Another LAW Failure

The third exception bit me.

I was sitting West for this hand, which was the last one played on a Saturday afternoon.


My partner opened 2 with what is a typical collection for him in the first seat at favorable vulnerability. I doubt that many of the other South’s bid 3, but the player at our table did. I anticipated that she would have much better hearts, which made me think that my king would be working. On the other hand, my diamond suit looked like three losers. I tempered my ambitions and bid a humble 3 to extend the preempt. North surprised me by bidding 4. The auction then came around to me. We definitely had nine spades. South probably had at least six hearts. I figured the opponents for ten in all. That would give my partner a singleton, and even if he had two, my king would probably cover one. I decided on 4, which was the final contract.

In the first round I would have doubled with South’s hand. If she had done so, North probably would have played 4, and I would have let her. If my partner could find the magical club lead, we would earn 200 points on the hand. As it was, we lost 50. There were nineteen trumps (ten for them and nine for us) but only seventeen tricks (eight for them and nine for us). The LAW says that this is unlikely in the extreme.

I was not disappointed with my partner’s inability to find ten tricks. I was disappointed to see that we had four certain tricks on defense. So, I should have passed or doubled.

There are three negative adjustments to the LAW: Negative purity, negative fit, and negative shape. These factors result in some holdings being better for defense than for offense.

The most common problem is reportedly negative purity, which is usually identified with broken honor holdings, especially in the trump suit. It is certainly true that both sides have such holdings in their respective trump suits. However, they do not result in defensive advantages. North-South can win only one spade trick on defense, and East-West has no heart tricks if that suit is trump.

In fact, after the first lead the hand plays itself. If North is declaring in hearts, and East finds the killer lead of the 10, East-West gets one spade, one diamond, and three clubs. If East is declaring in spades, North-south gets one spade, one heart, two diamonds, and no clubs. Aside from the trump suit, these are exactly the same results on both offense and defense.

The actual problem for North-South is clearly the third factor, negative shape. The two hands have identical distributions, which means that their ten trumps are not worth as much as they appear. Notice what happens if North trades a low diamond to South for a low club. They gain a trick in both suits if they are declaring in hearts, but they do not gain anything on defense. Furthermore, if North trades a low club for a low diamond, the same effect occurs!

Larry Cohen addresses this problem in To Bid or Not to Bid:

Patterns to be especially wary of are 4-3-3-3 and 5-3-3-2. If your distribution is flat, it becomes statistically more likely that the other players also are flat. If everyone’s distribution is balanced, it often depresses the number of tricks. However, flat distribution does not negatively affect the trick count as often as minor-honor problems do.

East-West has a similar problem. West’s trumps would be more valuable if they could be used for ruffing, but West has fewer cards in no side suit. My only ruffing value was in partner’s shortest suit.

What can be learned from this hand? The main thing is to be very careful about using the LAW on a hand with flat distribution. In this case, however, It was the opponents’ distributions that upset my calculation more than my own. In retrospect I asked myself “How weird (in percentage terms) was the perfect mirror imaging of the opponents’ hands?”

Start with the heart suit. South overcalled at the three level. She cannot have a solid suit; the king is in my hand. When I saw North’s raise to four, it seemed quite likely that South must have six pieces. The most likely reason for choosing a suit over a double in this position is because of an inability to support one of the suits. To be conservative, let us say that the chance of each of them having five hearts is no more than 40%. This is extremely conservative. If nothing was known about either hand other than the fact that the opponents had ten hearts between them, the probability would be 31.18%.

What about the other suits? There is really no good indication available of how any might split, so let’s consider random distributions. The probability of one of the three-card suits splitting 3-3 (assuming hearts are 5-5) is 39.16%. The probability of the other three-card suit splitting 3-3 (assuming hearts are 5-5 and the other suit is 3-3) is 47.62%. Of course, if all three of these suits split evenly, the fourth must also. The probability of all four splitting evenly is therefore no more than 7.46% under very conservative assumptions.

I made the right bid after all.

Judgment v. the LAW

Interpreting partner’s bids in a competitive auction can be tricky.

My favorite bridge book is Larry Cohen’s To Bid or Not to Bid. This best-selling tome promulgated the LAW of Total Tricks, a very clever technique for determining how high to bid in competitive auctions. The LAW had been developed in the sixties by an obscure Frenchman named Jean-Rene Vernes, who wrote an article about it in Bridge World. Cohen refined its use and introduced it to many thousands of players. This technique fundamentally differs from the previous approach, which relied on experience and judgment, in that it is based on actual research.

There is one very large problem with basing one’s decisions on the LAW. It depends upon being able to determine how many cards are held in the respective trump suits by all four players. There is not much that one can do about one’s opponents. Fortunately, as one moves up the bridge ladder, the percentage of opponents who are willing to bid 4S with only seven trumps decreases markedly.

Dealing with partners who are committed to using “judgment” is a different matter. I have a few partners who insist on using the competitive techniques of the days before the LAW was discovered. Here is an example of a hand that demonstrates how difficult it can be for a mixture of techniques to survive.

I was sitting West, and this agglomeration was burning my fingers:

A K J 3    A K 10 8 2    8    A Q 10 4

North dealt and opened 1. My partner passed, and South ventured a heart. I doubled. North bid 2, which was followed by two more passes. As I do in every competitive auction, I tried to count trumps and points. I had more than half of the high-card points in the deck, but both opponents had bid. They must surely have almost all of the missing honors.

I put six diamonds in North’s hand and no more than two in South’s. That left four for my partner. I expected him to have no more than three spades or clubs. The only possible eight-card fit was in hearts. We would never find it, and we would be facing a 4-1 split. Notrump was out of the question. If I doubled again, I would expect partner to bid one of his three card suits. I did not want to play in a 4-3 fit even at the two level opposite a hand with no entries at all. So, I passed.

