Monty Hall vs the Psychologists' Holy Grail

[ewmayer]

And Behind Door Number 1, A Fatal Flaw | New York Times

By JOHN TIERNEY
Published: April 8, 2008

The Monty Hall Problem has struck again, and this time it’s not merely embarrassing mathematicians. If the calculations of a Yale economist are correct, there’s a sneaky logical fallacy in some of the most famous experiments in psychology.

A.k.a. "Psychologists get exposed to the Full Monty, and experience that shrinking feeling."

[cheesehead]

In following the links from that article I eventually found this in Tierney's 1991 article "Behind Monty Hall's Doors: Puzzle, Debate and Answer?" at http://query.nytimes.com/gst/fullpag...pagewanted=all:

An earlier version, the Three Prisoner Problem, was analyzed in 1959 by Martin Gardner in the journal Scientific American. He called it "a wonderfully confusing little problem" and presciently noted that "in no other branch of mathematics is it so easy for experts to blunder as in probability theory."

I'd add that it's awfully easy for the rest of us to blunder when pondering probabilities, also.

Here's one way to get the Monty Hall problem right that I find useful. (I didn't see it stated this way in the articles themselves, but I haven't read the readers' comments.) It's a restating of the articles' explanation about the probabilities. I'm not claiming it reveals anything new about the problem, just that I find it a good way to keep the logic straight.

(BTW, this applies only to the version wherein Monty is required both to open a nonwinning door that you didn't choose, and to offer you the option of switching from your original choice to the other unopened door.)

Monty starts out with information you do not have, even about the doors he does not open, and his choice of door to open is a forced revelation of part of that extra information about the other doors.

Monty's choice tells you nothing about what's behind the door you chose (because Monty will never open that door, so his choice of one of the other doors cannot add anything to your information about the door you chose). That's why the probability that the car is behind the door you chose remains unchanged at 1/3 even after Monty opens another door.

Monty's choice does tell you something about both of the other doors, not just the one he opens. Since their sum of probabilities remains at 2/3, but the door he opens is revealed to have a probability of 0, then the entire remaining probability of 2/3 must be assigned to the door he did not open (and you did not originally choose).

- - -

"The Psychology of Getting Suckered" at http://tierneylab.blogs.nytimes.com/...ting-suckered/ has further explanation.

[wblipp]

Chen starts by saying the monkey has an unmeasureably small preference for one color of M&M, and then assumes the monkey always chooses the one color it prefers. That's what the sidebar is saying - the monkey is ALWAYS picking the one it prefers - so what is the meaning of "slight preference?"

If "slight preference" means the monkey will pick the one it prefers 53% of the time, then statistics has it picking green 51% of the time in the experiment - an even slighter preference.

[cheesehead]

Originally Posted by wblipp

Chen starts by saying the monkey has an unmeasureably small preference for one color of M&M,

Not exactly -- he's saying suppose the monkey has a slight preference that was not measurable during the first part of the experiment. If the preference was 1% different, then statistically the researchers would have to have performed N tests, say 1000, in order to reliably detect that preference. But if the researchers never considered how such a slight preference could affect their experimental findings, then they may have performed only, say, 300 tests, and that would have been enough to reliably detect a 3% preference but not a 1% preference.

(The numbers 1000 and 300 are just pulled out of the air -- I'm not looking up the formulas to find out how many are really required to detect 1% and 3%.)

So it's not that the monkey is necessarily always choosing a slightly-preferred one -- it's that if the monkey does that, it could explain the observed results without demonstrating cognitive dissonance.

So the researchers failed to eliminate a possible variable that could affect the explanation of their observed results, and thus did not prove the existence of cognitive dissonance.

[Brian-E]

Originally Posted by cheesehead

In following the links from that article I eventually found this in Tierney's 1991 article "Behind Monty Hall's Doors: Puzzle, Debate and Answer?" at http://query.nytimes.com/gst/fullpag...pagewanted=all:
....

When davar55 posed this same problem here a few months ago I had never heard of it before. I thought about it and asked whether it was agreed beforehand that the host would always open one of the other doors (without the prize) before the contestant made the final choice. My question was ignored and davieddy gave the correct answer assuming that it was indeed agreed beforehand. I thought then "oh well, silly question from me". So I'm quite gratified now that your link describes this question of agreement beforehand as a flaw in the puzzle as Marilyn vos Savant had stated it.

[cheesehead]

Originally Posted by Brian-E

I thought about it and asked whether it was agreed beforehand that the host would always open one of the other doors (without the prize) before the contestant made the final choice. My question was ignored and davieddy gave the correct answer assuming that it was indeed agreed beforehand. I thought then "oh well, silly question from me".

