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.

Defence to 3♠, part II

There was a small bug in the program I used in the last post. Some people might have noticed that the cross-IMPs didn't quite add up. It was to do with rounding errors but it's fixed now and I ran the whole test again. I also decided that I'd quite like to measure match point results after all, so those are included. Finally, we now have statistics for all four vulnerabilities and you'll see that it makes a substantial difference. I've also quoted some stats as percentages rather than raw numbers and it makes it much easier to read.

          Non-Vulnerable versus Non-Vulnerable

              Cross-IMPs/board     Makes%        MP %
HCP Freq     3NT    3SX    Pass   3SX  3NT  3NT  3SX Pass

0      1  -14.50   4.50   10.00   100    0    0   50  100
1     24  -13.02   3.48    9.54    83    0    2   56   92
2    183  -11.67   3.16    8.51    77    0    4   58   88
3    634  -10.21   2.38    7.83    74    0    8   56   86
4   1605   -8.74   1.61    7.14    74    2   13   53   84
5   2984   -7.82   1.27    6.55    72    4   17   52   81
6   5142   -6.47   0.92    5.56    67    8   23   52   76
7   8054   -5.39   0.70    4.69    63   11   27   52   71
8  11018   -4.50   0.66    3.84    57   15   31   53   66
9  14228   -3.49   0.51    2.99    53   21   35   54   61
10 16816   -2.69   0.42    2.27    50   25   39   54   57
11 19015   -1.90   0.61    1.29    43   31   43   56   51
12 19769   -1.38   0.78    0.60    39   36   45   58   46
13 19917   -0.63   0.82   -0.19    34   42   49   59   42
14 18692   -0.23   1.12   -0.89    30   47   51   62   37
15 16029    0.38   1.38   -1.76    25   53   54   64   32
16 13401    0.63   1.64   -2.27    22   57   55   66   29
17 10503    1.09   1.86   -2.94    18   62   57   67   25
18  7817    1.30   2.26   -3.56    15   66   58   70   22
19  5464    1.65   2.51   -4.17    12   71   60   72   18
20  3741    1.76   2.93   -4.69     9   75   61   74   16
21  2230    2.21   3.21   -5.42     6   81   62   76   12
22  1371    2.15   3.60   -5.74     5   82   61   79   10
23   778    2.31   4.06   -6.38     3   88   63   80    7
24   398    2.11   4.45   -6.56     3   89   62   82    6
25   186    2.31   4.63   -6.95     1   91   63   83    4

            Non-Vulnerable versus Vulnerable

              Cross-IMPs/board     Makes%        MP %
HCP Freq     3NT    3SX    Pass   3SX  3NT  3NT  3SX Pass

0      1  -13.50   3.00   10.50   100    0    0   50  100
1     24  -11.52   1.31   10.21    83    0    2   56   92
2    183   -9.64   0.74    8.90    77    0   10   53   87
3    634   -8.22   0.03    8.19    74    0   15   51   84
4   1605   -6.81  -0.66    7.47    74    2   21   48   81
5   2984   -6.04  -0.80    6.85    72    4   24   48   78
6   5142   -4.91  -0.88    5.80    67    8   29   48   73
7   8054   -4.18  -0.74    4.92    63   11   32   50   68
8  11018   -3.61  -0.43    4.04    57   15   35   52   63
9  14228   -2.94  -0.24    3.18    53   21   38   54   58
10 16816   -2.41  -0.06    2.47    50   25   40   55   54
11 19015   -2.07   0.58    1.49    43   31   42   59   49
12 19769   -1.89   1.08    0.81    39   36   43   63   45
13 19917   -1.49   1.47    0.03    34   42   44   65   40
14 18692   -1.44   2.09   -0.65    30   47   45   69   37
15 16029   -1.24   2.73   -1.50    25   53   46   72   32
16 13401   -1.26   3.25   -1.99    22   57   46   75   29
17 10503   -1.08   3.73   -2.64    18   62   47   78   25
18  7817   -1.19   4.44   -3.25    15   66   47   81   22
19  5464   -1.09   4.92   -3.83    12   71   48   83   19
20  3741   -1.24   5.54   -4.31     9   75   47   86   17
21  2230   -1.10   6.11   -5.01     6   81   49   89   13
22  1371   -1.30   6.61   -5.30     5   82   49   90   11
23   778   -1.38   7.21   -5.83     3   88   48   93    9
24   398   -1.77   7.77   -5.99     3   89   46   94    9
25   186   -1.63   8.01   -6.38     1   91   49   95    6

            Vulnerable versus Non-Vulnerable

              Cross-IMPs/board     Makes%        MP %
HCP Freq     3NT    3SX    Pass   3SX  3NT  3NT  3SX Pass

