Is going off in a game worse than missing an overtrick?

It's been a while since the last post, but I promised neither frequency nor regularity so won't apologise! I've just not been doing very much simulating recently. Somebody did ask me a question towards the end of last year so I might post on that subject at some point. Anyway, on to today's topic, which isn't even about simulations at all.

Your partner is declaring a game contract and has to tackle a suit of AKJx opposite xxxx. With zero information to go on other than the a priori odds, he plays for the drop and goes down one, -100. At your teammates' table, they take the finesse and make the contract, +620, lose 12 IMPs. Your partner has just carved the contract and cost your team 12 IMPs. What an idiot! And you tell him so.

Now, this isn't a piece about being polite to partner. You'd be better off keeping quiet and moving on to the next board, but some people can't keep schtum and have to say something. It's the magnitude that I'm questioning. You shouldn't tell him off for losing 12 IMPs, you should tell him off for losing 0.8 IMPs.

What am I talking about? He clearly lost 12 IMPs because there's a big 12 written in the minus column and it was all his fault! But this doesn't account for the fact that he might have been successful in his line on a luckier day, or it might have made no difference. He wasn't always going to lose 12 IMPs when he made this decision.

These are the probabilities of the various opposition holdings, courtesy of Richard Pavlicek's calculator tool:

    East  West  Ways   %
1   Qxxxx —      1    1.96
2   Qxxx  x      4   11.30
3   Qxx   xx     6   20.35
4   Qx    xxx    4   13.57
5   Q     xxxx   1    2.83
6   xxxx  Q      1    2.83
7   xxx   Qx     4   13.57
8   xx    Qxx    6   20.35
9   x     Qxxx   4   11.30
10  —     Qxxxx  1    1.96

It's quite straight-forward. For instance, line 1 shows that there is a 1.96% chance of East having all five missing cards. The blue lines are where it makes no difference which line you take. If the suit is 5-0, for example, both the finesser and the dropper will realise on the first round and make the same number of tricks. The red line is where the dropper will make while the finesser will go off. The green line is where the finesser will make while the dropper will go off. The finesser wins 20.35% of the time and the dropper 13.57% of the time. The ratio between these is why you see in books that the finesse is a 3:2 favourite.

Let's look at it in terms of IMPs. For all the blue cases — about two thirds of the time — it's a flat board. For the green case the finesser wins 12 IMPs. For the red case the dropper wins 12 IMPs. Thus, the finesser will win on average (0.6608 * 0) + (0.2035 * 12) + (0.1357 * -12) = 0.8136 IMPs.

As you can see, the 12 IMPs your partner cost the team is a mirage. He made a 0.8 IMP mistake — the rest of it was just bad luck. If his line had made it would still have been a 0.8 IMP mistake, even if it had gained IMPs. Now, 0.8 IMP mistakes are fairly bad as things go — if you make them on every board of a 32-board match you'll lose by a whole 26 IMPs — but there are far worse crimes at the bridge table.

Here's an example of a far worse crime. You're declaring 3NT and have 10 tricks on top. You merrily cash them away, lose concentration, don't realise you're actually squeezing somebody and your six of clubs is good. You've dropped an overtrick and your team loses 1 IMP. No probability calculations are needed — your play had no upside and you just took a 0% line for 11 tricks when you had a 100% line available. Your mistake was worth precisely 1 IMP, clearly greater than 0.8 IMPs. Next time your partner takes a view and plays for the drop instead of finessing, don't be so hard on him — especially if you dropped an overtrick earlier on!

Now one thing (as you may be shouting out now) which I've ignored here is variance. Given the choice of which mistake to make you might claim you'd still prefer to lose an overtrick, because if you're 5 IMPs up going into the last board of a knockout match then the overtrick error will never cost anything, whereas the failure to finesse will cost the match 20% of the time. And that might be true but it does require some fairly rigid assumptions. It goes out the window when you play a league match, or a multiple teams event, or it's early on in a sufficiently long knockout match. For the vast majority of situations, all you should need to worry about is maximising your expected number of IMPs.

One lesson you might learn from this is to not let yourself get bogged down in esoteric safety plays and squeeze chances and ignore simple basics like concentration and card counting. Very few of these plays will gain you more than half an IMP of advantage over the 'normal' line. It's all a waste of time as soon as you make a silly mistake and let through a no-play game costing 12 IMPs. Even if you only make one such mistake every 100 boards (and very few of us could say that), you're going to have to find the half-IMP brilliancy every 4 boards in order to compensate. The random deal just doesn't provide that kind of ammunition.

