Sacrificing for Dummies

It's ten years since I last wrote something here on the Law of Total Tricks. My goal this time is to come up with something really simple to remember when considering a save.

My thoughts on this were prompted by a recent hand:

My overcall of 1♦️opposite a passed hand was not a thing of beauty, I'll admit. But, I'm loath to make a sub-standard takeout double when our side is probably out-gunned. South's 3♣️ was described as "weak." What should West do here? I think a responsive double might work out best. If partner has four spades, we'll find it. If not, we'll likely be playing 3♦️ which can't be all bad. At the table(s), many pairs played 3♠️ either by East or West which mostly made given that N/S didn't find the double-dummy lead of ♦️K or ♦️T.

Over partner's 3♦️, North made a crazy leap to 5♣️. I could have been the hero by doubling (+300) but "knowing" that partner cannot bid higher (see Passed Hands may make only one Free Bid), I thought I'd allow him to pass or double, as appropriate. 5♦️ was completely unexpected and, as I'll show below, very unlikely to be the winning action. It's almost never right to take the last guess! And, it's OK to save with the ace of the enemy suit because it's likely to be of value at defense and offense. But kings, queens and jacks in their suit should be a red flag as they may be useful only on defense.

In fact, along double-dummy lines, N/S can make 3♣️, 2♥️, or 2NT. E/W can make 2♠️ or 3♦️.  21 total trumps. 18 total tricks. I would suggest that the shortfall in total tricks is due to the lack of useful shortness: each side has the (short) top honors in the other side's trump suit.

For the remainder of this article, we will consider entirely hypothetical situations. The following table shows the number of total tricks to make a sacrifice profitable at matchpoints, according to the levels of bidding involved:

Levels	Favorable	Equal Red	Equal White	Unfavorable
4/4	17	18	18	19
5/4	18	19	19	20
5/5	19	20	20	21
6/5	20	21	21	22
6♠️/6♥️	18	19	20	21
6/6m	19	20	20	21
7/6M	19	20	21	22
7♣️/6♦️	20	21	21	22
7/7	18	19	20	21

Note that it is assumed in all cases that the opposing contract is actually making. The requisite number of total tricks may be available but if they are distributed too evenly, the save will be a phantom.

Let's remind ourselves that the most common number of total tricks is 17. If the opponents bid 4♥️, and we have a good spade fit and are at favorable vulnerability, we can consider saving in 4♠️. How do we know if there will be 17 (or more) total tricks? The bidding will give clues as to the fits around the table. But, the simple number of tricks in each direction isn't really sufficient information (see "I Fought the Law"). A trick total of 17 will likely involve some shortness (singleton) somewhere at least. Do you have it? Did partner show shortness? Did one of the opponents? If so, you may try it. Otherwise, you might want to hold back until you think there are 18 total trumps.

There are several likely outcomes in 4♠️. Any time 4♥️ was not making, we will get a poor score, unless 4♠️ makes. Even if they didn't double, -100 instead of +100 (or 200 if we had doubled) will not usually score well.

But let's assume that 4♥️ was indeed making. If they didn't double 4♠️, we are guaranteed a good result. If they do double, as long as our estimate of 17 total tricks was accurate, we should be fine. Except when they could have made 650 and we are down four for -800. That's an all-too-common disaster. That's why, even in this situation, you really would like to have 18 total tricks.

And this is, according to the chart above, the most advantageous situation for taking a sacrifice (shown in green in the table).

There are three other situations where we might seriously consider a sacrifice (yellow rows in the table):

• at the 5-level over their 4-level game;
• 6♠️ over 6♥️;
• 7 over 7.

In each of these cases, we require 18 total tricks (not an uncommon situation) and of course favorable vulnerability. Each worsening of the vulnerability situation (see table) requires one additional total trick. Except in goulash-type hands, deals with 20 or more total tricks are rare. Also note that in the second and third of these situations, the all-white and unfavorable situations are particularly dangerous because they require 21 and 22 total tricks respectively (in each case, one more than the 5/4 sacrifices).

