Restricted Choice
(July 25, 2005: as a result of an e-mail commenting on legality of methods, I have clarified when you look at your cards when using the "pinochle rule" described below)
Do you remember the first time that you were introduced to the concept of "restricted choice" in a bridge setting? Did the explanation make complete sense, or did it seem that perhaps something was missing (for instance, the rationale for the odds figure discussed)? Or perhaps the explanation just did not make any sense at all?
Following is an explanation of "restricted choice" outside of a bridge setting, which can be used to demonstrate that the concept of "restricted choice" is completely valid. Newcomers to bridge should be receptive to the concept after working through the following scenario. Bridge teachers perhaps can use this example in a lesson on card play.
Monty Hall example
Remember the Monty Hall "Let's Make a Deal" TV show? A contestant is told that a substantial prize (a new car?) is behind one of three doors (named A, B, and C), and that small insignificant prizes (or nothing) are behind each of the other two doors. The contestant would choose a door, and then, after elaborate fanfare, Monty would open one of the other two doors to reveal an insignificant prize (a bar of soap) or nothing. Monty then would give the contestant a chance to change his/her mind about which door to choose: Stay with the original choice or switch? The studio audience would be screaming "Stay!" or "Switch" and the contestant would ultimately ...
What are the chances that the big prize is behind the other door? Sure thing? 50%? Some other number?
Did you ever receive the advice to go with your first impressions? For example, when answering multiple choice questions on some kind of test, did you ever go back and change some answers? Or did you go with the advice that some gave, "stick with your first answer?")
Well, research has indicated that students who go back and change answers IMPROVE their scores (when or why is not addressed here), and it is TWO to ONE that the big prize is behind the other door! The answer is SWITCH!!!
This conclusion can be reached in at least two ways. The first explanation does not really address the concept of restricted choice, but it is completely valid.
Explanation A: Careful Reasoning
When a contestant initially chooses a door, he/she has a probability of 1/3 of being correct; each door has a probability of 1/3 that the big prize is behind that door. But when Monty opens a door which has nothing behind it, the probability of the prize being behind that door is known to be ZERO. Now, which makes more sense? The 1/3 probability that was behind the door that was just opened "splits" between the other 2 doors, or should the 1/3 be assigned to the other door which the contestant did not choose? I hope that you realize that the contestant's probability of having originally chosen the correct door is STILL 1/3 (Monty's opening of a door does NOT affect this), and that therefore, the probability of the big prize being behind the other door is 2/3. SWITCH!!!
Explanation B: The concept of "Restricted Choice"
To simplify the analysis, we can assume that the contestant always chooses one door (and the prize could be assigned to any of 3 doors) or we can assume that the prize is always behind one door (and the contestant randomly chooses one of the 3 doors). You could analyze all 3x3=9 cases, but the end result will be the same. For convenience, assume that the contestant always chooses Door B, and that the prize can be behind either A, B, or C.
Three situations are possible:
- Situation 1: The prize is behind Door A. The only door that Monty Hall can open that has nothing behind it is Door C. His choice is RESTRICTED to Door C.
- Situation 2: The prize is behind Door B. Monty Hall can open EITHER Door A or Door C.
- Situation 3: The prize is behind Door C. Monty Hall is RESTRICTED to opening Door A.
In 2 out of 3 situations, the big prize is behind the other door. But wait, you say. In Situation 2, Monty could have opened either door. Doesn't that change the odds?
Well, the answer is NO. Let's assume that the scenario is repeated 300 times. Each time, the contestant chooses Door B, and the Prize is behind Door A 100 times, behind B 100 times, and behind C 100 times. Further, assume that Monty randomly chooses between A and C when the contestant has correctly picked Door B.
- 100 cases: Prize behind Door A, and contestant chooses Door B. Monty opens Door C to reveal nothing. Contestants correct decision is to SWITCH. Monty's decision to open Door C was a RESTRICTED choice.
- 100 cases: Prize behind Door C, and contestant chooses Door B. Monty opens Door A to reveal nothing. Contestants correct decision is to SWITCH. Same RESTRICTED choice decision.
- 100 cases: Prize behind Door B, and contestant should NOT switch. In half of these cases (50), Monty opens Door A, and opens Door C in the other 50 cases. Monty's choice as to which door to open is NOT restricted.
So, if the contestant chooses Door B, Monty will open Door A 150 times (100 times restricted and 50 times NOT restricted), and it will be correct to SWITCH in 100 of these cases. Monty will open Door C for the other 150 times, and it will be correct to SWITCH in 100 of these cases. Thus the ODDS are 2:1 in favor of switching.
The principle of RESTRICTED choice here is this: If Monty opens Door A to reveal nothing, the odds are 2 to 1 that he opened Door A because he HAD to; the prize is behind the other door. Same analysis if Monty opens Door C to reveal nothing.
