Never a Dull Deal

Probability is related to information. It changes with the play of each card. Different information sets lead to different probabilities. ‘Going with the odds’ entails ‘going with the information flow.’ Here is a simple example. Suppose South is a declarer looking for the ♥Q among 5 missing hearts and has available a two-way finesse. On the basis of the deal alone, there is a 50-50 chance that Her Majesty lies in either of the defenders’ hands The lead is the ♠2, from which declarer deduces that the spades are split 4 on the left and 3 on the right. This information leads to the conclusion that the ♥Q is more likely to have been dealt on the right in the ratio of the vacant places 10:9, so it is correct to finesse through East. Next suppose that at the other table North has become declarer in the same contract through a transfer sequence. The lead is the ♦2. By the same reasoning, North should finesse West for the ♥Q. He also is correct, but only one declarer is going to be successful. The information about one suit affects the probabilities in the other suits, but because the information is different, the deduction is also different.

Both the a priori odds and the vacant place calculation are based on the assumption that unknown cards can be placed randomly between the two defenders. That is an assumption of a condition of maximum uncertainty with regard to the placement of the unknown cards. It loses validity if there are indications one way or the other. Let’s next suppose that the South declarer doesn’t immediately take the heart finesse which he thinks is favored in the ratio of 10:9, rather he plays on diamonds and finds they are split 4-3 the other way. Now the vacant places are even at 6 and 6 between West and East. Due to this additional information the odds on the heart finesse have returned to their a priori value of 50-50. One feels that the more information one obtains the better will be the estimates probability and the better the chances of making the right decision.

If the contract were 6NT, declarer might postpone the heart finesse until some information is obtained from the club suit. So declarer safely ducks a club just to see what happens. If he can deduce the club split, he will be able to deduce the heart split and can play accordingly. The answer will not be a 50-50 decision anymore as one player will be assumed on the basis of the information available to hold 3 hearts. However, there are no guarantees as even a 60-40 finesse will lose 40% of the time. One has been taught to assure the contract against bad breaks at team play and to forget about overtricks. As a consequence one is discouraged from making an exploratory move if that might lead to the defeat of the contract on a bad split. That in turn results is an over-reliance on the a priori odds which, as we noted, are based on a condition of maximum uncertainty.

At matchpoints overtricks risking the contract for the sake of an overtrick can be the correct play. Exploratory moves for the purpose of gathering information make good sense. Let’s look at my misplay in a recent local Regional. It was a typical inexpert matchpoint game in which success often depends on the opponents not defending to best advantage. The deal was a routine 3NT across the whole field, so the play was all about overtricks. Some find such contracts dull, but I find even they contain elements of great interest. Besides which, our mistakes are stepping stones to improvement. Sure they are. First and foremost we must prepare ourselves to adapt to changing circumstances.

West	East	Bob	John
♠ K 7	♠ J T 9 4	1 NT	2 ♣
♥ A 7 2	♥ K J 10 3	2 ♦	3 NT
♦ J 8 7 3	♦ A Q	Pass
♣ A K 10 8	♣ Q 4 3	Lead ♠ 3

As West I covered the lead of the ♠3 with the ♠9 which won the trick as South, using standard carding, followed with the ♠2. How would you play this hand playing Teams where the primary objective is to make the contract? After you answer that, the next question is: how would you play the hand at matchpoints where the primary objective is to win as many overtricks as possible?

The simplest way to assure 9 tricks is to play the ♦Q at trick 2, establishing 2 tricks in the suit. A spade continuation will set up the ninth trick, so the defenders’ best return is a passive diamond to the now bare ace. No problem, as a 3rd heart trick is assured by riding the ♥J from dummy. At most declarer loses 2 spades and 2 tricks in the red suits. Giving up on the red suit finesses in this manner is playing the hand with extreme short-sighted pessimism as declarer has a double stop in every suit. There must be a reasonable limit to our fear of loss. However, if one plays that way, 3NT does indeed become a very dull deal. As with our lives, there are no dull deals but a lack of initiative makes them so.

At matchpoints there is a chance of 2 overtricks in a common contract, so declarer must go all out to obtain them if available. We want to find the ♦K located in the North and the ♥Q in the South. The opening spade lead gives us the immediate odds of 10:9 for the second condition. With this in mind I led the ♥J from dummy, hoping for a cover, passing it, and losing to the Queen. Unlucky? Or should I have followed Zia’s advice that if they don’t cover they don’t have it? More on this later. Next North cashed his 2 spade winners and led a low diamond. Should I finesse or go up with ♦A and rely on bringing in 4 club tricks?