From the red rows in the table, we can also see that we should never (well, hardly ever) even contemplate a sacrifice at the 6-level over a game contract, or 7♣️ over 6♦️, as these require at least 20 total tricks. Don't even think about these when not at favorable.

The other situations (amber in the table -- 5 over 5, 6 over 6 minor, 7 over 6 major) should generally be avoided too. To consider any of these at equal vulnerability--especially the last one when all white--is, well, just madness.

POD: Penalty-oriented double

Back in the early days of bridge there was the penalty double. However, people began to realize that it wasn't much use at low levels because you could rarely get rich even if they were psyching--they would just run to the real suit or otherwise escape justice. Thus the penalty double evolved into two clades: the penalty double and the informatory double (nowadays this is universally known as a takeout double).

It didn't take long for the takeout clade to further evolve into two sub-clades: pure takeout and "cooperative" doubles. The latter include all sorts of strangely named beasts: action, cooperative, competitive, etc.  The general rule (not very well described in the literature) is that the higher the level, the more tempted partner will be to leave it in. This latter form is at its most useful in matchpoint bridge because, if you can catch them speeding and get them down one doubled and vulnerable, it will beat any part-score you might be able to have made. 

I believe there are, however, two sub-clades of the penalty double:

1. pure penalty:  "don't take it out if you ever want to play with me again;"
2. penalty-oriented: "you're expected to leave it in if you have normal distribution, nothing that partner doesn't know about."

Actually, this latter type is perhaps more common than you think. It occurs any time you bid a game, the opponents sacrifice and the player in direct seat doubles. That player can never be sure that defending a doubled contract is the par result. What he means by the double is this: "From where I'm sitting, it looks like doubling is our best shot. Feel free to pull if your hand is more offensively oriented than I think it is."

But it might not be a clear sacrifice. Such situations may be somewhat rarer, but they are not unknown. An example came up just the other day in a pairs event. I will give you my partner's hand and the auction first: ♠KQJ8432 ♥T3 ♦Q ♣K53.

Both sides are vulnerable. Admittedly, not everyone will open 1♠ but I think it's the right bid, provided that you're willing to be a bit flexible later on in the auction. LHO overcalled 2♥ and partner raised to 2♠. RHO upped the ante with 3♥ and you made a preemptive raise to 3♠, showing six or more spades and presumably a hand that is on the weak side for an opening bid. LHO doesn't go quietly and bids 4♥. Partner doubles and it's back to you. Your call?

What do we know about the auction? The opponents are a self-described pickup pair and their profiles suggest intermediate. Do you think they have their bids? They might have stretched a bit, but nobody bids this way without something pretty good. What about partner's double? Is it a penalty double? I think it is. One of the "rules" that I like to go by is that once we've bid our suit three times, any double is for penalty. What could we be taking out into, realistically speaking?

So, what to do? Partner has 8-10 hcp and exactly three spades. He will never try to get a penalty in this situation knowing that our side has 10 spades. It's likely that partner has a relatively balanced hand, too, because with a singleton anywhere, he's going to bid 4♠ if he's at the max end of his box (5-10).

How many hearts do they have? Almost certainly nine. With a stiff (as noted), partner would have bid 4 himself (or passed). 

How many tricks to we have cashing? At most one spade and maybe a club. Partner should have a couple of sure tricks and maybe a third if he has AJ in, say, diamonds. Are we getting rich? At the very most, we might get 500 but 200 is more likely. Can we make game our way? Does partner have the spade ace? I think it's doubtful. With that card, and six points on the outside, I think double is an unlikely call. The spade ace will, essentially, be a bit of a waste, defensively speaking, given our own strong bidding in spades. 

