at hand. To determine which are those cards does require analysis that is challenging in some instances.
In bridge exposition, 'x' represents the 2 or any equivalent card in the same suit, whence it may be a great challenge to determine which cards 'x' represents in any specific diagram. Lacking such analysis 'x' nevertheless represents a card below any one that is specified. For example, the last of these five variations is equivalent to the original diagram because specification of the '5' implies that any unspecified 'x' is below the 5. The first variation is also equivalent to the original by convention. The middle three represent A in declarer's hand with three cards below 8, three cards below 7, and three cards below 6 respectively. To say which of those diagrams is equivalent to the original, and whether they are equivalent to each other, demands analysis of the play.
(For standard analysis of suit combinations, the 7 is generally different from 'x' where the longer holding (traditionally dummy) is four cards and the defenders hold two high cards such as the king and ten in this case. The 7 is irrelevant if one hand holds QJ98, however, and the 8 is irrelevant if one hand holds AQJ9. Furthermore, when the defenders do hold at least 7 and two higher cards, the cards below 7 become important.)
(As presented here, a diagram may be inconsistent. In the last of these variations on the given combination, there are eight 'x' in the diagram, whose lowest specified card is 9, but a suit contains only seven cards below 9. Put another way, a 12-card combination is given, with the king and ten nominally missing, but there are only thirteen cards in a suit.)
, then, the play of any deal is a zero-sum game.
At least since Crowhurst (1964), the analysis of suit combinations routinely makes further simplifications along the same lines. Most fundamental, the play of any suit combination is a zero-sum
game. In effect, the two sides agree on the relation of the suit to the entire hand so that their opposite entire objectives reduce to opposite objectives in the suit. (The double-dummy nature of the defense, below, makes this an important unexplored objective.*) The bottom line is that their opposite objectives can be expressed in terms of the number of tricks won and lost in the featured suit.
It is common to go two steps further with Crowhurst. First, a suit combination is a two-person zero-sum game. That means the two defenders play as one; they are of one mind. They know each other's cards and thereby, knowing the dummy, they know declarer's hand too. (That particular is properly called double-dummy defense.) One plan governs both their plays. If they choose to randomize their plays (see "Mixed strategy" below), they are able to randomize together.
Second, play of a suit combination amounts to a sequence of tricks with the lead always from dummy or from the closed hand
at declarer's option. In effect, the defenders always switch to a side suit when they win a trick, and declarer stops those side suits at least before discarding from the featured suit. Declarer is able to cross between hands using side suits; i.e. communication or
entry management is no problem.
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 KT876 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.
, 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 optimum plays for suit combinations were traditionally derived by hand, computerised anaysis is now available.
The optimal approach is to lead low toward the queen, a finesse
against the king. If the queen loses to the king, lead low toward the ten, a second-round finesse
against the jack. This wins two tricks about three-quarters of the time. The approximation is easy to see by considering the four possible lies of the king and the jack in the defending hands. You succeed in three of the four cases: both king and jack in east (24% chance), king alone in east (26% chance), and neither in east (24% chance). In the fourth case, king in west and jack in east (26%), you succeed if the jack is singleton (0.5% chance). Overall the probability of success is 74.5%.
Suppose two tricks are required from the next combination:
The optimal approach is to cash the ace and then lead low toward the jack. That fails only against KQxxx(xx) in east; that is the king, queen, and at least three of the five small hearts. In other words, it succeeds if West holds either honor or at least three spot cards. Overall the probability of success is 90.0%.
If three tricks are required, Lawrence recommends a different line of play. Cash the ace and then duck the second trick; that is, play low from both hands regardless of the defense. This succeeds when the suit is distributed 3-3 between the opponents and also when it splits 4-2 with one or both honors doubleton. (Against both honors doubleton, it wins four tricks. Against one honor doubleton it loses the second trick to that honor and the third trick to the other, winning the other three tricks.) Overall the probability of success is 64.6%.
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 judgment 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 exploitative 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.
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.
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%.
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.
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.
Card game
A card game is any game using playing cards as the primary device with which the game is played, be they traditional or game-specific. Countless card games exist, including families of related games...
contract bridge
Contract bridge
Contract bridge, usually known simply as bridge, is a trick-taking card game using a standard deck of 52 playing cards played by four players in two competing partnerships with partners sitting opposite each other around a small table...
, a suit combination is the holdings of one suit
Suit (cards)
In playing cards, a suit is one of several categories into which the cards of a deck are divided. Most often, each card bears one of several symbols showing to which suit it belongs; the suit may alternatively or in addition be indicated by the color printed on the card...
in declarer's and dummy's hands. The holdings in two opposing hands are unknown; one suit combination covers all possible lies of the remaining cards in those two closed hands.
