In each of the following problems, neither side is vulnerable. The auction goes:
The choices are obvious – double or bid 5♥. Your call?
1. ♠43 ♥AJ8765 ♦AQ4 ♣32
2. ♠432 ♥AJ8765 ♦AQ4 ♣3
3. ♠43 ♥AJ875 ♦AQ43 ♣32
4. ♠43 ♥AJ875 ♦AQ843 ♣2
5. ♠432 ♥AJ875 ♦AQ843 ♣–
6. ♠432 ♥AJ875 ♦AQ43 ♣2
Applying the Law of Total Tricks
Larry Cohen has made his living with these kinds of problems. He pictures his partner with some 1-4-4-4 hand, and counts up the trumps, basing his decision on the total tricks predicted. So, if there are 20 trumps, and so 20 tricks, it is clear to bid – one side or the other will make their contract. Similarly, if there are only 18 trumps, it is equally clear to defend – if we can make five hearts, eleven tricks, then we will hold them to seven for 500. Most of the time, both contracts will fail.
19 is a toss-up. Since we have the balance of power, we expect to set four spades. If we can make five hearts, 11 tricks, we will collect 300 from four spades, and lose four IMPs. If we are down at the five level, we could have collected 100 from four spades doubled, another four IMPs. We are gambling four IMPs either way, so bidding or defending seem equivalent. We will have to look at other factors to decide whether to compete or defend.
With that in mind, the Law sees 20 tricks on hand 1, making bidding absolutely clear. Likewise, there are only 18 trumps on hand 6, so that is an automatic double. This is true of hand five also, but that has so many plus features, the void, the double fit, that most players would bid on with hand 5, despite only 18 trumps. I agree, and so we will call that a Law five heart bid.
The other three hands have 19 trumps, and so need further analysis. The possible double fit in hand four clearly tips that towards bidding. The other two seem to be toss-ups. We’ll flip a coin and let our long heart suit come up heads, bidding on 2, defending on 3.
The Law defends on hands 3 and 6, bids with the rest. Hands 1, 4, and 6 are clear-cut. The others are close, and could go either way.
Applying SF and Loser Counts
Likewise, using my methods, we also envision partner with a 1-4-4-4 pattern, and count our losers, and the second fit opposite such hands. We will start with SF+2, and adjust where needed.
The two five-five hands have SF=18, and so at least 20 total tricks. Bidding on these is clear. The other four hands have SF=17, and so need further calibration. Recall, when SF=17 for a completely pure hand, there are SF+3, or 20 tricks, 6 losers, 3 on each side. On two of these hands, we have three spades and one club, and so only two short-suit losers, not three. On those hands, we are a full trick ahead of par. Admittedly, partner will likely have some wastage opposite our singleton, and we may get club ruffs, further lowering the trick total, but we are starting out with 21 total tricks before adjusting downward, and I can’t see losing two full tricks to wastage or ruffs, so bidding is clear here as well.
That leaves hands 1 and 3, where we have two doubletons. These hands seem to be the closest cases. Our hand is very pure, which suggests that we are closer to SF+3, or 20 tricks, than SF+2, so bidding seems correct for these as well. On these two hands, I bid, but only because of the relative purity of the hand. Give me, say, 43 AJ8765 A43 Q3, and I would defend.
So my methods suggest bidding on every hand, but, for two of the hands, we bid only because the hands are fairly pure.
Simulations
I ran simulations on all six hands[1], giving the opening bidder six spades, and 5-10 points, and giving partner 11+ points, with 1-4-4-4 shape. Bidding, as I anticipated, was correct on every hand. The table below gives the average IMPs won by bidding versus defending:
Table 6: IMPs Gained by Bidding Five Hearts on the Quiz Hands
Hand Number |
Hand |
IMPs gained by bidding |
1 |
♠43 ♥AJ8765 ♦AQ4 ♣32 |
3.87 |
2 |
♠432 ♥AJ8765 ♦AQ4 ♣3 |
2.93 |
3 |
♠43 ♥AJ875 ♦AQ43 ♣32 |
1.80 |
4 |
♠43 ♥AJ875 ♦AQ843 ♣2 |
4.45 |
5 |
♠432 ♥AJ875 ♦AQ843 ♣– |
2.39 |
6 |
♠432 ♥AJ875 ♦AQ43 ♣2 |
1.15 |
It is certainly gratifying that my methods got all six hands correct. However, a closer look at the numbers is less gratifying.
[1] These simulations were, as usual, double-dummy, with one exception. On hand 5, with our void, our double-dummy partner invariably led a club and gave us several ruffs. This skewed the results, so I redid this simulation by hand, forcing partner to make more earthly opening leads.
The Numbers Don’t Lie
There are two features worth pointing out in these numbers. First off, look at hands 1 versus 2, and hands 3 versus 6. My analysis suggests that bidding on is much clearer when we have a side-suit singleton, yet the numbers go the other way. Certainly we were more likely to make five hearts with the extra shortness, but that shortness also led to club ruffs, and increased the penalties against four spades doubled. This is due, in part, to the double-dummy nature of the simulation – we never let a potential club ruff get away, but the evidence also suggests that my loser-count evaluation may place too much weight on side-suit singletons. Clearly more research is needed here.
The other important number shows up in hand 1. Plus 3.87 IMPs by bidding is quite a lot. So defending with this hand would be a gross error. In my view, the purity of the hand means there are close to SF+3, or 20 total tricks, so bidding has to be correct. Law advocates would differ, and see 20 trumps, and so 20 total tricks. Both of us bid, but the rationale is quite different, and the distinction is incredibly important.
To clarify this, I ran one final simulation – giving South a 2-6-3-2 hand, with 10-12 high card points. That still leaves 20 trumps, and so, according to the Law, bidding should still be a huge winner. My belief, however, is that bidding was correct only because of the purity of hand 1. Very few of the arbitrary hands with that shape will be pure, and I would bet that defending will turn out to be the big winner.
We have a perfect test case. One theory suggests bidding, the other, defending. Who wins? Defending, by a landslide. Doubling four spades worked out to be 3.61 IMPs a hand better than bidding.
Now those are gratifying numbers.
Finis
This ends my Theory of Total Tricks series. There is still quite a lot of work and research to go, but I am quite convinced that counting short suit losers and using second fit gives a better estimate of total tricks than counting the number of trumps. Thanks for staying with me, and thank you all for the great comments.
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