## Monty Hall, again

April 8, 2008 at 7:31 am | Posted in math | 11 CommentsThere’s a NY Times article about cognitive dissonance in monkeys, and the Monte Hall problem.

http://www.nytimes.com/2008/04/08/science/08tier.html?_r=1&th&emc=th&oref=slogin

You have to log in to read it, but it’s free to create an account, and I don’t think I’ve gotten spam from doing it, not with the NY Times, anyway.

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The Monty Hall problem has always struck me as particularly interesting, because lots of people (even math professors) will get it wrong, and will violently insist that they are correct.

It’s also interesting to contrast this with “Deal or No Deal”. In that game show (for those who haven’t seen it), there are a lot of suitcases (26 in the US broadcast version), containing various amounts of money (all of which are visible on a big board, but without any indication of what amount is in what case). The contestant picks one at the start. He then opens various other cases, revealing what’s inside, removing those amounts from the board. At various times, the contestant is offered various amounts of money, which he can take (ending the game) or refuse (meaning he’ll open more cases.) Eventually, if he doesn’t take any deals, it comes down to two suitcases – his and one other – and he is given the choice of keeping his or taking what’s in the other one.

The reason you always want to swap in the Monty Hall problem is because Monty will never throw away the door with the car. Your decision before he throws one door away has 1-in-3 odds of being right. The odds of Monty having the winner was 2-in-3 before he threw away the loser, and they remain that way afterwards. (If Monty could randomly throw away the door with the car, then the odds of Monty having the winner would be 2-in-3 half the time and zero the other half, averaging to your odds of having it 1-in-3 times. Then, it wouldn’t matter if you swapped doors or not.)

In Deal Or No Deal, it’s different from the Monty Hall problem. The cases that get thrown away are random – the player could just as easily throw away the million-dollar case as he could the one-cent case. And the cases that are thrown away are opened, so the contestant always knows exactly which dollar amounts remain in play. The result is that every decision during the game is exactly the same as if the game had just started at that moment. The odds of the player’s case having a million dollars are 1-in-26 right up until the moment the million-dollar case is opened (which might be on the first move, or on the last move) after which, they go to zero or to certainty. And when there are only two cases remaining, it is exactly 50:50 which of the remaining dollar amounts are in which case.

Comment by David Charlap— April 8, 2008 #

I’ve seen that game show once. Besides the game show stress, bright lights, dramatic music, and Howie Mandel constantly reminding you what’s at stake, it’s interesting.

On the one hand, it’s a simple case of computing expected value. The person offering the money is “playing the game” every week for dozens of weeks, so all he has to do is offer enough so that, in the long run, they can pay the advertisers. The law of large numbers guarantees that they will make up for any short term losses in the long run; which is the same reason that casinos make money.

So this guy offers the contestant a little less than the expected value of the game at that state. It’s always a winning proposition for him.

But the contestant only plays the game once. They have to balance the opportunity to win the payoff of a lifetime, and the thrill of risking a “loss” (which does have its own subjective value). They don’t have a long run strategy. It’s interesting.

Comment by dereksh— April 8, 2008 #

At first I didn’t see the point of the show, but I must admit I’ve now watched it about 20 times. It is interesting to see how much risk to bear, and to watch the payoff curve rise and fall. Plus the human interest can be compelling sometimes.

one thing I notice is that they try to make the bankers offers seem so taylored and personal, based on his measure of the person, but I think they are computed straight off some actualrial formula. They practically admit this when they do the hypatheticals once a person has picked a case. You would have picked case X, the amount in that case was Y, and the offer would have been Z.

Another side note of sociology interest is the models. Totally gratuatous eye candy, right? Not necesarily. Just as some people have lucky/favorite numbers, I’ve seen some contestents with opinions about the models themselves. One guy the other week wasn’t selected cases by number, he was calling out the model’s name. Each time Howie would have to mention the case number so the audience would know what he had selected. Until then, I didn’t really realize that the models always stayed in the same place, but since then I can see how avid watches would come to learn the personalities of the various models and that could have meaning for them.

