A very ordinary hand

It has been said that the way to win at bridge is to make sure you get the ordinary hands right. In other words, squeezes, endplays, deceptions, coups of various sorts, etc. don't come up sufficiently often to give an advantage to the better player. But ordinary hands come up all the time.

Here's a hand from yesterday's STAC game:

I was sitting West and essentially took no part in the proceedings. Our opponents, two experts who are married (to each other) and sometimes--but not usually--play together, had the auction all to themselves. I might have opened 1C in third seat and perhaps I should have but I suspect that would have pushed them into a making 4H).

At my club, every N/S pair was in a heart part-score, mostly 2H by North but sometimes, as at our table, 3H by North and, once, 3H by South. My partner led the C4, clubs being the only safe-looking lead, although either red deuce would also be safe on this layout. Declarer ruffed out the spades, took two diamond finesses and emerged with 10 tricks pointing his way. A flat board, right?

Wrong. This was by no means a flat board. Here were the scores: 1 @ 200 (11), 6 @ 170 (7.5), 3 @ 140 (3) and 2 @110 (.5). I'll dispense with the anomalous scores of 200 and 110 and concentrate on the 170s and 140s.

At first, I couldn't see why our score was below average (3.5). We didn't put a foot wrong. How could declarer not take 10 tricks. And then I saw it: some declarers must have taken the spade finesse.

This hand of course is a perfect illustration of the (general) superiority of a 4-4 trump fit over, say, a 5-3 or 5-2 trump fit. You can usually use long cards in the long suit to discard losers from the other hand while using small trumps to ruff with, possibly ruffing out losers to establish the long suit.

But the main point here is that considering a suit, the spades in this case, in isolation may yield a different plan than considering the suit as part of a whole hand. If you were in a spade (or notrump) contract here, you would consider your play in the spade suit and opt for the 43% likelihood of the bringing in the suit with a finesse against the queen, versus the 18% chance of dropping a doubleton or singleton queen. Not even a close decision.

But here, you are in a suit contract and you have the luxury of being able to ruff a spade (we'll assume that trumps are 3-2, as here, for the sake of simplicity). The probability of dropping the queen after a ruff is: 36% for a 3-3 split, plus the same 18% chance that the queen would have dropped anyway. That's a total probability of 54%. The failure zone (46%) is made up of 32.3% for Qxxx, 12% for Qxxxx, and 1.5% for Qxxxxx.

So, again, it's not really a close decision if you know your probabilities (54% vs. 43%). Incidentally, it's a very common error to regard the finesse as a 50/50 shot. But when we can only finesse twice, as here, there will be holdings on our right that we can't pick up: half of the Qxxxx and Qxxxxx layouts we noted above, that's to say 6.75% of cases.

The conclusion is that the situation where you are missing six cards is a tricky one--and also a common one. It's worth spending some time to learn the probabilities.

Sometimes a bad split can be your friend

We all know, when we are declaring a contract, that bad splits are the enemy. It's nearly always harder to make our contract when suits don't split well. Nearly always!

Let's take the following contract as an example:

Clearly, this is going to be no problem if diamonds split 2-2. So, we draw two rounds of trumps and duck a diamond. Another heart comes back and you ruff. What do you pitch from dummy, by the way?

Now you play to the DA but unfortunately, RHO pitches a heart. Down one.

Not so fast! You aren't down yet. Cross back to your hand with the SK (RHO plays the Q) and, using vacant places and restricted choice you confidently run the nine. It wins and you are able to pitch your losing diamond on the fourth round of spades. You didn't pitch that little spade earlier, did you?

I wish I could tell you that I played like this. I could have. I should have! But I didn't. I just lamely conceded a diamond for down one and got on with the next board. As Eddie Kantar writes in his wonderful book Take All Your Chances, you should never just give up when there's even a glimmer of a hope. Notice that if spades had split a more normal 3-3 or 4-2, there would have been no hope at all. Well, it would make sense to run all your trumps and hope for a bad discard. But against good defenders there'd be no hope.

But the moral of the story is: sometimes a "bad" split is better than an even split. And, when there's a massive preempt at the table, this becomes even more likely.

The full layout:

If you're interested in the odds of the S9 winning the trick at T8 (as described above), then we know that East started with at least 8 hearts (based on the bidding and the opening lead), a diamond, two clubs and the spade Q. He has one vacant place for either the SJ or a ninth heart. West has shown 3H, 2C, 3D and so has five vacant places for the SJ. Taking restricted choice into account, the odds are 10 to 1 that running the nine will be successful.

