User:Martin Hogbin/Monty Hall problem/dissenters

This is a page to try to collate dissenting opinion on the Monty Hall problem (MHP) article.

As a relatively lighthearted approach to improving the MHP article am presenting some statements for editors to express their agreement or otherwise. Please sign in the appropriate section and only comment in the comment section. Please only sign in agree if you completely agree with the statement. Otherwise sign in disagree and comment if necessary. Please have any discussions on the questions and answers on the talk page.

The questions are in no particular order and have no hidden agenda, they just came off the top of my head as points that are sometimes argued over.

All editors of the page, those who want change and those who do not, are welcome to answer.

Simple problem

The Monty Hall problem is essentially a simple problem that most people get wrong.

Simple here means not requiring (for whatever reason) the use of conditional probability.

Agree

Martin Hogbin (talk) 15:56, 27 December 2009 (UTC)

Disagree

Nijdam (talk) 23:58, 27 December 2009 (UTC)

Just a puzzle

The Monty Hall problem is essentially a mathematical puzzle - its relationship to a TV show is not particularly relevant to its precise formulation

Agree

Martin Hogbin (talk) 16:23, 27 December 2009 (UTC)

Nijdam (talk) 23:58, 27 December 2009 (UTC)

Whitaker is definitive

Whitaker's question in 'Parade' is the definitive problem statement
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 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 switch your choice?

Disagree

Martin Hogbin (talk) 16:23, 27 December 2009 (UTC)

Nijdam (talk) 23:59, 27 December 2009 (UTC)

Whitaker is notable

Whitaker's question in 'Parade' is the most notable problem statement

Agree

Martin Hogbin (talk) 16:23, 27 December 2009 (UTC)

Comment

I'd say: a notable problem statement. Nijdam (talk) 12:58, 28 December 2009 (UTC)

Morgan is the most reliable source

The paper by Morgan et al is the most reliable source concerning the problem

Disagree

Martin Hogbin (talk) 16:23, 27 December 2009 (UTC)

Comment

I'd say: a reliable source. Nijdam (talk) 00:01, 28 December 2009 (UTC)

Always conditional

The Monty Hall problem always involves conditional probability

Agree

Nijdam (talk) 00:02, 28 December 2009 (UTC)

Disagree

Martin Hogbin (talk) 16:24, 27 December 2009 (UTC)

Deal with the simple case first

The article should deal comprehensively with the simple (not involving conditional probability) case first.

Agree

Martin Hogbin (talk) 16:31, 27 December 2009 (UTC)

Comment

The simple solution is actually also conditional. Nijdam (talk) 00:02, 28 December 2009 (UTC)

Deal only with the simple case

The article should deal only with the simple (not involving conditional probability) case.

Disagree

Martin Hogbin (talk) 16:31, 27 December 2009 (UTC)

Nijdam (talk) 00:03, 28 December 2009 (UTC)

K & W Statement is conditional

The unambiguous formulation given by Krauss and Wang must be treated conditionally
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. The car and the goats were placed randomly behind the doors before the show. The rules of the game show are as follows: After you have chosen a door, the door remains closed for the time being. The game show host, Monty Hall, who knows what is behind the doors, now has to open one of the two remaining doors, and the door he opens must have a goat behind it. If both remaining doors have goats behind them, he chooses one randomly. After Monty Hall opens a door with a goat, he will ask you to decide whether you want to stay with your first choice or to switch to the last remaining door. Imagine that you chose Door 1 and the host opens Door 3, which has a goat. He then asks you “Do you want to switch to Door Number 2?” Is it to your advantage to change your choice?

Agree

Nijdam (talk) 12:16, 28 December 2009 (UTC)

Disagree

Martin Hogbin (talk) 11:22, 28 December 2009 (UTC)

Comment

Martin, I'm sorry to find you here under "disagree", because it is impossible to find a solution not based on some conditional probability. But, then, please prove it to me. Nijdam (talk) 12:54, 28 December 2009 (UTC)

See this talk page for my reply. Martin Hogbin (talk) 15:17, 28 December 2009 (UTC)

Sources determine article structure

The structure of the article should be determined by what reliable sources say

Disagree

Martin Hogbin (talk) 11:27, 28 December 2009 (UTC)

All statements supported by sources

All significant statements made in the article should be supported by reliable sources

Agree

Martin Hogbin (talk) 11:27, 28 December 2009 (UTC)

Morgan not definitive

There are sufficient ambiguities, errors, and other problems with the paper by Morgan et al. for it not to be regarded as definitive

Agree

Martin Hogbin (talk) 11:28, 28 December 2009 (UTC)

Disagree

Nijdam (talk) 12:55, 28 December 2009 (UTC)

Comment

Comprehensive reasons for my opinion are given here. Martin Hogbin (talk) 11:48, 28 December 2009 (UTC)

My opinion

As I have withdrawn completely from this mediation now I have given my complete thoughts on the article below. Further discussion seems pointless as there is no sign of anyone changing their opinion or of a consensus emerging. Please do not intersperse comments but feel free to collapse this section.

