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Theory vs Practice
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"In theory, there is no difference between theory and practice."  - Yogi Berra

Let's look at a problem from BBO's excellent Bridge Master exercises in declarer play.  This one is #8 at the expert level.  Here is the problem as presented:

West
North
54
43
A32
KQ10987
East
South
AK32
AK2
KQJ10
32
W
N
E
S
2NT
P
4
P
4
P
6NT
P
P
P
D
1
6NT South
NS: 0 EW: 0
Q
3
6
A
3
1
0
2
5
Q
4
1
2
0
2
7
K
4
3
3
0
3
6
4

You can follow the play with the Next button on the handviewer.  You need club tricks to make your slam contract: you lead a club to dummy's king, which holds.  You return to your hand to lead towards dummy again, and West plays the last small club spot.  Is one choice better than the other?  Here is what the BBO Solution says about your decision:

South should begin by leading a club towards dummy. When West follows low, the K or Q should be played from dummy. It is a well-known defensive tactic for East to duck with A doubleton of clubs in this situation in order to leave declarer with a guess on the second round. When the Q holds the first trick, South has no inference as to the position of the A.

South returns to the closed hand and plays a second club. When West follows low it seems that South has a strict guess. Playing a low club will work when East began with doubleton A. Playing the K will work when East began with a doubleton J. The odds of these 2 holdings are the same.

This is actually not a guess. It is correct to play the K on the second round. When a declarer is faced with a problem such as this, he should formulate a plan and not deviate from that plan based on the defenders' early plays in the suit.

If delarer's plan is to play West for the J on the second round of the suit, it is irrelevant whether East wins the A on the first round when holding a doubleton. Declarer will always pick up that holding no matter how East defends.

If declarer's plan is to play West for the A on the second round of the suit, East, holding doubleton A, can always defeat the contract by ducking the first round. If East wins the A on the first round, South should play him for AJ doubleton.

The above club layout [East holding AJ doubleton] should only be picked up if declarer was planning on playing the K on the second round of the suit. This holding makes it correct to play the K on the second round regardless of whether East wins the A or the Q holds on the first round. 

I have played many BBO Bridge Master hands.  Almost all of them have Solutions that are marvelously clear and simple, and obviously correct.  I found this one both obscure and unconvincing.  You might want to consider whether you are convinced, one way or the other, before reading on.  (I'll tell you up front that when I ran the suit in SuitPlay, it agreed with BBO: it has exactly one line of play, matching the BBO Solution in every respect.  The Official Encyclopedia of Bridge has the same prescription, with an asterisk: "*But if East would not duck with Ax, finesse the ten if the king loses and lead the queen if the king holds.")

West
North
54
43
A32
KQ10987
East
South
AK32
AK2
KQJ10
32
W
N
E
S
2NT
P
4
P
4
P
6NT
P
P
P
D
1
6NT South
NS: 0 EW: 0
Q
3
6
A
3
1
0
2
5
Q
4
1
2
0
2
7
K
4
3
3
0
3
6
4

The key to this situation is whether - or how frequently - East might win the first club round, if holding Ax of clubs.  East has a clear optimum strategy here: always duck.  This reduces the declarer's maximum success rate to just over 53%, according to SuitPlay.  Declarer wins against singleton jacks, plus 3-2 divisions where West has xxx, Axx, AJx, AJ, Ax, or Jx.  No alternative defense by East does better; in particular, if East might win the first round holding Ax, the extra chance of jack in West is three times more likely than AJ doubleton with East.  

If East never wins the first round when holding Ax, then the BBO Solution assessment that declarer should play for AJ when East does win the first round is correct.  SuitPlay and the Encyclopedia use the same reasoning in arriving at the same conclusion.

How far does SuitPlay take this reasoning?  What if the dummy has only K Q 10 opposite declarer's 3 and 2 (which would make East's AJ doubleton less probable and Ax, Axx, Axxx, etc more so)?   Declarer leads to the queen, won by East's ace.  Declarer returns to hand to lead towards dummy again and  West plays low.  SuitPlay tells us that the still-possible distributions in which West has the jack total 23.99 percent.  The single distribution where East has AJ doubleton is 0.31 percent.  SuitPlay goes for that 0.31 % chance, because it knows that East would never win when not holding the jack, unless the ace was singleton, a mere 0.18%.  That is theory at work!  In practice - well I know I would give East a greater than 1 in 80 chance of winning the first round, holding the ace and not the jack.  At my club it's probably about 50-50.

More SuitPlay experiments:  I varied the K Q 10 experiment by putting the K Q 10 in South and the 3 2 in North.  As you might expect, SuitPlay's answer did not change.  I intended to suggest that the 3 and 2 were in the dummy, with the K Q 10 concealed in the declarer's hand.  In this case I would expect West to win the first round with Ax or Axx a relatively high percentage of the time, but SuitPlay has no way to account for this.  (Same result when dummy has K Q 9 and declarer has 10 2 and leads  the 2 on the first round.)  SuitPlay is a remarkable program, but it was not designed to take account of the information available to the defense, when formulating the plays the defense will make against various declarer strategies: the plays will always be those that most effectively limit declarer's chance of success.  There was probably a reason that the BBO problem had the KQ10987 of clubs exposed in dummy rather than hiddin in declarer's hand.  In theory there is no difference...

What is the effect of allowing for some deviation from the theoretically perfect defense?  The Encyclopedia has it right.  Considering only the original case of Ax with East, we could think of that as two distributions whose probabilities add to the original total.  In the first of the two East ducks the first round, and in the second East wins the ace.  The probability of the "ducks Ax" case is now less than that of the "held Jx" case, which means that when East does play low the first round, declarer should go up on the second.  It's essentially a restricted-choice line of reasoning: we assume that East did not have the option of winning.  So the BBO recommendation of going high on the second round when the queen won the first - apparently an even-money shot in theory - is the winner in practice.  It's the clear best play at any real bridge table.

What about that Solution statement, "When a declarer is faced with a problem such as this, he should formulate a plan and not deviate from that plan based on the defenders' early plays in the suit."  Does this really means anything?  What is "a problem such as this"?  We have seen a very similar problem (when East might win the first round holding Ax) in which it is right to finesse when East wins and not when East does not win.  As far as I can tell - in theory, at least, in a world where East never wins with Ax - the choice on the second round really is a toss-up.  The last paragraph of the BBO Solution feels like nonsense to me.  But I've been wrong before and I would welcome a clear argument on the other side.

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