What about "the law?" The strength appears to be well balanced between the two sides which is important for the law. We don't have a pure hand with good shortness (can't count the diamond queen for both), so it's possible that the law will be off a bit. Maybe 18 tricks instead of 19? If this is the case, and if it turns out that both sides can make exactly nine tricks, we should defend. What are the other possibilities (using my guesses for the probabilities)?

• 20 total tricks (15% likely):
• Both sides make game: par score +620 (pull)
• We make an overtrick, they an undertrick: +650 (pull)
• They make the overtrick: par score -650 (pull--unless we want to be -990)
• 19 total tricks (50% likely):
• We make an overtrick: par score +650 (pull)
• We make game: par score +620 (pull)
• They make game: par score -200 (pull)
• They make an overtrick: par score -650 (pull as before)
• 18 total tricks (30% likely):
• We make game: par score +620 (pull)
• Neither side makes game: par score +200 (pass)
• They make game: par score -500 (pull as before).
• 17 total tricks (5% likely):
• We make game: par score +800 (pass)
• We go down one: par score +500 (pass)
• We go down two: par score +200 (pass).

Just looking at the probabilities and following the LOTT, it looks like we want to pass 15% of the time and pull 85% of the time. 

Let's go back to partner's double. In the old days, we could distinguish between tentative penalty doubles and stand-up-on-your-chair-and-slam-the-red-card-down doubles (just kidding, of course). Is partner's double an absolute final decision? No, how can it be? The opponents have bid 4♥ strongly. They're not kidding around so they think they have a play for it. If partner has a heart trick coming, it must be available on offense, too. Why has partner doubled and not bid 4S himself? For the reasons given above: each side might have only nine tricks available, we almost certainly have to lose a spade and, likely, two hearts. Do we have the rest? Partner isn't sure. Basically, in this context, his "penalty" double simply says "I think this is our hand, I have a balanced hand, and they are probably going down." After all, it's very unlikely that opener is going to bid voluntarily again after this sequence. In other words, this is a classic POD situation.

It's decision time. Is there anything partner doesn't know about our hand that would justify pulling? Yes! We have a seventh spade! 

Partner's (my) hand: ♠T97 ♥J4 ♦AT63 ♣AT98. Leading the ♦A or underleading a club would result in down two for 500 and a 45% board. Leading our suit should have resulted in 200 and a 28% board. Neither of these would be total disasters. As it happened, we didn't set the contract, resulting in a 0% board.

So, there were 18 total tricks on the board. Not playing double-dummy defense would have resulted in 19 total tricks. 

Introducing four-card suits in high-level auctions

Have you ever found that the auction has progressed to a point where it is no longer comfortable to bid a four-card suit?  And yet it may be that you actually have a four-four fit.  Let's say partner has opened the bidding.  With a weak responding hand, we will typically simply bid a four-card major if we have one and there will be no further interest on our part in introducing a new suit.  But what if we have invitational strength (or better) and we want to bid our suits in natural order?  What if the opponent(s) intervene?  We can easily develop acrophobia as the auction gets close to or beyond the 3NT level.  The solution to this problem of course is the cooperative double.

Here's a case in point:

Some readers will say that the South hand could start with a negative double and then bid clubs.  I prefer to play so-called "negative free bids" (which, by the way, fit hand-in-glove with cooperative doubles) so that particular sequence becomes a game-force.  But even playing standard, don't you think that starting with a double distorts this hand significantly by marginalizing a good six-card suit?

Admittedly, you don't have much to spare after the sequence shown in the diagram. Yet it does appear that this hand "belongs" to us and, with an unannounced four-card heart suit why should we let them play 3♠  undoubled?  If partner doesn't have four hearts, he may be able to raise clubs, rebid diamonds or convert the double into penalties.  When they have a good fit, so have we.  Occasionally, we can catch them speeding and, as it turns out, partner is happy to pass.  Surprisingly, however, the only lead to actually set the contract is the singleton trump.  Let's hope we don't go wrong and lead a high club!

