Wednesday, October 22, 2008

Monty Hall Show(Discussion)

April 8, 2008 (NY Times)

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.

Even some of the smartest mathematicians initially come up with the wrong answer to the Monty Hall Problem. Perhaps the best way to understand it is to play the game yourself.

The economist, M. Keith Chen, has challenged research into cognitive dissonance, including the 1956 experiment that first identified a remarkable ability of people to rationalize their choices.

Dr. Chen says that choice rationalization could still turn out to be a real phenomenon, but he maintains that there’s a fatal flaw in the classic 1956 experiment and hundreds of similar ones. He says researchers have fallen for a version of what mathematicians call the Monty Hall Problem, in honor of the host of the old television show, “Let’s Make a Deal.”

Here’s how Monty’s deal works, in the math problem, anyway.

He shows you three closed doors, with a car behind one and a goat behind each of the others. If you open the one with the car, you win it. You start by picking a door, but before it’s opened Monty will always open another door to reveal a goat. Then he’ll let you open either remaining door.

Suppose you start by picking Door 1, and Monty opens Door 3 to reveal a goat. Now what should you do? Stick with Door 1 or switch to Door 2?

Before I tell you the answer, I have a request. No matter how convinced you are of my idiocy, do not immediately fire off an angry letter. In 1991, when some mathematicians got publicly tripped up by this problem, I investigated it by playing the game with Monty Hall himself at his home in Beverly Hills, but even that evidence wasn’t enough to prevent a deluge of letters demanding a correction.

Before you write, at least try a few rounds of the game, which you can do by playing an online version of the game. Play enough rounds and the best strategy will become clear: You should switch doors.

This answer goes against our intuition that, with two unopened doors left, the odds are 50-50 that the car is behind one of them. But when you stick with Door 1, you’ll win only if your original choice was correct, which happens only 1 in 3 times on average. If you switch, you’ll win whenever your original choice was wrong, which happens 2 out of 3 times.

Now, for anyone still reading instead of playing the Monty Hall game, let me try to explain what this has to do with cognitive dissonance.

For half a century, experimenters have been using what’s called the free-choice paradigm to test our tendency to rationalize decisions. This tendency has been reported hundreds of times and detected even in animals. Last year I wrote a column about an experiment at Yale involving monkeys and M&Ms.

The Yale psychologists first measured monkeys’ preferences by observing how quickly each monkey sought out different colors of M&Ms. After identifying three colors preferred about equally by a monkey — say, red, blue and green — the researchers gave the monkey a choice between two of them.

If the monkey chose, say, red over blue, it was next given a choice between blue and green. Nearly two-thirds of the time it rejected blue in favor of green, which seemed to jibe with the theory of choice rationalization: Once we reject something, we tell ourselves we never liked it anyway (and thereby spare ourselves the painfully dissonant thought that we made the wrong choice).

But Dr. Chen says that the monkey’s distaste for blue can be completely explained with statistics alone. He says the psychologists wrongly assumed that the monkey began by valuing all three colors equally.

Its relative preferences might have been so slight that they were indiscernible during the preliminary phase of the experiment, Dr. Chen says, but there must have been some tiny differences among its tastes for red, blue and green — some hierarchy of preferences.

If so, then the monkey’s choice of red over blue wasn’t arbitrary. Like Monty Hall’s choice of which door to open to reveal a goat, the monkey’s choice of red over blue discloses information that changes the odds. If you work out the permutations, you find that when a monkey favors red over blue, there’s a two-thirds chance that it also started off with a preference for green over blue — which would explain why the monkeys chose green two-thirds of the time in the Yale experiment, Dr. Chen says.

Does his critique make sense? Some psychologists who have seen his working paper answer with a qualified yes. “I worked out the math myself and was surprised to find that he was absolutely right,” says Daniel Gilbert, a psychologist at Harvard. “He has essentially applied the Monty Hall Problem to an experimental procedure in psychology, and the result is both instructive and counter-intuitive.”

