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Monty Hall Problem

Papers and Documents| Links| Simulation and Activities| Interesting Variants and Comments


 

The 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 columnist Marilyn 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 and 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.   

 

The Literature.       
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 irrational choices.  

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 out. Kluger 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: anakhoda@ucsc.edu
August, 2008

   

Papers and Documents:

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

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

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

Authors: Walter T. Herbranson and Julia Schroeder, Whitman College

 

Links:

 

Simulations and Activities:

Interesting Variants and Comments:


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: anakhoda@ucsc.edu

 


 

This page is being maintained by Aadil Nakhoda, under the supervision of Dr. Dan Friedman.