Restricted Choice – part 1

It’s funny how topics start cropping up in your life after lying dormant for a while.  It’s been years since I gave any thought to the Monty Hall Problem, and the bridge application of it, Restricted Choice.  But the May IBPA Bulletin just arrived (or a least a URL link to it did), and John Carruthers has written an article on just that subject.  Meanwhile, only last week, there was the same topic staring at me from a manuscript I was editing.

Allow me to digress for a moment.  We are planning to publish this Fall a collection of bridge writings by the late Frank Vine, a prolific contributor to The Bridge World and other publications in the 1970s and1980s.  I remembered some of Frank’s work, but it wasn’t until I started rereading it recently that I realized just what a fine writer he was. 

The particular piece I was working on was entitled ‘How I abolished the Rule of Restricted Choice’, and I’ll get into its theme shortly (actually mostly in my next blog).  For now, let me recap the Rule itself, and the related game show problem.

The best-known case is the one where you have 9 cards in a suit between the two hands, and are missing QJxx.  Everyone knows ‘8 ever, 9 never’, which expresses the fact that by the time you have played one round of the suit, and then led it a second time and had one opponent follow, you are slightly better to play for a 2-2 split rather than back the a priori favorite, the 3-1 break, and finesse.

But all this goes out of the window if an honor appears on the first round, leaving you a potential finessing position if it was a singleton.  Now your cases are singleton Q (say) and doubleton Q-J — and it turns out you are almost 2:1 on if you finesse, as opposed to playing for the drop (actually 1.84 to 1 — remember that for a paragraph or so).  The theory is that your opponent might have played either card from the Q-J doubleton, but had no choice about playing a singleton.

In the game show, ‘Let’s Make Deal’, host Monty Hall would offer you the choice of one of three doors.  Behind one of them was a prize, while the others concealed joke winnings, known as ‘zonks’.  The contestant selected a door.  Monty now opened one of the other doors, always revealing a zonk, and offered the contestant a chance to change his selection from the original pick to the other unopened door.  Classic restricted choice — Monty couldn’t open a door with the prize behind it, so if the contestant hadn’t picked the winner already the unopened door contained the prize.  Contestants rarely switched, but in fact the mathematics says they should have done so — by odds of 1.84 to 1, a familiar ratio to bridge players.

Back in the days of Canadian Master Point magazine, we ran a series of articles on Restricted Choice, variously by Chuck Galloway and Eric Sutherland, which later were republished in our anthology, Northern Lights. I remember we got a citation in a doctoral thesis penned by a graduate student in mathematics at Dartmouth College.  There were also letters to the editor from non-believers.  Personally, I love to play against people who don’t believe in mathematics, especially if there is money involved.

But back to Frank Vine.  In his story, he is playing a critical deal in a tight IMP match.  The familiar nine-card fit missing QJxx is part of the scenario, and RHO duly drops the queen behind the ace.  About to apply Restricted Choice and take the finesse, Frank is stopped by LHO who alerts his partner’s play.  ‘We always play the queen from QJ doubleton,’  he explains.

Declarer begins to think about this.  Obviously the odds have now changed, but to what?  Does Restricted Choice apply any more?  And does his opposite number at the other table have a mathematically better chance to make the hand than Frank does?  If so, does that make any sense?  I thought I knew the answers to these questions, but decided to consult some experts before going any further — my son Colin, a bridge expert with a degree in Combinatorics, and Bob MacKinnon, author of Samurai Bridge and my go-to guy on all matters involving probability and information theory.  I heard back very quickly from both of them — and I’ll tell you what they said in Part 2.