Other lessons you might learn are ones of partnership harmony. Don't be so hard on partner when his mistake appears to cost a game swing. At least consider whether his play had an upside or whether it would usually have made no difference at all. You've almost certainly made lots of marginally negative plays too, but they didn't happen to get highlighted by fate. And try to remember that just the other week you took a finesse to win 10 IMPs when you should have played for a 3-2 break and earned a flat board. It was still an error, despite the outcome, and partner said nothing.

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.

6NT

We missed a slam the other day.

          S: KJT2
          H: K42
          D: AKQ6
          C: K7


          S: Q95
          H: AQ83
          D: 73
          C: AJ64

The auction would bore you but North showed spades and diamonds with invitational values and South saw there was no fit and that the partnership had fewer than the traditionally required 33 points and so stopped in game. As you can see, though, slam is very good. It requires hearts 3-3 or a club finesse or one of several possible squeezes. Why is it so good? Undoubtedly, that 10 is valuable, but even if you take it away, the slam is still a decent spot.

So I endeavoured to find out. My plan was to delve into the details of what makes these hands worth 12 tricks and try to come up with a better evaluation method for 6NT contracts than pure high-card points. However, I didn't need to — points work fine. Over 2,000 tests where the partnership had 32 combined points, slam made a whopping 69% of the time! Since when did this change? We've always been told that you need 33 points to make 6NT and I've never heard anyone say anything to the contrary. Or is this common knowledge that all experts use but which has mysteriously remained a gap in my bridge learning? Post a comment calling me a naïve idiot if you like.

And don't give me any crap about double-dummy results being completely different from real life. I don't buy it. Yes, our solver will pick up any two-way guesses for a Queen or know which squeeze to play for, but in the real world the defence might underlead a Queen at trick one, obviating a guess, or they might have trouble knowing which four-card suit to discard from. And besides, I'm not saying that 69% of slams will make — only that 69% of slams are makeable and I for one would be very happy to bid them on this basis.

It doesn't stop there. With 31 points between the hands, over 2,000 deals, almost exactly 50% of them make 6NT. Who here is surprised by this? This means that you can bid slam more freely knowing that, at worst, it will be makeable 50% of the time.

Of course, if you're bidding these slams with lower point counts you're in more danger of having two cashing aces against you. If only there was some kind of convention which can be used to check for aces after a 1NT opening...

More gambling in third seat

Oops, I'm an idiot. When I did the gambling 3NT test before, I was actually giving North (our partner) 9-11 balanced. That was something which interested me too, but I had forgotten to take the condition out when I did the main test. So, as bad as those stats made it look to open a gambling 3NT in third seat, it's actually an awful lot worse! These are correct (hopefully):

Tricks   Num
0        355
1        482
2        680
3       1080
4       1307
5       1324
6       1323
7       1356
8       1554
9        510
10        29
11         0
12         0
13         0

That figure of 11% making 3NT goes down to 5.4%!

The reason I took a second look at the figures is because Steve asked:

What is recommended for a 3rd seat 3NT? I guess we could increase the hand strength until we reach a certain threshold. But what would that threshold be?

So I ran another 10,000 hands, giving South AKQJxxx but with no restriction on the rest of his hand apart from having no five card major. The following stats emerged:

Tricks   Num
 0        77
 1       136
 2       249
 3       398
 4       605
 5       785
 6      1073
 7      1254
 8      1955
 9      1359
10      1054
11       625
12       322
13       108

HCP      +        -        % 
 0       26      442      5.6
 1       34      355      8.7
 2       67      604     10.0
 3      137      805     14.5
 4      289      998     22.5
 5      282      833     25.3
 6      370      741     33.3
 7      478      659     42.0
 8      501      437     53.4
 9      385      309     55.5
10      324      169     65.7
11      262      109     70.6
12      144       37     79.6
13       79       26     75.2
14       56        6     90.3
15       23        1     95.8
16        5        1     83.3
17        4        0    100.0
18        2        0    100.0

Cont     +        -        % 
 0      244     2043     10.7
 1      361     1695     17.6
 2     1032     1720     37.5
 3      837      764     52.3
 4      650      250     72.2
 5      275       56     83.1
 6       55        4     93.2
 7       14        0    100.0

The first table shows the number of times we made a certain number of tricks. It tells us that, with no restriction on side values, we'll make 3NT only 35% of the time.

