Nathan Scandella (personal)

Monty Hall

I watched the movie 21 a while ago. I kind of like blackjack, and I'm a Techer, so I was happy to see a movie about MIT students. Something sat very wrong with me about the movie, and for some reason, I'm writing about it today. It is the discussion of the Monty Hall problem.

To summarize, the Monty Hall problem proposes a game show where the contestant gets to pick one of three doors to possibly reveal a prize. After choosing one door, the host (who knows where the prize is) opens up another door, which did not have the prize. He then asks the contestant if they'd like to change their mind, and go with the last of the three doors instead of the one they originally picked.

The common interpretation of this problem, and one employed by uber-genius Marilyn vos Savant, is that the contestant should always take the host up on their offer to switch choices, and that by doing so, they have a 2/3 chance of winning the prize.

This interpretation is wrong. That's right. It's wrong. The movie 21 was wrong. Marilyn vos Savant was wrong.

The reason is not in the math. It's in the logic used to interpret the problem. That logic assumes that because the host knows which door hides the prize, he will always open a door after the contestant makes their pick, and that he will always open a door without the prize. Where does this leap come from? This assumption is not justified based on the common statement of the problem (including the one given in 21, and in vos Savant's column).

I agree that if the game show host's strategy is as vos Savant, et al describe, then the probability of winning by switching goes to 2/3, and thus makes it a dominant strategy. No argument there. But decoding the strategy is the more difficult problem, and is where the common interpretation fails miserably.

First of all, in the problem statement, only one iteration of the game is described. How can it be assumed that what the host did in this round would be the exact same every single time? It can't. The host may choose to reveal doors every time, or only sometimes. They may reveal doors depending on whether the contestant correctly picks the right door on the first try, or not. The host's actions may be randomized. Nothing in the problem statement lends itself to the assumption necessary for the accepted answer to be mathematically correct.

Secondly, if the host is indeed following the assumed strategy, then (according to the vos Savant answer) they have provided a massive "clue" to the contestant. Why would they do this? Is there anything in the problem statement to suggest that the host is "on the contestant's side"? Not all games are setup to be easily won. Those damn basketball hoops at the county fair are smaller-than-regulation, specifically to cut down on the prize payout. People still play them, because they love games, and love a challenge (and are stupid, let's face it!).

Now, you may say "but this isn't a trivial problem, so the host's actions are only a massive clue to the super-brains among us". Not true. While the vast majority of the public would not be able to logically work through the math in order to arrive at the 2/3 probability, people of all intellectual levels are really good at seeing patterns. If the host actually adhered to the assumed strategy 100%, people who watched the show would pretty quickly learn that it usually would have been in the best interest for the contestant to switch. Most people's gut response to the problem would be that switching would give you a 1/3 or 1/2 probability of winning. Over time, if they witnessed that in actuality, about 2/3 of the contestants would have won by switching, that would become knowledge (even without understanding). People are good at that sort of pattern recognition, even if they aren't good at mathematical proofs. So, even if a particular contestant wasn't as smart as Marilyn, if they were in the habit of watching the show, they would know of the advantage in switching. Which takes us back to that meaning that the host has given the contestant a huge clue. And there's no reason to believe they would do that.

You could argue that the host might want the contestant to win, not because that meant their employer could keep more of the prizes, but because viewers love to see people win against the odds. Well, I don't really believe that either. For one, as I said, people would catch on to the pattern, and would then cease to believe a contestant had beaten the odds by switching and finding the prize. Secondly, people love baseball, even though the hitter only succeeds about 1/3 of the time. College hoops fans love the three-point line, even though shooters only make them about 1/3 of the time. I don't buy that there's any indication that the host's interests would be so aligned with those of the contestant.

Lastly, I will draw the analogy to poker. People (apparently) love to watch poker because of the mental gamesmanship between the competitors. In poker, there are set mathematical probabilities that govern each contestant's odds with each combination of cards they get. Knowing those are important. But, equally important are reading the actions, and reactions of the other players at the table. While it may be statistically advantageous for a player to fold on a given hand, they may try to bluff their way through the hand. They don't want their opponents to be able to perfectly correlate their response to a given set of cards. Sometimes, they may intentionally lose a hand, in the hopes of setting up a win later on with a bigger pot. So, the preferred strategy in poker involves a little of what might look like "randomness". If the game show host really wanted to make the show as interesting as possible (which can be assumed to be a goal), then they would be well served to play mind games with the contestant. Is he trying to help me? Maybe he knows I picked the correct door, and is trying to get me to lose? This type of strategy from the host would keep the game from becoming predictable for both the contestants and the audience, and is therefore a much better assumption for a host's strategy, given the information in the Monty Hall Problem.

I'm actually quite disappointed that someone as intelligent as Marilyn vos Savant would blow this one so badly. It's possible that she simply doesn't play games, and thus has a knowledge deficit in this area of gamesmanship. It certainly isn't limited to poker. I can find elements of the same types of strategies in basketball or tennis, for example.

Anyway, pick Marilyn's door if you want the majority of intellectuals to agree with you. Pick mine if you want to be right.

Posted at 05:48AM Dec 14, 2008 by Nathan in General  |  Comments[0]