Looks simple

It's the first hand of the evening at one of last week's STAC games (only we are vulnerable). After partner opens 3♦ in second seat, we find ourselves in 3NT with no opposition bidding. The lead is the ♥2 (fourth best) and this is what we see:

Dummy
♠	9 4 2
♥	10 7
♦	A Q 10 9 8 6 3
♣	6

My hand
♠	A K Q 8
♥	Q J 3
♦	5 2
♣	K Q 10 3

The opponents clear the hearts, we lead a small diamond, LHO follows with the four and it's decision time. Well, the first thing you notice about the dummy is that it's totally entryless outside diamonds (it's a great preempt, isn't it?).  The defenders have taken two tricks already and have a heart and a club to cash. So, if we're playing IMPs, we need to bring in the entire suit, so must finesse the Q to give us the best chance (about 26%) of making. At first glance, you might think that the chance of making is 50% but in practice, if we finesse the Q and it wins, we must also have a 2-2 split or see the J drop on our right.  Though even then, there's the chance of a dastardly defender deliberately throwing us off holding KJ.

In a diamond contract, there are at least 9 tricks so in effect we aren't competing with the declarers in diamonds. In any case, of 17 tables, only two played in diamonds. So, how do we maximize the number of tricks we will take, keeping in mind that the number may well be less than nine? Finesse the 10 (8 or 9) on the first round, in keeping with the Principle of Least Commitment. This play either breaks even or loses a trick to the Q play when the honors are split. But it enjoys a big win in the case where KJx is on our left and x is on our right (since we have no outside entries to dummy this layout will kill the suit stone dead if we finesse the Q). The 10 play also has a minor win in the unfortunate case where KJxx are all on our left, although this will not be a big comfort at the time. Nothing works of course when KJ are guarded on the right. See the table below for the details.

Layout	Cases	Probability	Q tricks	T tricks	Q expect	T expect
KJxx -	1	4.78%	2	3	0.096	0.143
KJx – x	2	12.44%	2	7	0.249	0.871
Kxx – J	1	6.22%	7	6	0.435	0.373
Kx – Jx	2	13.56%	7	6	0.949	0.814
Jxx – K	1	6.22%	6	6	0.373	0.373
Jx – Kx	2	13.56%	6	6	0.814	0.814
xx – KJ	1	6.78%	6	6	0.407	0.407
x – KJx	2	12.44%	1	1	0.124	0.124
Total	12	76.00%	53	61	4.536	5.157
Approx					4.42	5.08

When the "books" take a look at a suit like this, they assume that entries are available wherever needed.  However, with the excellent SuitPlay program by Jeroen Warmerdam, you can explicitly tell it how many entries accompany the suit, etc.

Since everything must be decided on the two leads towards dummy, we effectively have four realistic lines of play: Q or 10 on the first lead, A or the remaining honor on the second.  My analysis assumes that we will always guess right on the second lead but this will be easier sometimes than at other times.  In particular, if the Q won on the first trick, the Kx-Jx layout will be picked up easily.  But if the Q loses to the K, and LHO plays small to the second trick, we won't know whether he started with xx or Jxx.  Against most defenders, the 10 play may make the second guess easier because players tend to win as economically as possible. So, in a way, playing that 10 first is something of a safety play.

The best matchpoint line, starting with a finesse of the 10 and the one which I happened to choose at the table, did not give me the best chance of making the contract.  But it did give me the maximum expectation of diamond tricks.  As it happens, I lost the first trick to the J.  I was fortunate in that the defender who won the last heart trick didn't have the club ace and chose to lead a spade.  I therefore made my contract exactly.  This was worth only 5.5 out of 16 (approx 33%), however.  I shouldn't really be surprised.  At a club pairs, players perceive a premium on making contracts (and don't typically worry about minimizing the set). Perhaps in the Blue Ribbons I'd have had more company.

The table above has two columns each for the Q and the T plays.  The first is the number of tricks yielded.  The second is the expectation of tricks (the number times the probability).  By summing the expectations, we can compare the overall expectations of the two lines.  However, we poor humans cannot do these kinds of calculations at the table in practice (Chthonic would have no problem of course).  There's another, simplified, method of comparing the lines that Eric Rodwell describes in The Rodwell Files.  He suggests assuming that all possible layouts are equally likely.  This isn't quite accurate of course because, due to considerations of vacant places, KJ74 in one hand is not as likely as, say, KJ opposite 74.  That's because once the K and J have been "placed" in one hand, there are two fewer vacant places in that hand to take the 7, and if the 7 does go with the KJ, there are now three fewer vacant places to accommodate the 4 in the same hand.  However, to a first approximation, we can assume equal likelihood of each layout.  When there are n outstanding cards, there are 2^n possible layouts.  In this case, four missing cards so there are 16 possible layouts.

Once LHO plays a small card (the 7 or 4 in this case) to the first diamond lead, we can immediately eliminate four of the 16 cases. In the table above, the bottom row, labeled Approx, shows the expectations when we use this simplified method of calculation.  The two approximate values are 53/12 and 61/12.  As you can see, the numbers are very close to the accurate values.

However, for a complex situation like this one, even this amount of calculation is too much.  That's why I find myself frequently falling back on the principle of least commitment.