This was my partner’s hand:

Q 6 5 4 2    7 6 2    9 6 3    8 5 2

He did not bid 2 over the 2 rebid because he “did not want me to get too excited.” He was using forty years of bridge judgment, and his was a reasonable concern. I would in fact have raised him to 4. However, the opponents not only cannot beat this contract; they cannot stop us from taking eleven tricks.

Sitting in his chair I would have bid 2 with his hand without giving it a second thought. While it is true that the hand lacks the points for a traditional “free bid,” I would deem it mandatory to inform partner of a strong preference for one of his two suits. In fact, I would have bid even without the Q!

To be totally fair, if I had passed with his hand, my partner would not have passed the second time with mine. He would have had no clear idea of where we would have ended up, but he would have felt the need to tell me that he had much more than his original double indicated.

So, if we had switched chairs, we would have been fine. What’s more, either of us would have done better with a like-minded partner. I don’t like the idea of having to channel the ghosts of Charles Goren and Edgar Kaplan to interpret my partner’s bids, but I guess that I will need to do it or run the risk of more zeroes in situations like these.

In reality, North’s bidding was cagey. Most people would have opened her hand (which included seven diamonds, not six) 3, but that would have made it easy for us to find 4 irrespective of which chair I sat in.

There Oughta be a LAW

I forgot the “Corrections” to the Law of Total Tricks.

The Law of Total Tricks (the LAW for short) has been around for a few decades. It says that if the honor cards in a bridge hand are distributed roughly equally between the two teams, then the total number of tricks achievable by the two teams is equal to the total number of trump. If North-South is bidding spades, and East-West is bidding hearts, then the total number of potential tricks available to the two prospective declarers should be roughly equal to the number of spades held by North-South plus the number of clubs held by East-West. If North-South has nine spades, and East-West has eight hearts, the LAW predicts that seventeen total tricks are available, but it does not predict how many either side would take. If North-South can take ten tricks if spades are trump, then the prediction is that East-West can take seven if hearts are trump. If North-South can take only eight, then East-West should be able to garner nine.

In Wednesday’s game my partner and I tried to be LAWful, but we came a-cropper on one hand. Two green cards were in view as I examined this hand with neither side vulnerable:

75 QJT62 AQ84 Q5

This hand easily met my criteria for a third-seat opener, so I bid 1. RHO passed, and my partner bid 2 (three-piece Drury) with this aggregation:

9862 AK4 KJ6 T63

His bid said that he had 10-12 points and three hearts. I also knew that he did not have a singleton or void. A different bid is used for that holding. Seeing no chance for game I bid 2. After two more passes RHO ventured 2. I could see no extras in my hand, so I left it up to my partner. He elected to pass as well. The result was not good. They took eight tricks in the black suits. If hearts had been trump, we would have taken nine tricks in the red suits. We scored -110; East-West pairs that bid 3 made +140 points.

When I thought about this hand, I realized that the LAW had been violated. They had seven spades, and we had eight hearts. Fifteen trumps should generate fifteen tricks. However, a total of seventeen tricks were apparently available. Knowing that the LAW is very accurate, I felt a little cheated.

Well, that was not quite right. We could have set the spade contract by forcing declarer to trump the third round of diamonds. In fact, however, the opponents could have actually scored nine tricks if they had found their fit in clubs. So, the LAW’s prediction of sixteen tricks was still two short of what was really available. How could it be so far off?

I revisited the orignal article in The Bridge World by Jean-Rene Vernes. I found in a section called “Corrections” a list of mitigating factors:
  1. The existence of a double fit, each side having eight cards or more in two suits. When this happens, the number of total tricks is frequently one trick greater than the general formula would indicate. This is the most important of the “extra factors.”
  2. The possession of trump honors. The number of total tricks is often greater than predicted when each side has all the honors in its own trump suit. Likewise, the number is often lower than predicted when these honors are owned by the opponents. (It is the middle honors–king, queen, jack–that are of greatest importance.) Still, the effect of this factor is considerably less than one might suppose. So it does not seem necessary to have a formal “correction,” but merely to bear it in mind in close cases.
  3. The distribution of the remaining (non-trump) suits. Up to now we have considered only how the cards are divided between the two sides, not how the cards of one suit are divided between two partners. This distribution has a very small, but not completely negligible, effect.

In this case the first correction probably adds at least one to the the prediction. Our second suit only had seven cards, but it was solid. The opponents had only seven trump, but their best side suit, clubs, consisted of eight cards. What made this hand unusual was the fact that the opponents had all of the black honors except the Q, which was in a doubleton, and we had all of the red ones. So the first two factors combined to add two tricks to the total.

Is there any way that my partner or I could have divined this? I don’t think so, but I think that one of us should have bid 3 anyway. Here is why: Ordinarily, a bidder using the LAW knows that he is “protected” by the distribution if he bids up to the number of trump that his side holds. Our side held eight, so we were protected up to the two level. However, Vernes cited one exception in the last paragraph of his article: “This rule holds good at almost any level, up to a small slam (with only one exception: it will often pay to compete to the three level in a lower ranking suit when holding eight trumps).”

So, the factors in favor of bidding 3 were:
  • We certainly had eight trumps.
  • We had at least half of the honor values.
  • We were not vulnerable.

Unfortunately, neither I nor my partner knew whether the opponents had seven, eight, or nine spades. Since my honors were concentrated in two suits, I think that I should have just taken the bull by the horns and bid 3.

That brings up another question: would the opponents have bid 3? I doubt it. Neither of them knew about the club fit, and neither had anything extra.