The a priori odds that clubs will split evenly is just 36%, so it would appear at first glance that the diamond finesse is better. However, the location of the ♣10 gives an a priori chance for 4 club tricks of about 60%. To judge the probabilities at this late stage requires more information on the expectation of the splits in the other suits. If North is expected to hold 4 clubs, then the play of clubs for 4 tricks would lose its luster.

My thinking at the time turned away from card combinations to psychology and motivation. I viewed North’s strategy with suspicion. Why cash his spade winners and why not exit passively with a heart or even a spade? It would appear that I am committed to the diamond finesse regardless. It didn’t occur to me at the time that he had no devious plan in mind and was merely happy to be ahead in the game by taking a trick in hearts to which he was not entitled. In a confused state of mind I decided to take the diamond finesse on the grounds that, after all, every declarer in the field would be taking that finesse. Wrong! Not only on the play itself, but, more importantly, with the thinking behind it. We should always keep in mind that a defender is working with different information and can’t see the cards we hold, and vice versa. Here is the full deal.

Dealer: West Vul: All		North
		♠	A Q 5 3
		♥	Q 8 4
		♦	10 5 2
		♣	J 5 3
West				East
♠	K 7			♠	J 10 9 4
♥	A 7 2			♥	K J 10 3
♦	J 8 7 3			♦	A Q
♣	A K 10 8			♣	Q 4 3
		South
		♠	8 6 2
		♥	9 8 5
		♦	K 9 6 4
		♣	6 5 2

The Objective This hand provides a fine example of ‘do as I say, not as I did’. I indulged in the kind of fragmented thinking that goes along with the acceptance of arguments based on the a priori odds in each suit taken without reference to the deal as a whole. The first step in the process after the opening lead is to examine dummy and set a realistic target for the number of tricks one hopes to take. The lead gives us 2 tricks in spades, there are 3 tricks in clubs, and 2 finesses to be taken in the red suits. So a realistic target is one of 10 tricks. Of course, as this is matchpoints, declarer should strive for one trick more. The question arises as how one might negotiate to the desired ending without unduly risking what appears to be the normal result of making one overtrick.

The Deep Finesse Effect The program Deep Finesse is a wonderful tool as it provides us with the optimal results obtainable on the given lie of the cards. It knows all the cards so does not deal with probabilities. To obtain a perfect result usually declarer has to take any finesse that works. The wide-spread availability of this analysis acts as an encouragement to take every finesse in sight and hope it succeeds. As with references to the a priori odds, this treatment fragments one’s thinking into a suit-by-suit approach. It freezes the approach to be taken. In the above deal one focuses attention on the obvious finesses in the red suits hoping first and foremost to take the heart finesse in the right direction.

If a declarer makes the optimal number of tricks as determined by Deep Finesse, he is entitled to feel he has done his part well enough, but to insure a good score he may have to exceed the optimum. This usually comes about because the defence has erred, so a secondary aim of a declarer is to play the hand in such a way that the defenders won’t find the optimum defence. On this hand declarer aims to avoid the diamond finesse if possible. Why? Well, let’s not forget the club suit. As noted above the club suit is expected to provide 4 tricks about 60% of the time, which is a better percentage than the diamond finesse taken in isolation, so, although the doubleton ♦A Q stands out in the dummy like a sore thumb, declarer wants to blind North to its importance when he doesn’t hold the ♦K until it is too late. With a bit of luck in the club suit, the ♦Q may be discarded on the 4th club, and the finesse avoided. So timing becomes important. If the diamond finesse can’t be avoided, then reluctantly one has to take one’s chances along with everyone else.

If declarer were to play a spade at trick 2 to establish a second spade trick, North should become aware of the danger and lead a diamond before taking his second and last spade trick. More simply, North might be one of those stubborn folk who see that if declarer wants to play on spades, it must be correct on general grounds to switch suits. Taking the heart finesse immediately at trick 2 and hoping for a defensive error is better tactics. The question arises as to what is the best direction for that move. Preliminary evidence suggests that finessing through South has the greater chance of success (10:9 odds), but it appears ‘more natural’ to finesse through North. (My partner certainly thought so.) If South wins the ♥Q, she will undoubtedly return a spade, her partner’s suit. If North then cashes a second spade before returning a diamond, good timing will have been achieved. If he doesn’t take a second spade, declarer resorts to the diamond finesse as a matter of necessity, losing nothing in the process. The question arises as to how much is risked by taking the heart finesse ‘the wrong way’? Is it a good investment? To answer that we should look more closely at the possible distributions involved.