Do I hear any dissenters?  Yes, of course I do.  Some of you may feel that with such an "obvious" double, North should be able to double 3♠ for penalties?  But think about it for a moment.  How frequently is that going to come up?  At the three level, after the opponents have made a Law-of-total-tricks raise, the player sitting under the bidder will rarely have a holding that is good enough to double with confidence.  On this kind of auction, it's much more likely that it is our hand but we haven't yet found a fit.  Mel Colchamiro has a great name for a double by North here: the "balance-of-power" double.  "BOP 'em," he says.  You might argue that the 3♣ bid, which is so highly informative, should trigger penalty doubles.  That is a valid point of view.  Nevertheless, the opponents are jamming our auction and we have not yet had a good chance for either of us to show four hearts.  Transfer two of those diamonds into the heart suit and we have a decent shot at game, yet the auction so far would have proceeded just the same.

In general, it is my opinion that, playing against reasonably sane opponents who have announced a good fit, it is more useful to be able to find our best fit than to increase their penalty.  And, what if you have a great hand and a stack of trumps and are prevented from doubling?  It may be a blessing when partner is bust.  Those situations usually work better when the hand sitting over declarer has the trump stack while partner has the entries.  That avoids opening leader being end-played at every turn.

Yes, but how can you make a takeout double when there is only one suit to take out into?  I know some of you are thinking this.  The point here is that it is not a pure "takeout" double.  It's a cooperative double showing that we have the balance of power and asking partner his opinion.  One of the suits that partner might favor is the one that's already been bid and doubled – a penalty pass.

Going back to our example hand, the more observant of you may have noticed that we can make 3NT, despite having only 22 hcp.  That's another possibility that I didn't mention before – taking the double out into a notrump game.  Indeed, this is one of those fairly rare hands where notrump plays better than any suit.  Next to notrump our best fit is diamonds where we can take eight tricks.  But how much better it is to score 400 instead of 100?

For more blogs on this interesting topic, see for example Single-suit cooperative doubles or Using double to find out about fit.

Abiding by the law

My previous blog on pressure bidding prompted another look at the so-called Law of Total Tricks and, especially, I fought the law by Mike Lawrence and Anders Wirgren.

One of the first things I realized twenty years ago when I first read about the "law" was of course that at best it is a rule of thumb.  It is no more a true law (and perhaps even less so) than Bode's law.  Some people, if I am to believe the L&W book, seem to believe that it is absolutely true, in the same way that certain fundamentalist religious people think that their "book" is literally  true.  I say this because Lawrence and Wirgren devote a large part of the book to debunking the myths of the LOTT.  Would they bother if there weren't true believers?

Of course the "law" isn't literally true. Every bridge player knows that, including and especially Larry Cohen. And so do I.  Yet, one of my bridge friends still chides me for quoting the law when making general statements about competitive bidding, as if I was one of the devoted followers, believing every word of the gospel.

Nevertheless, the "law" does, in a very general sense, show how the total number of tricks available (ours at our best trump suit plus theirs at their best trump suit) increase with the combined lengths of the two best trump suits (ours plus theirs).  That's to say, on the whole, each additional trump (of ours) in one of our hands, or their trump in one of their hands, will result in an additional trick (either for us or for them).  It doesn't state this as a categorical fact.  It simply says that, in general, total tricks increases in step with total trumps.

Although it's never a true law, as mentioned above (in the majority of hands exact equality is the exception rather than the rule), it works best under the following conditions:

• high card points are more or less evenly divided between the two opposing pairs (the usual range is quoted as 17-23 but this is of course an arbitrary boundary);
• total trumps are nearer to 17/18 rather than the extremes of 14/21;
• shortness and length are more or less in harmony (long suits beget short suits/voids);
• you are willing to be off by one trick either way;
• if you have a choice of trump suits of equal length, you choose the best one;
• defense and declarer play at your table are perfect (double-dummy);
• the layout doesn't contain too many short-suit honors, such as K, Qx, Jxx.

