Fourth Seat Weak Twos

A team-mate writes:

Partner's view is that a fourth in hand weak two should be within a trick of making, i.e. 6.5 to 7.5 playing tricks. I gave him

4 A K Q 6 5 4 J 10 3 7 6 3

and his reaction was "Pass in a nanosecond. What else?"

I contend that this hand is worth 2 after three passes. My reasoning: on the evidence so far, my expectation would be that we have 8/9 hearts between us, and oppo have 8/9 spades. So by LTT this is likely to be a 17-trick hand. By virtue of the concentration of hearts in my hand, and the absence of a 2 opening by oppo, I judge we are more likely to make 3 than they are to make 3. So the par contract is either 3 tick, or 3X-1, depending on the vulnerability. Either way, we are more likely than not to go positive. Hence it’s better to open 2 than pass. A simulation should help prove or disprove this hypothesis.

Thanks for the question! This shouldn't be too difficult to do. We'll deal out three hands which wouldn't open the bidding and fix ours. For simplicity we'll assume that the opposition's third seat style is the same as their first seat style. Let's say they open all 12-counts and all 7-loser hands. As explained in this post, it doesn't matter much that this criteria will exclude some genuine hands and let through some false ones — the overwhelming majority will be miles from the borderline.

In order to gather this data I actually wrote some new functions for Deal which makes it all a lot simpler. This lets me produce lovely big tables like this puppy (from a sample size of 10,000 deals):

       E/W Spade Fit       N/S Heart Fit    DD Tricks

NSH  <8s  =8s  =9s  >9s  <8h  =8h  =9h  >9h  2H+  2S+  tot
7    478  371  172      1021                 595  430 1021
8   1597 2102 1313  105      5117           4221 3147 5117
9    896 1299  937  280           3412      3205 2229 3412
10    48  171  172   55                 446  438  396  446
11               3    1                   4    4    4    4

EWS  <8s  =8s  =9s  >9s  <8h  =8h  =9h  >9h  2H+  2S+  tot
5     14                   1   10    3         8        14
6    533                 136  281  115    1  392    3  533
7   2472                 341 1306  778   47 1983  441 2472
8        3943            371 2102 1299  171 3313 2908 3943
9             2597       172 1313  937  175 2349 2426 2597
10                  426       104  271   51  404  413  426
11                   15         1    9    5   14   15   15

Htks <8s  =8s  =9s  >9s  <8h  =8h  =9h  >9h  2H+  2S+  tot
5     12    6             15    3                   6   18
6    151   86   33    2  110  154    8            163  272
7    473  538  215   21  301  739  199    8       856 1247
8   1193 1341  811  102  378 2004  958  107 3447 2220 3447
9    872 1422 1032  216  185 1721 1430  206 3542 2217 3542
10   291  486  422   81   32  432  703  113 1280  655 1280
11    26   64   74   16        58  106   16  180   78  180
12     1        10    3         6    8        14   11   14

Stks <8s  =8s  =9s  >9s  <8h  =8h  =9h  >9h  2H+  2S+  tot
1     17                       14    3        16        17
2     14                        8    6         7        14
3      7    2              1    2    5    1    8         9
4     59                  13   15   30    1   49        59
5    319    4             79  131  105    8  265       323
6    976  107    4    2  187  555  331   16  938      1089
7   1183  922  167   11  311 1245  703   24 1999      2283
8    408 1857  957  111  279 1862 1082  110 2891 3333 3333
9     35  921 1089  201  133 1097  828  188 1811 2246 2246
10     1  124  340  100   18  173  283   91  432  565  565
11          6   38   14        15   33   10   43   58   58
12               2    2              3    1    4    4    4

     <8s  =8s  =9s  >9s  <8h  =8h  =9h  >9h  2H+  2S+  tot
tot 3019 3943 2597  441 1021 5117 3412  450 8463 6206  10k

The labels along the top should be self-explanatory. <8s means that East/West have less than 8 spades between them. 2H+ means that we can make 8 or more tricks in hearts, double-dummy.
Along the left side, NSH means North/South combined hearts; EWS means East/West combined spades; Htks means the number of tricks North/South can make in hearts and Stks is the number East/West can make in spades.

I started to do some analysis of when hearts isn't our best fit or when spades isn't the oppo's best fit but these are actually very unlikely and so I didn't pursue the matter. The fact is that we have a 10 count and nobody has opened yet. Therefore, the points are distributed fairly evenly between the other three hands. And since nobody has opened at the one level or with a preempt, it must mean that they don't have much distribution.

The good thing about nice big tables like this is I can just dump it to the screen and let you all do the conclusions! But I'll throw some points at you for free.

• (highlighted blue) 2 will make about 85% of the time. Can you really afford to pass this opportunity up?
• (red) Of the 6206 times where 2 makes, 3 will make 2961 times.
• (green) You'll make game in hearts a full 15% of the time!
• (purple) There's no guarantee that oppo have a spade fit at all. Even if they manage to find (and this is unlikely) every 5-3 and every 4-4 fit then it only comes to about 70% of cases. And well over half of these are 8 card fits where the suit breaks 4-1.

