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Determination of limit hands
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Delfí Querol, Francesc Bofill




The usual approach to determine whether a hand is limit to open or not the bidding, or to bid or not to bid Smilegame, or slam, has its origin on the point count system, which must and is nuanced with heuristic suppositions based, in much, on insufficiently justifiable personal feelings and convictions.

The advantage of such a procedure is its simplicity but it should be improved. If possible with verifiable tools. Quantifiable tools. Which might help this usual approach, and even or even (tuallySmile) substitute it.

Since always, and in this sense, there is a wide hole in bridge bidding.

In this paper we recall the only, as far as we know, previous attempt to fill the hole. (We apologize if we are missing some efforts unknown to us.) But this attempt revealed to be insufficient, imprecise and useless, as we will show.

So the hole persisted and it seems to us urgent to mitigate this historical lack. In the measure we feel able to contribute. And this is what mainly we deal with in this article.

We present another approach, quite far from point count systems, centred on direct determination of limit hands, through quantified statistical means.


In all the article we will only consider the following auction. And we will only deal with limit hands for game.

E       S       W       N

1     -        1      -

3*   -         ?

(3 *: Splinter: spade support and a heart singleton or void. Forcing only up to 3. Only with extra values will East rebid after any 3 or 4 sign-off from partner.)

(We are interested on this meaning of the 3 call because it allows us including non game forcing splinters. Generically we would have no objection to attend and evaluate meanings, uses or different preferences for 3. And try to derive if there is one that is the best. It would be very instructive, but, even though such issues are also susceptible of convenient and affordable statistical treatment, they fall far outside the scope of this article.)






Numerical limit for limit hands. Probability of game

(This PAGE doesn't need to be read accurately. It recalls the mentioned failed approximation to some quantifiable approach.)

It is known that, for Rubber, and IMPs, the convenience of playing game with a prescribed hand occurs whenever the value of the probability of making game with it is greater than a certain value q which is quite smaller than 0,5. We could consider this value a limit value for game, since a hand having q as its probability of making game with it, it would be a limit hand for game.

Let's find out this value of q. (1)

An approach to determine this value is to assume that the number of tricks that can be made is only 9 or 10. (Which it is not true because sometimes it will be 8 or less, although 9 or 10 are the most frequent cases when dealing with limit hands or hands close to limit hands). This assumption introduces an error, but it will be small and it simplifies, or even makes possible, a calculation. (2)

We make the calculation in the case Rubber, Vulnerable. For others it would be similar.

Let q be the probability of making a 4 game with some prescribed hand.

The points that we expect to win, in the long run, when playing 4 would be

Pts4 = q 620 - (1-q) 100 (*)

and, playing 3

Pts3 = q 170 + (1-q) 140 (**)

Since the hand in consideration is a limit hand Pts4 and Pts3 must be equal. Reading this values in (*) and (**) we have

q 620 - (1-q) q 100 = 170 + (1-q) 140

And we find.

q = 240/690 = 0,3478....

Which is the theoretical limit value we wanted to derive.

The statement corresponding to this result would be: "If the probability of game for a given hand is greater than 0.3478 then game must be bid, and not if it is smaller".

As you see the determination of this theoretical limit value is quite simple.

But it has as much of simple as of useless, because inferring, when receiving a particular hand, which is the probability of game with it, or simply to decide whether it will be higher or lower than this theoretical limit value 0,3478, and determine the appropriate action to do with it is completely unaffordable.

Could you tell, for instance, what is the probability of game for the Hand xxxx below? We do not. We feel sure to affirm that it will be less than the value 0,3478 we obtained, but we couldn't at all attempt a reliable approximation of its value.

And with the Hand Kxxx we will consider later, also in next paragraph? We, neither one thing nor the other.

The following table lists this limit value in other cases too

               Rubber         IMP

Vuln          0,3478        0,375

No vuln      0,4318        0,45


(1) As a matter of fact such a value q, valid for all limit hands, doesn't exist . But it would exist if the assumption of 9 10 tricks, which will follow, was true.

