Ones and Zeros: Monty Haul and Thomas Kuhn

Monty Haul and Thomas Kuhn

April 09, 2003

Danil brings up the Monty Haul Problem and wonders about the human tendency to prefer being right to being correct.

You find yourself on a game show called "Let's Make A Deal." The game is very simple. There are three doors: door #1, door #2, and door #3. Behind one door is a million dollars. The other two doors contain worthless joke prizes. All you have to do is pick which door you want to open, and you get whatever is behind it. But you only get to open one door. By simple math, then, you obviously have a 1 in 3 chance of picking the correct door and becoming an instant millionaire.

You pick a door. As soon as you tell Monty (the gameshow host) what door you want to open, he stops and says, "Okay, you've made your choice. Now, I'm going to do what we always do here on this game. I'm going to open one of the other two doors for you that I know has a booby prize." And he does so. Then he asks, "Okay, now, would you like to stay with your original guess, or would you like to switch to the other door that's still closed? You only get one shot, so do you want to stay with your original choice, or switch?"

The answer, unintuitively, is that you double your chances by switching. See this blog entry by Dean Esmay if you don't believe me. I also will bet significant cash money that if I use the 'switch' strategy exclusively, I will beat anyone using the 'stay' strategy 2 for 1. Dean is willing to bet $1000. I am, too. ) I started to describe how I explain this to people ("There are nine possibilities. You choose Door A,B or C and the producers choose A, B or C. Draw a grid and try both strategies..."), but my experiences are pretty much the same as Dean's. Some people won't change their mind about this, even in the face of evidence or "put your money where your mouth is" challenges. It occurred to me that what Danil (and Esmay) are talking about isn't the puzzle, but the paradigmatic shift required to move from "That's wrong/I don't get it" to "aha!" That's a much more interesting topic. So perhaps this is Kuhn's or Margolis' turf. The act of "getting it" is a little individual scientific revolution based on a personal body of 'normal science' knowledge, an experiment with results that don't fit the existing paradigm, and a list of semi-failed attempts to explain the new paradigm to those who still haven't made the jump. One wonders if Kuhn critic Steven Fuller would switch or stick. Danil is "still waiting for a convincing explanation of the correct answer." I say that because it requires a paradigm shift, there isn't such a thing and that the lack of such is what makes this interesting. In my ideal world, this would be the peppered moth research that made it clear on an intuitive level that Kuhn was fundamentally right.

.:Posted by Michael on April 9, 2003 2:03 AM:.

I think I read about this first in one of the John Allen Paulos books. The problem is that it is definitely non intuitive unless you use a grid or a tree diagram to show the possibilities, but its absolutely correct. It doesn't feel right to switch "why should it matter?" but the probabilities are with you, 2:1.

.:Posted by Jvstin ( total) on April 9, 2003 9:53 AM:.

This answer to the Monty Haul problem is correct when performed by robots. However, it ignores the human factor, ie. Monty is much more likely to pull this kind of trick when the contestant has already picked the right door, and is much less likely to pull it if the big prize remains unselected.

One can imagine some sort of a weighting factor -- a probabilistic equation developed by Monty to ensure that even if the problem is presented, the contestant still only has a 50-50 chance of winning. For instance, Monty might decide: if it's Monday, Tuesday or Wednesday, I'll pull the trick no matter what. If it's Thursday or Friday, I'll only pull the trick if the contestant has picked the right door already. Monty changes up his formula every week.

Still wanna bet that $1000?

This doesn't really attack the meat of posting, other than to say that sometimes the 'right answer' is relative.

.:Posted by HWRNMNBSOL ( total) on April 9, 2003 11:46 PM:.

Jvstin: Agreed, but I can show people a 3x3 grid of outcomes and still find that they don't see it.

HeeWho:

Probabilistic is a great word, almost worthy of our only President. It evokes mental images of Schroedenger's WMDs and Reagan's boxcar shuffle with the Minuteman IIs in the early 1980s.

