Since I am a bridge player, I recognize the corollary to "restricted
choice" in this classic question, and somehow it makes it easier to
understand the Monty Hall question. The two situations are not exactly
the same, however.
In the Monty scenario, we assume that Monty will ALWAYS offer the
switch, whether or not you chose the "right" door. Let's, for the sake
of argument, also assume that he intends to open a door that does NOT
conceal the "Big Prize" before getting your answer.
Assume you picked door #1. Now, if this is WRONG, Monty has no choice
regarding which other door to show you prior to asking you to switch;
it will have to be the remaining "wrong" one. Conversely, if it is
RIGHT, then Monty will be able to choose at random which of the
remaining two doors to show you. So let's say that you initially pick
#1 and Monty shows you #3 (50 lbs. of Purina Goat Chow); either
You picked "correctly" (1/3 chance). and Monty CHOSE #3 to show you
OR, You picked "incorrectly", and Monty had NO CHOICE but to show you
#3 (because #2 was in fact the winner)
It's therefore twice as likely that Monty revealed #3 because he HAD
to (and you picked wrong initially), than that he had a choice, and
CHOSE to pick #3 (and you picked right initially).
In the bridge scenario, we assume the declarer has nine cards in a
suit between him and the dummy; he is missing four cards in the suit,
which include the Q and the J. Now, he wishes to simply play the A
followed by the K on the assumption that the opposing cards are
divided 2-2; now, accordingly, he plays the A; and one opponent plays
the Q or the J. The basic assumption is that if each opponent had
started with two cards, the queen and another or the jack and another,
he would play the small card under the ace. Therefore, if someone
plays an honor card under the ace, he 1)Had that card singleton or 2)
Had both honor cards and CHOSE to play the one he did. Now, if you
presume (and this is where it gets sticky) that EVERYONE considers the
Q and J to be of EQUAL value (as a person holding both of them
SHOULD), then the Monty Hall dynamic applies. The declarer should
assume that it's twice as likely that the person who played the honor
card did so because he HAD to (singleton) than that he had both honors
and CHOSE to (QJ doubleton). However, if someone is ingrained to hold
on to their top cards in a suit until the very last, then that person
would never play the Q unless he HAD to (and would play the J if hw
had QJ). Therefore the play of the Q from this person would definitely
mark the J in the other hand; the play of the J would strongly dictate
playing the other hand for the Q as well (the only time the J-hand
would have the Q also would be if that hand had started with QJ
specifically). HOWEVER, a more experienced player would realize that
this would result in declarer's always playing the suit properly in
these cases; the next thing a beginner tries is to drop the QUEEN from
QJ (which would seem to preclude his holding the J as well), in an
effort to be deceptive. The ideal strategy is to randomize one's
choice in this case (which card to play from QJ).
OK, so how does this relate to Monty Hall?
You will recall that only one of the doors concealed a REAL "stinker"
(a "zonk", I believe it was called); the other non-big-prize door
still contained something decent, like a bedroom set or a
washer-dryer, stuff like that. So the burning question IS: when Monty
had a choice (i.e., the contestant indeed picked the right door), was
he biased toward showing you the "zonk", or biased toward showing you
the "nice prize", or choosing randomly, or (worse) choosing which to
show you according to how he "read" you? Consider: if Monty shows you
the "zonk", the potential gain or loss is the difference between the
"nice prize" and the "grand prize". However, if Monty shows you the
"nice prize", the gamble is bigger since the gain or loss from
switching is now the difference between the grand prize and NOTHING.
So a priori, it's a bigger gamble to switch when Monty shows you the
"nice prize" than when he shows you the "zonk" (ignoring the
restricted choice discussions above), because even if you switch and
you're wrong, you'll still get something decent.
Hmmm. This makes me want to stay up late watching Game Show Network
just to gather data on what Monty DID do in this regard.
The upshot of all this, and one that never gets mentioned by all the
mathematicians (which in not surprising since they don't inhabit the
real world), is that the principle of "restricted choice" is ONLY
valid when the person making the choice regards EACH choice as
equivalent, with NO bias.