Sleepless in Seattle (conclusion)

It's not often that you pick up a ten-card suit. My partner was dealt this beauty in a side game at the Seattle NABC: ♠AKJT987542 ♥– ♦873 ♣–. We were at favorable vulnerability and my LHO opened with a weak notrump. Partner bid 4♠ and that ended the auction. I produced quite a useful dummy as the ♣J was led: ♠– ♥KT97 ♦A954 ♣AQ872. Incidentally, do you think I should have raised or made some try for slam?

Now, the question is how to make the greatest number of tricks. There isn't much you can do about trumps. You can't take a finesse and they either split or they don't. And in the latter case, there's no possible way to pull off a trump coup given that you would need to ruff seven cards in your hand and get back to dummy yet again! So, are there any realistic chances for an extra trick in a side suit? A singleton ace of hearts seems a bit unlikely given that there are nine out against you and RHO opened with 1NT (suggesting at least two and fewer than six). What about diamonds? They might be three-three but even then, you have nothing to pitch on the thirteener. No, the only realistic chance is in clubs. How realistic? Given that you only have one outside entry to dummy, you are going to need the king to fall doubleton. Is LHO really likely to have led J from KJ doubleton? I don't think so either. Therefore, you need RHO to hold Kx. You weren't thinking of covering with the queen, were you?

When you rise with the ♣A, the 9 drops on your right.  There are a total of eight clubs out and the a priori probability of righty having precisely K9 doubleton is only 0.3%! Is it even worth bothering with?  You betcha!  The K is doubleton.

However, only three of 11 declarers actually made 12 tricks, not including my very experienced partner, unfortunately.  Of course, some of these others might not have been given helpful club leads.  There was also a -350 and a -150 our way.  I imagine that these were uncompleted transfers or, more likely, an ill-fated attempt at 4NT or 6NT.

To conclude my commentary on Seattle, I will simply observe that this tiny probability (0.3%) yielded an additional 46% of the matchpoints on the board.  Well worth trying for, especially given that there was negligible down side to the play.

A tale of two books

At a recent tournament, I was pleased to find two books that I hadn't seen before which interested me a lot.  One was Deceptive Defense: The Art of Bamboozling at Bridge by Barry Rigal.  The other was Bridge, Probability & Information by Robert F. MacKinnon.

The first of these is an excellent book, a must-read for anyone who wants to improve their game.  Whereas it's fun to pull off an advanced play such as a squeeze or endplay, it's even more fun to perpetrate a successful deception.  The look on the opponent's face is always worth it.

As a long-time student and enthusiast of probability theory as it applies to bridge, I've generally bought any book that I could find on the subject.  These range in quality from completely pointless (Frederick Frost's book) to totally brilliant (Kelsey and Glauert's Bridge Odds for Practical Players).  So, it was with great anticipation that I began reading MacKinnon's new book.  Especially given the allusion to information theory.  By the time I started to read the book a day or two later, I had seen a rave review in the ACBL Bridge Bulletin.

But I immediately found the style of the book somewhat annoying.  The book reads like a series of short essays on bridge probability.  They do not follow each other in logical order and each section is prefaced by a pithy, but typically totally irrelevant, quotation.  There are several really important concepts that MacKinnon puts forward.  But he seems to do so in such an off-hand manner, that the impact is very much lessened.  And he goes so far out of his way to ensure that the book does not read like a textbook that, where a little logical derivation of a result would be extremely helpful, it is usually missing entirely.  The author generally states these important results as facts or axioms without making it entirely clear how he derives the result.  The layout is not always as helpful as it might be, for example, the table he uses to demonstrate that the ratio of the number of combinations (and, therefore, probability) comes from the small number on the right divided by the large number on the left.  In this instance, the splits (large:small) are not aligned between the columns as suggested in the text.  Even the examples which he shows from actual play do not always seem to be entirely relevant to the ongoing argument.  I think that what this book craves most is a good editor.  The author definitely knows his stuff but, in my opinion, needs help in presentation.

This is definitely not a book for beginners, or even advanced players unless they have an abiding love of probability topics.  While it does "correct" some misconceptions suggested by other books, I do not think it will supplant Kelsey's book as the premier book on the topic.

Meanwhile, Rigal's book is, as always with this author, excellent.  It reads so well, and is sufficiently interspersed with relevant examples, that it is a hard to put down.  It concludes with a spectacular example of deceptive defense by the late great Maurice Harrison-Grey.  Grey's hand was ♠83 ♥9643 ♦AQ3 ♣KJ54.  His LHO opened 1D, partner bid 3S and RHO closed the auction with 3NT.  Grey led ♠8 and dummy tabled the following hand: ♠9 ♥AQT ♦KJ9852 ♣982.  Declarer held up his A until the third round (as the spade bidder could easily have held only six spades).  Put yourself in Grey's seat.  What do you discard on the third spade?  The hand went down one, by the way.

If you want to know the answer, you'll have to buy the book!  Or you could just ask me.