Would I Lie to Monty Hall?

Today in Tedium: Reader feedback and comments are one of the joys and frustrations of writing for a newsletter like this. A few weeks ago, Ernie shared an email with a couple of us Tedium writers that was a little bit of both. It was a simple correction to a piece that was ultimately incorrect. Though our dear, dear Tedium readers are often correct when pointing out the mistakes we make, sometimes they try to correct accurate information. And the inner petty pendant at the heart of most writers (at least this one) kind of loves when that happens. But that certainly is not the case for a lot of writers whose work strikes the wrong chord with the vile and unpleasant aspects of our society. As it so happens, this relationship between writers and readers is at the heart of something I’ve been wanting to write about for a while. And maybe, just maybe, it helped lead to a new variation of a contentious game theory problem. Today’s Tedium is talking about the Monty Hall problem and the columnist who got flack for giving the right answer. — Andrew @ Tedium

An episode of Let’s Make a Deal, just in case you forgot what this game looked like.

How getting it right can go so wrong

In September 1990, a reader proposed a seemingly simple question to Parade magazine columnist Marilyn vos Savant: “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?”

Her response was pretty straightforward, just another question from a curious but confused reader: “Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?”

Marilyn vos Savant was as good as any source to ask such a question in the pre-Internet era. After moving to NYC in her 30s to pursue a career in writing (one can relate), vos Savant began writing I.Q. quizzes for Omni magazine. Interestingly, the Guiness Book of World Records would recognize vos Savant as having the world’s highest recorded I.Q. until discontinuing the category in 1990. The distinction gave vos Savant a measure of fame that led to her Parade column in 1986, which she still writes today.

The problem that started it all. (via Newspapers.com)

Apparently the readers of Parade magazine were none too impressed with vos Savant’s pedigree and answered in volume to her solution to the Monty Hall problem. Letters from academics came from around the country. Robert Sachs, Ph.D of George Mason University wrote: “Since you seem to enjoy coming straight to the point, I’ll do the same. You blew it! 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.”

Another Ph.D, Barry Pasternack of the California Faculty Association, at least tried to offer some condolences in his rebuke of vos Savant’s solution: “Your answer to the question is in error. But if it is any consolation, many of my academic colleagues have also been stumped by this problem.”

Of course, there were also responses from people taking shots at Marilyn’s highest I.Q. claim. Scott Smith, Ph.D of the University of Florida wrote, “You blew it, and you blew it big! Since you seem to have difficulty grasping the basic principle at work here, I’ll explain. After the host reveals a goat, you now have a one-in-two chance of being correct. Whether you change your selection or not, the odds are the same. There is enough mathematical illiteracy in this country, and we don’t need the world’s highest IQ propagating more. Shame!”

And, inevitably, there were readers who couldn’t help but make sexist comments. Don Edward’s of Sunriver, Oregon commented: “Maybe women look at math problems differently than men.” Yes, I’m sure that’s the problem here.

After the initial uproar, vos Savant and her readers went back and forth addressing the issue periodically for months. She would offer additional reasoning for why her original solution was, in fact, correct. The fracas reached the point that The New York Times did an article on the situation, going so far as to interview the actual Monty Hall for his thoughts. He wasn’t at all surprised experts got it wrong as he’d watch contestants make the same mistake for years on Let’s Make a Deal, the game show from which his namesake paradox derived. He told the Times, “That’s the same assumption contestants would make on the show after I showed them there was nothing behind one door. 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.’ ”

Notice that bit about the money? It turns out that vos Savant was right about her solution but wrong about something else. Not that many of her irate readers caught it.

Monty Hall, accidental math teacher.

The problem with the problem

To combat the flood of incorrect solutions, vos Savant asked mathematicians and math teachers to conduct the Monty Hall experiment for themselves. To their surprise, she was right.

“After considerable discussion and vacillation here at the Los Alamos National Laboratory, two of my colleagues independently programmed the problem, and in 1,000,000 trials, switching paid off 66.7% of the time. The total running time on the computer was less than one second,” wrote G.P. Default, Ph.D.

Some even used her solution to win an easy bet: “I put my solution of the problem on the bulletin board in the physics department office at the Naval Academy, following it with a declaration that you were right. All morning I took a lot of criticism and abuse from my colleagues, but by late in the afternoon most of them came around. I even won a free dinner from one overconfident professor,” claimed Eugene Mosca, Ph.D.

Since it was the early 1990s and Times reporters had actual expense accounts back then, they went to Monty Hall’s Beverly Hills mansion and conducted the same experiment vos Savant asked her readers to conduct. After 20 trials proving vos Savant’s solution, Monty caught a problem with the original premise. After acknowledging vos Savant’s correct answer, Hall pointed out the premise doesn’t follow the rules of Let’s Make a Deal. The Times quoted this exchange between Hall and one of its reporters:

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.’