A bridge deal diagram usually shows dummy at the top, North, and declarer at the bottom, South. The given diagram shows a suit combination with seven hearts in dummy and four in declarer, or a "7–4 fit". The two opposing hands hold only two hearts, the king and ten. There are four possible lies of those two cards; the suit combination and its diagram implicitly include all four.
The term suit combination is also used for the sequence of plays from declarer and dummy, conditional on intervening plays by the opponents; in other words, the plan or strategy of play with a given two-hand holding. It may be convenient to attribute a plan or strategy of play to the opponents also.
For example dummy and declarer, North and South, may both hold four of the thirteen hearts as depicted. That pair of four-card holdings is a suit combination. So is a conditional plan such as "Lead the queen and play low unless the king covers the queen (thence play the ace). If the queen wins ... If the queen loses to the king ... If the queen is covered by king and ace ...".
Representation
Consider these five alternatives to the preceding diagram of a 4–4 fit.The first alternative differs from the original in replacing each of the 5, 4, 3, and 2 by 'x', which conventionally represents the 2 or any other card low enough to be equivalent to the 2 for the purpose at hand. To determine which are those cards does require analysis that is challenging in some instances.
In bridge exposition, 'x' represents the 2 or any equivalent card in the same suit, whence it may be a great challenge to determine which cards 'x' represents in any specific diagram. Lacking such analysis 'x' nevertheless represents a card below any one that is specified. For example, the last of these five variations is equivalent to the original diagram because specification of the '5' implies that any unspecified 'x' is below the 5. The first variation is also equivalent to the original by convention. The middle three represent A in declarer's hand with three cards below 8, three cards below 7, and three cards below 6 respectively. To say which of those diagrams is equivalent to the original, and whether they are equivalent to each other, demands analysis of the play.
(For standard analysis of suit combinations, the 7 is generally different from 'x' where the longer holding (traditionally dummy) is four cards and the defenders hold two high cards such as the king and ten in this case. The 7 is irrelevant if one hand holds QJ98, however, and the 8 is irrelevant if one hand holds AQJ9. Furthermore, when the defenders do hold at least 7 and two higher cards, the cards below 7 become important.)
(As presented here, a diagram may be inconsistent. In the last of these variations on the given combination, there are eight 'x' in the diagram, whose lowest specified card is 9, but a suit contains only seven cards below 9. Put another way, a 12-card combination is given, with the king and ten nominally missing, but there are only thirteen cards in a suit.)
Simplified setting
Optimal strategy in the play of one deal at the bridge table varies along with variation in declarer's objective; the opponents' information, skill, and objective; the contract and vulnerability; and the lie of the cards in four hands, which includes four suit combinations and their arrangement. In bridge exposition it is routine to suppose two partnerships with opposite objectives that incorporate the conditions of contest (scoring variant and tournament variant) and the contract and vulnerability. In terms of game theoryGame theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
, then, the play of any deal is a zero-sum game.
At least since Crowhurst (1964), the analysis of suit combinations routinely makes further simplifications along the same lines. Most fundamental, the play of any suit combination is a zero-sum
Zero-sum
In game theory and economic theory, a zero-sum game is a mathematical representation of a situation in which a participant's gain of utility is exactly balanced by the losses of the utility of other participant. If the total gains of the participants are added up, and the total losses are...
game. In effect, the two sides agree on the relation of the suit to the entire hand so that their opposite entire objectives reduce to opposite objectives in the suit. (The double-dummy nature of the defense, below, makes this an important unexplored objective.*) The bottom line is that their opposite objectives can be expressed in terms of the number of tricks won and lost in the featured suit.
It is common to go two steps further with Crowhurst. First, a suit combination is a two-person zero-sum game. That means the two defenders play as one; they are of one mind. They know each other's cards and thereby, knowing the dummy, they know declarer's hand too. (That particular is properly called double-dummy defense.) One plan governs both their plays. If they choose to randomize their plays (see "Mixed strategy" below), they are able to randomize together.
Second, play of a suit combination amounts to a sequence of tricks with the lead always from dummy or from the closed hand
at declarer's option. In effect, the defenders always switch to a side suit when they win a trick, and declarer stops those side suits at least before discarding from the featured suit. Declarer is able to cross between hands using side suits; i.e. communication or
entry management is no problem.
Upside down?
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.Limited scope of conventional objectives
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 KT876 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.
Deriving optimum suit plays
Within the simplified setting, declarer's optimal play of a suit combination may be derived using well-established game theoryGame theory
Game theory is a mathematical method for analyzing calculated circumstances, such as in games, where a person’s success is based upon the choices of others...
, 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 optimum plays for suit combinations were traditionally derived by hand, computerised anaysis is now available.
