The goat problem , three door problem , Monty Hall problem, or Monty Hall dilemma is a probabilistic problem . The question is whether a choice that was initially made by chance among three a priori equally probable possibilities should be changed when additional information is given.
The task is similar to the game show Let’s Make a Deal , hosted by Monty Hall , which in the German-speaking area has the variant Go on the whole! got known. Two terms mentioned above relate to the formulation of the problem in which the decision-maker awaits goats as consolation prizes behind two out of three doors if he has not chosen the door that stands for the main prize, a car.
Various views of the goat problem are often cited as an example of the tendency of the human mind to fall short when it comes to determining probabilities and have been the subject of long public debate.
The task at hand goes back to the biostatistician Steve Selvin , who introduced it to the American Statistician in a letter to the editor in 1975 . The problem became known to wider circles and the subject of a controversial debate in 1990 when it was published in Marilyn vos Savants' column "Ask Marilyn" in Parade magazine . This version was based on a letter to the editor vos Savant received from Craig F. Whitaker of Columbia, Maryland:
“Suppose you were on a game show with a choice of three goals. Behind one of the gates is a car, behind the others are goats. You choose a gate, say gate number 1, and the showmaster, who knows what's behind the gates, opens another gate, say number 3, with a goat behind it. He asks you: 'Would you like gate number 2?' Is it an advantage to change the choice of gate? "
The question in this form is under-determined; the correct answer depends on what additional assumptions are made. Vos Savant replied, “Yes, you should switch. The goal chosen first has a 1 ⁄ 3 chance of winning , but the second goal has a 2 ⁄ 3 chance of winning . ”Vos Savant's answer is correct, but only under the additional assumption that the showmaster, regardless of whether behind the candidate initially chosen Tor the car or a goat is, in any case, open an unelected gate with a goat and offer changing needs . Even with this additional assumption, it is counter-intuitive for many people that the chance of winning actually increases to 2 ⁄ 3 instead of just 1 ⁄ 2 . As a result, according to their own estimates, vos Savant received ten thousand letters, most of which doubted the correctness of their answer.
The experience-based answer
If you ask the question to people who have not yet dealt with the problem, they often assume that the chances of winning goals 1 and 2 are equal. The reason for this is often given that you don't know anything about the motivation of the showmaster to open gate 3 with a goat behind it and to offer a change. Therefore the principle of indifference applies .
The intuition in understanding the letter to the editor assumes that the problem is a description of a unique game situation. In addition, the answer testifies to a certain familiarity with game shows like Go All Out , in which the showmaster (presenter) plays an active and unpredictable role. In contrast to the problem variants in which the moderator is reduced to a “henchman” who is bound by fixed rules of conduct, it can realistically be assumed that he is completely free in his decisions (Monty Hall: “I am the host!”). This freedom can be illustrated with a few examples, whereby the car and goats behind the three goals were randomly distributed before each game. Because the contestants know this game show they have applied for as a participant, they are of course aware of the host's unpredictability.
- Game 1
- Candidate Alfred chooses gate 1, the moderator opens gate 1 with a goat behind it; Alfred loses.
- Game 2
- Candidate Berta chooses gate 1, the moderator opens gate 2 with a goat behind it and offers Berta to change her choice. Berta wants to change, but the moderator does not open a gate, but offers 5000 euros for Berta to stick with her first choice. This does not change her decision to change, and the moderator opens gate 3 with a goat behind it; Berta loses.
- Game 3
- Candidate Conny chooses gate 1, the moderator does not open a gate, but offers the candidate 1000 euros for not opening the gate; Conny takes the money and wins 1000 euros.
- Game 4
- Candidate Doris chooses gate 1, the moderator then opens gate 3 with a goat behind it and offers Doris to reconsider her choice ...
Given the moderator's various behavioral options, Doris should carefully weigh her chances of winning. If she thinks the moderator is being nice to her and is trying to discourage her from making the wrong first choice, then she should switch. However, if she thinks that the moderator is not well disposed towards her and just wants to distract her from her first, correct choice, then she should stick with Gate 1. If Doris cannot assess the moderator - even in the letter to the editor, no corresponding information is given - she has no way of correctly calculating her chances of winning. In particular, once the moderator has intervened, she can no longer invoke the principle of indifference, and the answer to the question "Is it an advantage to change the choice of gate?" Is in her case: "Not necessarily."