The Times article goes on to point out that this particular Let’s Make a Deal game also involved host participation with Monty Hall carrying a few thousand dollars in cash to try to persuade them to switch or not switch doors. After playing a few more rounds where the Times reporters kept winning goats, Monty Hall said, “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.”

A few very competent readers of vos Savant’s column noticed the same loophole Monty Hall did with one writing, “The problem is not well-formed, 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.” 

While vos Savant admits the ambiguity behind the original proposed question, she called her assumption about always being given the option to switch “minor” and no reason to excuse the error of her critics. She added, “I wouldn’t have minded if they had raised that objection because it would mean they really understood the problem. But they never got beyond their first mistaken impression. That’s what dismayed me.”

The Times finished its article with what “should be the last word on the subject” with Mr. Hall saying, “If the host is required to open a door all the time and offer you a switch, then you should take the switch. 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.”

As fascinating as vos Savant’s Monty Hall saga can be, it does lead to some questions about the real world application of probability theory. Can the Monty Hall problem be applied to other situations? Am I really about to watch 120 episodes of a British panel show just to find out?

The logic of the Monty Hall problem translates to panel shows pretty well.

Would I Lie to Monty Hall?

We’ve discussed Would I Lie to You (WILTY) before in the context of Britain’s unique culture of panel shows. Panel shows are similar to game shows in that a game is played for an audience’s amusement but panel shows typically feature celebrities who aren’t playing for prizes.

Would I Lie to You follows in the tradition of shows like To Tell the Truth where celebrities are asked to discern the veracity of various statements. Someone tells a story or makes a claim, and the other side has to determine if they are telling the truth or not. WILTY features two teams of three with each side being led by a captain that recurs every week, the entirety of the show the captains have been comedians David Mitchell and Lee Mack (with the exception of one episode where Greg Davies took over for Mack, which has been discounted from my study).

The majority of any given episode of WILTY involves one person reading a statement from a card with the other team attempting to guess whether it’s true or not. However, the show does include one particular game called “This Is My…“, where a non-playing person is introduced and each member of one team claims they have some relationship with that person. Only one of the three-member team is telling the truth.

Here is an excellent example that also happens to be hilarious.

And another with notoriously unreliable narrator Bob Mortimer.

A few comparisons to the Monty Hall problem should be immediately obvious, mainly that players have a one in three chance of guessing correctly at random. However, as we noted above, the fundamental aspect of the Monty Hall problem is the ability to switch your choice once you’ve eliminated one possibility. With WILTY, the game effectively gives this option in two ways.

First is simply to listen to the stories, over 103 episodes it’s pretty apparent one is almost always obviously a lie. Here’s Lee Mack screwing up just the introduction of his story:

Using this clip is a bit of cheating as it’s an outtake that wasn’t shown in the original broadcast. However, in the full clip, when the opposing team gets a chance to choose, they’ve eliminated Lee because of the outtake.

The second method to achieve the Monty Hall switch is actually to just eliminate the team captain. Over the course of 103 episodes, you would expect that David Mitchell and Lee Mack were telling the truth in approximately 34 episodes. In fact, they combine for just 11 total instances where they were telling the truth. Now this is an obviously imperfect method as double bluffs (seeming like you’re lying when you are actually telling the truth) are somewhat common on the show. However, this situation can also lead to a near certain situation if one of the guests is clearly lying, just pick the one that’s not the host.

To give a blank example, you are presented with two lies and one truth.

LIE LIE TRUE

Each says their story, giving the team their first chance to pick one at random, say, the first story. Eliminating the team captain after hearing the full stories and switching to the third option obviously yields the correct answer. After 14 series (what the Brits call seasons), both team captains seem acutely aware that they are rarely telling the truth while playing “This Is My…” David Mitchell has only told the truth 5 times and Lee Mack only selected him once when he was telling the truth. The remaining 4 times, Lee’s team selected the wrong choice.

However, there are times when something just feels right and suddenly the team captain is in play. As in this example involving changing the color of a BMW:

The important part of that clip really starts at the 8 minute mark, when Lee Mack’s team starts deliberating. Lee quickly points out the third option is problematic, giving them the realistic option of picking either David or contestant one. Lee ultimately agrees with his team and selects the first contestant … yielding the incorrect answer. If they had followed a Monty Hall approach, and switched when it was clear one of the guest contestants was lying, they would have had a better chance of selecting the team captain despite that option being the least likely over the course of the show.

In a direct comparison between the Monty Hall problem and the Would I Lie to You conundrum, the switch comes from game mechanics that aren’t always available given the individual situation and mood of the participants. But the overall comparison generally holds true. Pick an option, when one presents itself as a lie (hopefully not your original selection), switch and you should get the same results as the Monty Hall problem, i.e. being right about 66.7 percent of the time.

At least, I think that logic works out.