# 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.

The Monty Hall problem is 2:1 for switching, not 1.84:1. 2/3 of the time you picked the wrong door originally and you win by switching, and 1/3 of the time you picked the right door originally and lose by switching.

Correct – Monty Hall is 2:1 – Restricted Choice is 1.84 : 1 I think

I think the finesse and the drop have very similar odds now (1.09:1?), so I’ll look for clues in body language. If LHO provides the same information when RHO plays the J, I would be favouring the finesse.

However, if I were RHO holding QJ, and I know my partner is obliged to tell the opponents that we always play Q from QJ, I think I might falsecard. Therefore I will simply ignore the information given.

The movie 21 has a good explanation of the Monty Hall problem.

It is a psychology question now. If truly the players would never false card from QJ and would always say we play Q for QJ then playing for the drop is a slight favorite, isn’t it? What are the odds of QJ tight versus Q stiff on the RHO? 2-2 split happens about 41% of the time and QJ with RHO is 1/6 of the splits. 3-1 split happens 50% of the time and Q stiff with RHO is one of 8 such arrangements. 41/6 is around 7% and 50/8 is just over 6%.

Now going from the technical solution to the psychological one of does he ever false card and/or does his partner ever not announce the carding makes the problem near unanswerable. I think I’d take the finesse in this case counting on my actions matching the other table and counting on a little bit of psychology/false-card % to overcome the slight technical disadvantage.

I wonder what you have to say if partner falsecards when you explain this all to the opponents. We “almost” always play the Q from QJ doubleton. That would probably represents about 90% of the bridge population. Because most bridge players like to drop the higher honour more frequently I don’t think the mathematical odds are the true odds anyway.

Is a partnership allowed to have an agreement to always play the Q from QJ, and then to use this very agreement to its own advantage by violating it every time declarer has to choose between finesse and drop? Is declarer entitled only to knowledge of the agreement? Or also to knowledge of adherence to the agreement?