Although the question in the letter to the editor has already been answered, the suggestion was made to support Doris in her decision and to give her a real 50:50 chance of winning. To this end, it is assumed that she has the option of choosing one of the two remaining goals after flipping a fair coin. That way, she can ensure that her probability of winning is exactly 1 ⁄ 2 regardless of the moderator's intentions .
Response from Marilyn vos Savant
Due to Marilyn vos Savant's response to the letter to the editor, the problem also attracted a lot of international attention outside the specialist community and led to fierce controversy. Your answer was:
“Yes, you should switch. The first goal chosen has a 1 ⁄ 3 chance of winning , but the second goal has a 2 ⁄ 3 chance of winning . Here's a great way to imagine what's going on. Assume there are 1 million goals and you choose goal number 1. Then the moderator, who knows what's behind the goals and who always avoids the one goal with the prize, opens all goals except for goal number 777777. You would switch to this gate immediately, wouldn't they? "
Marilyn vos Savant does not consider a particular motivation of the moderator; According to a letter to the editor, it cannot be ruled out that the moderator will only open a goat's gate to distract the candidate from his first successful election. Instead, vos Savant obviously interprets the letter to the editor in such a way that the game show runs according to the same pattern over and over again:
- Course of the game show: The respective candidate chooses a goal, the moderator then always opens another goal with a goat behind it and then gives the candidate the choice between the two still closed goals. The candidate receives the car when it is behind the gate he last selected.
Thus, as a solution, she receives the average probability of winning all possible combinations of goals that are chosen by the respective candidates and can then be opened by the moderator. Because the first choice of a candidate is seen as random and the distribution of cars and goats behind the goals as random, each of the nine possibilities can be considered equally likely:
Gate 1 chosen Gate 2 Gate 3 Moderator opens ... Result when switching Result while keeping automobile goat goat Gate 2 or Gate 3 goat automobile goat automobile goat Gate 3 automobile goat goat goat automobile Gate 2 automobile goat Gate 1 Gate 2 chosen Gate 3 automobile goat goat Gate 3 automobile goat goat automobile goat Gate 1 or Gate 3 goat automobile goat goat automobile Gate 1 automobile goat Gate 1 Gate 2 Gate 3 chosen automobile goat goat Gate 2 automobile goat goat automobile goat Gate 1 automobile goat goat goat automobile Gate 1 or Gate 2 goat automobile
Three out of nine candidates win if they stick with their first choice, while six out of nine candidates get the car by switching. A candidate has an average chance of winning of p = 2 ⁄ 3 by switching .
This solution can also be illustrated graphically. In the pictures of the following table, the selected gate is shown arbitrarily as the left gate:
As a result, the perception of the gameplay of vos Savant can be reproduced in the following way:
- The respective candidate can choose two freely chosen gates, which the moderator has to open, and receives the car if it is behind one of these two gates.
Procedure taking into account the change strategy proposed by vos Savant:
- For example, if a candidate wants to open gate 2 and gate 3, they would first select gate 1, which remains closed. The candidate then switches to gate 2 when the moderator has opened gate 3, or vice versa. The candidate obviously has an average chance of winning of p = 2 ⁄ 3 . Accordingly, it would always be an advantage for a candidate to change goals.
Above all, the following main arguments lead to doubts about vos Savant's answer. While the first argument does not hold water and is based on improperly applied probability theory, the further arguments make it clear that the original problem allows a variety of interpretations:
- Provided that the showmaster follows the rules of the game outlined in the next section, a change of goal is not bad. The chance of winning the second goal is never 2 ⁄ 3 , but generally only 1 ⁄ 2 , because after opening a gate with a goat behind it, there are only two closed gates to choose from. The chances are therefore always equally distributed on both goals.
- The question in the letter to the editor does not contain any indications that the showmaster is following a certain rule of conduct. Such a rule could only be derived on the assumption that the game would be repeated several times under the same conditions: you choose any goal, the showmaster opens another goal with a goat behind it, and you are allowed to change your choice of goal. There is no mention of such a repetition of the game in the letter to the editor. So vos Savant's answer is based on additional assumptions that do not necessarily result in this form from the letter to the editor.
