Monty Hall Uses a Mixed Strategy

Abstract

Introduction Marilyn vos Savant posed a problem [12] simulating a popular TV game show hosted by Monty Hall. Her correct solution brought thousands of letters telling her that she was wrong. We quote her statement of the problem: Suppose you're on a game show, and you are given a choice of three doors. Behind one 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? Marilyns phrase say, No. is a little ambiguous. To clarify this, we note that if Doors 2 and 3 hide a goat and a car, then the host opens the door hiding the goat. On the other hand, if Doors 2 and 3 both hide goats then the host flips a coin to choose between these two doors. A number of interesting articles (e.g., [3], [7], and [8]) analyze games in which the host does not make a random choice in the latter case. In 1959, Martin Gardner [6] posed a problem equivalent to Marilyn's game involving three prisoners with one to be paroled. Gardner describes the game as a wonderfully confusing little problem. Another equivalent problem, involving three boxes, was posed by S. Selvin [10] in 1975. Ed Barbeau [1] has written a good review of the literature on Marilyn's game and related problems. A recent paper by Fernandez and Piron [4] considers Marilyn's game when the host influences the contestant to switch in certain situations so that the game is less predictable, and thus generates more audience interest. Two appealing solutions to Marilyn's game are: S1: After the host shows a goat, the contestant is looking at two closed doors with a car behind one of them. Thus there is a 50-50 chance with either door and there is no advantage in switching, S2: The probability of picking the car with her first choice was 1/3 and this does not change when the host shows a goat. Since the car is behind one of the two closed doors, the probability is 2/3 that the car is behind the other closed door and she should switch. In this note, we generalize the game by considering a set of N doors, with a car behind one of the doors and goats behind the rest. The sponsors of the show ask Monty to give away as few cars as possible. We consider three different generalizations, and in the third one, both the host and contestant use mixed strategies. In general, mixed strategy games can only be solved for given numerical values of the parameters involved, however in our game there is an explicit solution.