0      1  -16.50   5.50   11.00   100    0    0   50  100
1     24  -14.96   4.54   10.42    83    0    0   58   92
2    183  -13.80   4.37    9.43    77    0    1   61   88
3    634  -12.66   3.78    8.88    74    0    3   61   86
4   1605  -11.31   3.08    8.22    74    2    6   59   85
5   2984  -10.32   2.69    7.63    72    4    9   58   82
6   5142   -8.80   2.23    6.57    67    8   15   58   78
7   8054   -7.52   1.89    5.62    63   11   19   58   73
8  11018   -6.33   1.68    4.65    57   15   24   58   68
9  14228   -5.02   1.35    3.67    53   21   29   57   63
10 16816   -3.92   1.08    2.83    50   25   34   57   59
11 19015   -2.66   1.00    1.66    43   31   40   58   53
12 19769   -1.81   0.98    0.83    39   36   44   58   48
13 19917   -0.68   0.81   -0.13    34   42   48   58   43
14 18692    0.08   0.90   -0.98    30   47   52   59   39
15 16029    1.15   0.90   -2.05    25   53   57   59   34
16 13401    1.66   1.02   -2.68    22   57   59   60   30
17 10503    2.44   1.05   -3.49    18   62   63   61   26
18  7817    3.01   1.24   -4.25    15   66   66   62   23
19  5464    3.66   1.33   -4.99    12   71   69   62   19
20  3741    4.01   1.61   -5.62     9   75   70   63   16
21  2230    4.88   1.66   -6.54     6   81   75   63   12
22  1371    4.95   1.96   -6.91     5   82   75   65   10
23   778    5.40   2.28   -7.68     3   88   77   65    7
24   398    5.32   2.60   -7.92     3   89   78   65    7
25   186    5.70   2.69   -8.39     1   91   79   67    4 

              Vulnerable versus Vulnerable

              Cross-IMPs/board     Makes%        MP %
HCP Freq     3NT    3SX    Pass   3SX  3NT  3NT  3SX Pass

0      1  -15.00   3.50   11.50   100    0    0   50  100
1     24  -13.92   2.88   11.04    83    0    0   58   92
2    183  -12.47   2.52    9.96    77    0    2   60   88
3    634  -11.31   1.96    9.35    74    0    5   59   86
4   1605   -9.89   1.20    8.69    74    2   10   56   85
5   2984   -9.00   0.96    8.04    72    4   13   55   82
6   5142   -7.60   0.68    6.92    67    8   19   54   77
7   8054   -6.57   0.63    5.93    63   11   23   55   72
8  11018   -5.62   0.72    4.90    57   15   27   56   67
9  14228   -4.58   0.69    3.89    53   21   31   56   63
10 16816   -3.69   0.66    3.03    50   25   35   56   58
11 19015   -2.82   1.00    1.82    43   31   39   58   52
12 19769   -2.27   1.30    0.97    39   36   42   60   48
13 19917   -1.43   1.44   -0.01    34   42   46   61   43
14 18692   -1.00   1.86   -0.87    30   47   48   64   39
15 16029   -0.30   2.24   -1.94    25   53   51   66   33
16 13401   -0.03   2.61   -2.58    22   57   53   67   30
17 10503    0.50   2.89   -3.39    18   62   55   69   26
18  7817    0.74   3.43   -4.17    15   66   56   72   22
19  5464    1.15   3.75   -4.90    12   71   58   73   19
20  3741    1.25   4.26   -5.51     9   75   59   75   16
21  2230    1.80   4.63   -6.43     6   81   61   77   12
22  1371    1.68   5.10   -6.77     5   82   60   80   10
23   778    1.89   5.59   -7.47     3   88   62   81    7
24   398    1.58   6.11   -7.69     3   89   60   83    7
25   186    1.88   6.27   -8.15     1   91   62   84    4

Note the big difference when we're vulnerable and oppo aren't. It's now only a plus to double them in that narrow 12-14 point range — stronger hands do better to play in 3NT.

Defence to 3♠, part I

Ok, this is perhaps a crazy idea but let's see if we can prove it so using simulations.

Imagine your right-hand opponent opening a preemptive 3. Normal stuff, both sides are vulnerable. You can expect a 7 card suit and, say, 3-9 points with Qxxxxxx as a minimum suit quality. Not insane, but fairly aggressive.

Currently your methods are very likely to be that you double for takeout and overcall 3NT with a balancedish hand with a stop, say 16-22 points or thereabouts. What I want to find out is if there is any value in swapping these meanings about. That is, double with a balanced hand and bid 3NT with a takeout double shape. I told you it was crazy!