Restricted Choice in Bridge
OK, how can this concept apply to contract bridge? The concept typically (but not always) applies to the situation where an opponent plays a particular card, when the opponent MIGHT have played a different card. For example, assume your trump suit is AK654 opposite 10987 in dummy. The bidding and opening lead are unrevealing. Upon gaining the lead, you cross to dummy, and lead the 10. Your RHO plays the 2, you play the Ace, and LHO plays the Queen. You cross to dummy to lead another trump, and RHO plays the 3. Should you finesse or play for the drop?
Beginners often assume that this a guess; RHO either has or does not have the Jack. However, the odds strongly favor taking the finesse. Now an incorrect analysis sometimes goes like this. Ignoring all other suits and cards played, there are 12 "slots" available for the Jack in the hand of LHO, and there are 11 "slots" available for the Jack in the hand of RHO. These are about the same, so it seems just slightly more likely that LHO has the Jack (about 50-50 whether to play for the Jack to drop). However, this analysis ignores the situation where LHO plays the Jack on the first round. Now who has the Queen? It turns out that the likelihood of the LHO holding EITHER a stiff Jack or stiff Queen is approximately twice that of holding QJ tight. The correct assumption for declarer to make is that LHO played the Jack (or Queen) because he HAD to (RESTRICTED CHOICE)
For the record, the correct odds are (ignoring any inferences from the bidding, lead, or early play):
- Above scenario, missing 4 cards, QJ and 2 small cards; odds are 11:6 in favor of the finesse (not quite 2:1 odds)
- Missing 5 cards (e.g., AK765 opposite 1098), QJ and 3 small cards; odds are 10:6 or 5:3 in favor of the finesse
- Missing 6 cards, QJ and 4 small cards (e.g. AK76 opposite 1098); odds are 9:6 or 3:2 in favor of the finesse (this ignores the issue of misleading falsecards)
- Missing 7 or more cards. Although the odds could be computed, they likely would not be accurate because of signaling issues and/or inferences from the bidding
Defender's play from QJ tight
Which card should you play? Not analyzed here, but you should randomize in some manner. For example, choose the Queen about 50% of the time on the first lead. (Almost anything works; just do not always choose the Queen or Jack.) But how can you randomize? You are not allowed to use any artificial aids: this would include calculators, looking at the second hand on your watch, looking at the board number or a predetermined list of random numbers, etc. You are restricted to cards that you hold. So, here is a legal method for a 50% decision.
BEFORE looking at any card in your hand, shuffle and count your cards face down.
(Counting BEFORE looking at your cards is correct procedure according to the laws of bridge (Law 7B1); if you do not count, and do not have 13 cards, you might be subject to penalty. As far as I know, shuffling is NOT required.)
Then look at the top card in your hand (or the bottom card, or any random card, for that matter). If you later find that you have QJ tight, then, if the first card is BLACK, play the Queen if you must choose between Q or J from QJ. If the first card is RED, choose the Jack. I call this my "pinochle" rule for randomization. You can extend this to several cards if you want to have a objective method for making more than one 50% decision on a given hand. If you do not have QJ tight, then you can use the observation to make any other 50% decision that you might have.
In Closing
Correction of background: Actually, on the show hosted by Monty Hall,
the contestant did not have the option of choosing the other door. But many people,
including me when I first wrote this short teaching aide,
have often assumed so, and thus this type of situation is often discussed as the "Monty Hall" problem.
Note: There are 3 implicit assumptions in the scenario as described above:
1. Monty knows what door the valuable prize is behind
2. Monty MUST choose a door, and offer the contestant the chance to change his/her choice
3. Monty will never open a door that reveals the valuable prize
PLEASE READ: A book which I highly recommend is "The Drunkard's Walk: How Randomness Rules Our Lives" by Leonard Mlodinow (c2008 Pantheon Books New York). In it he reports what happened when Marilyn vos Savant was asked what the solution to the "Monty Hall" problem was (in her Parade September 9, 1990 column), and she answered (correctly, according to the implicit assumptions in the above paragraph). She received over 10,000 letters, of which 92% claimed that she was wrong. Around 1000 letters were from PhD's, many in mathematics and/or statistics, of which 65% (of the PhD's) claimed that she was wrong. From the comments received, she concluded that almost all of the respondents agreed with the implicit assumptions in the above paragraph. It is well established that most people, including highly intelligent people, do not understand probability very well. For more information, just google "Tversky and Kahneman" to find references to experiments where subjects misjudge probabilities (often VERY badly). In real life, these misjudgments can have serious adverse consequences, and I highly recommend that you read the book by Mlodinow so that you are better able to judge probabilities in circumstances that affect you (and, of course, there are other sources of information. GOOGLE!).
Final comment: There are "Restricted Choice" situations where the odds are exactly 2:1 as in the Monty Hall example. For more information, look up restricted choice in the Bridge Encyclopedia. There are also some very subtle restricted choice situations in the bidding and in leads, but that is beyond the scope of this article.
Stan Fuhrmann