So, what use is the "law"?  It can be useful when you have to make a competitive decision to bid on, pass or double and it is not completely obvious from bridge logic and experience what to do.  This generally involves higher-level contracts, because an experienced player will be familiar with the lower level situations (should I bid 3♠ over their 3♥, for instance).  Newer players can use it of course at all levels because they won't have developed the appropriate judgment yet.

As noted, it won't always be right.  But bridge, especially matchpoints, is a game where we try to maximize our score in the long run without worrying too much about the occasional hand where things don't work out. For example, with no enemy bidding, no special avoidance requirements, and holding three small trumps in dummy (with outside entries) and AQT9x in our hand we will play low from dummy, playing the 9 if RHO plays low.  Generally speaking, we expect to lose one trick only.  But in fact, this favorable outcome will only occur 76% of the time.  Do we feel hard done by if occasionally we lose two tricks?  Of course not.  Bridge is not normally a game of absolutes.

Now, you might be wondering what "long suits beget short suits/voids" was supposed to mean.  What I mean by that is that as you get longer suits in your hand, you expect, or at least hope for, short suits to go with them, if not in your hand, then perhaps in partner's.  For example, you pick up a 7-card suit.  If the other suits are 2-2-2, this feels a little unnatural, and a bit of a disappointment, doesn't it?  You'd be much less surprised by the 7-3-2-1 pattern.  In fact, the 7-3-2-1 shape is three-and-one-half times more common than 7-2-2-2.  And, as Lawrence and Wirgren tell us, it is really shortness rather than length that is most indicative of the number of tricks we can take.  All things being equal, we'll take more tricks with a 7-3-2-1 hand than a 7-2-2-2 hand.  But the "law" only takes into account the length of the trumps, in this case, presumably the seven-card suit.  That's one of the obvious reasons why the law can't possibly be the gospel truth.

So, going back to my earlier question "what use is the law?", I do believe that it can help you decide what to do in a high-level competitive auction.  Here's an example from a recent club game.  You pick up in second seat, vulnerable versus not, ♠T7 ♥93 ♦KQT4 ♣97653.  RHO passes, as do you, and LHO opens the proceedings with 2♠.  Partner bids 4♣ (showing a good hand and at least 5-5 in hearts and clubs) and RHO bids 4♠.  Now what?  Let's try to do a "law" calculation.  Given that LHO opened a weak two at favorable vulnerability in third seat, he probably has only five spades.  I doubt if RHO has five spades with shape because he'd probably have jumped straight to 5♠.  But he might have five spades without any shortness.  It looks like we have a ten-card club fit and they have a nine-card spade fit so we're guessing 19, or possibly only 18 tricks.  We don't have much in the way of points, but it does seem to be our hand, based on the auction so far, though perhaps not by a lot.  If they can make 4♠ (-420), we could be -500 or even -800 in 5♣ so that doesn't look good.  OTOH, if we can make 5♣ (600), they might be only -500 or -300 when doubled.  We decide to pass smoothly (ah, there's the rub!) hoping that given partner's quasi-game-forcing bid, our pass will be considered forcing.

Partner comes in now with 5♥ which strongly suggests 6-5 in the round suits since he'd probably just double with the 5-5 he originally promised.  RHO takes the push to 5♠ (perhaps he did have five spades after all) and we revise our idea of the hand now.  It looks like they have 10 spades and we have 10 clubs with an 8-card heart fit.  So I'm guessing we might have 20 total tricks.  However, given that all our honors are in what appears to be a short suit in partner's hand (at most a doubleton), I'm going to be conservative and estimate 19 total tricks as before.  So we decide to double expecting to gain 100, 300 or 500 as against (for 6♣) -800, -500, or -200.  If we're wrong and there are 20 total tricks, we will be spectacularly wrong in the two extreme cases: they make 850 (pass would be best) or we could make 1370; but still doing fine in the middle (and perhaps more likely) cases: 100, 300 instead of -500, -200.