Possible advantages of doing it this way:

1. We catch oppo in a penalty more often.
2. We can be less worried about the wide range of our 3NT overcall — there's less need to stretch to invite a slam because if we can make 6NT we have a good chance of 1100 from 3X.
3. If partner bids 3NT anyway then it's often being played from the right side e.g. if he has Qx opposite your Ax.
4. We can act with weaker hands than we might have acted with before (see below).

Disadvantages:

1. Partner can't convert a takeout double to penalties (but sitting under the preemptor, remember).
2. You can't double and bid a new suit with a very strong hand because partner might pass 3NT. You'll have to add some more system to handle those hands.
3. If partner passes your 'takeout' 3NT then the contract will probably be wrong-sided.

You might look at these lists and decide that the disadvantages suck horribly and the advantages, while useful, aren't a major incentive. And, heck, you're probably right. But I think it would be interesting to look into it some more. So let's get simulatin'.

Modelling The Hands

East has 7 spades, 3-9 HCP, 5.5 losers or worse, 2+ HCP in spades, less than 4 hearts. Nothing too controversial there, I hope.

South has a balanced hand with less than 26 HCP, his spade length + spade HCP is 5 or more i.e. Ax, Kx, Qxx, Jxxx or xxxxx (remember, we're balanced).

West has less than four spades and less than seven hearts. He's undefined other than that. The reason he has less than four spades is because with four spades he'd almost always bid 4 over whatever we do so our choice will be irrelevant. We restricted his hearts too to make sure oppo don't miss many biddable heart games and skew the results.

North has a six loser hand or worse. With better, he'd be pushing on towards slam or a suit game contract. He also has less than six hearts and no seven card minor either (with six hearts we'd usually play in 4 rather than try for a penalty or declare 3NT). Finally, our combined HCP is 32 or less. This is so that we don't miss too many slams and thereby unfairly hinder the results of the people playing in 3NT.

Here's a hand that this produces:

          S: KJ5
          H: K752
          D: A752
          C: K8
 S: T              S: Q976432
 H: QJ8            H: A64
 D: KQJ86          D: ---
 C: 6532           C: JT4
          S: A8
          H: T93
          D: T943
          C: AQ97

Yeah, I know South has a 10 count! We'll be doing experiments over the full range of strengths. Hopefully we'll prove that passing is right with a balanced 10 count, but who knows? On this hand 3NT makes 9 tricks. However, 3X goes off 3 for +800. Here's another one:

          S: JT
          H: T653
          D: A74
          C: J943
 S: 82             S: Q976543
 H: Q8             H: A7
 D: QJT6           D: 852
 C: A7652          C: T
          S: AK
          H: KJ942
          D: K93
          C: KQ8

19 points this time. Here, 3NT makes 9 tricks again (double-dummy, you get the hearts right). 3X is only off one trick, though and your +200 is crap compensation for your missed +600. One more hand as a tie-breaker:

          S: T
          H: K73
          D: AQ8765
          C: KJ5
 S: ---            S: A876532
 H: AJT652         H: 9
 D: J94            D: KT
 C: Q974           C: T83
          S: KQJ94
          H: Q84
          D: 32
          C: A62

These are random, I promise! This is obviously a rather unlikely layout, though. 3NT here makes 10 tricks but 3X doesn't fare too well and gets whacked for +1100.

On that last example, we'd probably extract our pound of flesh anyway by passing and seeing partner reopen. Fair enough. I haven't thought about protective auctions in the slightest — for now all I'm looking at is how 3NT fares in comparison to 3X.

Running a Tournament

What we're going to do is run a tournament. We have three pairs with three different strategies for handling balanced hands over an opposing 3 opening.

Pair 3NT always overcall 3NT. Their partner never pulls because he never gets dealt the sort of hands where he would want to pull. Sometimes he'll think about pulling if he's doubled when he holds a six card minor, but he'll always decide to stick it out. West has great judgement. If 3NT is going three off he'll always double it. If it's going exactly two off then he's not quite so sure and he'll double it half the time.

Pair X always doubles to show a balanced hand. West doesn't ever get greedy by redoubling and partner always leaves it in.

Pair P always pass, regardless of their hand. Their partner passes too. West, with his renowned judgement, will always bid 4 if it's makeable and pass otherwise.

Let's look at some more examples:

          S: 3
          H: K85
          D: 953
          C: K97632
 S: KT             S: A987652
 H: AQT            H: 93
 D: AQ842          D: 76
 C: J85            C: Q4
          S: QJ4
          H: J7642
          D: KJT
          C: AT

This isn't so good for Pair 3NT. It goes off 4 and West doubles it for -1100 to your side. In reality, North might pull to 4 but that doesn't appear to do much better. Pair X will concede -930 while passive old Pair P will quietly lose -620. On the cross-IMPs, Pair P have +8.5, Pair X have -1.0 and Pair 3NT have -7.5.

