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.

A couple of difficult hands from Auburn

In an otherwise decent effort Kim and I had a couple of tricky hands at the CMBA sectional in Auburn, MA.  First, a defensive problem.  Your hand is ♠– ♥JT864 ♦KQ ♣Q87654.  None vulnerable and partner deals and opens 1♠.  You bid a forcing 1NT.  LHO, a player who has never met a hand on which she could not find an overcall, bids 2♦.  Partner passes and it comes back around to you.  Double is primarily for take-out (but with the expectation after the pass that it might well be converted to penalty).  Bidding 2♥ and 3♣ both seem somewhat flawed.  So, let's say you do double and partner leaves it in.  Game for us seems unlikely, so 300 would be a top and even 100 might get most of the matchpoints.  In any case, you have to defend assuming that we are in the right contract.

Partner leads the ♠A (Ace from AK) and dummy comes down with an undeserved trick for declarer: ♠T8642 ♥9752 ♦3 ♣AT2.  Dummy follows low and you are at the cross-roads.  Partner won't be expecting your hand, that's for sure.  Maybe something like ♠93 ♥AT84 ♦72 ♣KJ765 or maybe ♠9 ♥AT864 ♦Q2 ♣K8765.  If you had one or two small trumps, you'd like to ruff a spade early so somehow you'd like to persuade partner to play a small one, if any, before your trumps get drawn.  You might do this by playing a low club then a low heart.  On the other hand, with your actual hand, you don't particularly want to waste any trumps on ruffing partner's losers.  Rather, you want partner to get dummy's entry off the table before the high spade could become good.  So, although this might typically suggest you have the king, I think the right card at trick one is the ♣8.

Unfortunately, neither of us defended optimally and on this occasion, declarer's hand was just good enough to take advantage and score 8 tricks for 180.  This wasn't an absolute bottom for us, but it was a low score.  Actually, it turns out that the normal contract was 3♣ by our side making exactly, so even +100 would not have been a good matchpoint score.

Here's a poor result that was entirely my fault, but is interesting theoretically, nonetheless.  I picked up ♠AK7 ♥98 ♦Q542 ♣KT85 in fourth seat at favorable vulnerability. Partner opened 1♥ and I responded a forcing 1 notrump.  Partner now rebid 2♥ which, in our system practically guarantees six pieces and tends to show a minimum hand strength-wise.  Obviously, I was going to bid game, but which game?  I felt that it might be advantageous to have the lead come up to my hand, especially on a minor suit lead, and bid 3NT – but I neglected three important factors.

First of all, partner's hand might be short on entries given the auction (or alternatively have a poor heart suit).  Both of these factors argue in favor of playing in a major suit game.  Secondly, the choice of notrump versus a major suit tends to work better with a 5-3 fit rather than a 6-2 fit.  Finally, choosing notrump over any 8-card major suit fit should generally only be considered with a plethora of high-card points, something like in the range 27-30.

So, to my contract of 3NT, a fourth-best deuce of spades was led.  Dummy came down much as expected with
♠J6 ♥AKJ764 ♦K8 ♣J97.  Obviously, I was going to try the J.  If it held, my judgment would be vindicated and I would likely make the same number of tricks as the heart declarers.  Unfortunately, the J was covered by the Q and I won with the Ace.  Now, I was definitely behind the heart declarers.  Any lead from the other defender would have likely given away a trick.  Not only that but I now had to be quite careful.  If the ♦K proved not to be an entry, it would be highly embarrassing to leave several hearts stranded in the dummy.

So, I turned to a couple of guidelines.  One was that if hearts were 3-2 I was destined to score badly.  The heart declarers would always score 20 points better than me.  If hearts were 4-1 offside, I'd be just as badly off, probably even worse.  That didn't bear thinking about.  But what if hearts were 4-1 on-side?  The heart declarers would all likely finesse the J and then try to drop the Q or T.  A first-round finesse was obviously called for, but which finesse?