No, not silly at all!

That just further illustrates the trickiness of reasoning about probabilistic situations: One single, seemingly nit-picky addition (or omission) that's easy to overlook or dismiss can completely change the outcome probabilities!

(Or perhaps we could view it as a typical peril of translating word problems into numeric equations.)

[wblipp]

Originally Posted by cheesehead

So it's not that the monkey is necessarily always choosing a slightly-preferred one -- it's that if the monkey does that, it could explain the observed results without demonstrating cognitive dissonance.

So the researchers failed to eliminate a possible variable that could affect the explanation of their observed results, and thus did not prove the existence of cognitive dissonance.

Yes, that's what he is saying.

What I'm saying is that his mathematics is wrong. I'm saying that the effect he notes cannot explain the observed results. I'm saying that his error - the only way he can get to the conclusion (wrong conclusion) that the effect can explain the results is to make the error of assuming a slight preference in the pre-trial becomes an absolute preference in the trial.

In particular, the sidebar shows 3 scenarios and shows that the monkey prefers green in two of them - hence the monkey prefers green 2/3 of the time. I'm saying the monkey will pick green 53% of the time in the two scenarios he prefers green, and 47% of the time in the one scenario he prefers blue, hence will pick green 51%. The 53% preference cannot explain picking green 2/3 of the time. In fact, any preference becomes diminished in the trial, not augmented.

William

[cheesehead]

Originally Posted by wblipp

I'm saying that his error - the only way he can get to the conclusion (wrong conclusion) that the effect can explain the results is to make the error of assuming a slight preference in the pre-trial becomes an absolute preference in the trial.

I think I understand why you're saying that, but also why that is not the "only way".

I think some more details of the experiment need to be explained. Here is my understanding/guess, based on what's in the articles: It is not necessary that the monkey always have an absolute preference for red over blue -- only that it has that preference for the short duration of the time needed to make its first of two successive choices. (At some other time, say 10 minutes later, the monkey could have a preference for blue over red.) What is necessary is that when the researchers offer the second choice, quickly after the first, they do not include the color the monkey chose the first time -- no matter what color that is.

In particular, the sidebar shows 3 scenarios and shows that the monkey prefers green in two of them - hence the monkey prefers green 2/3 of the time.

Not exactly -- it shows that the monkey prefers green in two out of those three possible scenarios. If all of those three scenarios existed an equal number of times and they were the only possible scenarios, then you can say the monkey prefers green 2/3 of the time.

But those are not the only three possible scenarios -- they're just the three in which the monkey prefers red over blue.

I'm saying the monkey will pick green 53% of the time in the two scenarios he prefers green, and 47% of the time in the one scenario he prefers blue,

Shouldn't that be more like 100% and 0%, giving an average over the three scenarios, assuming equal numbers of presentations, of 67% green and 33% blue, but those percentages apply only in the scenarios in which the monkey first chooses red over blue?

BTW, I lost track -- where did your 53% came from in the articles? Or if it's not from the articles, would you please define exactly how it was originally derived in your example?

[davieddy]

Originally Posted by Brian-E

When davar55 posed this same problem here a few months ago I had never heard of it before. I thought about it and asked whether it was agreed beforehand that the host would always open one of the other doors (without the prize) before the contestant made the final choice. My question was ignored and davieddy gave the correct answer assuming that it was indeed agreed beforehand. I thought then "oh well, silly question from me". So I'm quite gratified now that your link describes this question of agreement beforehand as a flaw in the puzzle as Marilyn vos Savant had stated it.

I would have acknowledged your question, but I thought that
Davar55's reply to my preceding one had settled it unambiguously.

David

BTW the only way to guarantee a response in these forums is
to say something wrong. Thankyou (and indirectly cheesehead)
for confirming that my solution was correct:)

[wblipp]

Originally Posted by cheesehead

Shouldn't that be more like 100% and 0%, giving an average over the three scenarios, assuming equal numbers of presentations, of 67% green and 33% blue,

That's what Chen is saying. The experiments show 2/3, and Chen says this is why. But it's 100% and 0% only if the "undetectable small preference" means the monkey will ALWAYS make a particular choice. My contention is that such a preference is easy to detect, and that monkey would never have passed the pre-test.

Originally Posted by cheesehead

BTW, I lost track -- where did your 53% came from

I made it up to give a numerical example of "small preference," selected as the smallest integer preference that also gives an integer average.

[davieddy]

Originally Posted by ewmayer

And Behind Door Number 1, A Fatal Flaw | New York Times


A.k.a. "Psychologists get exposed to the Full Monty, and experience that shrinking feeling."

There are lies, damned lies, statistics and psychology