- Marilyn vos Savant's interpretation does not refer to the goals specifically named in the question, and thus ignores any preferences that the moderator might have with regard to individual goals. That is why she receives a 2 ⁄ 3 probability of winning by switching, which is not valid for every moderator behavior. Accordingly, the above table, which only illustrates average probabilities, does not correctly depict such preferences.
The Monty Hall Standard Problem
Because the task formulated in Whitaker's letter to the editor did not seem unambiguously solvable to some scientists, they proposed a reformulation of the goat problem. This reformulation, known as the Monty Hall standard problem, which is supposed to lead to the same solution as that of Marilyn vos Savant, provides certain additional information that invalidates the experience-based answer and, unlike the interpretation of vos Savant, also takes into account the specific game situation :
“Suppose you are on a game show and have a choice of three goals. There is a car behind one gate and a goat behind the others. The car and goats were randomly placed on the gates before the show and you have no information about the location of the car. The rules are: After you have chosen a gate, it remains closed for the time being. Showmaster Monty Hall, who knows what's behind the gates, must now open one of the two remaining gates. There must be a goat behind the gate he has opened. After Monty Hall opens a gate with a goat, he asks you if you want to stick with your first choice or switch to the last remaining gate. Suppose you choose gate 1, and the show host opens gate 3 with a goat. He then asks you: 'Would you like to switch to Gate 2?'. Is it beneficial to change your choice? "
In particular, the moderator has the opportunity to freely decide which gate to open if he has the choice between two goat gates (so you chose the auto gate first). Divided into individual steps, the following rules of the game result, which are known to the candidate who can win a car:
- A car and two goats are randomly assigned to three gates.
- At the beginning of the game, all gates are locked so that the car and goats are not visible.
- The candidate, to whom the position of the car is completely unknown, chooses a gate, which remains closed for the time being.
- Case A: If the candidate has chosen the gate by car, the moderator opens one of the other two gates, behind which there is always a goat. The moderator is free to choose. In the following solutions, however, additional assumptions are made about the type of selection process that the moderator uses.
- Case B: If the candidate has chosen a gate with a goat, then the moderator must open that of the other two gates behind which the second goat is standing.
- The moderator offers the candidate to reconsider his decision and choose the other unopened gate.
- The gate ultimately chosen by the candidate will be opened and he will receive the car if it is behind this gate.
Significance of the additional assumption on the moderator's behavior:
- For case A , in which the moderator can choose between two goals with a goat, additional assumptions are possible as to how the moderator makes his decision: For example, the moderator can choose between the two goals with the same probability. Or he chooses the goal with the highest number.
With such an additional assumption, a different problem arises, which can lead to different chances of winning the candidate's goal selection. It is always assumed that the candidate is familiar with the decision-making procedure under the moderator's authority.
The balanced moderator
For this solution, the following additional assumption is made:
- Case A: If the candidate has chosen the gate with the car, the moderator will open one of the other two gates, chosen at random, with the same probability , behind which there is always a goat.
How should the candidate decide in the penultimate step if he first selected gate 1 and then the moderator opened gate 3 with a goat behind it?
The probability of the car is 1 ⁄ 3 behind the first goal chosen by the candidate 1. Because of the symmetry in the rules, especially because of rules 4 and 5, this probability is not influenced by opening another gate with a goat behind it. Therefore, after opening gate 3, the car is behind gate 2 with a 2 ⁄ 3 probability, and a change leads to success with a 2 ⁄ 3 probability .
For the purpose of the declaration, it is assumed that the candidate initially selected goal 1 and then changes his mind. An analogous explanation applies to situations in which the candidate has chosen gates 2 or 3 and the moderator opens other gates accordingly. Six cases have to be considered in order to be able to model the equal probability of opening gates 2 and 3 by the moderator according to rule 4. This corresponds to a random experiment in which the two goats can be distinguished from one another and any distribution of car and goats behind the three gates is equally likely ( Laplace experiment ).
To evaluate the table, the cases in which the moderator opens gate 3 must now be considered (this is the condition). These are cases 2, 4 and 5. You can see that in two of these three cases the candidate wins by switching. Assuming that the candidate first chose gate 1 and the moderator opens gate 3 with a goat behind it, the car is behind gate 2 in two thirds of the cases. The candidate should therefore change his choice in favor of gate 2. It can also be seen from the table that if the moderator opens gate 2 instead of gate 3, the candidate also wins the car in two out of three cases by switching to gate 3.
