To evaluate the viability of Losing Trick Count and the Bergen method of counting points, I selected hands in which there was at least an eight-card fit and a makeable (according to Deep Finesse) suit contract. I then determined the optimal bidding level for both North-South and East-West using LTC v. dummy points plus declarer Points. For the latter I used the following scale:

Bergen
PointsOptimal
Level

The table below shows what I found. The Level column shows the level of the most lucrative contract for the team as calculated by Deep Finesse. The numbers in columns two through seven are percentages. The second through fourth columns use Bergen points. The fourth through seventh columns use Losing Trick Count.

———-Bergen—————–Losing Trick Count——-LevelExact1-1.52+Exact1-1.52+

1-1.5 means that the specified level by Bergen or TLC was at least one level higher or lower than the best Deep Finesse level but not as much as two levels. 2+ means that the specified level was at least two levels higher or lower than the best Deep Finesse level.

So, the last line of the table (Level 7) indicates that in the 149 hands in which Deep Finesse determined that a grand slam was possible, the Bergen method recommended bidding it 61% of the time, but it was off by one or one and one-half tricks 34% and by two or more tricks almost 5%. LTC was right almost 42%, missed by one 33+% and by two or more 25%. By the way, if the LTC produced a result of more than 13 tricks, I treated that as “Exact.”

In general, Bergen Points appears to be far superior for contracts of level five and above. Maybe this should not be considered surprising for a system that was introduced in a book about bidding slams.

LTC produced more consistent results and had the edge on lower-level hands. In most of those cases Bergen points yielded too high of a bid. Underbidding was rare. Incidentally, the number of hands with low-level contracts is deceptive. In a considerable number of cases the opponents would have dominated the bidding. For example, in many Level 1 boards the opponents would be able to bid game or even slam. Therefore, the fact that Bergen points indicated a higher contract than could possibly be made would be irrelevant. I ran a test on Level 1 hands, however, with those hands excluded, and the results did not change much.

The optimal level calculated by Deep Finesse assumes that both sides play the hand “double dummy.” So, the opening lead is assumed to be the best one possible, and neither the declarer nor the defense make any mistakes. In the real world, of course, people do not play the hands perfectly. Even so, I can think of no better objective way to evaluate the bidding systems.

The other caveat is that some bidding mistakes are much worse than others. A conservative system that recommends bidding at too low of a level will at least produce a positive score. On the other hand, the rewards for accurate bidding at the game level or higher are substantial, especially in total points or IMPs scoring.

In the lecture cited in the first post, Ron Klinger claimed that LTC was “estimated to be at least 80% effective.” The data in this study do not seem to support that claim unless the meaning of the term “effective” is much different from the results of the “Exact” column. It is certainly true that a bid of two spades would be “effective” if nine tricks are available, in the sense that the amount of points won with a two-level bid are the same as with a three-level bid. On the other hand, most bidders would certainly like to know if that ninth trick is likely when the opponents bid three clubs.

I am no expert, but it seems to me that people who depend on LTC — and there are a lot of them — should also consider the adjustments that Bergen recommends, at least on hands with slam potential.

No issue has received greater scrutiny in bridge theory than the quantification of the trick-taking value of a hand, or, since bridge is a team game, a pair of hands. Since Charles Goren’s books began dominating the bridge world after World War II, the primary method has been to rely on point count, where an ace is worth four points, a king three, a queen two, and a jack one. A hand with twelve or thirteen points was considered good enough to open.

A hand with some long suits and some short suits can usually take more tricks. Originally this was accommodated by adding points for short suits. At some point in the last few decades it was found that it was more accurate to add points for length rather than shortness.

The crucial numbers for evaluating the worth of a pair of hands are these:

• Game in no-trump: 25.
• Game in a major suit: 26.
• Game in a minor suit: 29.
• Small slam: 32 or 33.
• Grand slam: 35-37.

Everyone with any experience at all knew that this method had its shortcomings. For example, a queen-jack doubleton was evaluated as three points, but it might not take any tricks at all. Some people systematically adjusted for various things. For example, some subtracted one for no aces or for 4-3-3-3 distribution. Others simply advocated the use of “judgment.”

Ron Klinger and others have championed an entirely different way of determining the trick-taking ability of a pair of hands. This method is called “Losing Trick Count” or LTC for short. If two hands were found to have an eight-card fit, or if one of the hands had a self-sufficient trump suit, LTC claimed that the number of tricks that could be taken by the two hands together could be closely approximated by subtracting the number of “losers” from the magic number of 24.

Counting losers is not difficult: In each hand each suit can have up to the lesser of three and the number of cards in the suit. Aces and kings are never losers (unless the king is a singleton).

• Three or more pieces: Aces and kings are not losers. Queens are not losers if accompanied by another honor: Qxx is 2.5 losers. Jxx or worse: three losers.
• Doubletons: AQ is .5 losers.
• Singleton Ace is no losers; all others are one.
• Void: no losers.

You can estimate your partner’s LTC based on the bidding.

• An opening hand generally has no more than seven losers.
• A level two hand (15-17 points) has around six.
• A jump-shift hand has about five.
• A strong two-club opener has four or fewer.
• A simple raise (6-9) requires nine or so.
• An invitational hand (10-12) has around eight.

A much more detailed explanation is available here.

Note that LTC is, according to Klinger, only applicable when a fit has been found. He recommends using point count when looking for a no-trump contract. Although some people use LTC in determining the value of a hand for purposes of opening, Klinger himself concentrates on using it for determining the optimal level of bidding.

Marty Bergen, on the other hand, has recommended a large number of adjustments to the point count method. He supplied three different algorithms:

• “Starting points” are used until a fit is established.
• “Dummy points” are used when you can support partner’s suit.
• “Bergen points” (I much prefer the term “Declarer points,” and I think that he should have saved his eponymous term for the sum of the two hands) are used when partner has supported your suit.

Bergen uses the same scale for determining the value of the two hands that Goren recommended. In a no-trump contract, both players are expected to use starting points. In suit contracts in which a fit has been established, the declarer uses declarer points and the dummy uses dummy points.

Bergen has published his adjustments in the book Slam Bidding Made Easier and in a series of columns in The Bridge Bulletin. Calculation of starting points begins with high card points, and then makes the following four adjustments:

1. Quacks: QJ=the number of queens and jacks (bad); AT=the number of aces and tens (good). Calculate (AT – QJ) / 3 and round down. Add that number to your count. If QJ is three or more greater than AT, you will end up subtracting (adding a negative number).
2. Quality: Add one for each suit in which you have two of the top three honors or three of the top five.
3. Long: In each suit add one for each card after the fourth one.
4. Dubious: Subtract one for each unprotected honor.

The total is your starting points. Dummy points are starting points with three adjustments for shortness:

1. Doubleton: Add one if you have two or more trump.
2. Singleton: Add two if you have two or more trump; add another one if you have at least four trump.
3. Void: Add the lesser of five and the number of trump that you hold.

Declarer points also start with starting points, but the adjustments are different:

1. Length: Add one for each trump after the fifth one. So, a seven card trump suit is worth five extra points (three in the starting Length adjustment and another two here).
2. Short: Add one for one or more doubletons, two for each singleton, and four for a void.
3. Side suit: Add one if your second-longest suit has at least four cards.

Bergen’s method is designed to be used both for the purposes of determining whether to bid and for collectively finding the optimal level. Since the latter objective is, for suit contracts in which a fit was established, exactly the same as the objective of LTC, comparison of the two approaches made for an interesting application of the Hand Analysis project. The results will be in the next post.