The Distribution of Sides

If one accepts that the deduction that the spades are split 4-3, the most common distribution of the NS cards are as follows:

	I	II	III	IV	V	VI
	N S	N S	N S	N S	N S	N S
	♠ 4 – 3	♠ 4 -3	♠ 4 – 3	♠ 4 – 3	♠ 4 – 3	♠ 4 – 3
	♥ 3 – 3	♥ 3 – 3	♥ 2 – 4	♥ 2 – 4	♥ 4 – 2	♥ 3 – 3
	♦ 3 – 4	♦ 4 – 3	♦ 4 – 3	♦ 3 – 4	♦ 3 – 4	♦ 2 – 5
	♣ 3 – 3	♣ 2 – 4	♣ 3 – 3	♣ 4 – 2	♣ 2 – 4	♣ 4 – 2
Weights	100	75	75	56	56	45

 

The relative weights are based on the number of card combinations that result from a random deal of the other 3 suits. From the weights we can be a rough estimation of the probability of each condition having been dealt. The assumption is the same as that behind the vacant place calculation, but here we display only the most common combinations of splits in hearts, diamonds, and clubs. This focuses the mind on what is most probable given the information available so far from the bidding and the opening lead. It is of interest to note that the single most likely distribution (Condition I) is the actual distribution encountered at the table, the flattest of flat 4-3-3-3 on both sides of the table. If declarer had to choose just one condition to play for, that would be it.

If we were to add more possibilities, we would be adding more uneven splits, but the bidding has not indicated there are extreme distributions of concern. Also, if there North were to hold two 4-card suits, one can imagine that the opening lead might have been at least part of the time a passive lead in the other 4-card suit, rather in a suit headed by the ♠A Q which might give away a trick immediately. Even so, it appears that the ♥Q is more likely have been dealt to South than to North, the odds being much the same as the vacant place odds.

However, holding the ♥Q, South may cover the ♥J as often as 1 time in 10, balancing the odds as to the location of that card. Playing the ♥J from dummy and overtaking with the ♥A when it is not covered appears to be a dangerous and unreliable way to collect information, but if North doesn’t follow to the second heart, declarer wins the ♥K in dummy and leads a spade. The odds for the diamond finesse have improved, as North’s most likely shape is now 4=1=4=4, and the ♥10 3 still stand guard if the diamond finesse loses. There is another consideration at matchpoints, which is this: how will the majority of declarers play the hand? I imagine that most will take the ‘natural’ heart finesse, playing to the ♥A in hand and finessing through North, in part to protect against an early diamond switch. It risks little to go this route even if the odds are against it. The strategy is to play for an average and hope the defenders make a mistake along the way.

My error was not so much in trying for a cover of the ♥J, but in not following through and playing the ♥A when South played low. When North won the ♥Q and cashed his spade winners, my luck changed and I was presented with the very mistake for which I should have been hoping all along. As the reader can see, when North cashes both his spade winners, he set up my ninth trick (2 spades, 3 hearts, 1 diamond and 3 clubs), so it was now a matter of the overtrick. Fearing that North might hold 4 clubs was misguided because if North held the ♦K and 4 clubs he would be squeezed when I ran off the hearts and the ♠J in the dummy from this position:

West	North
♠ —	♠ J	The lead of the ♣ J executes the squeeze.
♥ —	♥ —
♦ J	♦ Q
♣ A K 10 8	♣ Q 4 3

With the diamond finesse still at the forefront of my thinking, I didn’t follow through by rejecting the diamond finesse and playing for the better odds of 4 tricks in clubs. I should have been focused from the very beginning on the most probable conditions with regard to both minors taken simultaneously. (Just envisioning Condition I would have been a good start.) Taking the ♦A and playing off the hearts would determine the split in the heart suit to have been 3-3, so the remaining possibilities are Conditions I, II, and VI, and one may observe some discomfort on the play of the last spade under Condition II when South holds the ♦K and 4 clubs to the CJ. As at worst only one trick remains to be lost, there is an average score to gain and nothing to lose by refusing the diamond finesse. With all hands revealed, failure to count becomes the most glaring of errors.

The bidding and the play are important factors in determining probabilities. Generally speaking the a priori odds represent a reasonably accurate guess of one’s chances during the play as long as even splits are encountered. That may change significantly if some splits are known to be extreme. There was no evidence that such was the case on this deal as bid and played. If one calculates from Conditions I, II and VI the odds for obtaining 4 club tricks by playing ♣A – ♣Q – ♣K, one finds it to be 70%, much better than the a priori odds, because of the squeeze possibilities that have evolved.

NEXT!