Dr. Gilbert, however, says that he has yet to be persuaded that this same flaw exists in all experiments using the free-choice paradigm, and he remains confident that the overall theory of cognitive dissonance is solid. That view is shared by Laurie R. Santos, one of the Yale psychologists who did the monkey experiment.

“Keith nicely points out an important problem with the baseline that we’ve used in our first study of cognitive dissonance, but it doesn’t apply to several new methods we’ve used that reveal the same level of dissonance in both monkeys and children,” Dr. Santos says. “I doubt that his critique will be all that influential for the field of cognitive dissonance more broadly.”

Dr. Chen remains convinced it’s a broad problem. He acknowledges that other forms of cognitive-dissonance effects have been demonstrated in different kinds of experiments, but he says the hundreds of choice-rationalization experiments since 1956 are flawed.

Even when the experimenters use more elaborate methods of measuring preferences — like asking a subject to rate items on a scale before choosing between two similarly-ranked items — Dr. Chen says the results are still suspect because researchers haven’t recognized that the choice during the experiment changes the odds.

“I don’t know that there’s clean evidence that merely being asked to choose between two objects will make you devalue what you didn’t choose,” Dr. Chen says. “I wouldn’t be completely surprised if this effect exists, but I’ve never seen it measured correctly. The whole literature suffers from this basic problem of acting as if Monty’s choice means nothing.”

July 21, 1991(NY Times)

Perhaps it was only an illusion, but for a moment here it seemed that an end might be in sight to the debate raging among mathematicians, readers of Parade magazine and fans of the television game show "Let's Make a Deal."

They began arguing last September after Marilyn vos Savant published a puzzle in Parade. As readers of her "Ask Marilyn" column are reminded each week, Ms. vos Savant is listed in the Guinness Book of World Records Hall of Fame for "Highest I.Q.," but that credential did not impress the public when she answered this question from a reader:

"Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the other doors, opens another door, say No. 3, which has a goat. He then says to you, 'Do you want to pick door No. 2?' Is it to your advantage to take the switch?"

Since she gave her answer, Ms. vos Savant estimates she has received 10,000 letters, the great majority disagreeing with her. The most vehement criticism has come from mathematicians and scientists, who have alternated between gloating at her ("You are the goat!") and lamenting the nation's innumeracy.

Her answer -- that the contestant should switch doors -- has been debated in the halls of the Central Intelligence Agency and the barracks of fighter pilots in the Persian Gulf. It has been analyzed by mathematicians at the Massachusetts Institute of Technology and computer programmers at Los Alamos National Laboratory in New Mexico. It has been tested in classes from second grade to graduate level at more than 1,000 schools across the country.

But it was not until Thursday afternoon that a truly realistic simulation of the problem was conducted. The experiment took place at the Beverly Hills home of Monty Hall, the host of 4,500 programs of "Let's Make A Deal" from 1963 to 1990. In his dining room Mr. Hall put three miniature cardboard doors on a table and represented the car with an ignition key. The goats were played by a package of raisins and a roll of Life Savers.

After Mr. Hall allowed this contestant 30 independent attempts to win the car, two conclusions seemed clear:

Ms. vos Savant's vitriolic critics, including the mathematics professors, are dead wrong. But Ms. vos Savant is not entirely correct either, because there is a small flaw in her wording of the problem that was detected not only by Mr. Hall but also by some of the experts. Despite her impressive analysis and 228-point I.Q., she was not as quick as Mr. Hall in understanding the psychological dimensions of the problem. 'So Easy' to Blunder

A few mathematicians were familiar with the puzzle long before Ms. vos Savant's column. They called it the Monty Hall Problem -- the title of an analysis in the journal American Statistician in 1976 -- or sometimes Monty's Dilemma or the Monty Hall Paradox.

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."

The experts responded in force to Ms. vos Savant's column. Of the critical letters she received, close to 1,000 carried signatures with Ph.D.'s, and many were on letterheads of mathematics and science departments.

