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Monty Hall Problem
When the TV game show Let’s Make a Deal first aired on NBC in 1963, nobody could have predicted its impact on the academic fields of economics, mathematics, psychology and statistics. The show had various fun-filled segments, but the most interesting was called “Big Deal of the Day.” Each episode would end with this segment, in which a participant would be shown three doors. Behind two were goats (or other worthless prizes called “zonks”) and behind one was the grand prize, often a new car. The participant chose the door and received the prize behind it.
As recounted by Barry
Nalebuff in 1987, Monty Hall,
the host of the television show, would always give the participant the
option to switch from his initial choice, after Monty opened one of the
unchosen doors to reveal a zonk. Nalebuff and later writers said that
participants hardly ever accepted the option.
Should the participant switch in such a situation? This “Monty
Hall Problem” was presented (as a hypothetical situation) to popular
vos Savant in 1990, and she said Yes. She explained why
switching doubled the probability of winning the valuable prize.
Several readers, including PhD mathematicians, told Marilyn that she
was ignorant or worse, and that participants had a 50-50 chance of
winning whether or not they switched. Variants on her
explanation can be found here
Of course, Marilyn was right in the situation just described.
But in slightly different situations the answer is very
different. If Monty had opened the door at random (and it happened to
be unchosen and to not have a goat behind it) then Marilyn’s critical
readers would have been right. If Monty had only offered the option to
switch when the participant had initially chosen the prize door, then
the participant should never switch, since then he would surely lose
the valuable prize.
Which situation best describes the actual game show? In
correspondence with us, Monty Hall said that he rarely (if ever)
offered the switch option. Carol Andrews, Monty Hall’s longtime
assistant, asserts that Monty never offered the switch option to the
final guest. In correspondence with us, Nalebuff recalls that Monty
Hall did offer the switch option although he cannot recollect whether
the option was contingent on whether the participant initially chose
the right door. We may never know the actual fact of the matter,
because very few tapes of the show (which ran daily from 1963 to 1968,
and then weekly until as late as 1986, with 3,800 shows altogether) are
available to the public. Due to a dispute over residual rights, the
tapes are said to be deteriorating in a Hollywood garage.
Whatever its true historical basis, the Monty Hall Problem has been studied and debated intensively by a large and eclectic set of economists, mathematicians and psychologists, as early as 1975. The purpose of this webpage is to collect and provide documents, links, simulations and activities on the Monty Hall Problem. We hope that it is useful to students, professors and journalists alike.
Even before Marilyn vos Savant, the mathematician Steve Selvin first posed and solved the Monty Hall problem in 1975. Eisenhauer(2000) shows students how to construct matrices to calculate the probabilities in this puzzle. Puza, et al (2005), provide a Bayesian analysis of several variants of the problem. Bailey (2000) generalizes the Monty Hall Problem to include N doors, implementing theories such as the Minimax theorem to solve for the problem. Boumans (2009) discusses various interpretations of the problem as well as other apparent anomalies.
It took a long time before anyone investigated the Monty Hall
problem empirically. Do people really refuse an option that would
double their chances of winning a valuable prize? It seems that this paper
by one of us was the first laboratory test. After reviewing the earlier
literature, the paper shows that people really are reluctant to switch.
In the baseline treatment, the switch rate begins at 10%, rises with
experience to around 40% and then for some reason retreats to about
30%. The paper goes on to show that the switch rate rises to
well over 50% when people have good opportunities to learn from
experience, and makes some general points about the nature of
Several later investigators also looked for more effective
ways to help people solve the problem. Scott Page (1998) tried the 100
Door variant first suggested by Marilyn vos Savant, and found
that people indeed are far more likely to switch when 98 out of the 99
unchosen doors were opened to reveal no prize. But people had a hard
time transferring the lesson back to the original 3 door problem.