That's when I turned to my Principle of Least Commitment for guidance.  This is the lazy man's way of avoiding having to learn all 656 suit combinations from the Bridge Encyclopedia.  In this case, least commitment suggests running the 9.  The advantage of running the 9 is that if RHO wins with the Q, you know where the T is (unless RHO is very devious indeed).  If you run the 9 and it loses to the T of course, you know nothing about the Q and if you finesse the J and it loses to the Q, you know nothing of the T. 

If entries to dummy were not a problem (or if hearts were trumps), then the best play is to cash a high heart, cross over and finesse the J.  You'll make 6 tricks 37% of the time and 5 tricks 88% of the time.  But if we assume no outside entry to dummy, then we essentially want to duck a trick to maintain our link.  Again, this suggests running the 9, which is what I did.  It lost to the T.

Another way of looking at it is this: if indeed there is no further entry to dummy, running the 9 first will result in either 2 or 5 tricks in the suit, assuming that the hearts are distributed unfavorably: Q532–T or T532–Q.  Finessing the J first will result in either 3 or 2 tricks.

A spade came back and now I had another decision to make.   So far, my strategy was not panning out.  The heart declarers would finesse the J and see the T come up on their left.  Then they'd bang down the top hearts and hope to drop the Q.  If that happened, I'd lose.  Was there a way to win?  Yes: take another finesse in hearts.  But wait!  If that lost to the Q, I might be in the ignominious position of not taking any heart tricks at all and going down quite a few.

Here's where I goofed.  I got scared.  I didn't "stay with the program."  I couldn't bear the thought of looking so silly so I played off the A and K.  The Q failed to appear.  She was exactly where I needed her for a good board.  What an idiot!  I ended up with -50 while my competition were all +420.  I might still have ended up with 400 which would have been good for slightly over average.

The Principle of Least Commitment

Do you know all 656 of the suit combinations in the Bridge Encyclopedia?  Neither do I. Can you always visualize exactly what's out against you and evaluate every line in terms of its percentage success rate?  Neither can I.

So begins the article that I've been working on for several years which might eventually make it into some bridge magazine if I can ever perfect it.  This is the current state of the article. It seems to me to be a valuable principle, as much so as the principle of restricted choice, to which it is related, for example.  But I've never heard anyone mention anything like it.  Am I missing something?  Is it so blindingly obvious that I'm the only person to think it worth writing down?


I was reminded of it last night at the bridge club because there were two PLC transgressions at our table, at least that I noticed.  Here's one: you are in a 3NT contract with 24 hcp and you have the following suit to play: AQT94 in dummy opposite 65 in the closed hand.  You have the tempo and sufficient entries to both hands.  How do you play the suit to maximum advantage?  Well, you finesse the 9/T.  If the K and J are split then you are simply guessing.  If they're both guarded offside you're doomed to lose two tricks in the suit regardless.  But here's the case where it matters: KJx on your left and xxx on your right.  By finessing the T first, you pick up the entire suit.  If you finesse the Q first, you must give up a trick.  Least commitment.  As it happened, KJxx was on the left so it didn't matter but the declarer didn't give himself quite the best chance.


Here was the second case: You're in 3S and your trump suit is 632 in the dummy and AQT874 in the closed hand.  At first sight you might say, aha, just like last time, let's finesse the T first (as the actual declarer did, losing to Jx).  But here there are only four cards out, as opposed to the six in the last example.  You should expect to be finessing once only (not twice as before).  This despite the fact that you have an extra card in the short hand with which to finesse.  In "normal" layouts of the suit (2-2 or 3-1 splits), the cards T and below are essentially irrelevant here.  Correct play is to take the obvious finesse of the Q which has a 27% chance of picking up the entire suit (essentially, you need the K onside and a 2-2 split or some other fortuitous event like singleton J offside).


In this case, there were three losers outside the trump suit.  Our opponents had stopped in 3S where some might have been in game.  Thus, there might be something to be said for taking the safety play for five tricks.  However, as is often the case when we have no sequences of our own (here, they have none either), the "least commitment" strategy is to bang down the Ace (that takes no guesswork at all!).  Now, you increase your chance of taking 5 tricks to 83%.


If you can offer any suggestions for my description of the PLC, I'd appreciate it.