Comment by SCD— April 9, 2008 #

That’s an interesting observation about the models. What would an avid watcher’s strategy, then? If they have favorites, would you want to save them for later, or pick them sooner? Do you want one of your favorites to have the million, or do you want one of your favorites to open up their briefcase to a small number when you pick them, thus raising your chances for the million, thus raising the payoff the banker offers you? We like to associate good things with good, but I’m not sure which way people would go.

The next obvious question is, do the producers of the show know these general preferences, and do they factor that into their decision (or pseudo-random algorithm) of which models start out with which dollar amounts? Would they try to create perceptions that certain models always (or at least often) have low dollar amounts (good news to the contestant who picks them)? I wonder…

Comment by dereksh— April 9, 2008 #

Good questions. Of course the show, and Howie swear up and down that the numbers are TRULY random, and that he, the banker, the models, the producers, etc., don’t know which is which. If they do know, or even have an impact on what amounts go where, it would be scandulous. Nonetheless, my wife too suspects that the “popular” numbers are weighted to more good amounts.

Of course, it’s a bit of a catch 22 if you were trying to stack the deck. If you put the high numbers in the “good/popular” places, whether it be by model or number, there is a better chance of knocking out high values, but a better chance of having one in your case too. LIkewise the question of do people keep what they like or pick what they like.

Thus I don’t think they could gain much by trying to “stack” the deck. Too many human nature variables.

Comment by SCD— April 10, 2008 #

I agree. Best to make a simple random assignment and trust in the natural variability of human preferences and the law of large numbers.

Comment by dereksh— April 10, 2008 #

I’m pretty sure the allocation of the cases is random. If it’s not, then the secret will get out, and it will ruin the show. (On the other hand, shows have been ruined in the past over things like this, so maybe producers just don’t care.)

WRT the amount that the banker offers, there have been a lot of discussions in various forums about his algorithm. It is based on the expected value, but with a bias. Towards the beginning of the game, the amount offered is significantly less than the expected value. As the game progresses, the offers approach the expected value. If a contestant holds out to the very end, the last few offers sometimes actually exceed the expected value by a small amount. The idea here is that they don’t want contestants taking the early deals (since it makes for a boring show), and they want the contestants to have to make really hard choices near the end of the game (the agonizing over decisions, and the resulting conversations keeps things interesting for the audience.)

There is, however, a certain amount of randomness to the biassing. Identical configurations of the board do not result in identical offers. I think this is purely random, just to keep the game unpredictable, but some believe that the variation is deliberate, in order to keep the more-interesting contestants on TV longer than the less-interesting ones.

The algorithm used for the TV show is not known, but the algorithm used for the NBC on-line version is well documented. It was a fixed non-random formula until recently. Today, it’s a random percentage of the expected value (with the random-percent envelope increasing as more cases are removed.) You can see the details here.)

(No, I don’t watch the show that much, but my brother frequents several math and gaming newsgroups, and this was a frequent topic for discussion when the show was first on the air.)

Comment by David Charlap— April 10, 2008 #

Here’s a question. Why 26 cases? Whay not a “rounder” 25 or since that would screw up the board symetry, why not 24? Is there some mathematical principal (law of large numbers or otherwise) that says that 26 is close to the minimum needed for proper effect?

Or is that just what fit easily on their board with the desired font size?

Comment by SCD— April 11, 2008 #

Fascinating!!! I am intrigued!

Comment by Renee— April 11, 2008 #

Nothing about 26 that I know of. (*shrug*)

Comment by dereksh— April 11, 2008 #

According to Wikipedia, different productions of the show (in different countries) have different numbers of cases. They range from 20 to 26.

I suspect the number is somewhat arbitrary. You want an even number, so the board will look nice. I suspect the amount of floor space in the studio for the models is also a factor.

Comment by David Charlap— April 15, 2008 #