How did we do in practice?  Both sides can take 10 tricks in their black suit, so it was right not to let them play 4♠. Their RHO made the all-too-common error of going to the well twice and falling down it the second time. We score +100 for about average (somewhat as we'd expect).  However, there were no other pairs our way scoring 100 for 5♠X.  Some pairs managed to set 5♠X two tricks but this must have been careless declarer play.

I think the law was helpful here, keeping us at average despite a difficult competitive board.  It was difficult because a) our balanced and poorly fitting hand suggested fewer total tricks than there were; and b) the other side can make game on only 17 high-card points.  For the full hand, look up board 18 from the Newton-Wellesley game on Christmas Eve.  I was not playing, by the way, so all my thinking described above was fictional, but reasonable.

Those tough high-level decisions

One of the trickiest aspects of bridge is when the opponents jam the auction.  Here's an example.  Nobody is vulnerable and you are fourth to speak. You need to have your thinking cap on from the moment you pick up and sort your cards: ♠K9754 ♥JT962 ♦K85 ♣ –.  LHO opens 1♣, partner bids 1♠ and RHO bids 5♣.  Quick, you have 10 seconds to think about this before you start imparting unauthorized information to partner.

You can spend some of the time trying to think about what partner has for his overcall.  But I think that's not going to help much.  Overcalls are, by their very nature, wide-ranging, especially a space-gobbling 1♠ over 1♣.  At all-white, partner could have ♠QJT62 ♥A87 ♦742 ♣32 or ♠AQJT2 ♥AQ7 ♦A742 ♣2 or anything in between.  How can you tell whose hand it is and how high to bid?

Fortunately, you really don't have to.  With all its faults, there is no guide to this sort of situation like the law of total tricks.  So, how many total tricks do we think there are?  We apparently have 10 spades and the opponents surely have 10 clubs.  That's 20 tricks, more or less.  But is it more or is it less?  If it's only 19 tricks, suggesting an "impure" layout, we will be right to bid on only if we can make slam but if we can just make game, then we will be better off doubling (500 vs. 450).  This doesn't really look like a slam situation so we might double (takeout-oriented and maybe partner can pass with wastage in clubs).

What if the total tricks are 20.  This is the toughest problem because if each side can make 10 tricks, we should pass (100 vs. -100).  However, the other divisions of 20 tricks all favor bidding: 450 vs 300, -300 vs -400.

What if the total tricks are 21 (or even more).  Then all situations favor bidding on: 980 vs. 300, 450 vs. 100, -300 vs. -920.  With 22 tricks, bidding at least one more is a no-brainer.

So, it comes back to guessing how many total tricks there are.  Voids tend to increase tricks, short-suit honors tend to reduce tricks.  Again, we don't know partner's hand but if he's a disciplined bidder, he probably won't have Qxx, Jxx or xxx in clubs (three losers) unless he has a very good hand otherwise.  He might have Jxxx or Txxx in clubs but that seems unlikely on the bidding.  So, he might have a wasted A or K in clubs but probably nothing else.  I think in this case, I would estimate more rather than fewer total tricks.  Let's say 21.  Therefore, I would probably bid 5♠.  Even if there are only 20 total tricks, bidding on will be right most of the time, as noted above.

On this particular hand (board 17 from Friday's world-wide pairs), there were 22 total tricks because RHO also had a void (in spades), the opponents had 11 clubs between them and there was essentially no wastage.  In practice, we let them play (and make) 5♣ unmolested for a below-average result.  I'm not saying that pass was wrong, just that I probably wouldn't pass.  It could have been exactly right.  Essentially, RHO made a good bid because it made life very difficult for us.  That's why good players make this kind of bid: it causes problems.