I Fought the Law of Total Tricks
“The key is…not in the number of trumps, but in the distribution."
Larry Cohen’s groundbreaking book To Bid or Not To Bid, published in 1992, introduced most of the bridge world to The Law of Total Tricks (LoTT). Larry didn’t invent the Law—Jean-René Vernes came up with it in the 1950s and published an article in The Bridge World in 1969. But Larry’s book brought it to the masses and has been hugely influential in modern bidding theory. “The Law” is used as the reason—or justification—for many a competitive bid.
Simply stated, the LoTT says that on most deals, the total number of tricks that can be taken by both partnerships in their best trump suits will be equal to the total number of trumps the two partnerships have in their best suits.
If N/S have 8 spades and E/W have 8 hearts, the total number of trumps is 16. The LoTT says that most of the time there will be 16 tricks available as well. That might mean that N/S and E/W can each take 8 tricks in their suit. Or maybe N/S can take 9 tricks in spades and E/W can take 7 in hearts. Or vice versa. It could be 10 and 6.
You can’t always tell how many tricks each side is going to make, because much of that depends on the location of crucial cards. But a card that is well placed for N/S is poorly placed for E/W, so the trick total stays the same. For example, if North has the ace-queen of clubs, the location of the king will determine whether they get 1 or 2 tricks in the suit. But it will also determine how many tricks E/W get: if their king is in front of the ace-queen, they have 2 losers, but if it’s behind the ace they have a winner.
The location of that king changes how many tricks your side can take, but it doesn’t change your correct competitive decision. Say you’re E/W and deciding whether to bid 3♥ over 2♠. If the king is well placed for N/S, their finesse works and they take 8 tricks. That means you also take 8 tricks and go down 1 in 3♥. But -50 or -100 is better than -110. (Unless you are vulnerable and they double.) If the king is in the right place for you, N/S only make 7 tricks in 2♠, but you are now taking 9 tricks in hearts and make your 3♥. So either way, it’s right for you to bid 3H.
There are two important extrapolations from the LoTT that make it a very simple and useful tool to guide competitive bidding decisions:
It is usually right to compete to the level of your fit. (So if you have 9 trumps, you should usually compete to the 3-level, where you are contracting for 9 tricks.)
It’s usually wrong to let the opponents play at the level of their fit. (So if they have 8 trumps, don’t let them play at the 2-level. This is why it’s right to bid 3H in the scenario above.)
Larry goes through all the math that leads to these conclusions. If you’re into that, I highly recommend the book. Otherwise, just take his word for it.
The word “Law” in the name is a misnomer. The LoTT is a guideline, and it has proved to be a very helpful one. It’s the reason that after 1♠ (X) we routinely jump to 3♠ with a weak hand and 4 spades and to 4♠ with 5 of them. It’s the basis for Bergen Raises.
Like any guideline, the LoTT is not perfect. Larry freely acknowledges this. In fact, he wrote a follow-up book—Following the Law—about some of the adjustments needed to get the most out of the LoTT.
I Fought the Law is not a condemnation of the LoTT by any means. Rather, it is suggesting an alternate method to add judgment to the LoTT.
I asked Larry about the book, and he didn’t disagree with the authors at all. He said I Fought the Law is basically an extra chapter of To Bid or Not To Bid—it’s taking the framework of the LoTT and adding some useful pieces to improve its efficacy and reliability.
I knew that the LoTT was far from perfect and that some adjustments needed to be taken into account. Pure hands—hands with honors concentrated in long suits—tend to take more tricks than expected; in contrast, hands with lots of honors in short suits take fewer than the LoTT would predict. Double fits increase the trick count while misfits reduce it.
That’s all fine, but I assumed the LoTT was still right a sizeable majority of the time—maybe 60-70%. I have a memory of reading To Bid or Not To Bid in my 20s and going through a hand record and being amazed how often it was right—the number of tricks double dummy said each side could make equaled the number of total trumps.
So I was quite surprised to read that the LoTT is right less than 50% of the time. According to Wirgren’s analysis, which was based on double-dummy analysis, the LoTT is right 55.6% of the time when there are 14 total trumps, 42% of the time with 15 total trumps, and 44.08% of the time with 16 total trumps. For all other trump totals, it is under 30%. Sometimes there are more tricks than the LoTT predicts, sometimes fewer. Overall, the LoTT was exactly right about 40% of the time.