Formal math solution
The events are defined:
- : The profit is behind goal ( )
- : The moderator has opened the gate ( )
The situation is as follows: The candidate has chosen gate 1 and the moderator has then opened gate 3. Is it worth changing for the candidate? What is the probability that the car is behind Gate 2? We are looking for the conditional probability that the car is behind gate 2 if it is known that it is not behind gate 3. One can find this probability with Bayes' theorem .
Based on the task (rules 1, 4 and 5 and the choice of candidate), the following requirements apply:
Applying Bayes' theorem then gives:
The candidate should therefore switch to double his chances of winning from 1 ⁄ 3 at the beginning to 2 ⁄ 3 now .
The lazy moderator
For this solution, the following additional assumption is made:
- Case A: The moderator, who does not like to travel long distances, prefers to open gate 3 because he is located there as the showmaster. So if the car were behind Gate 1, chosen by the candidate, it would definitely open Gate 3, but definitely not Gate 2.
For the following explanation it is assumed that the candidate initially selected gate 1. An analogous explanation applies to situations in which the candidate has chosen gates 2 or 3 and the moderator opens other gates accordingly. Although it would be sufficient here to consider the first three game situations, a distinction is made between six cases in order to be able to model the problem in a manner comparable to the above table solution for the balanced moderator. Each game situation is therefore considered twice. This corresponds to a random experiment in which the two goats can be distinguished from one another and any distribution of car and goats behind the three gates is equally likely ( Laplace experiment ).
To evaluate the table, the cases in which the moderator opens gate 3 must now be considered (this is the condition). These are cases 1, 2, 4 and 5. You can see that only in two of four of these cases does the candidate win by switching. His probability of winning is therefore only p = 1 ⁄ 2 . It can just as easily be seen from the table that if the moderator opens gate 2, the candidate will surely win if he switches to gate 3.
Formal math solution
The situation is as follows: The candidate has chosen gate 1, and the moderator has then opened gate 3. The following mathematical relationships then apply, taking into account the event sets defined above:
Applying Bayes' theorem then gives the conditional probability that the car is behind Gate 2:
However, the conditional probability that the car is actually behind Gate 1 also applies
The profit behind goal 2 is just as likely as the profit behind goal 1. In this case, the candidate can stay with goal 1 as well as switch to goal 2. If the moderator has opened gate 3, his chance of winning is independent of the decision 1 ⁄ 2 .
The unbalanced moderator
This solution makes the following additional assumption:
- Case A: If the moderator has the opportunity to choose a goal from two goals with a goat behind each (the candidate has chosen the goal with the car behind it), then he opens the goal with the highest possible number with the probability and the goal with the lower number with the probability .
Then the following mathematical relationships apply, taking into account the event sets defined above:
Applying Bayes' theorem then gives the conditional probability that the car is behind Gate 2:
This calculation describes the general case from which the “balanced moderator” ( ) and the “lazy moderator” ( ) can be derived as special cases.
The general solution
From the consideration of the unbalanced moderator the probability of winning a goat gate can be derived that independently of its respective preference for a particular Ziegentor by changing after opening always at least 1 / 2 , on average even 2 / 3 is. As long as the moderator is forced according to the rules of the game to always open an unselected goat goal and then offer a change, a candidate should change his choice of goal in any case.
Clarification attempt by the New York Times in 1991
In an article on the first page of the Sunday edition of the New York Times in 1991, an attempt was made to clarify the debate on solving the "Monty Hall problem", which had been going on for 10 months at the time. The following four people were asked to contribute to this attempt at clarification: Martin Gardner , Persi Diaconis , Monty Hall and Marilyn vos Savant . After Monty Hall had read the task carefully, he played the game with a test candidate in such a way that he always lost when he changed, by only offering the change if the candidate had chosen the winning goal in the first step.
Gardner confirmed this variant with the words: "The problem is not well formulated if it is not made clear that the moderator always opens a goat door and offers a change." Otherwise the moderator could only offer the change if it was his The advantage would be to let the candidate change, which would reduce the chances of a change to zero. This ambiguity can be eliminated by the moderator promising beforehand to open another door and then offer a change.
Vos Savant confirmed this ambiguity in their original problem statement and that this objection, had it been raised by their critics, would have shown that they really understood the problem; but they would never have given up their first misconception. In her book, published later, she writes that she had also received letters from readers who pointed out this lack of clarity. However, these letters were not published.