The problem as stated says "I'm going to do what we always do here on this game." Inserting volition into Monty's patter breaks it. It takes the exercise from being an interesting question about the nonintuitive nature of conditional probability to a question about a game show. Your example is creation science.

Would it bother you to have me describe your reaction as an engineer's response? I think you're trying to change the problem from a mathematical one to a real world one. In the real world, I prefer the Structural Engineer to the Architect. According to this blog entry, Architects "are encouraged to draw buildings which are fatter at the bottom than at the top, and told that if they do that, the engineers will probably be able to make them stand up."

When are you going to update your blog?

Either of you have any personal experiences with how you "got it" on the correctness of switching? I saw several comments on Dean Esmay's blog about programmers who wrote test apps...

.:Posted by Michael Croft ( total) on April 10, 2003 7:00 AM:.

To be honest, this problem has far less to do with conditional probability than with information theory. When Monty shows you a loser, he has increased the information you have to make your choice from a pure 1-in-3 guess to a pure 1-in-2 guess. It is only by switching, however, that you can take advantage of that extra information; inaction leaves you at your original 33.3% chance of winning.

(This question is really popular among actuaries, as we tend to consider oursevles as having great conditional probability instincts... but at least 1 in 4 gets it wrong because they fail to reflect that Monty *knows* which one is the winner and which two are losers, so picks with certainty not at random.)

Of course, this question is not true to the original show, where there was one BIG prize, one good prize and one lousy prize (more often than not). With that, the problem (and the choice) requires knowing a bit about the relative value of the BIG and good prizes.

.:Posted by Jack ( total) on April 10, 2003 12:40 PM:.

Jack,

The probability of switch being the correct choice and stay being the correct choice need to add up to 100%. If I read your response correctly, "stay" leaves you with your original 33.3% chance, and switch is a pure 1-in-2 guess, so you think that's 50%. Correct me if I've misread, but I don't see where you account for the missing 16.6%. However, you don't have a pure 1-in-2 guess, you have a 2-in-3 guess (B or C was the location of the prize). if you switch you're chosing "not A", so your probability is "1 - 33.3%"

To get what you describe, Monty must eliminate one door and then reshuffle. You should still switch, but the advantage of doing so is half the advantage if he doesn't reshuffle.

.:Posted by Michael Croft ( total) on April 10, 2003 2:22 PM:.

You misunderstand my point. The 1/3 and 1/2 don't show the relative probalities, they show WHY the decision is obvious. The 1-in-2 pick MUST be better than the 1-in-3 pick.

Initially, it's a 1-in-3 choice (that we can all agree on). Once information is added, there is an AVAILABLE 1-in-2 choice, but you are restricted in how you can access it. You *can't* stay where you are and get any advantage from the 1-in-2 choice... Your choice was 1-in-3 and Monte showing you that one of the other doors was a loser told you nothing, because that was a certainty (with two other doors, one HAD to be a loser, and Monte knows which!). Therefore, you MUST switch to get to the better situation of picking 1-in-2. This means while it's a "1-in-2" pick, it's a choice between non-equal options.

Your odds aren't 1-in-2 if you switch, because if they were the universe of possibilities would be 1/3 + 1/2, which doesn't sum to 1. So, since we know staying is 1/3, switching MUST be 2/3rds, as the sum MUST be one (or zero if there's a chance to always lose, but in this case there is not such a chance).

That's an approach to this answer that never enters conditional probability methodology, just the fact that information is added and basic probability rules (sum of probabilities must equal one or zero).

Note that if Monte *doesn't* know which is the winner, his choice could be the winner, in which case switching is no advantage to you... your chance of winning is now always 0. But if his random choice was a loser, we revert to switching being a good strategy again.

Now, for another interesting math question...

We have 2 glasses, not necessarily the same size, each half-full of liquid, one coffee and one milk.

Pour SOME milk from the milk glass into the coffee. Mix completely.