Examples
Two tricks are required from the following combination:The optimal approach is to lead low toward the queen, a finesse
Finesse
In contract bridge and similar games, a finesse is a technique which allows one to promote tricks based on a favorable position of one or more cards in the hands of the opponents....
against the king. If the queen loses to the king, lead low toward the ten, a second-round finesse
Finesse
In contract bridge and similar games, a finesse is a technique which allows one to promote tricks based on a favorable position of one or more cards in the hands of the opponents....
against the jack. This wins two tricks about three-quarters of the time. The approximation is easy to see by considering the four possible lies of the king and the jack in the defending hands. You succeed in three of the four cases: both king and jack in east (24% chance), king alone in east (26% chance), and neither in east (24% chance). In the fourth case, king in west and jack in east (26%), you succeed if the jack is singleton (0.5% chance). Overall the probability of success is 74.5%.
Suppose two tricks are required from the next combination:
The optimal approach is to cash the ace and then lead low toward the jack. That fails only against KQxxx(xx) in east; that is the king, queen, and at least three of the five small hearts. In other words, it succeeds if West holds either honor or at least three spot cards. Overall the probability of success is 90.0%.
If three tricks are required, Lawrence recommends a different line of play. Cash the ace and then duck the second trick; that is, play low from both hands regardless of the defense. This succeeds when the suit is distributed 3-3 between the opponents and also when it splits 4-2 with one or both honors doubleton. (Against both honors doubleton, it wins four tricks. Against one honor doubleton it loses the second trick to that honor and the third trick to the other, winning the other three tricks.) Overall the probability 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 advisable 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
The Official Encyclopedia of Bridge
The Official Encyclopedia of Bridge presents comprehensive information on the card game contract bridge with limited information on related games and on playing cards...
, 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 judgment 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 exploitative 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.
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.
This suit combination is an example.
Two tricks are required.
The line of play claimed by The Official Encyclopedia of Bridge
The Official Encyclopedia of Bridge
The Official Encyclopedia of Bridge presents comprehensive information on the card game contract bridge with limited information on related games and on playing cards...
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-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. 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
Finesse
In contract bridge and similar games, a finesse is a technique which allows one to promote tricks based on a favorable position of one or more cards in the hands of the opponents....
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
Finesse
In contract bridge and similar games, a finesse is a technique which allows one to promote tricks based on a favorable position of one or more cards in the hands of the opponents....
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. A well-known result in game theoryNash equilibrium
In game theory, Nash equilibrium is a solution concept of a game involving two or more players, in which each player is assumed to know the equilibrium strategies of the other players, and no player has anything to gain by changing only his own strategy unilaterally...
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
Finesse
In contract bridge and similar games, a finesse is a technique which allows one to promote tricks based on a favorable position of one or more cards in the hands of the opponents....
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
Finesse
In contract bridge and similar games, a finesse is a technique which allows one to promote tricks based on a favorable position of one or more cards in the hands of the opponents....
over the jack. A similar analysis as in the previous example shows that the line of play that starts with a deep finesse
Finesse
In contract bridge and similar games, a finesse is a technique which allows one to promote tricks based on a favorable position of one or more cards in the hands of the opponents....
in 31.43% of the cases leads to more tricks than the line of play starting with a finesse
Finesse
In contract bridge and similar games, a finesse is a technique which allows one to promote tricks based on a favorable position of one or more cards in the hands of the opponents....
. 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
Transitive relation
In mathematics, a binary relation R over a set X is transitive if whenever an element a is related to an element b, and b is in turn related to an element c, then a is also related to c....
: 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.
See also
- FinesseFinesseIn contract bridge and similar games, a finesse is a technique which allows one to promote tricks based on a favorable position of one or more cards in the hands of the opponents....
- Principle of restricted choice (bridge)Principle of restricted choice (bridge)In contract bridge, the principle of restricted choice states that play of a particular card decreases the probability its player holds any equivalent card. For example, South leads a low spade, West plays a low one, North plays the queen, East wins with the king...
- Safety playSafety playSafety play in contract bridge is a generic name for plays in which declarer maximizes the chances for fulfilling the contract by ignoring a chance for a higher score. Declarer uses safety plays to cope with potentially unfavorable layouts of the opponent's cards...
- Bridge probabilitiesBridge probabilitiesIn the game of bridge mathematical probabilities play a significant role. Different declarer play strategies lead to success depending on the distribution of opponent's cards. To decide which strategy has highest likelihood of success, the declarer needs to have at least an elementary knowledge of...
Further reading
- J.M. Roudinesco, The Dictionary of Suit Combinations.
- Eric Crowhurst, "Suit Combinations". In The Official Encyclopedia of BridgeThe Official Encyclopedia of BridgeThe Official Encyclopedia of Bridge presents comprehensive information on the card game contract bridge with limited information on related games and on playing cards...
(authorized by the ACBL), 1964 and many following editions. - Mike Lawrence, How to Play Card Combinations.
- Alan Truscott, Standard Play of card Combinations.
- M. Kosmulski, Rozgrywka pojedynczego koloru, Warsaw, 1990 (in Polish).