"Our math department had a good, self-righteous laugh at your expense," wrote Mary Jane Still, a professor at Palm Beach Junior College. Robert Sachs, a professor of mathematics at George Mason University in Fairfax, Va., expressed the prevailing view that there was no reason to switch doors.

"You blew it!" he wrote. "Let me explain: If one door is shown to be a loser, that information changes the probability of either remaining choice -- neither of which has any reason to be more likely -- to 1/2. As a professional mathematician, I'm very concerned with the general public's lack of mathematical skills. Please help by confessing your error and, in the future, being more careful."

The criticism has continued despite several more columns by Ms. vos Savant defending her choice. "You are utterly incorrect," insisted E. Ray Bobo, a professor of mathematics at Georgetown University. "How many irate mathematicians are needed to get you to change your mind?" 'The Henry James Treatment'

Mr. Hall said he was not surprised at the experts' insistence that the probability was 1 out of 2. "That's the same assumption contestants would make on the show after I showed them there was nothing behind one door," he said. "They'd think the odds on their door had now gone up to 1 in 2, so they hated to give up the door no matter how much money I offered. By opening that door we were applying pressure. We called it the Henry James treatment. It was 'The Turn of the Screw.' "

Mr. Hall said he realized the contestants were wrong, because the odds on Door 1 were still only 1 in 3 even after he opened another door. Since the only other place the car could be was behind Door 2, the odds on that door must now be 2 in 3.

Sitting at the dining room table, Mr. Hall quickly conducted 10 rounds of the game as this contestant tried the non-switching strategy. The result was four cars and six goats. Then for the next 10 rounds the contestant tried switching doors, and there was a dramatic improvement: eight cars and two goats. A pattern was emerging.

"So her answer's right: you should switch," Mr. Hall said, reaching the same conclusion as the tens of thousands of students who conducted similar experiments at Ms. vos Savant's suggestion. That conclusion was also reached eventually by many of her critics in academia, although most did not bother to write letters of retraction. Dr. Sachs, whose letter was published in her column, was one of the few with the grace to concede his mistake.

"I wrote her another letter," Dr. Sachs said last week, "telling her that after removing my foot from my mouth I'm now eating humble pie. I vowed as penance to answer all the people who wrote to castigate me. It's been an intense professional embarrassment." Manipulating a Choice

Actually, many of Dr. Sachs's professional colleagues are sympathetic. Persi Diaconis, a former professional magician who is now a Harvard University professor specializing in probability and statistics, said there was no disgrace in getting this one wrong.

"I can't remember what my first reaction to it was," he said, "because I've known about it for so many years. I'm one of the many people who have written papers about it. But I do know that my first reaction has been wrong time after time on similar problems. Our brains are just not wired to do probability problems very well, so I'm not surprised there were mistakes."

After the 20 trials at the dining room table, the problem also captured Mr. Hall's imagination. He picked up a copy of Ms. vos Savant's original column, read it carefully, saw a loophole and then suggested more trials.

On the first, the contestant picked Door 1.

"That's too bad," Mr. Hall said, opening Door 1. "You've won a goat."

"But you didn't open another door yet or give me a chance to switch."

"Where does it say I have to let you switch every time? I'm the master of the show. Here, try it again."

On the second trial, the contestant again picked Door 1. Mr. Hall opened Door 3, revealing a goat. The contestant was about to switch to Door 2 when Mr. Hall pulled out a roll of bills.

"You're sure you want Door No. 2?" he asked. "Before I show you what's behind that door, I will give you $3,000 in cash not to switch to it."

"I'll switch to it."

"Three thousand dollars," Mr. Hall repeated, shifting into his famous cadence. "Cash. Cash money. It could be a car, but it could be a goat. Four thousand."

"I'll try the door."

"Forty-five hundred. Forty-seven. Forty-eight. My last offer: Five thousand dollars."