I'm sure lots more things can be said about this (and please do, in the comments) but I think the evidence strongly favours bidding with this hand.

Update: Another Table

Sorry, I forgot to run stats on when both sides make exactly 9 tricks — a good time to pass this out. Here is the full double-dummy cross table (using 10,000 new deals).

Htks  2H-   2S-   2H=   2S=   3H=   3S=   4H+   4S+   tot
5      26    10           8           5           3    26
6     230    95          54          46          35   230
7    1348   450         385         355         158  1348
8          1138  3350  1012         917         283  3350
9          1361        1290  3535   748         136  3535
10          614         451         202  1303    36  1303
11          117          58          15   192     2   192
12                       13           3    16          16

Stks  2H-   2S-   2H=   2S=   3H=   3S=   4H+   4S+   tot
1       1    14     4           8           1          14
2       3    11     7           1                      11
3       1     9     3           2           3           9
4      12    55    12          14          17          55
5      49   335    92         140          54         335
6     149  1067   316         385         217        1067
7     340  2294   704         811         439        2294
8     447        1012  3271  1290         522        3271
9     406         917         748  2291   220        2291
10    179         255         122          37   593   593
11     16          28          13           1    58    58
12      1                       1                 2     2

      2H-   2S-   2H=   2S=   3H=   3S=   4H+   4S+   tot
tot  1604  3785  3350  3271  3535  2291  1511   653   10k

The important bit is highlighted red. This shows that, of the 3535 times when 3 made exactly, 3 made 884 times — about a quarter of the time.

Responding 3NT to a Preempt

Three hands with a similar theme have cropped up recently:

1.        S: 963
          H: 9
          D: K7654
          C: AKQ3
 S: AKT2           S: J84
 H: K53            H: QJT842
 D: AQT2           D: 9
 C: 52             C: JT6
          S: Q75
          H: A76
          D: J83
          C: 9874

North opened 1, I overcalled 3 as East and partner bid 3NT to play. Oppo started off with a couple of rounds of clubs but then got bored and switched and partner wound up with an overtrick. You may not agree with my choice of overcall, but give me an extra heart and it makes no difference to the outcome — 4 is substantially the better contract.

2.        S: KJ98765
          H: 9
          D: 42
          C: 743
 S: A              S: 42
 H: KJT852         H: Q74
 D: Q6             D: KJT983
 C: T952           C: AK
          S: QT3
          H: A63
          D: A75
          C: QJ86

At our team-mates' table, North opened 3 in first, East passed and South tried 3NT. Not a great success as you can see. 4 has no chance either but it's a lot fewer undertricks.

3.        S: 9743
          H: AJ98752
          D: K
          C: 9
 S: A8             S: QT52
 H: 43             H: 
 D: A9642          D: QJT5
 C: Q652           C: KJ843
          S: KJ6
          H: KQT6
          D: 873
          C: AT7

At a friend's table, North opened 3 in first, East passed and South tried 3NT rather than the raise. It worked out fine on a low diamond lead, but could have gone horribly wrong with 4 excellent.

Bad luck or bad judgement? Let's give our partners some likely hands for their preempts and simulate how the two strains compare.

On the first hand, with LHO opening 1 and partner overcalling 3, I ran 1000 tests. On 498 of those, both contracts failed or both contracts made. However, on every single one of the remaining 502 deals, 3NT went off with the major game making. Never was 3NT better. In terms of IMPs (and I assumed that we were vulnerable for all tests and never doubled), playing in hearts scores you a whopping 7.4 IMPs/board. The reasoning, I suppose, goes as follows: if partner has good enough hearts to run then it's highly unlikely that he's able to stop the club suit.

Next, the spade preempt and again 1000 tests were carried out. This was a lot closer. In fact, the raise to 4 won by only a single case. On 187 occasions the major game was making with 3NT failing, while on 186 the reverse occurred. The rest of the time, both contracts made or both went off. The IMPs score was more favourable to those playing in suit contracts, though, with an average gain of 1.9 IMPs/board. This reflects that, while it's fairly even in terms of purely making your contract, 3NT is likely to go off more and those vulnerable undertricks can add up.

And finally, the other heart preempt when we had that four-card support. This time, of the 1000 tests, 40 resulted in a game swing for 3NT, 330 for raising to the heart game, while the other 630 were more neutral. In all, you gain an average of 5.5 IMPs/board by playing in hearts.

Obviously, the oppo may well have a blind lead. On hand 1, a club lead might not be obvious from a lot of North's holdings and 3NT will sneak home. Nevertheless, I think that a figure of 7.4 IMPs/board is pretty convincing, as is 5.5 IMPs/board for hand 3. Hand 2 is much closer.

Generalisations to follow. Some day. If you're lucky.