Diaconis said about the task: "The strict argument is that the question cannot be answered without knowing the moderator's motivation." This was in complete contrast to the publications, which based their solution on exact mathematics in contrast to "intuition" founded.
Monty Hall himself gave the following advice: “If the moderator always has to open a door and offer a change, then you should change. But if he has the choice of offering a change or not, pay attention: no guarantee! Everything depends on his mood. "
Paul Erdős and the goat problem
Andrew Vázsonyi describes how the famous mathematician Paul Erdős reacted in 1995 to the goat problem and the claim of the 2 ⁄ 3 solution. After Vázsonyi first heard about the problem from a friend, directly based on vos Savant's original version, he solved it with a decision tree and could hardly believe the 2 ⁄ 3 solution that resulted. When he then presented Erdő's problem and solution, "one of the greatest experts in probability theory" said: No, that is impossible. There is no difference. Vázsonyi describes the reaction to the solution with the decision tree as follows: To my amazement, that did not convince him. He wanted a simple solution with no decision trees. I gave up at that point because I don't have a common sense explanation. It is "hopeless" for someone who is not familiar with decision trees and Bayes' theorem to understand the solution. When Vázsonyi was asked by Erdős an hour later to tell him the reason for the change, he finally showed him a computer simulation. According to Vázsonyi, Erdős objected that he still did not understand the reason, but was reluctantly convinced.
A few days later, according to Vázsonyi, Erdős announced that he had now understood the solution after the mathematician Ron Graham had given him the reason for the answer. However, Vázsonyi writes that he himself did not understand this reasoning.
In his book on Paul Erdős, Paul Hoffmann gives Graham's reasoning: “The key to the Monty Hall problem is knowing in advance that the moderator will always give you the option to choose another door. That is one of the rules of the game and must be included in the considerations. "
At the end of his article, Vázsonyi writes in the “Marilyn Knows Best” section that he later got a deeper insight into the problem through an article on the subject in the Skeptical Inquirer from 1991. In this article, through which Gero von Randow also encountered the problem, the exact original task vos Savants from Parade magazine is presented .
The older Monty Hall problem
In February 1975, the American Statistician academic journal published a letter to the editor from Steve Selvin, then an assistant professor of biostatistics at the University of California at Berkeley. In this letter, titled "A Problem in Probability," he suggested a word problem as an exercise in probability. The solution he gives is similar to the table presented in the section on vos Savant's answer. Another letter from the same author, entitled "On the Monty Hall Problem," appeared in August of that year, referring to his first letter and responding to reader objections to his proposed solution. It was at this point in time that the term “Monty Hall Problem” appeared in the media for the first time.
In his second letter, Selvin presented further arguments in favor of his solution, including a formal mathematical calculation using conditional probabilities. He added that his calculations were based on certain, non-explicit, assumptions about the behavior of the moderator Monty Hall. He also quoted a reader who pointed out that the critical assumptions regarding moderator behavior were necessary in order to be able to solve the problem at all, and that the initial distribution was only part of the problem, while this was a subjective decision problem .
It makes sense to regard this early Monty Hall problem as a forerunner of the question known today as the goat problem, including the dispute about the then already controversial additional assumptions regarding the moderator's rules of conduct.
Overview of the specialist literature on "the" goat problem
References to literature
In the publications on the goat problem (Monty Hall problem), different questions and models are examined, sometimes even within one publication.
Authors such as Gill and Krauss & Wang as well as Krauss & Atmaca base their solution from Savant's original text and only make their additional assumptions explicit in the course of their analysis. The correctness of vos Savant's solution, which triggered the violent controversy, is expressly emphasized.
In the appendix to vos Savant's book, Donald Granberg writes that there is consensus that vos Savant's answer is essentially correct, provided seven "highly plausible" assumptions are made. Below this is the assumption that the moderator is required to open an unselected goat door after the first choice, as well as the assumption that the moderator is honest.
