Dominance in the Monty Hall Problem

Elementary decision-theoretic analysis of the Monty Hall dilemma shows that the problem has dominance. This makes possible to discard nonswitching strategies, without making any assumptions on the prior distribution of factors out of control of the decision maker. A path to the Bayesian and the minimax decision-making environments is then straightforward. Suppose there is a set of ‘doors’ D with at least three elements. One door θ ∈ D is a winning door. You will be asked to choose a door x ∈ D, and then all doors except one door y ∈ D \ {x} will be revealed as not winning. Then you will be asked to guess if x = θ (action match with door x) or x 6= θ (action switch to door y). You win if your guess is correct. The case of three doors is widely known as the Monty Hall problem. The observation of this note is that the problem has dominance, which makes sense of the question ‘to switch or not to switch’ even in the situations when no prior distributions are assigned to the parameters out of control of the decision maker. The idea is quite simple: if the guessing strategy chooses x and in some situation plays match, so does not switch to y, then another strategy which chooses y and plays switch all the time is at least as good as the first whichever θ ∈ D. This simplistic view is basically right, modulo subtleties involved in the formal definition of ‘strategy’ and ‘situation’ A strategy is a pair (x, ax) where x ∈ D and ax is a function of y ∈ D \ {x} with values ax(y) ∈ {match, switch}. For constant functions we call (x, match) and (x, switch) single-action strategies, and call (x, switch) always-switching strategy. ∗Postal address: Department of Mathematics, Utrecht University, Postbus 80010, 3508 TA Utrecht, The Netherlands. E-mail address: [email protected]