The game show problem has received considerable attention since it appeared in Marilyn vos Savant's column in the Parade Magazine. I consider in the note the player's optimal decision strategy and the probability of winning the prize in a generalized version of the game show problem. By means of the information economics approach, we can easily (1) represent the host's various strategies as a simple matrix form, (2) extend the problem to more-than-three-door cases, (3) incorporate the player's prior information, and (4) consider the problem in which the player's choice behavior depends not on the probability of winning the prize, but on the expected utility in the generalized game show problem. The intricacies of this wonderfully confusing little problem make it an excellent tool in teaching probability and statistics courses.

Key Words: Information economics; Decision analysis; Probability models; Bayes rule.

* Young H. Chun is an Associate Professor of Decision Science, E. J. Ourso College of Business Administration, Louisiana State University, Baton Rouge, LA 70803-6316. The author is grateful to the Associate Editor and a referee for their valuable comments. The research was partially funded by a two-year grant from the Research Competitiveness Subprogram of the Board of Regents Support Fund, LEQSF(1998-00)-RD-A-04.

A simple probability problem has provoked a series of disputes between a columnist said to be the holder of the world's highest IQ score and numerous Ph.D.s in academia since it appeared in the "Ask Marilyn" column in the September 9, 1990 issue of Parade Magazine. The original three door game show problem appeared in the column is stated as follows: "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? - D. Craig F. Whitaker, Columbia, Md."

Ms. Marilyn vos Savant, listed in the "Guinness Book of World Records Hall of Fame" for "Highest IQ" in the world, replied that, "Yes; you should switch. The first door has a one-third chance of winning, but the second door has a two-thirds chance." When she innocently printed the reply in the magazine supplement to many Sunday newspapers, she had no idea that it would provoke a national controversy. She received thousands of letters, nearly all insisting that, because two options remained, the chances were even. The most vehement criticism has come from statisticians and scientists, who have alternated between gloating at her and lamenting the nation's innumeracy. As she received more letters, she defended her original claim repeatedly in the December 2, 1990 issue, February 17, 1991 issue, and July 7, 1991 issue of the same magazine.

The debate raging among statisticians, readers of Parade Magazine, and fans of the television game show, "Let's Make a Deal", was also reported in the July 21, 1991 issue of the New York Times. Recently, the game show problem was aired in Tom and Ray's CarTalk radio show in October 25, 1997 and, since then, they reportedly have received "satchels full of mail." They decided to put the puzzle and its solution on the CarTalk Web page, http://cartalk.com/about/monty, saying "what could be more fun than proving a bunch of pompous academics wrong!!"

Note that the same problem with different names had been discussed in academic journals before it appeared in Ms. vos Savant's column in 1990. Selvin ( 1975 ), for example, considered the probability of winning the prize behind one of three curtains in the TV game show hosted by Mr. Monty Hall. He called it the Monty Hall Problem in the American Statistician and concluded that switching doubles the chances of winning the prize from 1/3 to 2/3. Nalebuff ( 1987 ) also proposed the TV game show problem as a puzzle in Economic Perspectives and gave the same answer. Later, Kotz and Stroup ( 1983 ) and Siegel (1990 ) used the puzzle for instructional purposes in their introductory statistics books.

After the game show problem appeared in the vos Savant's column, several attempts to end the debate were also reported in academic journals and newsletters. Based on a decision tree with conditional probabilities, for example, Chun ( 1991 ) illustrated in OR/MS Today that it is the player's advantage to take the switch. In the subsequent issues of the newsletter, the Editor published a representative sample of the letters he received, saying that "no subject has inspired more letters to the Editor of OR/MS Today than the Game Show Problem."

Saying that the host's strategy is not clearly explained in the original game show problem, Morgan et al. ( 1991 ) discussed a variety of host's strategies and concluded that "in the vos Savant scenario, we can state that it is always better to switch." In their concluding remarks, Morgan et al. ( 1991 ) suggested three possible extensions to the vos Savant's game show problem: (1) "allowing the host the option of immediately opening the player's chosen door," (2) considering "n doors as does vos Savant in the September article," and (3) incorporating "prior information on the part of the player as to the location of the car."

In response to their suggestions, a generalized version of the game show problem is considered in this note and the player's optimal decision strategy that maximizes the probability of winning the prize is found. By means of an information economics approach proposed by Marschak (1971), we can easily (1) represent the host's various strategies as a simple matrix form, (2) extend the problem to more-than-three-door cases, and (3) incorporate the player's prior information. I show that the vos Savant's game show problem can be easily modeled without any ambiguity and solved as a special case of the generalized problem.

According to Seymann ( 1991 ), "A surprisingly large number of undergraduate students, when faced with the inescapable conclusion that switching will double their chance of winning [ in the vos Savant's game show problem ], still maintain they wouldn't switch. Their reason for this unexpected decision may be summarized by the statement, 'If I switched and lost, I'd kick myself for having switched.' " Nalebuff ( 1987 ) also concluded that "since many if not most individuals choose to stay with their original choice, does this suggested we should look for alternatives to Bayes rule?" As an alternative, we propose in this note the information economics model that maximizes the player's expected utility, rather than the probability of winning the car, effectively explaining why so many players are reluctant to switch doors.

As pointed out by Seymann ( 1991 ), the confusion and controversy sparked by this simple problem may be attributed mainly to the definition and the solution of the problem. It appeared to the author that most statisticians and mathematicians viewed the game show problem as a simple probability problem and tried to solve it from the probabilistic perspective. As a decision scientist, however, I believe that the problem can be best treated as a decision problem and the information economics approach is the most effective way for modeling this type of decision problems.

Thus, I hope that the intricacies of this wonderfully confusing little problem provide an excellent opportunity for the information economics approach to be introduced to the readers of the American Statistician, to be used along with the Bayesian rule as a teaching tool in probability and statistics courses, and to be applied to a wide variety of different situations arising in decision science applications such as the acceptance sampling plans in Ronen ( 1994 ).