Krauss & Wang add several assumptions to the vos Savants task, which they refer to as the “standard version”, so that the vos Savants solution can be derived precisely. Krauss & Atmaca also begins with the original problem from Savants, whereby the moderator, before opening the goat door, says, according to Gero von Randows, I'll show you something. According to Steinbach, these words of the moderator are "nonsensical" from the candidate's point of view if, based on the rules of the game, he expects to be shown a goat anyway. Henze also lets the moderator say in his task formulation, before he opens the goat door, should I show you something? , and, after presenting the Savants solution as correct, writes: In all of these considerations, it is of course crucial that the moderator must keep the car door secret, but is also obliged to open a goat door. In a lecture in the summer semester of 2014, he wrote this addition at the beginning of the task and pointed out in detail that vos Savant was right.
Lucas uses a problem formulation that dictates certain rules of behavior for the moderator from the outset. In assessing the violent reactions to vos Savant's solution, however, it does not matter to Lucas that these rules of conduct were not formulated in the problem presented by vos Savant.
Morgan et al. and Gill, in turn, do not address the fact that in vos Savants' original question there was no rule that the moderator is obliged to open a non-selected goat door after the first choice and offer a change. Morgan et al. See the only flaw in Savants' solution. in that she did not explicitly assume that if the candidate has chosen the car door, the moderator will open both possible goat doors with equal probability. Only after their explanations of the task and solution do Morgan et al. and Gill other ways of playing the game. Morgan et al. now even assume that the moderator may also open the car door without informing the player, which in a "plausible scenario" leads to the "popular answer 1 ⁄ 2 " in the event that he opens a goat door that has not been selected. They even write that the moderator's perspective requires not following the “vos savant scenario” in order to prevent players from always switching. Allowing the moderator to immediately open the door chosen by the candidate is what they call a “generalization” that does not change the considerations of the conditional probabilities.
Götz (2006) sees “the famous 'goat problem'” as “sufficiently discussed”. In his description of the problem it says: Now comes the crucial point. The game master asks the candidate whether she would like to stick to her original choice of door or change to the other, still closed door. As a solution, he writes that the “change” strategy leads to a car with a probability of 2 ⁄ 3 . After various approaches, he mentions that “R. Grothmann (2005) ”pointed out that it must be clear whether the game master has to open a door that has not been selected or can also open the selected one .
“The controversial goat problem known from the media” is “completely analyzed and solved” by Steinbach. He is based on Gero von Randow's formulation of the problem. Steinbach suspects that the different answers to the original question can be traced back to the fact that the proponents of the 2 ⁄ 3 solution take the perspective of the "thinker", the proponents of the solution 1 ⁄ 2 that of the candidate: Just from the words of the moderator and the When looking at the goat, the candidate cannot tell whether any rule of the game applies - and certainly not which one. […] All that remains is to flip a coin: this is how the candidate gets caught - regardless of the moderator's behavior! - with probability 1 ⁄ 2 the right door.
Gigerenzer and Grams point out that much of the debate on the goat problem stems from the authors' insufficient distinction between “decision when risk” and “decision when uncertainty”: “Among the thousands of articles that have been written about the Monty Hall Problem were published, the difference between risk and uncertainty was practically ignored ”(Gigerenzer).
The question is qualitative, not quantitative
In terms of the various solutions, as they were reproduced above, says Götz "CHANGE IS NEVER WORSE THAN STAY!" ( Capitals of Reference). Morgan et al., The “discoverers” of the solutions based on additional assumptions about moderator behavior, had already drawn attention to this fact in 1991. Despite these qualitative consistency and the fact that the problem "Is it advantageous to change the choice of the gate?" To an action and not asking for a chance are the assumptions that different probability values lead, always the subject of fierce Discussions. The bibliography of the book The Monty Hall Problem by Rosenhouse , published in 2009, contains over a hundred publications.
Georgii initially comes under the assumption that the moderator is obliged to open an unelected goat door after the first choice of the candidate, immediately to the probability of winning 2 ⁄ 3 in the event of a door change. The "triviality" of this solution, which corresponds exactly to the answer from Savants , is, according to Georgii, due to the fact that we have set the moderator to a fixed behavior, so that he always performs the game as described . He sees the “deeper reason” for this determination in the fact that we have implicitly assumed a frequentist interpretation of the conditional probabilities, which presupposes the repeatability of the process and therefore fixed rules . Following the observation by Morgan et al. Georgii also writes that the moderator's perspective requires not following the “vos savant scenario”: Now the moderator will not play the game regularly. From this point of view, the “subjective interpretation” is more appropriate. As an example, he then cites the variant with a winning probability of 1 ⁄ 2 , in which the moderator opens one of the two goat doors with the same probability before the change offer, regardless of which door the player has chosen. (The moderator can therefore also open the goat door chosen by the player.) After this explanation, he draws the following conclusion: Similar to the Bertrand paradox , the various answers are based on a different interpretation of a fuzzy task. [...] The philosophical uncertainty about the meaning of conditional probabilities makes it even more difficult.
Influence of moderator behavior when choosing the car door
Georgii's remark that it depends on “how the player assesses the behavior of the moderator” can also be applied to the question of the likelihood of the moderator opening a certain goat door if the candidate has chosen the car door. Most textbook authors, however, do not consider such a subjective assessment of moderator behavior. Specifically, they assume that the moderator acts in a balanced way , that is, that he selects the goal according to an even distribution . This makes this approach the most common explanation in the literature that a goal change leads to a win with a probability of 2 ⁄ 3 . This winning probability of 2 ⁄ 3 in the event of a goal change refers explicitly to the time after the moderator has opened a goal.
Studies in which the candidate assesses the moderator as not being equally likely to make his goal selection were first carried out in 1991 by Morgan et al. and published independently in 1992 by Gillmann. Morgan et al. vos Savant's task changed so that the question related exactly to the door numbers mentioned, which were only used as explanatory examples at vos Savant. The variant vos Savants with one million doors was described by Morgan et al. as a "dubious analogy". Applying the Morgan et al. without additional assumptions, this variant delivers the same result as with only three doors, namely a value between 1 ⁄ 2 and 1 - compared to 99.9999% with vos Savant.
In their reply to Morgan et al. vos Savant points out the abridged version of both her question and her answer, the full version of which she reproduces in her reply letter. Morgan et al. in turn respond that there is no reference in this representation that the question originated from a "reader in Columbia, Maryland". This is important because the restriction "that the moderator has to show a goat" was added by vos Savant himself. Vos Savant herself has indicated that she felt that this "most significant" limiting condition was not sufficiently emphasized in the original reader question and that she therefore added it in her answer.
With regard to the other prerequisites, which were not formulated in her original question, she sticks to her view that they do not appear to her to be important for a general understanding of the problem, since events are regarded as "random" by default. Steinbach also shares this view, who describes these assumptions as “tacit, but indisputable and irrelevant” before he mathematically examines them under the heading “splitting hairs”.
According to Georgii, the different points of view on the "fuzzy task" are reduced to the question of whether it is part of a fixed rule of the game that the moderator has to open an unselected goat door and offer a change.
While with Georgii the question of the likelihood of the moderator opening a certain goat door if the candidate has chosen the car door is not discussed and does not play a role in his solution, Götz refers to two “different probability terms that underlie the respective approaches . ”The“ classic solution ”without considering this moderator behavior should be interpreted“ frequentist ”and empirically verified. In contrast, a “ Bayesian solution [...] provides the basis for evaluating an individual situation. How should the candidate hic et nunc behave after the game master has opened a door? [...] So you ask about the probabilities of the state or the probabilities of knowledge (and not about the probabilities of future random events ). ”In other words: After the moderator has opened the gate, the candidate makes the assessment of his two options for action dependent on which basic behavior she assumes the moderator will. The extreme case of a lazy moderator is characterized by the answer to the following question: "If the moderator, after seeing my decision for a goal, had selected the goal he had just opened under all other circumstances, if only possible - no car behind it - would have been? "
If the candidate doesn't know anything about the moderator's preferences, "changing", according to Götz, "brings a chance of success of 2 ⁄ 3 ". Good estimates for the unknown parameter p can be obtained by observing the behavior of the game master in the appropriate situation when the car is behind door 1 and the candidate has (initially) chosen this door .
Bayesian studies were first published by Morgan et al. carried out based on their results, in which the moderator randomly selects the gate to be opened according to assumed a priori probabilities for it .
Comparison of the different solutions
Numbering of the gates
The characterization of the behavior of a lazy moderator carried out in the last section shows that a solution in this regard is not tied to a numbering of the gates (usually “candidate chooses gate 1. Moderator opens gate 3 whenever possible”).
Empirical review of a solution related to moderator behavior
If, for example, the 50:50 solution found for the variant of a lazy moderator is to be empirically tested, it must be taken into account that the statement derived on this basis relates to a conditional event. In a test series of 300 game shows, which are carried out according to the additional assumption of lazy moderators , about 100 shows do not go through the event that is the subject of the investigation. The specific reason for this is that if a car is hidden behind gate 3, the moderator is forced to open gate 2. Such courses of play lie outside the investigation area, so that the winnings always achieved after a goal change have to be disregarded in the evaluation of the test series.
Decision-making situations with different chances of winning
The goal change strategy, which is set “globally” for all possible decision-making situations, has an overall 2: 1 advantage. However, an asymmetrical course of the game can lead to decision-making situations in which a goal change is more or less promising than the average. Such effects are evident with regard to an asymmetrical probability distribution in the drawing of the winning goal, but they can, as the results for the lazy moderator show, also be caused by an asymmetrical moderator behavior. With moderator behavior, however, the possible deviations for the probability of winning a goal change from the a priori value of 2 ⁄ 3 downwards are limited, since the value cannot be fallen below 1 ⁄ 2 , because "changing is never worse than staying" - see above.
The two solutions predicting a 2: 1 advantage
Even if the “classic” vos savant solution, consistent with the solution for the balanced moderator, predicts a 2: 1 advantage for a goal change, their viewing angles and arguments are very different: One time there is an a priori probability for the situation specified immediately before the moderator's decision on a gate to be opened. The other time, the probability relates to the point in time when the moderator has already opened "his" gate, although the additional assumption is made that the moderator has made his selection with the same probability. The fact that both approaches deliver the same probability of winning results from a symmetry analysis that derives the a posteriori value from the a priori value.
Game theory approach
With different assumptions about the probability with which the moderator will open a certain goat door if the candidate has chosen the car door, different odds of winning can be calculated for the respective individual case. This aspect has been taken by some authors as the starting point for game theoretical investigations of the goat problem. The additional assumption about this probability is understood as a mixed strategy in the sense of a two-person game , which even has the character of zero-sum . Hiding the car, which is rated as the first move of the moderator, is also included in the sequential course of the game . With a simple argument, using strategies that are obvious to both players and symmetrical with regard to goals, Gill was able to show that the minimax value is 2 ⁄ 3 .
The set of minimax strategies for both players was determined by Gnedin. The candidate only has a single minimax strategy in which he draws his first chosen goal according to an equal distribution and then always changes the goal. The statement is remarkable insofar as it does not require a priori assumption about the moderator's behavior and still makes statements for each individual decision-making situation that occurs in the game. An even stronger argument for the candidate never to keep the goal initially chosen arises from Gnedin's dominance analyzes for strategies.
Further mathematically examined variants
In addition to the interpretations of "the" goat problem presented above, there are other variants that have been examined in the specialist literature. In general, it should be noted that the authors - as already with regard to the interpretations presented above - do not have a consensus as to which mathematical model corresponds to “the” goat problem and its question. Sometimes the models only serve the purpose of an explanatory comparison:
The moderator can also open the gate by car
Lucus, Rosenhouse, Madison and Schepler and Morgan et al. analyze, among other things, the variant in which the moderator randomly chooses his gate from the two remaining gates and, if necessary, opens the gate by car. A short calculation confirms the intuitively obvious assumption that for this variant in the event that a gate with a goat is opened, the probability of winning when changing is 1 ⁄ 2 .
Moderator can also open the gate selected first
In one of the two variants he investigated, Georgii also allows the moderator to open the gate first selected by the player with a goat. If the moderator randomly chooses between the two goat gates with the same probability, the probability of winning a change in accordance with the "response of the critics" is 1 ⁄ 2 even if he opens a goat gate that was not chosen.
The goat problem in the media
Jamie Hyneman and Adam Savage investigate the goat problem in episode 177 Mythen Without End of their documentary series Mythbusters . It confirmed the two claims that (1) people tend to stick to their first choice and (2) that changing the original decision increases the chance of winning significantly.
Influence of Wikipedia
While working on Wikipedia , W. Nijdam and Martin Hogbin found an error in 2010 in the work by Morgan et al. According to this, if a non-informative a priori distribution is used as a basis for the moderator behavior, the probability of winning when changing goals is 2 ⁄ 3 and not , as Morgan et al. had calculated. Morgan et al. Used the confirmation of this fact to publish for the first time the original question from Craig F. Whitaker's letter to Marilyn vos Savant.
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