(2) The real lay down is that, without the 9-10 tricks assumption, to different limit hands it would correspond different values of q.


Limit hands. Direct determination

With the auction in the introduction, in Page 1. If West hand was:


Hand xxxx

Q  9 4 2

K J 9

7 2

8 5 3 2


It seems clear that he must sign-off to 3. And it seems clear too that this hand is not a limit hand for game.

Let's check it. We use the usual meaning of limit hand for game: It is a hand that even only slightly improved by any change of one of its cards it deserves a game try, while any slight change to worse demands the contrary.

The file "Bloc xxxx" (3), contains a block of 100 numbered random deals that match the auction and in which the West hand is always Hand xxxx above. In addition, since we want to determine whether the Hand xxxx is a limit hand, we impose that East has a minimum splinter. (4)

There is game in only 6 hands in the block. And indeed we must stop at 3.

We list them for the interested reader: 12, 19, 52, 55, 72 and 85. And we reproduce one of them:


52       107

            8 7 5 4 2

            A Q 3

            K 9 4

 Q 9 4 2              A J 8 5

 K J 9                  -

 7 2                       K 106 5 4

8 5 3 2               A Q 7 6

            K 6 3

            A Q 106 3

            J 9 8

            J 10


We are now going to repeatedly modify this Hand xxxx of West to gradually approach a new hand that can be considered a limit hand.  We change only some cards in clubs. No clubs length, that will always 4. We won't neither touch the other suits.

(This could help the reader if he wants to realize how the modification in a single suit -in which partner will have less than 5 cards- would affect. It seems clear, for example, that the consequences of changes in hearts would have less impact. But this is beyond our current goals.)

The present holding in hearts is especially deliberated to make less plausible a successful defence in hearts by opponents. And allow us to avoid considering this case in our current discussion.

Let's start by changing 8c by Ac.

 Hand Axxx

Q 9 4 2

K J 9

7 2

A 5 3 2


 The file "Bloc Axxx" (which can be obtained, like the others, through the reference at the end of this article) contains 1000 deals for this hand. We chose the value 1000 instead of 100 to get more accurate statistical results.

Instead of writing here the list of numbers of each of the deals for which there is game, we have included it at the end of "Bloc Axxx".

We also remit the reader to the spreadsheet "Full Axxx". In this spreadsheet  in column A is the number of deal and at the right, in column B, the number of tricks in spades for East-West corresponding to the previous deal, with the best play of the hand of both sides. That is to say

Col A    Col B 

  1            8       (in deal 1 in "Bloc Axxx" there are exactly 8 tricks in spades for E-W) 

  2           10      (in deal 2 in "Bloc Axxx" there are exactly 10 tricks in spades for E-W) 

  3            9       (in deal 3 in "Bloc Axxx" there are exactly 9 tricks in spades for E-W).....       ..... etc.

From "Full Axxx" we can easily obtain the percentages or frequencies that we will present. It is interesting to compare the files "Bloc Axxx" and "Full Axxx".


CASE 1. Matchpoints.


We note that there is game in 513 cases (over 1000), that is, a 51'3%, or with a frequency of 0,513. We will accept that the actual probability of game, with Hand Axxx, somewhat hidden by inherent statistical imprecisions, will however differ little from 0,513. (We made some recalculations that allow us making this assertion.) At matchpoints the most likely option has to be played, so, with Hand Axxx, game must be bid.

Also we will later see how greatly a small change in a hand, or concretely in Hand Axxx would modify this value 0,153.

Thus we believe admissible to consider Hand Axxx a limit hand at matchpoints, because 0,513 is very close to 0,5, which would be the ideal value for a limit hand.


CASE 2. IMPs or Rubber (score of Chicago).


The exactness of the procedure we are going to propose is also subject to the inaccuracies inherent to statistical methods. And it may require recalculating certain values from time to time. But clearly advantages the approach in the previous Page in two aspects:

1. It takes into account the fact that the number of tricks may be other than 9 or 10.

2. Instead of trying to determine limit probability values, which was useless and inconsistent as we showed, we try to determine limit hands. We seek concrete limit hands to compare with.

In the calculations that follow, we assume that there are no doubled contracts. We shall see later that this inexactness is not significant and, instead, it facilitates calculations.

Inside the file "Full Axxx" and in all the subsequent "Full ..."   that we use from now on, we have included an additional table, which allows appropriately summarize the adequate action according with the form of punctuation and vulnerability.

And also it lets us infer how far or how close the hand can be considered a limit hand. We reproduce here the table there, in "Full Axxx", in order to explain it. (We recall that all the cases we consider refer to the 1000 hands in "Bloc Axxx" where West hand is always the Hand Axxx.)



         1          2         3           4         5             6          7              8           9          10         11 

Matchpts  Tricks   Freq    Playing 4s           Playing 3s                         IMPs

    0.513        6         1          -400  -0.4          -300   -0.3                        -100      -3    -0.003

                     7        14         -300   -4.2         -200    -2.8                       -100       -3    -0.042

                     8       124        -200  -24.8        -100    -12.4                      -100      -3     -0.372

                     9      348        -100   -34.8         140     48.72                    -240      -6     -2.088

                   10      408         620    252.96     170     69.36                     450      10       4.08

                     11       94          650     61.1          200     18.8                       450      10      0.94

                   12        11            680    7.48         230     2.53                       450      10      0.11

                   13        0

                           1000                   257.3                   123.9      133.4                            2.62

NO VULNERABLE                       186.84                 131.66     55.18                           1.06


At first reading, the reader may spend small attention to what it is italicized.

 Under  1 we find the value 0,513, that we calculated above, and which is the ratio between the number of deals in the block in which there is game in spades (there are 513), and the total of deals (1000). (In percentage this would be 51.3%.)

This value is > 0.5 and the conclusion is that, at matchpoins, 4 must be played. Both vulnerable and non-vulnerable. 

Under 3 the number 14 of the second row indicates the number of deals that have 7 tricks in spades. 7 is the number under 2 just at the left side of 14.

Now we calculate the punctuation we would obtain if we played 4 at every deal in the block knowing that the number of tricks is that in column 2. (We assume that we will not be doubled). Actually we want the points, in average, per deal. That is, we should calculate the sum of the points that we would get playing the whole set of deals and divide it by 1000. This division by 1000 can be anticipated at each step. So:

The number -4'2 in column 5 is obtained by multiplying -300 (which is the score that corresponds to seven tricks, i.e., 3 down) by 14 which is the number of deals in which we would go exactly 3 down. We would obtain the value -4200 that divided by 1000 is the value -4'2 in the table.

The number 257'3 in bold is the sought number. That is, what we would gain on average per deal playing 4 in every deal in the block.

Under 6 and 7 we find the same calculations for the case 3 was played at every deal of the bloc.

Under 8 we find the difference 257'34 - 123'91 = 133'4 which is what we would gain, on average, playing 4 instead of 3. If this number was negative it would mean that 3 would be preferable to 4.

We conclude that in Rubber Vulnerable with Hand Axxx, 4 is to be played.

Finally we consider the IMP scores:

Under 9 they are differences playing 4 or 3 and under 10 the IMPs corresponding to this difference. Under 11 we give the result of the same calculation under 5 but changing points with IMPs of who plays 4.

 The value 2,625 value under 11 indicates the IMPs obtained on average per deal playing 4 against 3 of opponents in the other room.(We assume that in both rooms the same tricks are made. Those there can be done. And that contracts are not doubled)

That is, also in IMPs, 4 must be played.

If non-vulnerable the conclusions are, for this case, that not always, the same. The values remain positive but now they are lower than when vulnerable.


Hand Axxx is a limit hand at matchpoints. But it doesn't seem to be the case for the other two modalities, since in both clearly game should be played. Let's worsen it to search for a limit hand, for Rubber and for IMPs.

In Hand Axxx we change the Ac with Kc

Hand Kxxx

Q 9 4 2

 K J 9

 7 2

 K 5 3 2

The file "Bloc Kxxx" contains 1000 deals fitting the initial auction and in which West hand is always the Hand Kxxx.

In the spreadsheet "Full Kxxx" we find again a first column with the deal number in "Bloc Kxxx" and another column with its corresponding tricks in spades.

Now we won't transcribe the entire table in "Full Kxxx" but we show the most significant results.

At matchpoints: the frequency of game is 0.35. As expected ( Hand Kxxx was limit at matchpoints) you have to play only 3.

                    Rubber            IMPs for who plays 4

Vuln                34,54             0,308

No vuln          -2,96             -0,442


We recall and interpret what the numbers mean:

In Rubber: 34,54 is the points difference, on average, playing 4 instead of 3. As the number is positive we should play 4.

-2.96 Is negative. That is, not vulnerable it is better to play 3 than 4.

Vulnerable 4 must be played and no vulnerable 3.

IMPs:The numbers 0.308 and -0.442 are the IMP are earned, on average, by 4 against 3. Vulnerable 4 must be played and no vulnerable 3.

We review changes when moving from Hand Axxx to Hand Kxxx.

Changing At by Kt (a little more than 1 point in a likely no long in East) gives the transitions

133.43 ---> 34.54

55.18 ---> - 2.96

2,625 ---> 0.308

 1.06 ---> - 0.442

(The results for these two hands are also in column 1 and column 3 of the summarizing table below.)


In view of the above table and as -2.96 is a value close to 0 compared with 420, and 140, -50, -100, ... that are the magnitudes handled in each particular deal, we can consider the Hand Kxxx as a limit 4 hand in Rubber vulnerable. We do not think Hand Kxxx to be limit for other cases.

We need 3 limit hands for the remaining 3 cases. We would have to continue adjusting previous hands, improving them or making them worse to get a more close to 0 value; positive or negative. For the reason at the beginning of this article we will only do changes in the club suit. We exempt the reader of this adjustment work.

Here is a summary table of the results we have obtained. The reader can check in the corresponding files in the link at the end.

 Q 9 4 2

 K J 9

 7 2

The head of each column indicates clubs suit

                                    Axxx            K10xx            Kxxx            QJ10x            QJxx

Matchpoints             0,513           0,383              0,35           0,272           0,235

Rubber Vuln            13,34             53,53            34,5 4         -8,08           -28,43

Rubber No vuln        55,18           7,78                -2,96          -26,8            -37,18

Imps Vuln                2,625         0,755              0,308           -0,7              -1,81

Imps No vuln            1,06            -0,16             -0,442          -1,06             -1,365


 Bold numbers are the closest to 0 in each row. We believe that their proximity to 0 is indicative enough and therefore we state



Matchpoints: Axxx is a limit hand

Rubber: QJ10x is a limit hand vulnerable and Kxxx is a limit hand no vulnerable.

IMPs: Kxxx is a limit hand vulnerable and K10xx is a limit hand no vulnerable.




(3) This file, and all other files used in this article, may be downloaded at the link at the end of this paper.

(4) In this block, and in the other we will use, the hands of North and South and East are drawn and accepted or re-drawn simultaneously depending on if they may or may not be considered to fit the auction and other requirements, if any. Hands are not forced to fit, but are freely drawn and accepted if they are admissible. Doing it in this way we may consider acceptable the statistical results we derive.

Appendix. Opponents can hardly double (5)

What we are going to do now is something we don't like toFrown: instead of only verifying assertions and assumptions, we sometimes just argue with heuristic suppositions based, in much, on insufficiently justified personal feelings and convictions.