Slembeck and Tyran (2004) got virtually everyone to switch. They used a
powerful combination of social
learning and social pressure, featuring contests between
groups playing the game. Kluger
and Wyatt (2004) find that asset market prices "solve" the
Monty Hall Problem when at least two individual traders have figured it
and Friedman (2006) find that merely framing the problem in
terms of a carefully chosen asset helps individuals solve it. Franco-Watkins
et al (2003) use various different experimental techniques to
study the subjects’ choice behavior and probability judgment. Peter
Mueser and Donald Granberg(1999) experiment with different
protocols Monty Hall has to follow, but find that subjects stick to
their original choice in almost all cases. Granberg
(1999) used different cases of equal and unequal probabilities of
winning the prize. In the case of unequal probabilities, he assigned
different probabilities that each door would have a prize. His results
were similar to other researchers who found that the number of subjects
who preferred to switch was less than expected.
Howard Margolis compares the Monty Hall problem to other game theory puzzles, and points out analogies to optical illusions. Keith Chen (2008) connects tests of cognitive dissonance and rationalization to the Monty Hall problem, and points up some methodological flaws in the cognitive dissonance literature.
Written by Dan Friedman and Aadil Nakhoda
contact info for Aadil: email@example.com
Please Note: To access the following documents, you may require access to the respective sites/servers they are hosted on.
Author: Herb Bailey, Rose-Hulman Institute of Technology, Indiana
Author: Joseph G. Eisenhauer, Canisius College, New York
Authors: Ana Franco-Watkins, University of Maryland, College Park
Peter Derks, College of William and Mary
Michael Dougherty, University of Maryland, College Park
Author: Daniel Friedman. Department of Economics, UC Santa Cruz
Authors:Daniel Friedman, Department of Economics, UC Santa Cruz
Brian Kluger , University of Cincinnati, Cincinnati, OH
This article is forthcoming in the Journal of Behavioral Finance.
Author: Donald Granberg, Department of Sociology, University of Missouri, Columbia
- Are Judgment Errors Reflected in Market Prices and Allocations? Experimental Evidence Based on the Monty Hall Problem
Authors: Brian D. Kluger, University of Cincinnati, Cincinnati, OH
Steve B. Wyatt, University of Cincinnati, Cincinnati, OH
Author: Howard Margolis, University of Chicago
Authors: Andrea Morone, University of Bari
Annamaria Fiore, University of Bari
- The Monty Hall Dilemma Revisited:Understanding the Interaction of Problem Definition and Decision Making
Authors: Peter R Mueser, Department of Economics, University of Missouri-Columbia
Donald Granberg, Department of Sociology, University of Missouri-Columbia
Author: Barry Nalebuff, Princeton University, Princeton, New Jersey
Author: Scott Page, Department of Economics, University of Iowa
Authors:Borek D. Puza, David G.W. Pitt,Terrence J. O'Neill
Australian National University, Canberra, Australia
Steve Selvin, School of Public Health, University of California, Berkeley
Authors: Tilman Slembeck, Department of Economics, University of St. Gallen and Zurich University of Applied Sciences, Winterthur, Switzerland
Jean-Robert Tyran, Department of Economics, University of St. Gallen, Bodanstrasse, Switzerland
Author: Marcel Boumans, University of Amsterdam
- Are Birds Smarter Than Mathematicians? Pigeons (Columba livia) Perform Optimally on a Version of the Monty Hall Dilemma
Authors: Walter T. Herbranson and Julia Schroeder, Whitman College
- Entry in Wikipedia
- A webpage by Keith M Ellis. Provides a simple solution to the Monty Hall problem.
- An explanation provided by The New York Times. They also let you play a game in order to experience the problem itself.
- Here you can find some discourse on the Monty Hall problem along with some simulations and programs.
- Discussion of the question asked to Marily vos Savant and her interpretation. An interesting simulation is also included.
- Some links that solve and discuss the problem:
- Watch a video on Youtube explaining the Monty Hall problem
- A videoclip from the movie '21' explaining the Monty Hall problem.
- Monty Hall problem explained in the TV program 'Numbers'
- Car and Goats Game
- Simple Monty Hall
- Here is an interesting problem suggested by Harvey Rubinstein of Hudson County Community College in New Jersey, and a suggested answer by us.
Hope you enjoyed learning about the Monty Hall Problem.
Please feel free to contact us if you have any queries or suggestions.
Aadil's email address: firstname.lastname@example.org