The second table shows us the number of side HCPs we hold and how often we make 3NT (+) or go off in it (-). This shows that with 0-7 points on the side we're a fair underdog in our contract, with 8-9 it's pretty close and we need 10+ before we're favourite to make. That's a 20 count, folks. HCPs aren't perhaps a very useful measure so the third table breaks it down by controls (A=2, K=1). This says that with 2 controls (an Ace or two Kings) we're unlikely to make 3NT, with 3 controls it's close and with 4 controls we're a solid favourite.

In other words, we need a pretty massive hand before we can hope to actually make 3NT and if we do get this, we're hardly crippling ourselves by opening at the one level, or 2. My advice is to find another use for your 3NT opening entirely. Over 1 million hands, with North and East passing, I found myself holding the following hands:

7 solid clubs with 3 or more controls on the side:   162 *
6-5 in the minors or longer and 0-12 points:         932
6-5 or 7-4 in the minors or longer and 0-12 points: 1382
7+ minor/4 major and 0-12 points:                    892

These are just suggestions but as you can see they are vastly more frequent than a gambling 3NT opening which you can actually expect to make. The 6-5 hand seems quite enticing, especially if you can lower this to 5-5 when non-vulnerable. Just 932 per million amounts to about once every 1000 hands where you're in third seat — maybe once a year if you play a lot. Better than never, anyway.

* Of course, we should double this figure as solid diamond suits count too.

Defence to 3♠, part III

One of the negative points of using 3NT as a takeout double shape is that when partner passes it (with a few values, no long heart suit and a decent spade stopper) you will often wrong-side the contract. The preemptor's partner is now on lead and can play a spade through the stopper at trick one. This could be expensive, but how expensive?

Today's simulation gives East the same hands he held before. South holds a takeout double hand, about 13+ points with spade shortage and some support for the unbid suits. And North holds a decent spade stopper (KJx or better), 0-3 hearts and no 7 card minor.

This isn't the most wonderful bit of modelling in the world ever, but it doesn't matter much. We're comparing the times when we make 3NT as North with the times when we make it as South. A few deals will slip through where 3NT isn't a realistic contract but it's unlikely that which hand plays it makes much of a difference in these cases.

Over 10,000 deals there were 285 occasions where 3NT made by North but not by South and, interestingly, 193 occasions where 3NT made by South but not by North. On the other 95% of boards, 3NT made from both hands or it went off from both hands.

In terms of IMPs, we lose 0.075 IMPs/board by playing from the South hand.

I must admit that I am a little surprised at these findings and had expected something a fair bit higher, but perhaps I shouldn't have. After all, why should partner's stoppers always be positional? Why shouldn't he be able to duck enough rounds to cut off the preemptor anyway? Here's one of the deals we looked at:

          S: AQ4
          H: KT4
          D: QT6
          C: KJ76
 S: 76             S: KJT9853
 H: QJ6            H: 95
 D: K9753          D: J8
 C: 843            C: 52
          S: 2
          H: A8732
          D: A42
          C: AQT9

Obviously the play is much more comfortable when you have two spade stoppers instead of one, but you'll make 3NT easily enough from either hand. East might have one of the red suit guards, of course, but then you can still make it by guessing which. I'm not suggesting that in any way we're happy to play this contract from the South hand rather than the North hand, just that if we do it's not the end of the world.

So there you have it — wrong-siding 3NT contracts when an opponent has preempted is nothing like as costly as you might have thought.

Gambling in third seat

A friend of mine has been playing a Gambling 3NT opening in third seat, thinking it was the same as first or second seat. I commented that the chances of making 3NT opposite a passed hand were pretty slim and you need to have significantly more to take a shot. That was an understatement. Say you hold:

74 84 72 AKQJT63

Partner is a passed hand, RHO is a passed hand. What odds do you give yourself of making 3NT? Over 10,000 deals, we made the following numbers of tricks:

Tricks   Num
 0        17
 1        78
 2       241
 3       573
 4       759
 5      1071
 6      1452
 7      1925
 8      2753
 9      1098
10        33
11         0
12         0
13         0

You'll make your contract a massive 11% of the time. You'll go 5 off or worse almost 17% of the time. In real life you'll do better than this as oppo have a blind lead and if they get it wrong you can often rattle off 9 tricks, but even if we doubled our odds we're still well short of the kind of percentage we need.

If you happen to be playing a third seat 3NT the same as a first seat one then stop. Right now.