There are other factors that influence the total number of tricks likely to be taken at the table. The LoTT assumes you will play in your best fit—which means the fit in which you can take the most tricks. It’s not uncommon that you have two 8-card fits and can take a trick or two more in one or the other. Maybe you suffer a ruff in one and not the other, or the 4-4 fit produces an extra trick compared to the 5-3 because you get a pitch or two. Sometimes you need to play in your 8-card fit instead of the 9- or 10-card fit to maximize your tricks. You might be able to make an extra trick in an obscure 7-card fit that no one would ever find. It also assumes that you always play from the right side. Mere mortals do not always choose the right strain and play from the right side, so in reality, the trick total will be less in real life than in theory.
And since the total number of trumps is often hard to know for sure, the LoTT can sometimes be hard to utilize. Take a common auction where you have a 3-over-3 decision to make:
Say you have 3 spades. It’s likely your side has only 8 spades—partner presumably would have made the “Lawful” bid of 3♠ with 6 of them. How many hearts do E/W have? Did West bid 3♥ because they have 6? Or were they just keeping you from playing at the 2-level? If there are only 16 total trumps the Law says it’s right to pass; but if there are 17, it could be right to bid on. The LoTT is only going to be so helpful here.
Even when you know the trump total, the number of trumps often isn’t the deciding factor. Lawrence and Wirgren go through example after example where what you would expect from a simple application of the LoTT is wrong. Adding a trump doesn’t necessarily produce an extra trick unless you can do something useful with it.
To illustrate this point, they give this example hand with several possible dummies.
♠ QT97 ♥ AQJ8 ♦ T65 ♣ T5
♠ AK86 ♥ K97 ♦ 742 ♣ 832
♠ AK862 ♥ K97 ♦ 72 ♣ 832
♠ AK862 ♥ K9 ♦ 742 ♣ 832
♠ AK862 ♥ K97 ♦ 742 ♣ 83
♠ AK86 ♥ K973 ♦ 74 ♣ 832
♠ AK862 ♥ K987 ♦ 742 ♣ 8
♠ AK8632 ♥ K98 ♦ 742 ♣ 8
On the first layout, there are 5 minor-suit losers and you can take 8 tricks in spades. Hand 2 adds a trump, and now you can take 9 tricks. Is it the additional trump that makes the difference? Hands 3 and 4 also have a 9th trump but still have 5 top losers. The key to hand 2’s success wasn’t just the additional trump, but where it came from—turning the three losing diamonds into a doubleton added a trick, where taking away a heart or a club in hands 3 and 4 didn’t. Hand 5 demonstrates that even without adding the 9th trump, you add a trick when you remove a diamond. Hand 6 also adds a trick, but it requires removing 2 clubs. The extra trump in hand 7 adds nothing to the hand.
The authors’ conclusion is that the fundamental teaching of the LoTT—that the number of total trumps is the primary factor on the number of total tricks available—is flawed.
Their conclusion is that “The number of tricks available to both sides when playing in their best suit is a function of distribution. The number of trumps held by each side is a secondary issue”(110).
They propose a formula based on two metrics: Short Suit Total—assigning value to voids, singletons, and doubletons—and Working Points—high cards that are taking tricks. This works very well looking at all four hands, but is often somewhere between difficult and impossible to assess at the table. Their definition of Working Points assigns 3 points to a king if the ace is onside but none if it’s behind you. It counts AKxxx opposite xxxx as 10 working points if the suit splits 2-2, since it’s just as good as if you had the queen and jack.
Of course their evaluation method works very well when you get to take the location and distribution of the opponents’ cards into account when counting points. Still, it’s a nifty framework they have come up with, and in the example deals they give, using SST and WP to estimate your side’s trick-taking capacity did prove helpful. Note that they are not interested in the total number of tricks—only how many your side can take.
I found this book enlightening, in that I had assumed the LoTT a better guideline that it actually is. The authors’ argument that distribution is more important than the number of trumps is compelling. I’m curious to see how applying their method of estimating SST and WP works at the table. I’ll let you know.
In all, it’s certainly an interesting read, especially if you’re a LoTT devotee. How useful it proves at the table is still TBD.