Proposal for the article

I believe that the article should give simple solutions to the MHP first, citing some of the many sources that give such solutions.

These solutions should not include any prominent 'health warnings' stating that they are incomplete or answer the 'wrong question'. I would not object strongly to careful wording of the simple solutions in order to reduce or remove what some see as technical errors in them, provided this is done in a non-obtrusive way that will not obstruct their understanding and acceptance by the average reader.

The simple solutions should be followed with a section explaining, with reference to the simple solutions (and maybe discreetly to the unconditional formulation) why the answer is 2/3 and why it matters that the host knows where the car is. This is what everybody has difficulty with and gets wrong.

Following this I would support a scholarly and comprehensive discussion of all aspects of the problem, including criticism of the simple solutions, responses to that criticism, and game theory, arranged in any logical manner.

Rationale

There are no 'bomb-proof' arguments, including mine, regarding the simple solutions to the MHP (despite the protestations of some). What I propose to give is a collection of reasons, from varying perspectives, why the article should be as I propose. I assume throughout this discussion the standard game rules, that the host always opens an unchosen door to reveal a goat and always offers the swap. I also use the word 'answer' to mean the value of the probability that the player will win the car by switching in the specified circumstances.

The MHP is a problem is an unconditional problem

By far the most notable and well known statement of the MHP is the question by Whitaker to vos Savants column. This was a question by a member of the general public to a regular column in a popular general-interest magazine. The question was changed by vos Savant to make it easier to explain by the addition of door numbers.

It is far from clear whether the intention was to ask a conditional or an unconditional question. What did Whitaker actually want to know? Although Whitaker actually asked if it is better to switch, as there is no dispute about this point, I have extended his question to ask by how much. Was it:

1) I am on a quiz show in which the standard rules apply. I have chosen a door and I have just seen another door opened to reveal a goat. Given this information, and taking into account the specific door opened by the host, what is the probability of winning if I switch. (Conditional formulation)

2) There is a quiz show in which the standard rules apply. Is it better for a contestant to switch or stick and by how much? (Unconditional formulation)

Is it really possible that Whitaker meant to ask question 1)? Would a member of the public be asking a conditional probability question in which the door number opened by the host is of significance, to a popular magazine. Hardly. It was some ten years after vos Savant's reply before anybody even proposed this interpretation. (Although Selvin, in response to comments to his problem even earlier, had noticed that possibility and quickly acted to nullify it. However, neither vos Savant nor the original critics, Morgan et al, seemed to be aware of the previous discussion).

There is no doubt in my mind that what Whitaker actually wanted to know was the answer to 2) In fact had he meant to ask 1) it is quite remarkable that he did not immediately respond to vos Savant's solution with an argument along the lines of Morgan. There is no sign that he did anything like this.

Even if given a condition we are entitled to ignore it

In Whitaker's statement we are told that the host says, 'Do you want to pick door No. 2?'. We are not told that he always uses these words. Consider for a moment what the situation would be if we were specifically told that sometimes the host says, Do you want to pick door No. 2?' and other times he says, 'Do you want to choose door No. 2?'. We would then need to take the host's words as a significant condition of the problem. We would need to consider introducing a parameter such as s, the probability that the host will say 'choose' given that the car is behind door 1. As it is, the situation is not so clear cut and people makes the decision to ignore what could be a vital clue as to the actual location of the car. They do so at their peril, however. In a show in which the host is permitted to influence the outcome, it is quite possible that he might give strong clues to the player as to the present location of the car.

Why then do we ignore this possibility. Because common sense tells us that it is not intended to be an important condition of the problem. Common sense also tells us that the door number opened by the host is not meant to be important. Even though we have been given that information, we should ignore it.

Suppose the problem is conditional

How does the host choose between goat-doors when he has a choice

Let us be a little perverse then. Let us assume that Whitaker actually meant to ask the conditional version of the problem. What is the correct solution.

We now have to ask the question of how the host chooses between the legal goat-doors. Does he choose uniformly or might he have some bias?

If we are subjectivists we may well decide to apply the principle of indifference and take the unknown distribution to be uniform. If, on the other hand, we are frequentists we might observe that a random choice by the host is a requirement of quiz shows and again take the distribution to be uniform.

Regardless of what our view of probability is we might conclude that, if we are consistent in our approach, and do not take the unknown distributions in Whitaker's statement to be uniform, then the problem is rather boringly insoluble. On any reasonable basis we must take the host's legal goat-door choice to be uniform; no source which addresses this issue does otherwise.

Solving the symmetrical conditional problem

Having decided, as we must, that the host is unbiased we now ask ourselves what the difference is between the two formulations, one in which a unknown door is opened by the host and the other in which a known, but irrelevant, door is opened. Sure, there are two subtly different concepts, an unconditional case in which no special condition is given, and a conditional one in which a specific door is opened. One requires our sample space to be conditioned and the other does not. However, even that distinction is not that clear cut. In what we generally call the unconditional case, there is a condition, that the host opens an unchosen door to reveal a goat. It is only by convention that, because that event is part of the game rules, and therefore happens with certainty, we do not use it as a condition.