          S: ---
          H: JT64
          D: JT74
          C: A9865
 S: AJ             S: Q765432
 H: K873           H: 95
 D: K8             D: 932
 C: QT742          C: J
          S: KT98
          H: AQ2
          D: AQ65
          C: K3

Mmm, nice preempt! 3NT makes exactly, while 3X is only two off. Pair 3NT win 6 IMPs, pair X win 2 IMPs and pair P lose 8 for their measily +200.

Results

Hopefully that all fits well enough. All that now remains is to sit back for a day and let the computer get on with it. Over the course of 200,000 hands, our tournament yields the following results, broken down into individual point counts for South.

               Cross-IMPs/board      Makes       MP Best
HCP   Freq     3NT    3SX    Pass   3SX   3NT    3SX   3NT

0.       1  -13.50   0.50   13.00     1     0      1     0
1.      32  -12.20   2.11   10.09    25     0     30     2
2.     165  -11.73   2.04    9.63   124     1    150    15
3.     690  -11.26   1.47    9.70   537     6    611    79
4.    1574  -10.10   1.49    8.64  1150    43   1344   230
5.    2993   -8.96   1.24    7.67  2097   128   2421   572
6.    5114   -7.72   1.06    6.74  3396   374   3877  1237
7.    7976   -6.98   1.03    5.94  5006   793   5849  2127
8.   11075   -5.56   0.96    4.66  6360  1734   7589  3486
9.   14042   -4.62   1.12    3.64  7373  2853   9228  4814
10.  17141   -3.70   1.24    2.74  8338  4350  10784  6357
11.  19148   -2.82   1.65    1.56  8195  5964  11654  7494
12.  19732   -2.21   1.82    0.86  7736  7076  11709  8023
13.  19448   -1.47   2.15    0.04  6684  8075  11167  8281
14.  18421   -0.90   2.51   -1.49  5544  8679  10435  7986
15.  16201   -0.49   2.97   -2.62  4081  8537   9148  7053
16.  13327    0.03   3.44   -3.55  2834  7705   7617  5710
17.  10573    0.56   3.65   -4.21  1949  6537   6021  4552
18.   7927    0.99   4.11   -5.09  1152  5331   4554  3373
19.   5662    1.27   4.38   -5.70   662  4044   3281  2381
20.   3694    1.33   4.62   -5.99   371  2736   2208  1486
21.   2398    1.56   5.07   -6.66   168  1901   1484   914
22.   1330    1.41   5.55   -6.88    65  1084    861   469
23.    761    1.87   5.82   -7.64    23   659    496   265
24.    385    2.50   5.69   -8.24     8   355    248   137
25.    190    1.81   6.22   -8.09     2   173    129    61

To explain the figures, let's take those balanced 10-count hands which we said earlier we hoped would do well by passing. Out of the 200,000 tests we ran, 17,141 gave South a 10-count. Pair 3NT lost an average of 3.70 IMPs/board on the cross-IMPs, Pair X gained 1.24 IMPs/board but Pair P did the best with 2.74 IMPs/board. Thank goodness for that, I was worried that I'd end up recommending we all start getting involved with these hands! Of those 17,141 hands, 3X made 8,338 times (about 49% of the time) and 3NT made 4,350 times (25%). So it's not surprising that Pass won out. The last two columns count up the number of times bidding 3NT wins over doubling. Here, doubling has the edge by 10,784 - 6,357. I didn't count matchpoint wins for passing.

The results at the top and bottom of the table shouldn't be taken too seriously as the sample size won't be large enough. Figures in the 5-21 range, though, will generally be fairly accurate.

The most important thing to note is that doubling oppo in 3 does better than bidding 3NT yourself for every single point count. And it's not marginal, either — you win stacks more IMPs by playing for a penalty. The second thing to notice is that passing is only the best option up until the 10 HCP mark. At 11 HCP it's close, at 12 HCP there's only one IMP difference but after that doubling is miles better. This is what I said earlier about being able to act with weaker hands — doubling is better than pass (just) with an 11 count, but you need a 14 count before overcalling 3NT is better than pass.

It should be noted, too, that despite the fact that doubling with a balanced 11 count is net positive on the IMPs, it's super high-variance. Your team-mates might not be best pleased if you come back with -730 or worse a full 43% of the time!

Conclusions

So should we all start changing our system now? Hell no! First of all, we need to look at the downsides. It may well turn out that the cost of bidding 3NT with a takeout double shape is far too expensive. But even if that wasn't too prohibitive, we still haven't really proved anything. You need to look at the system as a whole, not just one decision by one half of the partnership. If we have to guess what to do then doubling is a better guess than 3NT, but it may transpire that passing and expecting partner to do something sensible is better still. We shall see.

It is interesting, though, and I'll keep looking into it. I'd be very interested to hear of other people's thoughts on this.