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The Student, the Explorer, and the Perverse Explorer
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There are a number of bridge hands that are well beyond my skill level, hands that I will never fully comprehend.  I am reminded of this sad fact every time I watch a set of hands.  Consider this simple-looking problem from the current USBF Trials.  With East-West vulnerable, you arrive in six hearts after a fairly uninformative auction, and West tracks the ace of spades:

North
K7
KQ543
Q10
KJ107
South
QJ10
AJ1096
AK
A98

The contract hinges on finding the queen of clubs, a fact obvious to you and to everyone at the table.  Whatever line you choose will work out if someone has a singleton queen, so we'll rule that out.  Likewise, if spades are 6-2, or trumps 3-0, you might have enough distributional clues to help your guess, but no, trumps are 2-1 and everyone will follow to the third spade.

How do we increase the odds on this 50-50 guess?  There seem to me to be three approaches to this problem.

 

                                             The Student


The Student is an intermediate player who has read all the books and memorized all the probability tables.  He will carefully play the third round of spades, trying to learn something.  When nothing interesting develops, he will play the defender with the short trump holding for the club lady.

This is not a bad approach, since the queen will be with the singleton heart 12/23 of the time, around 52%.  Of course, you and I are experts, and we won't settle for the 52% play.  We will try to learn as much as we can ...

                                            The Explorer


The expert Explorer will cash all of his winners, coming down to three clubs in each hand, KJ10 opposite A98.  Against many typical opponents, if clubs were originally three-three, everyone will be down to three clubs, leaving our Explorer with a blind guess.  However, any time a defender has four or more clubs, they are forced into a revealing club discard, and the Explorer will play that hand for the club queen.

So, our Explorer, against such naive defenders, will make the hand whenever clubs are 5-1 or 6-0, two-thirds of the time when clubs are 4-2, and half of the 3-3 club splits.  Success rate - 66%.  

                                          Saboteurs


Unfortunately, most defenders are not this naive, and when they sense the Explorer's strategy, they are likely to resort to a simple counter - whenever clubs are 3-3, the player without the club queen will discard a club.  Against such terrorists, our Explorer will never pick up the club queen when clubs are three-three, dropping the success rate to 48%.  That's not so good. 

We might even run up against some Super Saboteurs, who discard a club from three small, and, when a player has Qxxx in clubs, a winning case for Explorer, both defenders will discard a club.  Explorer will suspect the layout, but won't be able to tell who has the club length, and so will guess these only half the time.  Against these high-tech trouble-makers, our Explorer will only find the club queen 32% of the time!


                                          The Perverse Explorer


The Perverse Explorer plays exactly the same way, reducing each hand to three clubs, but then plays the hand who did not discard a club for the queen.  Essentially, the Perverse Explorer figures out what Explorer would do, and plays the other way.  Since our Explorer succeeded on 48% of the hands against Saboteurs, Perverse Explorer finds the queen on 52% of those hands.  And Perverse Explorer has a wonderful time against Super Saboteurs, landing the slam 68% of the time.

Obviously, if you think you are defending against a Perverse Explorer, you will discard a club whenever you hold the queen ...

At the table, both teams in Milner vs. Fleisher bid to six hearts, played by North.  At one table, both North and South had shown balanced hands, which encouraged a passive opening lead, and East chose to lead a spade from 9862.  At the other table, North opened one notrump(!), South bid 2C Stayman, and then 4C, keycard for hearts.  Here too, a passive lead seemed best, and East also led a spade.  As the play progressed, both defenders threw their other spade, so declarer knew that spades were originally 4-4, and that the lead was from 9862.  

There are certainly hands where a passive choice in one suit suggests that the defender has the missing queen - with nothing in either suit, East might have led a club.  That seems doubtful here.  North and South have all the club spots, so, given a choice between, say, 9862, and 6432, the spade seems right.  So, both our declarers faced a more-or-less straight guess for the club queen.  Would the experts play like the Student?  Of course not.  They both played out all their winners.

Schermer, for Milner, was an Explorer.  Fleisher turned out to be a Perverse Explorer.  This time Explorer got it right.

                                      For the Defense


How should the defenders approach this problem?  The naive defense is very poor, and gets killed by an Explorer.  Likewise, the Super Saboteurs are over-doing it, and get murdered by Perverse Explorers.  The simple Saboteurs seem to have it about right.  A player holding club length should discard a club, and occasionally two clubs, and a player holding three small clubs should also discard a club, or occasionally two clubs. 

The defense can improve on this slightly.  On rare occasions, when clubs are 3-3, the player holding the queen should discard a club.  If they do this around 5% of the time, then, in the three card club-only ending, declarer will have a complete 50-50 guess.  Both the Explorer and the Perverse Explorer will land their slam half the time.  

You might point out, a bit smugly, that Student made the hand 52% of the time.  So what?  What expert would ever settle for the percentage play when I can show off my card-reading skills and do worse?  

Besides, only a true expert could find a way to go down two tricks in this heart slam. 

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