So are the simple solutions valid?

The simple solutions solve the unconditional problem and, by symmetry, the conditional problem must have the same answer as the unconditional problem. Thus the simple solution solves the conditional formulation.

The common objection to this solution is that neither we nor any of the sources include this symmetry argument in their simple solutions thus they are incomplete. However, no solution is totally complete. In particular the conditional solution shown in the article is far from complete. The player is stated to have chosen door 1, but that is not a rule of the game, it does not happen with certainty, so our diagram should show the ample space including all the possible original choices that the player might have made and then, in some way, condition it based on the choice given. For most people there is no need to do this because it is quite obvious to them, because of the complete symmetry with respect to door number, that we need only show one case that the player initially chooses door 1. The situation is similar with the simple solution. We should state that the simple solution gives the answer to the unconditional formulation and that this formulation has exactly the same answer as the conditional case, but we have no need to say this because it is quite obvious.

How to present the solutions

Let us now be perverse and pedantic. Whitaker intended to ask the conditional question and the simple solution does not solve it. How should we present this in an encyclopedia intended to be accessible to the general public. Let me first give two approaches that I consider wrong:

1) Ignore the fact. This is only an encyclopedia for ordinary people, they will not notice a minor technical error.

2) Hit them with it. Give only complete and rigorous solutions right from the start or, at least, add prominent 'health warnings' to the simple solutions pointing out that they are, in fact, wrong. If the readers cannot understand the 'proper' solution that is just too bad, they do not deserve to understand the subject if they cannot do it properly.

I do not support either of the above approaches and neither do most good text books and encyclopedia articles. What they do is start with a simple version of the subject, sometimes with certain technical difficulties glossed over, and then, when the main issues have been understood (in our case that the answer is 2/3 and it does matter that the host knows where the car is) go through the details. It is very common to see in text books things like 'Equation 1 is not strictly correct but actually relies on the instated assumption... We will now look at that in more detail...'. To do things in this way is not structural bias but writing an article so people can understand it It is true that some authors do insist on getting things correct right from the start but they are usually writing for well defined technical audiences. We are writing for a wide range of readers, with the vast majority being non-expert.

In summary I do not think that a perverse and pedantic approach should be allowed to spoil the primary function of this article which is to inform our readers.

Comments

Simple Or Conditional

Yes, simple solutions are valid. And yes, this actual game show has rules, but no one knows whether this special show has ever been or will ever be repeated. The simple solution gives the answer "switch" (Pws on average 2/3), and as long as no "correct conditional solution" can give another advice than "switch", you don't need them. But nevertheless they could be shown in the Lemma, in odds-form. Conditional solutions even can show the Worst Case Scenario (Pws within the range of at least 1/2 to 1), but never can give any "correct closer" result for any "actual game" than 2/3.

In my opinion, besides the simple solution (Pws on average 2/3), you also should show the Worst Case Scenario. Richard says that the conditional solution can show that, and I think you can show this worst case scenario also as follows, e.g.:

The question asked is presenting an already given actual situation you are confronted to, a firm actual state, a status quo. And you should make your decision.

You know that the odds on the car originally were 1/3:1/3:1/3, and further you know that in any case the host did dispose of at least one goat to show, and that in 1/3 of cases he even got both goats, and only in this 1/3 switching will loose the car. In other words, in the remaining 2/3 switching will definitely win. And the question says that the host already had opened one of his two doors, showing a goat. So you should accept his offer and switch.

In assessing a possible decision it is wise to also consider the worst case scenario, and so you should indeed pay regard to all imaginable variants of the procedural method of the host in showing a goat. What, if the host just uses to never open that second still closed door, under all circumstances and if ever possible? Then the probability to win by switching could only be at least 1/2 (Falk)  in that worst case, but never less. And, on the other hand, what if he uses to always open that second still closed door, under all circumstances and if ever possible, but just actually he didn't? Then you would know that his second closed door is very likely to hide the car, so actually you definitely should switch. This answer is correct for any single game, if this game show should ever be repeated.

You know that, for the actual game, it is impossible to give a better result for Pws than "2/3", and no other one can ever give a "correct" closer result. So the simple solution is enough, it is "more than correct."   :)

Please stay in the mediation, we need you! Regards, Gerhardvalentin (talk) 15:43, 22 January 2011 (UTC)

Thanks for your comments and support. The mediation is essentially nonexistent. I was hoping that the mediators would take a more active role, not joining in the argument but getting everyone to state their case more clearly. As it is we are all rehearsing the same arguments over and over again with no sign of cooperation. My plan is to make my case and leave it at that. I do not expect that I will have much to add to it. Martin Hogbin (talk) 19:30, 23 January 2011 (UTC)