Now pour coffee/milk mixture back into the milk glass until both glasses are at their original fill level and mix.

Is there now more milk in the coffee or coffee in the milk?

(Actuaries do way too many math puzzles...)

.:Posted by Jack ( total) on April 10, 2003 6:51 PM:.

M: Probabilistic is a great word, almost worthy of our only President. It evokes mental images of Schroedenger's WMDs and Reagan's boxcar shuffle with the Minuteman IIs in the early 1980s.

Me: It would be a more silly-sounding word, yet more economical on the letters, to call it 'probalistic'. However, this sounds dangerously close to 'somebody in favor of guns'.

M: The problem as stated says "I'm going to do what we always do here on this game." Inserting volition into Monty's patter breaks it. It takes the exercise from being an interesting question about the nonintuitive nature of conditional probability to a question about a game show. Your example is creation science.

Me: Fine, you bastard, dirty up my convenient worldview with reality. I swear, it's smartypants answers like this that make me wonder whether it's even worthwhile to cook up my own smartypants answers.

M: Would it bother you to have me describe your reaction as an engineer's response? I think you're trying to change the problem from a mathematical one to a real world one. In the real world, I prefer the Structural Engineer to the Architect. According to this blog entry, Architects "are encouraged to draw buildings which are fatter at the bottom than at the top, and told that if they do that, the engineers will probably be able to make them stand up."

Me: It wouldn't bother me, but it also wouldn't be particularly accurate. Engineers, as a general rule, are fairly anal about using mathematical tools to arrive at the correct solutions, so in a sense my response is kind of the anti-engineer answer. If anything, it's a contractor's response. Contractors are always looking to cut corners. By shaking the problem loose from its conventional framework, I can potentially look all smart and stuff, and possibly screw you out of some money in the bargain.

M: When are you going to update your blog?

Me: That's the Rainy Day blog. We're only supposed the update the Rainy Day blog when it hasn't been updated in a really long time. It's only been half a year. We're not sure how Rainy a Day it needs to be to justify updating the Rainy Day blog, but we're not there yet. Seriously, I've given up on blogging; it was too much like work (which I'm all about avoiding), and too little like throwing eggs at stupid people and then running away (which is basically how I've interacted with the internet all these years). Hence, I blog through your comments fields. Pretty lazy, hah? look out, you got some egg on you.

.:Posted by HWRNMNBSOL ( total) on April 10, 2003 9:35 PM:.

Michael asks whether I have any stories on getting the Monty Haul problem. I sort of do.

I first encountered it in college. I was one of the people who didn't believe it when I first heard it. I was talking with one of the Bray brothers, who of course got it right away. I had to be convinced. We did this by breaking out a deck of cards and doing a quick game of three-card monte. A few minutes were sufficient to show me the light.

I then observed that, interestingly, if the Game Show Host isn't informed in advance of which door has the prize, the odds of switching collapse back to 50-50. This, in turn, confused Mr. Bray. Another game of three-card monte managed to convince *him*.

Three-card monte: Undervalued Educational Tool.

.:Posted by HWRNMNBSOL ( total) on April 10, 2003 9:41 PM:.

What I find amusing about this question is the warblogger Renaissance Man who got it wrong some long time ago.

HeeHoo: Screw political blogging, update the dang gaming blog.

.:Posted by Ginger ( total) on April 10, 2003 10:49 PM:.

Kuhn inhales his food!

.:Posted by ChuckEye ( total) on April 11, 2003 3:33 AM:.

Kuhn inhales his food!

Not recently...

.:Posted by Michael ( total) on April 11, 2003 4:06 PM:.

Heewho: I just wanted to say thank you for the opportunity to conflate your views and creation science. That's the kind of thing that I get out of a good blog comment discussion.

If Probalistic sounds dangerously close to 'someone who likes guns', does that make Cabalistic dangerously close to "a senator from Texas who likes guns?"

.:Posted by Michael Croft ( total) on April 11, 2003 4:58 PM:.