Much  complete quantifications are possible but they would take, at least, as much efforts as were needed to quantify all we did till now. And we believe that presenting what we already did would not deserve a delay. Perhaps we will come back to it in the future. 

In the calculations we have assumed that there are no doubled contracts. This made things quite more simple.

We cause inaccuracy but it seems to us that it is of small relevance. Because the feasibility of a successful penalty double is extremely indiscernible. And, in the long run, it might even go against the doublers. There are many deals that could be successfully doubled, but it is very difficult to detect, by a single opponent, which are they. (Below we provide resources so that the reader can check by himself about what we are saying.)

By now we give some reasons in support of the correctness of our assertion. 

1. It is not reasonable for North to double West's 3 or 4 sign-offs.  Because East has not limited his hand and can still have extra strength. And we saw how small variations of the force of one hand will greatly change the chances shown trough the values that we may find in the tables we derived.

(So a double from North, and perhaps also by South, shouldn't be for penalties. At most it should be collaborative. The resources given till now, and those below, could tangentially help the interested reader to check the goodness of some of other alternative agreements for a double. But argue this is not our present intention)

And further North is in front of the strong hand. (Probably a double from South of East's 3 should transmit the message of the plausibility of a final double, rather than or together with a defence in hearts. But argue this is no more our present intention today). 

Almost systematically North cannot double. The chances of a successful double are restricted to half.

2. In the case of an hypothetical 4 closing, South's doubles tend to be dissuaded because the upper limit of force of West has a fairly wide margin, and, as we saw, even small variations of the force of, in this case, the hand of West  will greatly change the chances ...

3. In the case of a 3 closing, when bidding reaches South, opponents have already limited their hands, but the level is one less. Besides, a failed double concedes more than game.

4. If South has 5 trumps he could double quite more frequently but the odds of receiving such a hand are about 0,015 (1.5%)

From now on we assume that we are not in the case 5-0.

5. West may have 5 or even 6 trumps.

6. As we pointed out, detection of a reasonable double is virtually inexistent.

The resources, that we provide now, may help to test these assertions.

Let us visually inspect a few of the deals of two of the files we have generated for the case. They are the files with 100 deals "Bloc Supralimit 4p" in which the West hands are now also drawn and are better than limit, and "Bloc Sublimit 3p" in which they are worse.

Specifically, we have translated "sublimit"  through the conditions:

.) 6hd+ and hd - hh <8, where 6hd+ indicates 6 or more distribution points and hh honours in hearts.

And for "Bloc Supralímit 4p" we have translated supralímit condition by:

.) 15hd- and hd - hh> 7.

We impose 15hd- because otherwise we would be in slam area, and the auction would continue and be another.


These files serve to test options for South to double 4 and 3, respectively. Take into account that, in those two blocks we imposed East to have a minimum splinter, which wouldn't be necessarily the case in caseSmile of a 4 from partner. and, as we saw, small variations of the force of ...So in this cases, the odds for a double still diminish.

For this checking, and to ease avoiding the influence of seeing the 4 hands, we extracted two files that let you see only the hands of South. They are called "CartesSud Supralimit 4p" and "CartesSud Sublimit 3p".

We invite the reader to inspect these two files and try to guess for which deals a double would be successful. At the end of the file we put the number of deals for which it would be so.

If none of opponents has 4 or more trumps, the detection of a viable double is virtually impossible. 

So, to infer the difficulty of detecting the convenience of a double, when south has 4 or less trumps, it is enough to try to detect it in the case where South has exactly 4. We have imposed this condition in the blocks not to have to discard from inspection those deals where it does not have them.


(5) Here we need some grammatical help from any of you, native english friendly readers. To mean that it will not be easy for opps to double, as we meant to mean Smile, which is correct between:  can hardly, or can't hardly, or neither of both?

Link to download files


One of our nicknames at BBO. It is that thatSmile corresponds to our bidding-oriented affinity facet:


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