"Let's open the door."

"You just ended up with a goat," he said, opening the door. The Problem With the Problem

Mr. Hall continued: "Now do you see what happened there? The higher I got, the more you thought the car was behind Door 2. I wanted to con you into switching there, because I knew the car was behind 1. That's the kind of thing I can do when I'm in control of the game. You may think you have probability going for you when you follow the answer in her column, but there's the pyschological factor to consider."

He proceeded to prove his case by winning the next eight rounds. Whenever the contestant began with the wrong door, Mr. Hall promptly opened it and awarded the goat; whenever the contestant started out with the right door, Mr. Hall allowed him to switch doors and get another goat. The only way to win a car would have been to disregard Ms. vos Savant's advice and stick with the original door.

Was Mr. Hall cheating? Not according to the rules of the show, because he did have the option of not offering the switch, and he usually did not offer it.

And although Mr. Hall might have been violating the spirit of Ms. vos Savant's problem, he was not violating its letter. Dr. Diaconis and Mr. Gardner both noticed the same loophole when they compared Ms. vos Savant's wording of the problem with the versions they had analyzed in their articles.

"The problem is not well-formed," Mr. Gardner said, "unless it makes clear that the host must always open an empty door and offer the switch. Otherwise, if the host is malevolent, he may open another door only when it's to his advantage to let the player switch, and the probability of being right by switching could be as low as zero." Mr. Gardner said the ambiguity could be eliminated if the host promised ahead of time to open another door and then offer a switch.

Ms. vos Savant acknowledged that the ambiguity did exist in her original statement. She said it was a minor assumption that should have been made obvious by her subsequent analyses, and that did not excuse her professorial critics. "I wouldn't have minded if they had raised that objection," she said Friday, "because it would mean they really understood the problem. But they never got beyond their first mistaken impression. That's what dismayed me."

Still, because of the ambiguity in the wording, it is impossible to solve the problem as stated through mathematical reasoning. "The strict argument," Dr. Diaconis said, "would be that the question cannot be answered without knowing the motivation of the host."

Which means, of course, that the only person who can answer this version of the Monty Hall Problem is Monty Hall himself. Here is what should be the last word on the subject:

"If the host is required to open a door all the time and offer you a switch, then you should take the switch," he said. "But if he has the choice whether to allow a switch or not, beware. Caveat emptor. It all depends on his mood.

"My only advice is, if you can get me to offer you $5,000 not to open the door, take the money and go home."


Akrajen said...

Consider the following semantic argument: Assuming that you picked one of the 'goat doors' to be eliminated in the first round, then:

1) Many in the audience were also speculating on which door has the prize, so those who chose the other unknown door did so while there was still a 1/3 probability. By switching from your original choice to get the better odds (1/2) you're really only switching to a decision that someone else made while the odds were still at 1/3 probability.

2) Once one door has been eliminated, you can negate the 1/3 probability of your original choice by choosing anew - to switch or not switch - it's still a choice, this time made under the conditions that there are only two possibilities. At this point, you might also think of the new choice as a bet about the validity of your original selection in which there is only the binary option of it being correct or incorrect. You can now choose the original selection again, assigning it the binary 'correct' value, or switch to the other door by assigning it the 'correct' value. Each has a 1/2 probability of being true.

Akrajen said...

better actually to say:
You can now choose the original selection again, assigning it the binary 'correct' value, or switch to the other door by assigning your original choice the 'incorrect' value. Each has a 1/2 probability of being true.

Akrajen said...

The previous comments assume that the contestant picked the door, having no idea where the prize is and not the host. It also assumes that the contestant can lose the game by eliminating the prize door in the first round.

Taro said...

This problem is on Baysian theorem: if you choose one of the three boxes and the host opens the other box and show you no prize in it, you will face the prob of which box has the prize.

Taro said...

Then, do you stick to the previous choice, or swap it?

The answer of this prob is 'swap'. The reason is simple:

Taro said...

See below: