The Monty Hall problem is fascinating not because of its mathematical details but because a large percentage of people give the wrong answer to it. This problem
has managed to fool even mathematicians and scientists. An accepted version of the problem is the following :
"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"
The correct answer is that you must switch doors. This is because the new information provided by the host of the game impacts prior probabilities in the following way: The initial door choice has probability 1/3 to
have a car behind it, given the game definition above. However, after the host opens a door that has a goat behind it, there is new and valuable information that was not available at the time the initially probability
was calculated. One of the initial choices is now eliminated. The new fact does not affect the probability assigned initially to the door selected to have a car behind it, but it increases the probability that the car
is behind the remaining two doors.
The problem for most people starts when they think that the change in probability is uniformly assigned so that the remaining two choices get 1/2 of 1/3 each, and thus each has a
probability of 1/2 to have the car. Under this assumption, switching doors does not make any difference. But there is a subtle important detail due to the rules of the game, namely that the host will never open the door
initially selected but only one of the remaining doors that has a goat and if both have a goat behind them, he will choose one at random. This means that the action of the host is not uniform with respect to all doors
and due to this fact the probability change is also not distributed uniformly. Since the distribution occurs only for the two doors not chosen and the value distributed is 1/3, then the probability of the door not
selected increases by that much, specifically from 1/3 to 2/3, since the probability that the opened door has a car behind it is now 0. Therefore, switching to the other door is equivalent to the act of increasing the
probability of getting a car from 1/3 to 2/3. This can be proved formally using Bayes' Theorem .
Undoubtedly, the solution obtained by applying Bayes' Theorem is correct and that is corroborated by computer
simulations . I also had doubts about the correct solution when I first came across this problem. I ran the simulation several times and for a large sample of trials, only to find out that when one stayed with the
initial choice the success rate converged to 1/3 but when a switch was made the rate increased to 2/3.
2. Fear of Freedom
The real issue with the Monty Hall problem is why most people stick to
their original choice and do not take into account correctly how the new information, namely the opening of a door by the host, impacts their probability of success.
Most social psychologists will argue that this
behavior can be explained by cognitive dissonance, a theory that - in an oversimplified way for our purposes here - claims that when people make a choice between alternatives, they tend to devalue what they did not
choose. But Dr. Keith Chen argues against this theory as an explanation of the choice people tend to make in the Monty Hall problem .
I argue that the deeper reason as to why most people stick to their original
choice is because of fear when freedom of choice is offered. This is related to the controversial ideas presented by Erich Fromm in his book The Fear of Freedom
. Fear prevents human intuition from taking into account all the facts and reaching for the correct conclusion. But why fear? The answer to this may be simpler than most people think. For example, when one is given the freedom to switch to an alternative choice after a choice is already made, this raises the overall level of uncertainty, fear kicks in, rational decision making in inhibited and most people tend to stick to the choice already made.
The refusal to switch in the Monty Hall problem may thus be a normal reaction of people who believe that everyone is out to get them rather than to benefit them and when they are let free to change their choice by
someone they view it with suspicion. Thus, the reason I think that the majority of people do not make the switch in the Monty Hall problem, as dictated by rigorous scientific thinking, is that fear, induced by freedom
of choice, prevents them from doing so. The decision to stick to original choice is then justified a posterior
on the basis of uniform probability and the emotion disappears like it never existed. As a matter of fact, this type of fear is so common that people do not even recognize it as such.
One must differentiate between
what truly drives people to stick to their initial choice and the explanation they offer for their action before or after they learn the correct answer. It may be true that most people will defend their choice not to
switch based on equal probabilities but this is a posterior
easy justification that fits their behavior and it just happens to be available to them. They may not understand the true cause of their behavior but the 50/50 model just happens to fit it optimally.
In a similar
manner, we must differentiate between those that actually play the game and those that read about it or it is presented to them as a math problem. The decision of the first group is primarily influenced by fear of
freedom. I argue that a high proportion of the subjects in the second group are also driven by fear of freedom because the prize they receive may not be tangible but they perceive it as a confirmation of their ability
to solve problems. Insecurity about own ability and fear of failure can have the adverse effect in solving problems that involve freedom of choice.
If experiments were to be performed when the subjects involved were
told that they cannot switch after the host opens a door with no prize behind it, their instant intuitive reaction may be that they would have better chances if they were able to do so. Not being free to choose may work
in their favor, as opposed to when offered freedom of choice that I argue it works against them. It remains for someone to do the experiment to find out if my conjecture is correct.
Conjecture: It is due to
freedom of choice that people tend not to value correctly alternatives in light of new information.
4. Theories vs. Phenomena
Cognitive dissonance is a phenomenon, and in more precise terms it is an
epiphenomenon, not an explanatory hypothesis or a theory that explains phenomena and can be tested experimentally. Basically, the experiments performed by psychologists attempt to replicate a phenomenon and
have nothing to do with testing a theory that explains the phenomenon. As a consequence, the results obtained is the phenomenon itself. But there is more into this and especially with the danger
when a phenomenon is used as a theory. I will attempt to explain this below.
Ice formation due to temperature drop is a phenomenon, not a theory. One needs some theory to explain why ice is formed and then test
it. Now, if one cools water and gets ice, any measurements before and after cannot be used alone to explain why ice is formed. Specifically, if one takes measurements of temperature and water hardness and
insists that he has a theory about ice formation that ice is formed because the temperature drops during the experiment from 20 deg C to 0 deg C, he is confirming the phenomenon, not discovering a theory.
The explanation of the phenomenon is much more involved and deeper. It has to do with a first order phase transition involving nucleation and crystal growth, in a nutshell.
phenomenologists and make phenomenological generalizations. This is one reason some philosophers, like the Vienna Circle logical Positivists, rejected psychology and its methods as scientific. Some of
the experiments replicate the phenomenon they are trying to study. Maybe they report only the experiments that work that way. They are biased seriously to that extend because the phenomenon is real and their
setups will see it if the conditions are right. However, what they observe may have other causes, like initial preference. Thus, they may see a phenomenon they expect but this is really an epiphenomenon, meaning a
secondary phenomenon running in parallel with a primary phenomenon that has specific causes.
I argue that in the Monty Hall class of problems, the epiphenomenon is the devaluation of the choices not made
through post decision rationalization but the primary phenomenon is fear of changing the initial choice, which is caused by freedom of choice in a counterfactual sense - If freedom of choice were not
to be offered, no insecurity and fear would be caused about changing beliefs. The initial choice is not merely done based on probabilities in the Monty Hall problem. Since the probability that each door hides the car is
1/3 then there is no reason for the player to choose one door over another. The choice is made after it is forced. That causes the player to invoke belief and luck to make a choice. No random generator is offered
to the player to make his choice. Such choice cannot be altered easily based on additional information, especially if that information is related to an increased long-term success rate as opposed to a single
Therefore, I think the problem is that initial forced choice is based more often on belief and when freedom of choice is offered by the host that requires abandoning belief and adopting a
rational probabilistic account. Most people will stick to their beliefs, which is what is more important to them traditionally. Cognitive dissonance does no play a role in this because the initial selection
was not random but based on belief. However, it appears there is cognitive dissonance, as an epiphenomenon, when one is not aware of the initial preference. Thus, cognitive dissonance, which is a phenomenon,
when turned into a theory in a wrong way, will predict what people tend to choose in the Monty Hall game. But it is really an epiphenomenon and it is hardly a theory with explanatory content.
There is more to Monty Hall problem that meets the eye. The way people tend to respond to this problem may not be directly related to poor knowledge of statistics, cognitive dissonance, or other
psychological factors but may be due to the fact that they are free to choose. Freedom of choice causes fear to them although this is such a common experience that it is not recognized as such. The effect of fear is
sticking to choices already made regardless any new information that affects their probability of success. The justification offered based on equal probabilities just happens to be a model that fits a behavior they do
not understand. Cognitive dissonance is an epiphenomenon and cannot be used in the place of a theory with explanatory content about why people tend to stick to their initial choices in the Monty Hall problem.